Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T01:13:40.883Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  29 June 2018

Philipp Fleig
Affiliation:
Max-Planck-Institut für Dynamik und Selbstorganisation, Germany
Henrik P. A. Gustafsson
Affiliation:
Stanford University, California
Axel Kleinschmidt
Affiliation:
Max-Planck-Institut für Gravitationsphysik, Germany
Daniel Persson
Affiliation:
Chalmers University of Technology, Gothenberg
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Eisenstein Series and Automorphic Representations
With Applications in String Theory
, pp. 527 - 558
Publisher: Cambridge University Press
Print publication year: 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlén, O. 2016. Global Iwasawa-decomposition of SL(n, AQ). arXiv:1609.06621 [math.NT].Google Scholar
Ahlén, O., Gustafsson, H. P. A., Kleinschmidt, A., Liu, B., and Persson, D. 2017. Fourier coefficients attached to small automorphic representations of SLn (A). arXiv:1707.08937 [math.RT].Google Scholar
Alexandrov, S. 2009. D-instantons and twistors: Some exact results. J. Phys., A42, 335402.Google Scholar
Alexandrov, S. 2013. Twistor approach to string compactifications: A review. Phys. Rep., 522, 157.CrossRefGoogle Scholar
Alexandrov, S., and Banerjee, S. 2013. Modularity, quaternion-Kähler spaces, and mirror symmetry. J. Math. Phys., 54, 102301.CrossRefGoogle Scholar
Alexandrov, S., and Banerjee, S. 2014a. Dualities and fivebrane instantons. JHEP, 11, 040.Google Scholar
Alexandrov, S., and Banerjee, S. 2014b. Fivebrane instantons in Calabi-Yau compactifications. Phys. Rev., D90(4), 041902.Google Scholar
Alexandrov, S., and Banerjee, S. 2015. Hypermultiplet metric and D-instantons. JHEP, 02, 176.Google Scholar
Alexandrov, S., Banerjee, S., Manschot, J., and Pioline, B. 2017. Multiple D3-instantons and mock modular forms I. Commun. Math. Phys., 353(1), 379411.CrossRefGoogle Scholar
Alexandrov, S., Manschot, J., Persson, D., and Pioline, B. 2015. Quantum hypermultiplet moduli spaces in N=2 string vacua: A review. Proceedings of Symposia in Pure Mathematics, 90, 181212.CrossRefGoogle Scholar
Alexandrov, S., Manschot, J., and Pioline, B. 2013. D3-instantons, mock theta series and twistors. JHEP, 04, 002.Google Scholar
Alexandrov, S., Persson, D., and Pioline, B. 2011a. Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces. JHEP, 03, 111.Google Scholar
Alexandrov, S., Persson, D., and Pioline, B. 2011b. On the topology of the hypermultiplet moduli space in type II/CY string vacua. Phys. Rev., D83, 026001.Google Scholar
Alexandrov, S., and Pioline, B. 2012. S-duality in twistor space. JHEP, 08, 112.Google Scholar
Alexandrov, S., Pioline, B., Saueressig, F., and Vandoren, S. 2009. D-instantons and twistors. JHEP, 03, 044.Google Scholar
Alexandrov, S., Pioline, B., and Vandoren, S. 2010. Self-dual Einstein spaces, heavenly metrics and twistors. J. Math. Phys., 51, 073510.CrossRefGoogle Scholar
Alexandrov, S., and Saueressig, F. 2009. Quantum mirror symmetry and twistors. JHEP, 09, 108.Google Scholar
Alvarez-Gaumé, L., Moore, G. W., and Vafa, C. 1986. Theta functions, modular invariance and strings. Commun. Math. Phys., 106, 140.CrossRefGoogle Scholar
Angelantonj, C., Florakis, I., and Pioline, B. 2012a. A new look at one-loop integrals in string theory. Commun. Num. Theor. Phys., 6, 159201.CrossRefGoogle Scholar
Angelantonj, C., Florakis, I., and Pioline, B. 2012b. One-loop BPS amplitudes as BPS-state sums. JHEP, 06, 070.Google Scholar
Angelantonj, C., Florakis, I., and Pioline, B. 2013. Rankin-Selberg methods for closed strings on orbifolds. JHEP, 07, 181.Google Scholar
Angelantonj, C., Florakis, I., and Pioline, B. 2015. Threshold corrections, generalised prepotentials and Eichler integrals. Nucl. Phys., B897, 781820.CrossRefGoogle Scholar
Apostol, T. M. 1976. Introduction to Analytic Number Theory. Springer, New York, London.Google Scholar
Apostol, T. M. 1997. Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics, vol. 41. Springer, London.Google Scholar
Arakawa, T., and Moreau, A. 2016. Joseph ideals and lisse minimal W-algebras. J. Inst. Math. Jussieu, 121.Google Scholar
Argurio, R. 1998. Brane Physics in M-Theory. Ph.D. thesis, Free University of Brussels. arXiv:hep-th/9807171 [hep-th].Google Scholar
Arthur, J. 1989. Orbites unipotentes et représentations, II (Unipotent automorphic representations: Conjectures). Astérisque, 171–172, 1371.Google Scholar
Arthur, J. 2003. The principle of functoriality. Bull. Amer. Math. Soc. (N.S.), 40(1), 3953.CrossRefGoogle Scholar
Arthur, J. 2005. An introduction to the trace formula. Pages 1–263 of Arthur, J., Ellwood, D., and Kottwitz, R. (eds), Harmonic Analysis, the Trace Formula, and Shimura Varieties. Clay Mathematics Proceedings, vol. 4. American Mathematical Society, Providence, RI.Google Scholar
Aspinwall, P. S. 1997. K3 surfaces and string duality. Pages 421–540 of Efthimiou, C., and Greene, B. (eds), Fields, Strings and Duality (Boulder, CO, 1996). World Scientific Publishing, River Edge, NJ.Google Scholar
Aspinwall, P. S. 2001. Compactification, geometry and duality: N = 2. Pages 723–805 of Harvey, J. A., Kachru, S., and Silverstein, E. (eds), Strings, Branes and Gravity, TASI 99 (Boulder, CO). World Scientific Publishing, River Edge, NJ.Google Scholar
Atiyah, M. 1987. The logarithm of the Dedekind η-function. Math. Ann., 278 (1–4), 335380.CrossRefGoogle Scholar
Aurich, R., Lustig, S., Steiner, F., and Then, H. 2005. Can one hear the shape of the Universe? Phys. Rev. Lett., 94, 021301.Google Scholar
Bachas, C. 1996. D-brane dynamics. Phys. Lett., B374, 3742.CrossRefGoogle Scholar
Bachas, C., and Porrati, M. 1992. Pair creation of open strings in an electric field. Phys. Lett., B296, 7784.CrossRefGoogle Scholar
Bala, P., and Carter, R. W. 1976a. Classes of unipotent elements in simple algebraic groups, I. Math. Proc. Cambridge Philos. Soc., 79(3), 401425.CrossRefGoogle Scholar
Bala, P., and Carter, R. W. 1976b. Classes of unipotent elements in simple algebraic groups, II. Math. Proc. Cambridge Philos. Soc., 80(1), 117.CrossRefGoogle Scholar
Banerjee, S., and Sen, A. 2008. Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on T**6. JHEP, 03, 022.Google Scholar
Bao, L., Bielecki, J., Cederwall, M., Nilsson, B. E. W., and Persson, D. 2008a. U-duality and the compactified Gauss-Bonnet term. JHEP, 0807, 048.Google Scholar
Bao, L., and Carbone, L. 2013. Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories. arXiv:1308.6194 [hep-th].Google Scholar
Bao, L., Cederwall, M., and Nilsson, B. E. W. 2008b. Aspects of higher curvature terms and U-duality. Class. Quant. Grav., 25, 095001.CrossRefGoogle Scholar
Bao, L., Kleinschmidt, A., Nilsson, B. E. W., Persson, D., and Pioline, B. 2010. Instanton corrections to the universal hypermultiplet and automorphic forms on SU(2,1). Commun. Num. Theor. Phys., 4, 187266.CrossRefGoogle Scholar
Bao, L., Kleinschmidt, A., Nilsson, B. E. W., Persson, D., and Pioline, B. 2013. Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons. J. Phys. Conf. Ser., 462(1), 012026.CrossRefGoogle Scholar
Basu, A. 2007. The D**10 R**4 term in type IIB string theory. Phys. Lett., B648, 378382.CrossRefGoogle Scholar
Basu, A. 2008a. The D**4 R**4 term in type IIB string theory on T**2 and U-duality. Phys. Rev., D77, 106003.Google Scholar
Basu, A. 2008b. The D**6 R**4 term in type IIB string theory on T**2 and U-duality. Phys. Rev., D77, 106004.Google Scholar
Basu, A. 2011. Supersymmetry constraints on the R4 multiplet in type IIB on T2. Class. Quant. Grav., 28, 225018.CrossRefGoogle Scholar
Basu, A. 2014. The D6R4 term from three loop maximal supergravity. Class. Quant. Grav., 31(24), 245002.CrossRefGoogle Scholar
Basu, A. 2016a. Poisson equation for the three loop ladder diagram in string theory at genus one. Int. J. Mod. Phys., A31(32), 1650169.Google Scholar
Basu, A. 2016b. Proving relations between modular graph functions. Class. Quant. Grav., 33(23), 235011.CrossRefGoogle Scholar
Becker, K., Becker, M., and Strominger, A. 1995. Five-branes, membranes and nonperturbative string theory. Nucl. Phys., B456, 130152.CrossRefGoogle Scholar
Beilinson, A., and Drinfeld, V. 1991. Quantization of Hitchin’s integrable system and Hecke eigensheaves. www.math.uchicago.edu/∼mitya/langlands/hitchin/BD-hitchin.pdf.Google Scholar
Beineke, J., Brubaker, B., and Frechette, S. 2011. Weyl group multiple Dirichlet series of type C. Pacific J. Math., 254(1), 1146.CrossRefGoogle Scholar
Beineke, J., Brubaker, B., and Frechette, S. 2012. A crystal definition for symplectic multiple Dirichlet series. Pages 37–63 of Bump, D., Friedberg, S., and Goldfeld, D. (eds), Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol. 300. Birkhäuser/Springer, New York.Google Scholar
Bekaert, X. 2011. Singletons and their maximal symmetry algebras. Pages 71–89 of Dragovich, B., and Rakic, Z. (eds), Modern Mathematical Physics: Proceedings, 6th Summer School, Belgrade Serbia, September 14–23, 2010. arXiv:1111.4554 [math-ph]Google Scholar
Belinsky, V. A., Khalatnikov, I. M., and Lifshitz, E. M. 1970. Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys., 19, 525573.CrossRefGoogle Scholar
Belinsky, V. A., Khalatnikov, I. M., and Lifshitz, E. M. 1982. A general solution of the Einstein equations with a time singularity. Adv. Phys., 31, 639667.CrossRefGoogle Scholar
Berenstein, A., and Zelevinsky, A. 2001. Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math., 143(1), 77128.CrossRefGoogle Scholar
Berkovits, N. 1998. Construction of R(4) terms in N=2 D = 8 superspace. Nucl. Phys., B514, 191203.CrossRefGoogle Scholar
Bhargava, M. 2004. Higher composition laws. I: A new view on Gauss composition, and quadratic generalizations. Ann. of Math. (2), 159(1), 217250.CrossRefGoogle Scholar
Birch, B. J., and Swinnerton-Dyer, H. P. F. 1965. Notes on elliptic curves, II. J. Reine Angew. Math., 218, 79108.CrossRefGoogle Scholar
Blasius, D. 1994. On multiplicities for SL(n). Israel J. Math., 88(1–3), 237251.CrossRefGoogle Scholar
Blumenhagen, R., Lüst, D., and Theisen, S. 2013. Basic Concepts of String Theory. Theoretical and Mathematical Physics. Springer, Heidelberg.CrossRefGoogle Scholar
Bodner, M., and Cadavid, A. C. 1990. Dimensional reduction of type IIB supergravity and exceptional quaternionic manifolds. Class. Quant. Grav., 7, 829pp.CrossRefGoogle Scholar
Bogomolny, E. B., Georgeot, B., Giannoni, M. J., and Schmit, C. 1997. Arithmetical chaos. Phys. Rep., 291, 219324.CrossRefGoogle Scholar
Borcea, C. 1992. Calabi-Yau threefolds and complex multiplication. Pages 489–502 of Yau, S. T. (ed.), Essays on Mirror Manifolds. International Press, Hong Kong.Google Scholar
Borcherds, R. E. 1992. Monstrous moonshine and monstrous Lie superalgebras. Invent. Math., 109(2), 405444.CrossRefGoogle Scholar
Borcherds, R. E. 1998. Automorphic forms with singularities on Grassmannians. Invent. Math., 132(3), 491562.CrossRefGoogle Scholar
Borel, A. 1966. Introduction to automorphic forms. Pages 199–210 of Borel, A., and Mostow, G. D. (eds), Algebraic Groups and Discontinuous Subgroups. Proceedings of Symposia in Pure Mathematics, vol. 9. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Borel, A. 1972. Représentations de Groupes Localement Compacts. Lecture Notes in Mathematics, vol. 276. Springer, Berlin, New York.CrossRefGoogle Scholar
Borel, A. 1979. Automorphic L-functions. Pages 27–61 of Borel, A., and Casselman, W. (eds), Automorphic Forms, Representations and L-functions. Proceedings of Symposia in Pure Mathematics, vol. 33. American Mathematical Society, Providence, RI.Google Scholar
Borel, A. 1991. Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 126. Springer, Berlin, New York.CrossRefGoogle Scholar
Borel, A. 1997. Automorphic Forms on SL(2, R). Cambridge Tracts in Mathematics, vol. 130. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Borel, A., and Tits, J. 1972. Compléments à l’article: ‘Groupes réductifs’. Inst. Hautes Études Sci. Publ. Math., 41, 253276.CrossRefGoogle Scholar
Borho, W. 1977. Berechnung der Gelfand-Kirillov-Dimension bei induzierten Darstellungen. Math. Ann., 225(2), 177194.CrossRefGoogle Scholar
Borho, W., and Brylinski, J.-L. 1982. Differential operators on homogeneous spaces, I: Irreducibility of the associated variety for annihilators of induced modules. Invent. Math., 69(3), 437476.CrossRefGoogle Scholar
Born, M., and Infeld, L. 1934. Foundations of the new field theory. Proc. Roy. Soc. Lond., A144, 425451.Google Scholar
Bossard, G., Cosnier-Horeau, C., and Pioline, B. 2017a. Four-derivative couplings and BPS dyons in heterotic CHL orbifolds. SciPost Phys., 3, 008.CrossRefGoogle Scholar
Bossard, G., Cosnier-Horeau, C., and Pioline, B. 2017b. Protected couplings and BPS dyons in half-maximal supersymmetric string vacua. Phys. Lett., B765, 377381.CrossRefGoogle Scholar
Bossard, G., and Kleinschmidt, A. 2015. Supergravity divergences, supersymmetry and automorphic forms. JHEP, 08, 102.Google Scholar
Bossard, G., and Kleinschmidt, A. 2016. Loops in exceptional field theory. JHEP, 01, 1.Google Scholar
Bossard, G., and Pioline, B. 2017. Exact ∇4R4 couplings and helicity supertraces. JHEP, 01, 050.Google Scholar
Bossard, G., and Verschinin, V. 2014. Minimal unitary representations from supersymmetry. JHEP, 1410, 008.Google Scholar
Bossard, G., and Verschinin, V. 2015a. E4R4 type invariants and their gradient expansion. JHEP, 03, 089.Google Scholar
Bossard, G., and Verschinin, V. 2015b. The two ∇6R4 type invariants and their higher order generalisation. JHEP, 07, 154.Google Scholar
Braverman, A., and Gaitsgory, D. 2002. Geometric Eisenstein series. Invent. Math., 150(2), 287384.CrossRefGoogle Scholar
Braverman, A., Garland, H., Kazhdan, D., and Patnaik, M. 2014. An affine Gindikin-Karpelevich formula. Pages 43–64 of Etingof, P., Khovanov, M., and Savage, A. (eds), Perspectives in Representation Theory. Contemporary Mathematics, vol. 610. American Mathematical Society, Providence, RI.Google Scholar
Braverman, A., and Kazhdan, D. 2011. The spherical Hecke algebra for affine Kac-Moody groups I. Ann. of Math. (2), 174(3), 16031642.Google Scholar
Braverman, A., and Kazhdan, D. 2013. Representations of affine Kac-Moody groups over local and global fields: A survey of some recent results. Pages 91–117 of Latata, R., Ruciński, A., Strzelecki, P., Swiatkowski, J., Wrzosek, D., and Zakrzewski, P. (eds), European Congress of Mathematics: Proceedings of the 6th Congress (6 ECM), Held at the Jagiellonian University, Kraków, July 2–7, 2012. European Mathematical Society, Zürich.Google Scholar
Braverman, A., Kazhdan, D., and Patnaik, M. M. 2016. Iwahori-Hecke algebras for p-adic loop groups. Invent. Math., 204(2), 347442.CrossRefGoogle Scholar
Breckenridge, J. C., Michaud, G., and Myers, R. C. 1997. More D-brane bound states. Phys. Rev., D55, 64386446.Google Scholar
Brekke, L., and Freund, P. 1993. p-adic numbers in physics. Phys.Rept., 233, 166.CrossRefGoogle Scholar
Breuil, C., Conrad, B., Diamond, F., and Taylor, R. 2001. On the modularity of elliptic curves over Q: Wild 3-adic exercises. J. Amer. Math. Soc., 14(4), 843939.CrossRefGoogle Scholar
Broedel, J., Mafra, C. R., Matthes, N., and Schlotterer, O. 2015. Elliptic multiple zeta values and one-loop superstring amplitudes. JHEP, 07, 112.Google Scholar
Broedel, J., Matthes, N., and Schlotterer, O. 2016. Relations between elliptic multiple zeta values and a special derivation algebra. J. Phys., A49(15), 155203.Google Scholar
Broedel, J., Schlotterer, O., and Stieberger, S. 2013. Polylogarithms, multiple zeta values and superstring amplitudes. Fortsch. Phys., 61, 812870.CrossRefGoogle Scholar
Broedel, J., Schlotterer, O., Stieberger, S., and Terasoma, T. 2014. All order α′-expansion of superstring trees from the Drinfeld associator. Phys. Rev., D89(6), 066014.Google Scholar
Brown, F. 2014. Single-valued motivic periods and multiple zeta values. SIGMA, 2, e25.Google Scholar
Brown, F. C. S. 2006. Multiple zeta values and periods of moduli spaces M0,n. arXiv:math/0606419 [math.AG].Google Scholar
Brubaker, B., Bump, D., Chinta, G., Friedberg, S., and Hoffstein, J. 2006a. Weyl group multiple Dirichlet series, I. Pages 91–114 of Friedberg, S., Bump, D., Goldfeld, D., and Hoffstein, J. (eds), Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory. Proceedings of Symposia in Pure Mathematics, vol. 75. American Mathematical Society, Providence, RI.Google Scholar
Brubaker, B., Bump, D., Chinta, G., and Gunnells, P. E. 2012. Metaplectic Whittaker functions and crystals of type B. Pages 93–118 of Bump, D., Friedberg, S., and Goldfeld, D. (eds), Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol. 300. Birkhäuser/Springer, New York.Google Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2006b. Weyl group multiple Dirichlet series, II: The stable case. Invent. Math., 165(2), 325355.CrossRefGoogle Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2008. Twisted Weyl group multiple Dirichlet series: The stable case. Pages 1–26 of Gan, W. T., Kudla, S. S., and Tschinkel, Y. (eds), Eisenstein Series and Applications. Progress in Mathematics, vol. 258. Birkhäuser Boston, Boston, MA.Google Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2009. Gauss sum combinatorics and metaplectic Eisenstein series. Pages 61–81 of Ginzburg, D., Lapid, E., and Soudry, D. (eds), Automorphic Forms and L-functions I: Global Aspects. Contemporary Mathematics, vol. 488. American Mathematical Society, Providence, RI.Google Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2011a. Eisenstein series, crystals, and ice. Notices Amer. Math. Soc., 58(11), 15631571.Google Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2011b. Schur polynomials and the Yang-Baxter equation. Comm. Math. Phys., 308(2), 281301.CrossRefGoogle Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2011c. Weyl group multiple Dirichlet series, Eisenstein series and crystal bases. Ann. of Math. (2), 173(2), 10811120.CrossRefGoogle Scholar
Brubaker, B., Bump, D., and Friedberg, S. 2011d. Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory. Annals of Mathematics Studies, vol. 175. Princeton University Press, Princeton, NJ.Google Scholar
Brubaker, B., Bump, D., Friedberg, S., and Hoffstein, J. 2007 . Weyl group multiple Dirichlet series, III: Eisenstein series and twisted unstable Ar. Ann. of Math. (2), 166(1), 293316.CrossRefGoogle Scholar
Brubaker, B., and Friedberg, S. 2015. Whittaker coefficients of metaplectic Eisenstein series. Geom. Funct. Anal., 25(4), 11801239.CrossRefGoogle Scholar
Bruhat, F., and Tits, J. 1967. Groupes algébriques simples sur un corps local. Pages 23–36 of Springer, T. A. (ed.), Proceedings of a Conference on Local Fields (Driebergen, 1966). Springer, Berlin.Google Scholar
Bruhat, F., and Tits, J. 1972. Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math., 41, 5251.CrossRefGoogle Scholar
van der Geer, G. 2008. Siegel modular forms and their applications. Pages 181–245 of Ranestad, K. (ed.), The 1–2–3 of Modular Forms: Lectures from the Summer School on Modular Forms and their Applications, Held at Nordfjordeid, Norway, June 2004. Universitext. Springer, Berlin.Google Scholar
Brylinski, R., and Kostant, B. 1994. Minimal representations, geometric quantization, and unitarity. Proc. Nat. Acad. Sci. U.S.A., 91(13), 60266029.CrossRefGoogle ScholarPubMed
Bump, D. 1998. Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge.Google Scholar
Bump, D. 2005. The Rankin-Selberg method: An introduction and survey. Pages 41–73 of Cogdell, J. W., Jiang, D., Kudla, S. S., Soudry, D., and Stanton, R. J. (eds), Automorphic Representations, L-functions and Applications: Progress and Prospects. Ohio State University Mathematics Research Institute Publications, vol. 11. de Gruyter, Berlin.Google Scholar
Bump, D., Friedberg, S., and Goldfeld, D. (eds) 2012. Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol. 300. Birkhäuser/Springer, New York.CrossRefGoogle Scholar
Bump, D., and Nakasuji, M. 2010. Integration on p-adic groups and crystal bases. Proc. Amer. Math. Soc., 138(5), 15951605.CrossRefGoogle Scholar
Bump, D. W. 1982. Automorphic Forms on GL(3,R). Lecture Notes in Mathematics, vol. 1083. Springer, Berlin, Heidelberg.Google Scholar
Candelas, P., Horowitz, G. T., Strominger, A., and Witten, E. 1985. Vacuum configurations for superstrings. Nucl. Phys., B258, 4674.CrossRefGoogle Scholar
Carbone, L., Lee, K.-H., and Liu, D. 2017. Eisenstein series on rank 2 hyperbolic Kac-Moody groups. Math. Ann., 367(3–4), 11731197.CrossRefGoogle Scholar
Cardy, J. L. 1989. Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys., B324, 581596.CrossRefGoogle Scholar
Cardy, J. L. 2004. Boundary conformal field theory. arXiv:hep-th/0411189 [hep-th].Google Scholar
Carnahan, S. 2010. Generalized moonshine, I: Genus-zero functions. Algebra Number Theory, 4(6), 649679.CrossRefGoogle Scholar
Carnahan, S. 2012. Generalized moonshine, II: Borcherds products. Duke Math. J., 161(5), 893950.CrossRefGoogle Scholar
Carnahan, S. 2012. Generalized moonshine, IV: Monstrous Lie algebras. arXiv:1208.6254 [math.RT].CrossRefGoogle Scholar
Carter, R. W. 1993. Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley Classics Library. John Wiley & Sons, Chichester. Reprint of the 1985 original.Google Scholar
Casselman, W., and Shalika, J. 1980. The unramified principal series of p-adic groups, II. The Whittaker function. Compos. Math., 41(2), 207231.Google Scholar
Cheng, M. C. N. 2010. K3 surfaces, N=4 dyons, and the Mathieu group M24. Commun. Num. Theor. Phys., 4, 623658.CrossRefGoogle Scholar
Cheng, M. C. N., Duncan, J. F. R., Harrison, S. M., and Kachru, S. 2017. Equivariant K3 Invariants. Commun. Num. Theor. Phys., 11, 4172.CrossRefGoogle Scholar
Cheng, M. C. N., Duncan, J. F. R., and Harvey, J. A. 2014a. Umbral moonshine. Commun. Num. Theor. Phys., 08, 101242.CrossRefGoogle Scholar
Cheng, M. C. N., Duncan, J. F. R., and Harvey, J. A. 2014b. Umbral moonshine and the Niemeier lattices. Res. Math. Sci., 1, Art. 3, 81.CrossRefGoogle Scholar
Cheng, M. C. N., and Verlinde, E. 2007. Dying dyons don’t count. JHEP, 09, 070.Google Scholar
Cheng, M. C. N., and Verlinde, E. P. 2008. Wall crossing, discrete attractor flow, and Borcherds algebra. SIGMA, 4, 068.Google Scholar
Chinta, G., and Gunnells, P. E. 2007. Weyl group multiple Dirichlet series constructed from quadratic characters. Invent. Math., 167(2), 327353.CrossRefGoogle Scholar
Chinta, G., and Gunnells, P. E. 2010. Constructing Weyl group multiple Dirichlet series. J. Amer. Math. Soc., 23(1), 189215.CrossRefGoogle Scholar
Chinta, G., and Gunnells, P. E. 2012. Littelmann patterns and Weyl group multiple Dirichlet series of type D. Pages 119–130 of Bump, D., Friedberg, S., and Goldfeld, D. (eds), Multiple Dirichlet Series, L-Functions and Automorphic Forms. Progress in Mathematics, vol. 300. Birkhäuser/Springer, New York.Google Scholar
Cogdell, J. W. 2004. Lectures on L-functions, converse theorems, and functoriality for GLn. Pages 1–96 of Cogdell, J. W., Kim, H. H., and Ram Murty, M. (eds), Lectures on Automorphic L-Functions. Fields Institute Monographs, vol. 20. American Mathematical Society, Providence, RI.Google Scholar
Cogdell, J. W. 2005. Converse theorems, functoriality, and applications. Pure Appl. Math. Q., 1(2, Special Issue: In memory of Armand Borel. Part 1), 341367.CrossRefGoogle Scholar
Cogdell, J. W., Kim, H. H., Piatetski-Shapiro, I. I., and Shahidi, F. 2001. On lifting from classical groups to GLN . Inst. Hautes Études Sci. Publ. Math., 93, 530.CrossRefGoogle Scholar
Cogdell, J. W., and Piatetski-Shapiro, I. I. 1994. Converse theorems for GLn. Inst. Hautes Études Sci. Publ. Math., 79, 157214.CrossRefGoogle Scholar
Cogdell, J. W., and Piatetski-Shapiro, I. I. 1999. Converse theorems for GLn, II. J. Reine Angew. Math., 507, 165188.CrossRefGoogle Scholar
Cohen, H. 1992. Elliptic curves. Pages 212–237 of Waldschmidt, M., Moussa, P., Luck, J. M., and Itzykson, C. (eds), From Number Theory to Physics. Springer, Berlin.Google Scholar
Collingwood, D. H., and McGovern, W. M. 1993. Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold, New York.Google Scholar
Conrad, B., Diamond, F., and Taylor, R. 1999. Modularity of certain potentially Barsotti-Tate Galois representations. J. Amer. Math. Soc., 12(2), 521567.CrossRefGoogle Scholar
Conway, J. H., and Norton, S. P. 1979. Monstrous moonshine. Bull. London Math. Soc., 11(3), 308339.CrossRefGoogle Scholar
Conway, J. H., and Sloane, N. J. A. 1999. Sphere Packings, Lattices and Groups. Springer, New York.CrossRefGoogle Scholar
Cremmer, E., and Julia, B. 1978. The N=8 supergravity theory, 1: The Lagrangian. Phys. Lett., B80, 48.Google Scholar
Cremmer, E., and Julia, B. 1979. The SO(8) supergravity. Nucl. Phys., B159, 141212.CrossRefGoogle Scholar
Cremmer, E., Julia, B., Lu, H., and Pope, C. 1998. Dualization of dualities, 1. Nucl. Phys., B523, 73144.CrossRefGoogle Scholar
Dabholkar, A. 2005. Exact counting of black hole microstates. Phys. Rev. Lett., 94, 241301.CrossRefGoogle Scholar
Dabholkar, A., Denef, F., Moore, G. W., and Pioline, B. 2005. Precision counting of small black holes. JHEP, 0510, 096.Google Scholar
Dabholkar, A., Gomes, J., Murthy, S., and Sen, A. 2011. Supersymmetric index from black hole entropy. JHEP, 04, 034.Google Scholar
Dabholkar, A., and Harvey, J. A. 1989. Nonrenormalization of the superstring tension. Phys. Rev. Lett., 63, 478.CrossRefGoogle ScholarPubMed
Dabholkar, A., Murthy, S., and Zagier, D. 2012. Quantum black holes, wall crossing, and mock modular forms. arXiv:1208.4074 [hep-th].Google Scholar
Damour, T., and Henneaux, M. 2000. Chaos in superstring cosmology. Phys. Rev. Lett., 85, 920923.CrossRefGoogle ScholarPubMed
Damour, T., and Henneaux, M. 2001. E(10), BE(10) and arithmetical chaos in superstring cosmology. Phys. Rev. Lett., 86, 47494752.CrossRefGoogle Scholar
Damour, T., Henneaux, M., and Nicolai, H. 2002. E(10) and a ‘small tension expansion’ of M theory. Phys. Rev. Lett., 89, 221601.CrossRefGoogle Scholar
Damour, T., Henneaux, M., and Nicolai, H. 2003. Cosmological billiards. Class. Quant. Grav., 20, R145–R200.CrossRefGoogle Scholar
de Graaf, W. A. 2011. Computing representatives of nilpotent orbits of θ-groups. J. Symbolic Comput., 46(4), 438458.CrossRefGoogle Scholar
De Medts, T., Gramlich, R., and Horn, M. 2009. Iwasawa decompositions of split Kac-Moody groups. J. Lie Theory, 19(2), 311337.Google Scholar
Deitmar, A. 2012. Automorphic Forms. Universitext. Springer, London.CrossRefGoogle Scholar
Deligne, P. 1974. La conjecture de Weil, I: Inst. Hautes Études Sci. Publ. Math., 43, 273307.Google Scholar
Denef, F., and Moore, G. W. 2011. Split states, entropy enigmas, holes and halos. JHEP, 1111, 129.Google Scholar
D’Hoker, E. 1999. Perturbative string theory. Pages 807–1011 of Deligne, P., Kazhdan, D., Etingof, P., Morgan, J. W., Free, D. S., Morrison, D. R., and Witten, E. (eds), Quantum Fields and Strings : A Course for Mathematicians. American Mathematical Society, Providence, RI.Google Scholar
D’Hoker, E., and Duke, W. 2017. Fourier series of modular graph functions. arXiv:1708.07998 [math.NT].Google Scholar
D’Hoker, E., and Green, M. B. 2014. Zhang-Kawazumi invariants and superstring amplitudes. Journal of Number Theory, 144, 111150.CrossRefGoogle Scholar
D’Hoker, E., Green, M. B., Gurdogan, O., and Vanhove, P. 2017. Modular graph functions. Commun. Num. Theor. Phys., 11, 165218.CrossRefGoogle Scholar
D’Hoker, E., Green, M. B., Pioline, B., and Russo, R. 2015a. Matching the D6R4 interaction at two-loops. JHEP, 01, 031.Google Scholar
D’Hoker, E., Green, M. B., and Vanhove, P. 2015b. On the modular structure of the genus-one type II superstring low energy expansion. JHEP, 08, 041.Google Scholar
D’Hoker, E., Gutperle, M., and Phong, D. H. 2005. Two-loop superstrings and S-duality. Nucl. Phys., B722, 81118.CrossRefGoogle Scholar
D’Hoker, E., and Phong, D. H. 2002. Lectures on two loop superstrings. Conf. Proc., C0208124, 85–123. (Proceedings of International Conference on Superstring Theory, Hangzhou, China, August 12–15, 2002.)Google Scholar
D’Hoker, E., and Phong, D. H. 1986. Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys., B269, 205234.CrossRefGoogle Scholar
D’Hoker, E., and Phong, D. H. 2005. Two-loop superstrings, VI: Nonrenormalization theorems and the 4-point function. Nucl. Phys., B715, 390.CrossRefGoogle Scholar
Di Francesco, P., Mathieu, P., and Senechal, D. 1997. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York.Google Scholar
Diamond, F. 1996. On deformation rings and Hecke rings. Ann. of Math. (2), 144(1), 137166.CrossRefGoogle Scholar
Diamond, F., and Shurman, J. 2005. A First Course in Modular Forms. Graduate Texts in Mathematics, vol. 228. Springer, New York.Google Scholar
Dijkgraaf, R., Verlinde, E. P., and Verlinde, H. L. 1997. Counting dyons in N=4 string theory. Nucl. Phys., B484, 543561.CrossRefGoogle Scholar
Dirac, P. A. M. 1931. Quantized singularities in the electromagnetic field. Proc. Roy. Soc. Lond., A133, 6072.Google Scholar
Dirac, P. A. M. 1962. An extensible model of the electron. Proc. Roy. Soc. Lond., A268, 5767.Google Scholar
Dixon, L. J., Ginsparg, P. H., and Harvey, J. A. 1988. Beauty and the beast: Superconformal symmetry in a monster module. Commun. Math. Phys., 119, 221241.CrossRefGoogle Scholar
Dixon, L. J., Kaplunovsky, V., and Louis, J. 1991. Moduli dependence of string loop corrections to gauge coupling constants. Nucl. Phys., B355, 649688.CrossRefGoogle Scholar
Ðoković, D. Ž. 2001. The closure diagram for nilpotent orbits of the split real form of E7. Represent. Theory, 5, 284316.CrossRefGoogle Scholar
Ðoković, D. Ž. 2003. The closure diagram for nilpotent orbits of the split real form of E8. Cent. Eur. J. Math., 1(4), 573643.CrossRefGoogle Scholar
Donagi, R., and Witten, E. 2013. Supermoduli space is not projected. Pages 19–72 of Donagi, R., Katz. S., Klemm, A., and Morrison, D. R. (eds), String-Math 2012. Proceedings of Symposia in Pure Mathematics, vol. 90. American Mathematical Society, Providence, RI.Google Scholar
Douglas, M. R. 1995. Branes within branes. Pages 267–275 of Baulieu, L., Kazakov, V., Picco, M., Windey, P., Di Francesco, P., and Douglas, M. R. (eds), Strings, Branes and Dualities: Proceedings, NATO Advanced Study Institute, Cargèse, France, May 26-June 14, 1997. NATO ASI Series C, vol. 520. Springer, Dordrecht.Google Scholar
Drinfeld, V. G. 1980. Langlands’ conjecture for GL(2) over functional fields. Pages 565–574 of Lehto, O. (ed.), Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Academia Scientiarum Fennica, Helsinki.Google Scholar
Drinfeld, V. G. 1983. Two-dimensional l-adic representations of the fundamental group of a curve over a finite field and automorphic forms on GL(2). Amer. J. Math., 105(1), 85114.CrossRefGoogle Scholar
Duff, M. J., Lu, H., and Pope, C. N. 1996. The black branes of M theory. Phys. Lett., B382, 7380.CrossRefGoogle Scholar
Duncan, J. F. R., Griffin, M. J., and Ono, K. 2015. Proof of the umbral moonshine conjecture. Res. Math. Sci., 2, Art. 26, 47.CrossRefGoogle Scholar
Duncan, J. F. R., Mertens, M. H., and Ono, K. 2017a. O’Nan moonshine and arithmetic. arXiv:1702:03516 [math. NT].Google Scholar
Duncan, J. F. R., Mertens, M. H., and Ono, K. 2017b. Pariah moonshine. Nat. Commun. 8(1). arXiv:1709.08867 [math.RT].CrossRefGoogle ScholarPubMed
Dvorsky, A., and Sahi, S. 1999. Explicit Hilbert spaces for certain unipotent representations, II. Invent. Math., 138(1), 203224.CrossRefGoogle Scholar
Dynkin, E. B. 1952. Semisimple subalgebras of semisimple Lie algebras. Mat. Sbornik N.S., 30(72), 349462.Google Scholar
Eguchi, T., and Hikami, K. 2009. Superconformal algebras and mock theta functions 2: Rademacher expansion for K3 surface. Commun. Num. Theor. Phys., 3, 531554.CrossRefGoogle Scholar
Eguchi, T., and Hikami, K. 2011. Note on twisted elliptic genus of K3 surface. Phys. Lett., B694, 446455.CrossRefGoogle Scholar
Eguchi, T., and Hikami, K. 2012. Twisted elliptic genus for K3 and Borcherds product. Lett. Math. Phys., 102, 203222.CrossRefGoogle Scholar
Eguchi, T., Ooguri, H., and Tachikawa, Y. 2011. Notes on the K3 surface and the Mathieu group M24. Exper. Math., 20, 9196.CrossRefGoogle Scholar
Eichler, M. 1954. Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math. (Basel), 5, 355366.CrossRefGoogle Scholar
Eichler, M., and Zagier, D. 1985. The theory of Jacobi forms. Progress in Mathematics, vol. 55. Birkhäuser, Basel.CrossRefGoogle Scholar
Englert, F., Houart, L., Kleinschmidt, A., Nicolai, H., and Tabti, N. 2007. An E(9) multiplet of BPS states. JHEP, 0705, 065.Google Scholar
Enriquez, B. 2014. Elliptic associators. Selecta Math. (N.S.), 20(2), 491584.CrossRefGoogle Scholar
Falbel, E., Francsics, G., Lax, P. D., and Parker, J. R. 2011. Generators of a Picard modular group in two complex dimensions. Proc. Amer. Math. Soc., 139(7), 24392447.CrossRefGoogle Scholar
Fernando, S., and Günaydin, M. 2016. Massless conformal fields, AdSd+1/CFTd higher spin algebras and their deformations. Nucl. Phys., B904, 494526.CrossRefGoogle Scholar
Ferrara, S., and Günaydin, M. 1998. Orbits of exceptional groups, duality and BPS states in string theory. Int. J. Mod. Phys., A13, 20752088.CrossRefGoogle Scholar
Ferrara, S., and Maldacena, J. M. 1998. Branes, central charges and U duality invariant BPS conditions. Class. Quant. Grav., 15, 749758.CrossRefGoogle Scholar
Figueroa-O’Farrill, J. M., and Papadopoulos, G. 2003. Maximally supersymmetric solutions of ten-dimensional and eleven-dimensional supergravities. JHEP, 03, 048.Google Scholar
Figueroa-O’Farrill, J. M., and Simon, J. 2004. Supersymmetric Kaluza-Klein reductions of AdS backgrounds. Adv. Theor. Math. Phys., 8(2), 217317.CrossRefGoogle Scholar
Flath, D. 1979. Decomposition of representations into tensor products. Pages 179–183 of Borel, A., and Casselman, W. (eds), Automorphic Forms, Representations and L-functions: Proceedings of Symposia in Pure Mathematics, vol. 33. American Mathematical Society, Providence, RI.Google Scholar
Fleig, P., and Kleinschmidt, A. 2012a. Eisenstein series for infinite-dimensional U-duality groups. JHEP, 1206, 054.Google Scholar
Fleig, P., and Kleinschmidt, A. 2012b. Perturbative terms of Kac-Moody-Eisenstein series. Pages 265–275 of Donagi, R., Katz, S., Klemm, A., and Morrison, D. R. (eds), Proceedings, String-Math 2012, Bonn, Germany, July 16–21, 2012. Proceedings of Symposia in Pure Mathematics, vol. 90. American Mathematical Society, Providence, RI.Google Scholar
Fleig, P., Kleinschmidt, A., and Persson, D. 2014. Fourier expansions of Kac-Moody Eisenstein series and degenerate Whittaker vectors. Commun. Num. Theor. Phys., 08, 41100.CrossRefGoogle Scholar
Florakis, I., and Pioline, B. 2017. On the Rankin-Selberg method for higher genus string amplitudes. Commun. Num. Theor. Phys., 11, 337404.CrossRefGoogle Scholar
Forte, L. A. 2009. Arithmetical chaos and quantum cosmology. Class. Quant. Grav., 26, 045001.CrossRefGoogle Scholar
Freed, D. S. 1986. Determinants, torsion, and strings. Commun. Math. Phys., 107, 483513.CrossRefGoogle Scholar
Freedman, D. Z., and Van Proeyen, A. 2012. Supergravity. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Frenkel, E. 2007. Lectures on the Langlands program and conformal field theory. Pages 387–533 of Cartier, P., Moussa, P., Julia, B., and Vanhove, P. (eds), Frontiers in Number Theory, Physics and Geometry, II: On Conformal Field Theories, Discrete Groups and Renormalization. Springer, Berlin, Heidelberg.Google Scholar
Frenkel, E. 2009. Gauge theory and Langlands duality. arXiv:0906.2747 [math.RT].Google Scholar
Frenkel, E., Gaitsgory, D., Kazhdan, D., and Vilonen, K. 1998. Geometric realization of Whittaker functions and the Langlands conjecture. J. Amer. Math. Soc., 11(2), 451484.CrossRefGoogle Scholar
Frenkel, E., Gaitsgory, D., and Vilonen, K. 2001. Whittaker patterns in the geometry of moduli spaces of bundles on curves. Ann. of Math. (2), 153(3), 699748.CrossRefGoogle Scholar
Frenkel, E., Gaitsgory, D., and Vilonen, K. 2002. On the geometric Langlands conjecture. J. Amer. Math. Soc., 15(2), 367417.CrossRefGoogle Scholar
Frenkel, E., and Witten, E. 2008. Geometric endoscopy and mirror symmetry. Commun. Num. Theor. Phys., 2, 113283.CrossRefGoogle Scholar
Frenkel, I., Lepowsky, J., and Meurman, A. 1988. Vertex Operator Algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic Press, Boston, MA.Google Scholar
Frey, G. 1986. Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sarav. Ser. Math., 1(1), iv+40.Google Scholar
Friedan, D., Martinec, E. J., and Shenker, S. H. 1986. Conformal invariance, supersymmetry and string theory. Nucl. Phys., B271, 93.Google Scholar
Friedberg, S., and Zhang, L. 2015. Eisenstein series on covers of odd orthogonal groups. Amer. J. Math., 137(4), 9531011.CrossRefGoogle Scholar
Friedberg, S., and Zhang, L. 2016. Tokuyama-type formulas for characters of type B. Israel J. Math., 216(2), 617655.CrossRefGoogle Scholar
Friedlander, H., Gaudet, L., and Gunnells, P. E. 2015. Crystal graphs, Tokuyama’s theorem, and the Gindikin-Karpelevič formula for G2. J. Algebraic Combin., 41(4), 10891102.CrossRefGoogle Scholar
Fulton, W. 1997. Young Tableaux. London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge.Google Scholar
Fulton, W., and Harris, J. 2008. Representation Theory: A First Course. Graduate Texts in Mathematics, vol. 129. Springer, London.Google Scholar
Gaberdiel, M. R., Hohenegger, S., and Volpato, R. 2010a. Mathieu moonshine in the elliptic genus of K3. JHEP, 10, 062.Google Scholar
Gaberdiel, M. R., Hohenegger, S., and Volpato, R. 2010b. Mathieu twining characters for K3. JHEP, 09, 058.Google Scholar
Gaberdiel, M. R., Hohenegger, S., and Volpato, R. 2012. Symmetries of K3 sigma models. Commun. Num. Theor. Phys., 6, 150.CrossRefGoogle Scholar
Gaberdiel, M. R., Persson, D., Ronellenfitsch, H., and Volpato, R. 2013. Generalized Mathieu moonshine. Commun. Num. Theor Phys., 07, 145223.CrossRefGoogle Scholar
Gaiotto, D., Moore, G. W., and Neitzke, A. 2010. Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys., 299, 163224.CrossRefGoogle Scholar
Gaitsgory, D. 2004. On a vanishing conjecture appearing in the geometric Langlands correspondence. Ann. of Math. (2), 160(2), 617682.CrossRefGoogle Scholar
Gaitsgory, D. 2010. Notes on geometric Langlands: The extended Whittaker category. www.math.harvard.edu/∼gaitsgde/GL/extWhit.pdf.Google Scholar
Gaitsgory, D. 2016. Recent progress in geometric Langlands theory. arXiv:1606.09462 [math.AG].Google Scholar
Gan, W. T. Lecture slides on automorphic forms and representations. www.math.nus.edu.sg/∼matgwt/.Google Scholar
Gan, W. T. 2000a. An automorphic theta module for quaternionic exceptional groups. Canad. J. Math., 52(4), 737756.CrossRefGoogle Scholar
Gan, W. T. 2000b. A Siegel-Weil formula for exceptional groups. J. Reine Angew. Math., 528, 149181.Google Scholar
Gan, W. T., Gross, B., and Savin, G. 2002. Fourier coefficients of modular forms on G2. Duke Math. J., 115(1), 105169.CrossRefGoogle Scholar
Gan, W. T., Qiu, Y., and Takeda, S. 2014. The regularized Siegel-Weil formula (the second term identity) and the Rallis inner product formula. Invent. Math., 198(3), 739831.CrossRefGoogle Scholar
Gan, W. T., and Savin, G. 2005. On minimal representations: Definitions and properties. Represent. Theory, 9, 4693 (electronic).CrossRefGoogle Scholar
Gannon, T. 2002. Boundary conformal field theory and fusion ring representations. Nucl. Phys., B627, 506564.CrossRefGoogle Scholar
Gannon, T. 2016. Much ado about Mathieu. Adv. Math., 301, 322358.CrossRefGoogle Scholar
Gannon, T., and Lam, C. S. 1992. Lattices and theta function identities, 2: Theta series. J. Math. Phys., 33, 871887.CrossRefGoogle Scholar
Garland, H. 2001. Certain Eisenstein series on loop groups: Convergence and the constant term. Pages 275–319 of Dani, S. G., and Prasad, G. (eds), Proceedings of the International Conference on Algebraic Groups and Arithmetic, Mumbai, 2001. Narosa Publishing House, New Delhi, for Tata Institute of Fundamental Research, Mumbai.Google Scholar
Garland, H. 2006. Absolute convergence of Eisenstein series on loop groups. Duke Math. J., 135, 203260.CrossRefGoogle Scholar
Garland, H. 2011. On extending the Langlands-Shahidi method to arithmetic quotients of loop groups. Pages 151–167 of Adams, J., Lian, B., and Sahi, S. (eds), Representation Theory and Mathematical Physics. Contemporary Mathematics, vol. 557. American Mathematical Society, Providence, RI.Google Scholar
Garland, H., Miller, S. D., and Patnaik, M. M. 2017. Entirety of cuspidal Eisenstein series on loop groups. Amer. J. Math., 139(2), 461512.CrossRefGoogle Scholar
Garrett, P. 1999. Satake parameters versus unramified principal series. www.math.umn.edu/∼garrett/m/v/satake_urps.pdf.Google Scholar
Garrett, P. 2014. Transition: Eisenstein series on adele groups. www.math.umn.edu/∼garrett/m/mfms/notes_2013–14/12_2_transition_ Eis.pdf.Google Scholar
Gatti, V., and Viniberghi, E. 1978. Spinors of 13-dimensional space. Adv. Math., 30(2), 137155.CrossRefGoogle Scholar
Gauntlett, J. P., Gutowski, J. B., Hull, C. M., Pakis, S., and Reall, H. S. 2003. All supersymmetric solutions of minimal supergravity in five dimensions. Class. Quant. Grav., 20, 45874634.CrossRefGoogle Scholar
Gaussent, S., and Rousseau, G. 2014. Spherical Hecke algebras for Kac-Moody groups over local fields. Ann. of Math. (2), 180(3), 10511087.CrossRefGoogle Scholar
Gelbart, S. 1984. An elementary introduction to the Langlands program. Bull. Amer. Math. Soc., 10(2), 177219.CrossRefGoogle Scholar
Gelbart, S., Jacquet, H., and Rogawski, J. 2001. Generic representations for the unitary group in three variables. Israel J. Math., 126, 173237.CrossRefGoogle Scholar
Gelbart, S., and Piatetski-Shapiro, I. 1984. Automorphic forms and L-functions for the unitary group. Pages 141–184 of Herb, R., Kudla, S., Lipsman, R., and Rosenberg, J. (eds), Lie Group Representations, II (College Park, MD, 1982/1983). Lecture Notes in Mathematics, vol. 1041. Springer, Berlin, New York.Google Scholar
Gelbart, S., and Shahidi, F. 1988. Analytic Properties of Automorphic L-functions. Perspectives in Mathematics, vol. 6. Academic Press, Boston, MA.Google Scholar
Gelbart, S. S. 1975. Automorphic Forms on Adele Groups. Annals of Mathematics Studies, vol. 83. Princeton University Press, Princeton, NJ.CrossRefGoogle Scholar
Gelbart, S. S. 1976. Weil’s Representation and the Spectrum of the Metaplectic Group. Lecture Notes in Mathematics, vol. 530. Springer, Berlin, New York.CrossRefGoogle Scholar
Gelbart, S. S., and Miller, S. D. 2003. Riemann’s zeta function and beyond. Bull. Amer. Math. Soc., 41(1), 59112.CrossRefGoogle Scholar
Gelbart, S. S., and Rogawski, J. D. 1991. L-functions and Fourier-Jacobi coefficients for the unitary group U(3). Invent. Math., 105(3), 445472.CrossRefGoogle Scholar
Gelfand, I., Graev, M., and Piatetski-Shapiro, I. 1968. Representation Theory and Automorphic Functions. Saunders Mathematics Books. Saunders, Philadelphia, PA.Google Scholar
Gelfand, I. M., and Cetlin, M. L. 1950. Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.), 71, 825828.Google Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2006. Givental integral representation for classical groups. arXiv:math/0608152.Google Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2009. On Baxter Q-operators and their arithmetic implications. Lett. Math. Phys., 88(1–3), 330.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2010a. On q-deformed gll+1-Whittaker function, I. Comm. Math. Phys., 294(1), 97119.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2010b. On q-deformed gll+1-Whittaker function, II. Comm. Math. Phys., 294(1), 121143.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2011a. On q-deformed gl+1-Whittaker function, III. Lett. Math. Phys., 97(1), 124.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2011b. Parabolic Whittaker functions and topological field theories, I. Commun. Num. Theor. Phys., 5, 135202.CrossRefGoogle Scholar
Gerasimov, A., Lebedev, D., and Oblezin, S. 2014. Baxter Operator Formalism for Macdonald Polynomials. Lett. Math. Phys., 104, 115139.CrossRefGoogle Scholar
Gerasimov, A. A., Lebedev, D. R., and Oblezin, S. V. 2012. New integral representations of Whittaker functions for classical Lie groups. Uspekhi Mat. Nauk, 67(1(403)), 396.Google Scholar
Gibbons, G. W., Green, M. B., and Perry, M. J. 1996. Instantons and seven-branes in type IIB superstring theory. Phys. Lett., B370, 3744.CrossRefGoogle Scholar
Ginzburg, D. 2006. Certain conjectures relating unipotent orbits to automorphic representations. Israel J. Math., 151, 323355.CrossRefGoogle Scholar
Ginzburg, D. 2014. Towards a classification of global integral constructions and functorial liftings using the small representations method. Adv. Math., 254, 157186.CrossRefGoogle Scholar
Ginzburg, D., and Hundley, J. 2013. Constructions of global integrals in the exceptional group F4. Kyushu J. Math., 67(2), 389417.CrossRefGoogle Scholar
Ginzburg, D., Rallis, S., and Soudry, D. 1997a. On the automorphic theta representation for simply laced groups. Israel J. Math., 100, 61116.CrossRefGoogle Scholar
Ginzburg, D., Rallis, S., and Soudry, D. 1997b. A tower of theta correspondences for G2. Duke Math. J., 88(3), 537624.CrossRefGoogle Scholar
Ginzburg, D., Rallis, S., and Soudry, D. 2003. On Fourier coefficients of automorphic forms of symplectic groups. Manuscripta Math., 111(1), 116.CrossRefGoogle Scholar
Givental, A. 1997. Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture. Pages 103–115 of Khovanski, A., Varclienko, A., and Vassiliev, V. (eds), Topics in Singularity Theory. American Mathematical Society Translations, Series 2, vol. 180. American Mathematical Society, Providence, RI.Google Scholar
Goddard, P., Nuyts, J., and Olive, D. I. 1977. Gauge theories and magnetic charge. Nucl. Phys., B125, 1.Google Scholar
Goddard, P., and Olive, D. 1986. Kac-Moody and Virasoro algebras in relation to quantum physics. Internat. J. Modern Phys. A, 1(2), 303414.CrossRefGoogle Scholar
Goddard, P., and Thorn, C. B. 1972. Compatibility of the dual pomeron with unitarity and the absence of ghosts in the dual resonance model. Phys. Lett., B40, 235238.CrossRefGoogle Scholar
Godement, R. 1962. Domaines fondamentaux des groupes arithmétiques. Pages 201–225 of Séminaire Bourbaki, Vol. 8, Exp. No. 257. Société Mathématique de France, Paris.Google Scholar
Godement, R. 1995. Introduction à la théorie de Langlands. Pages 115–144 of Séminaire Bourbaki, Vol. 10, Exp. No. 321. Société Mathématique de France, Paris.Google Scholar
Goldfeld, D. 2006. Automorphic Forms and L-functions for the Group GL(n, R). Cambridge Studies in Advanced Mathematics, vol. 99. Cambridge University Press, Cambridge.Google Scholar
Goldfeld, D., and Hundley, J. 2011a. Automorphic Representations and L-Functions for the General Linear Group. Volume I. Cambridge Studies in Advanced Mathematics, vol. 129. Cambridge University Press, Cambridge.Google Scholar
Goldfeld, D., and Hundley, J. 2011b. Automorphic Representations and L-Functions for the General Linear Group. Volume II. Cambridge Studies in Advanced Mathematics, vol. 130. Cambridge University Press, Cambridge.Google Scholar
Gomes, J. 2017. U-duality invariant quantum entropy from sums of Kloosterman sums. arXiv:1709.06579 [hep-th].Google Scholar
Gomez, H., and Mafra, C. R. 2013. The closed-string 3-loop amplitude and S-duality. JHEP, 1310, 217.Google Scholar
Gomez, R., Gourevitch, D., and Sahi, S. 2016. Whittaker supports for representations of reductive groups. arXiv:1610.00284 [math.RT].Google Scholar
Gomez, R., Gourevitch, D., and Sahi, S. 2017. Generalized and degenerate Whittaker models. Compos. Math., 153(2), 223256.CrossRefGoogle Scholar
Goncharov, A., and Shen, L. 2015. Geometry of canonical bases and mirror symmetry. Invent. Math., 202(2), 487633.CrossRefGoogle Scholar
Gourevitch, D., and Sahi, S. 2013. Annihilator varieties, adduced representations, Whittaker functionals, and rank for unitary representations of GL(n). Selecta Math. (N.S.), 19(1), 141172.CrossRefGoogle Scholar
Gourevitch, D., and Sahi, S. 2015. Degenerate Whittaker functionals for real reductive groups. Amer. J. Math., 137(2), 439472.CrossRefGoogle Scholar
Graham, R., and Szepfalusy, P. 1990. Quantum creation of a generic universe. Phys. Rev., D42, 24832490.Google Scholar
Gran, U., Gutowski, J., and Papadopoulos, G. 2010a. Classification of IIB backgrounds with 28 supersymmetries. JHEP, 01, 044.Google Scholar
Gran, U., Gutowski, J., and Papadopoulos, G. 2010b. M-theory backgrounds with 30 Killing spinors are maximally supersymmetric. JHEP, 03, 112.Google Scholar
Gran, U., Papadopoulos, G., and von Schultz, C. 2014. Supersymmetric geometries of IIA supergravity, I. JHEP, 05, 024.Google Scholar
Gran, U., Papadopoulos, G., and von Schultz, C. 2015. Supersymmetric geometries of IIA supergravity, II. JHEP, 12, 113.Google Scholar
Gran, U., Papadopoulos, G., and von Schultz, C. 2016. Supersymmetric geometries of IIA supergravity, III. JHEP, 06, 045.Google Scholar
Green, M. B. 1995. A gas of D instantons. Phys. Lett., B354, 271278.CrossRefGoogle Scholar
Green, M. B. 1999. Interconnections between type II superstrings, M theory and N=4 supersymmetric Yang-Mills. Pages 22–96 of Cevesole, A., Kounnas, C., Lüst, D., and Theisen, S. (eds), Quantum Aspects of Gauge Theories, Supersymmetry and Unification: Proceedings, 2nd International Conference, Corfu, Greece, September 20–26, 1998. Lecture Notes in Physics, vol. 525. Springer, Berlin, New York.Google Scholar
Green, M. B., and Gutperle, M. 1997. Effects of D instantons. Nucl. Phys., B498, 195227.CrossRefGoogle Scholar
Green, M. B., and Gutperle, M. 1998. D particle bound states and the D instanton measure. JHEP, 01, 005.Google Scholar
Green, M. B., Gutperle, M., and Kwon, H.-H. 1998. Sixteen fermion and related terms in M theory on T**2. Phys. Lett., B421, 149161.CrossRefGoogle Scholar
Green, M. B., Gutperle, M., and Vanhove, P. 1997. One loop in eleven-dimensions. Phys. Lett., B409, 177184.CrossRefGoogle Scholar
Green, M. B., Kwon, H.-H., and Vanhove, P. 2000. Two loops in eleven-dimensions. Phys. Rev., D61, 104010.Google Scholar
Green, M. B., Miller, S. D., Russo, J. G., and Vanhove, P. 2010. Eisenstein series for higher-rank groups and string theory amplitudes. Commun. Num. Theor. Phys., 4, 551596.CrossRefGoogle Scholar
Green, M. B., Miller, S. D., and Vanhove, P. 2015a. SL(2, Z)-invariance and D-instanton contributions to the D6R4 interaction. Commun. Num. Theor. Phys., 09, 307344.CrossRefGoogle Scholar
Green, M. B., Miller, S. D., and Vanhove, P. 2015b. Small representations, string instantons, and Fourier modes of Eisenstein series. J. Number Theory, 146, 187309. With an appendix by D. Ciubotaru and P. Trapa.CrossRefGoogle Scholar
Green, M. B., Russo, J. G., and Vanhove, P. 2008a. Low energy expansion of the four-particle genus-one amplitude in type II superstring theory. JHEP, 02, 020.Google Scholar
Green, M. B., Russo, J. G., and Vanhove, P. 2008b. Modular properties of two-loop maximal supergravity and connections with string theory. JHEP, 07, 126.Google Scholar
Green, M. B., Russo, J. G., and Vanhove, P. 2010a. Automorphic properties of low energy string amplitudes in various dimensions. Phys. Rev., 81(8), 086008.Google Scholar
Green, M. B., Russo, J. G., and Vanhove, P. 2010b. String theory dualities and supergravity divergences. JHEP, 1006, 075.Google Scholar
Green, M. B., Schwarz, J., and Witten, E. 1987. Superstring Theory. Vols I & II. Cambridge University Press, Cambridge.Google Scholar
Green, M. B., and Sethi, S. 1999. Supersymmetry constraints on type IIB supergravity. Phys. Rev. D, 59(4), 046006.CrossRefGoogle Scholar
Green, M. B., and Vanhove, P. 1997. D instantons, strings and M theory. Phys. Lett., B408, 122134.CrossRefGoogle Scholar
Green, M. B., and Vanhove, P. 2006. Duality and higher derivative terms in M theory. JHEP, 0601, 093.Google Scholar
Gritsenko, V. A., and Nikulin, V. V. 1997. Siegel automorphic form corrections of some Lorentzian Kac–Moody Lie algebras. Amer. J. Math., 119(1), 181224.Google Scholar
Gross, B. H., and Kudla, S. S. 1992. Heights and the central critical values of triple product L-functions. Compos. Math., 81(2), 143209.Google Scholar
Gross, B. H., and Wallach, N. R. 1994. A distinguished family of unitary representations for the exceptional groups of real rank = 4. Pages 289–304 of Kostant, B., and Brylinski, J.-L. (eds), Lie Theory and Geometry. Progress in Mathematics, vol. 123. Birkhäuser Boston, Boston, MA.Google Scholar
Gross, B. H., and Wallach, N. R. 1996. On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math., 481, 73123.Google Scholar
Gross, B. H., and Zagier, D. B. 1986. Heegner points and derivatives of L-series. Invent. Math., 84(2), 225320.CrossRefGoogle Scholar
Gross, D. J., and Witten, E. 1986. Superstring modifications of Einstein’s equations. Nucl. Phys., B277, 1.Google Scholar
Gubay, F., Lambert, N., and West, P. 2010. Constraints on automorphic forms of higher derivative terms from compactification. JHEP, 1008, 028.Google Scholar
Gukov, S., and Witten, E. 2006. Gauge theory, ramification, and the geometric Langlands program. arXiv:hep-th/0612073 [hep-th].CrossRefGoogle Scholar
Günaydin, M., Koepsell, K., and Nicolai, H. 2001a. Conformal and quasiconformal realizations of exceptional Lie groups. Comm. Math. Phys., 221(1), 5776.CrossRefGoogle Scholar
Günaydin, M., Koepsell, K., and Nicolai, H. 2001b. The minimal unitary representation of E8(8) . Adv. Theor. Math. Phys., 5(5), 923946.CrossRefGoogle Scholar
Günaydin, M., Neitzke, A., Pavlyk, O., and Pioline, B. 2008. Quasi-conformal actions, quaternionic discrete series and twistors: SU(2, 1) and G2(2) . Commun. Math. Phys., 283, 169226.CrossRefGoogle Scholar
Günaydin, M., Neitzke, A., Pioline, B., and Waldron, A. 2006. BPS black holes, quantum attractor flows, and automorphic forms. Phys. Rev. D, 73(8), 084019.CrossRefGoogle Scholar
Günaydin, M., and Pavlyk, O. 2005. Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups. JHEP, 01, 019.Google Scholar
Günaydin, M., and Pavlyk, O. 2006. A unified approach to the minimal unitary realizations of noncompact groups and supergroups. JHEP, 09, 050.Google Scholar
Gurevich, N. 2013. The twisted Satake isomorphism and Casselman-Shalika formula. arXiv:1307.7510 [math.RT].Google Scholar
Gustafsson, H. P. A., Kleinschmidt, A., and Persson, D. 2016. Small automorphic representations and degenerate Whittaker vectors. J. Number Theory, 166, 344399.CrossRefGoogle Scholar
Hamel, A. M., and King, R. C. 2002. Symplectic shifted tableaux and deformations of Weyl’s denominator formula for sp(2n). J. Algebraic Combin., 16(3), 269300 (2003).CrossRefGoogle Scholar
Harish-Chandra, . 1968. Automorphic Forms on Semisimple Lie Groups. Notes by Mars, J. G. M.. Lecture Notes inMathematics, vol. 62. Springer, Berlin, New York.CrossRefGoogle Scholar
Harish-Chandra, . 1978. Admissible Invariant Distributions on Reductive p-adic Groups. Pages 281–347 of Rossmann, W. (ed.), Lie Theories and Their Applications: Proceedings of the 1977 Annual Seminar of the Canadian Mathematical Congress, Queen’s University, Kingston, Ontario, 1977. Queen’s Papers in Pure and Applied Mathematics, vol. 48. Queen’s University, Kingston, ON.Google Scholar
Harris, M. 2002. On the local Langlands correspondence. Pages 583–597 of Tatsien, L. (ed.), Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol. II. Higher Education Press, Beijing.Google Scholar
Harris, M., and Kudla, S. S. 1991. The central critical value of a triple product L-function. Ann. of Math. (2), 133(3), 605672.CrossRefGoogle Scholar
Harris, M., and Taylor, R. 2001. The Geometry and Cohomology of Some Simple Shimura Varieties. Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton, NJ. With an appendix by V. G. Berkovich.Google Scholar
Harvey, J. A., and Moore, G. W. 1996. Algebras, BPS states, and strings. Nucl. Phys., B463, 315368.CrossRefGoogle Scholar
Harvey, J. A., and Moore, G. W. 1998. On the algebras of BPS states. Commun. Math. Phys., 197, 489519.CrossRefGoogle Scholar
Hashizume, M. 1982. Whittaker functions on semisimple Lie groups. Hiroshima Math. J., 12(2), 259293.Google Scholar
Hasse, H. 1933. Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F. K. Schmidtschen Kongruenzzetafunktionen in gewissen elliptischen Fällen. Nachr. Gesell. Wissen. Göttingen, 42, 253262.Google Scholar
Hecke, E. 1937a. Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, I. Math. Ann., 114(1), 128.CrossRefGoogle Scholar
Hecke, E. 1937b. Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, II. Math. Ann., 114(1), 316351.CrossRefGoogle Scholar
Helgason, S. 2001. Differential Geometry, Lie Groups and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Henneaux, M., Persson, D., and Spindel, P. 2008. Spacelike singularities and hidden symmetries of gravity. Living Rev. Rel., 11, 1.CrossRefGoogle ScholarPubMed
Henniart, G. 2000. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math., 139(2), 439455.CrossRefGoogle Scholar
Hickey, T. J. 1986. On the Fourier-Jacobi Coefficients of Certain Eisenstein Series on a Unitary Group. Ph.D. thesis, University of Chicago.Google Scholar
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., and Zaslow, E. 2003. Mirror Symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence, RI.Google Scholar
Horowitz, G. T., and Strominger, A. 1991. Black strings and P-branes. Nucl. Phys., B360, 197209.CrossRefGoogle Scholar
Howe, P. S., and West, P. C. 1984. The complete N=2, D=10 supergravity. Nucl. Phys., B238, 181220.CrossRefGoogle Scholar
Howe, R. 1974. The Fourier transform and germs of characters (case of Gln over a p-adic field). Math. Ann., 208, 305322.CrossRefGoogle Scholar
Howe, R. 1979. θ-series and invariant theory. Pages 275–285 of Borel, A., and Casselman, W. (eds), Automorphic Forms, Representations and L-Functions. Proceedings of Symposia in Pure Mathematics, vol. 33. American Mathematical Society, Providence, RI.Google Scholar
Howe, R. 1985. Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons. Pages 179–207 of Flato, M., Sally, P., and Zukerman, G. (eds), Applications of Group Theory in Physics and Mathematical Physics (Chicago, 1982). Lectures in Applied Mathematics, vol. 21. American Mathematical Society, Providence, RI.Google Scholar
Howe, R. 1990. Another look at the local θ-correspondence for an unramified dual pair. Pages 93–124 of Gelbart, S., Howe, R., and Sarnak, P. (eds), Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I (Ramat Aviv, 1989). Israel Mathematical Conference Proceedings, vol. 2. Weizmann Science Press, Jerusalem.Google Scholar
Huang, J.-S., Pandžić, P., and Savin, G. 1996. New dual pair correspondences. Duke Math. J., 82(2), 447471.CrossRefGoogle Scholar
Hull, C., and Townsend, P. 1995. Unity of superstring dualities. Nucl. Phys., B438, 109137.CrossRefGoogle Scholar
Humphreys, J. E. 1975. Linear Algebraic Groups. Graduate Texts in Mathematics, vol. 21. Springer, Heidelberg.CrossRefGoogle Scholar
Humphreys, J. E. 1980. Arithmetic Groups. Lecture Notes in Mathematics, vol. 789. Springer, Berlin, New York.CrossRefGoogle Scholar
Humphreys, J. E. 1997. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, London.Google Scholar
Ion, B., and Sahi, S. 2015. Double affine Hecke algebras and congruence groups. arXiv:1506.06417 [math.QA].Google Scholar
Ishii, T., and Stade, E. 2007. New formulas for Whittaker functions on GL(n, R). J. Funct. Anal., 244(1), 289314.CrossRefGoogle Scholar
Ishikawa, Y.-H. 1999. The generalized Whittaker functions for SU(2, 1) and the Fourier expansion of automorphic forms. J. Math. Sci. Univ. Tokyo, 6, 477526.Google Scholar
Ivashchuk, V. D., and Melnikov, V. N. 1995. Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity. Class. Quant. Grav., 12, 809826.CrossRefGoogle Scholar
Iwaniec, H. 2002. Spectral Methods of Automorphic Forms. Second edn. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Jacquet, H. 1967. Fonctions de Whittaker associées aux groupes de Chevalley. Bull. Soc. Math. France, 95, 243309.CrossRefGoogle Scholar
Jacquet, H. 1984. On the residual spectrum of GL(n). Pages 185–208 of Herb, R., Kudla, S., Lipsman, R., and Rosenberg, J. (eds), Lie Group Representations, II (College Park, MD, 1982/1983). Lecture Notes in Mathematics, vol. 1041. Springer, Berlin, New York.Google Scholar
Jacquet, H., and Langlands, R. P. 1970. Automorphic Forms on GL(2). Lecture Notes in Mathematics, vol. 114. Springer, Berlin, New York.CrossRefGoogle Scholar
Jiang, D., Liu, B., and Savin, G. 2016. Raising nilpotent orbits in wave-front sets. Represent. Theory, 20, 419450.CrossRefGoogle Scholar
Jiang, D., and Rallis, S. 1997. Fourier coefficients of Eisenstein series of the exceptional group of type G2. Pacific J. Math., 181(2), 281314.CrossRefGoogle Scholar
Jiang, D., and Soudry, D. 2004. Generic representations and local Langlands reciprocity law for p-adic SO2n+1. Pages 457–519 of Hida, H., Ramakrishnan, D., and Shahidi, F. (eds), Contributions to Automorphic Forms, Geometry, and Number Theory. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Jones, G., and Jones, J. 1998. Elementary Number Theory. Springer Undergraduate Mathematics Series. Springer, London.CrossRefGoogle Scholar
Joseph, A. 1974. Minimal realizations and spectrum generating algebras. Comm. Math. Phys., 36, 325338.CrossRefGoogle Scholar
Joseph, A. 1976. The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. École Norm. Sup. (4), 9(1), 129.CrossRefGoogle Scholar
Joseph, A. 1985. On the associated variety of a primitive ideal. J. Algebra, 93(2), 509523.CrossRefGoogle Scholar
Joyce, D., and Song, Y. 2012. A theory of generalized Donaldson-Thomas invariants. Mem. Amer. Math. Soc., 217(1020), iv+199.Google Scholar
Julia, B. 1981. Group disintegrations. Pages 331–350 of Hawking, S. W., and Roček, M. (eds), Superspace and Supergravity (Proceedings of Nuffield Workshop, Cambridge, June 16 to July 12, 1980). Cambridge University Press, Cambridge.Google Scholar
Kac, V. G. 1990. Infinite Dimensional Lie Algebras. Third edn. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Kapranov, M. 1998. Double affine Hecke algebras and 2-dimensional local fields. arXiv:math/9812021.Google Scholar
Kapustin, A., and Witten, E. 2007. Electric-magnetic duality and the geometric Langlands program. Commun. Num. Theor. Phys., 1, 1236.CrossRefGoogle Scholar
Kashiwara, M. 1990. Crystalizing the q-analogue of universal enveloping algebras. Comm. Math. Phys., 133(2), 249260.CrossRefGoogle Scholar
Katz, S., Klemm, A., Pandharipande, R., and Thomas, R. P. 2016. On the motivic stable pairs invariants of K3 surfaces. Pages 111–146 of Faber, C., Farkas, G., and van der Geer, G. (eds), K3 Surfaces and Their Moduli. Progress in Mathematics, vol. 315. Birkhäuser Boston, Boston, MA.Google Scholar
Kawai, T. 1996. N=2 heterotic string threshold correction, K3 surface and generalized Kac-Moody superalgebra. Phys. Lett., B372, 5964.CrossRefGoogle Scholar
Kawazumi, N. 2008. Johnson’s homomorphisms and the Arakelov-Green function. arXiv:0801.4218 [math.GT].Google Scholar
Kazhdan, D., Pioline, B., and Waldron, A. 2002. Minimal representations, spherical vectors, and exceptional theta series. Commun. Math. Phys., 226, 140.CrossRefGoogle Scholar
Kazhdan, D., and Polishchuk, A. 2004. Minimal representations: Spherical vectors and automorphic functionals. Pages 127–198 of Dani, S. G., and Prasad, G. (eds), Algebraic Groups and Arithmetic. Narosa Publishing House, New Delhi, for Tata Institute of Fundamental Research, Mumbai.Google Scholar
Kazhdan, D., and Savin, G. 1990. The smallest representation of simply laced groups. Pages 209–223 of Gelbart, S., Howe, R., and Sarnak, P. (eds), Festschrift in honor of II Piatetski-Shapiro on the Occasion of His Sixtieth Birthday, Part I (Ramat Aviv, 1989). Israel Mathematical Conference Proceedings, vol. 2. Weizmann Science Press, Jerusalem.Google Scholar
Kim, H. H. 1996. The residual spectrum of G2. Canad. J. Math., 48(6), 12451272.CrossRefGoogle Scholar
Kim, H. H., and Lee, K.-H. 2012. Quantum affine algebras, canonical bases, and q-deformation of arithmetical functions. Pacific J. Math., 255(2), 393415.CrossRefGoogle Scholar
Kim, H. H., and Shahidi, F. 1999. Symmetric cube L-functions for GL2 are entire. Ann. of Math. (2), 150(2), 645662.CrossRefGoogle Scholar
Kim, H. H., and Shahidi, F. 2000. Holomorphy of Rankin triple L-functions: Special values and root numbers for symmetric cube L-functions. Israel J. Math., 120, 449466.CrossRefGoogle Scholar
Kim, H. H., and Shahidi, F. 2002. Functorial products for GL2 × GL3 and the symmetric cube for GL2. Ann. of Math. (2), 155(3), 837893. With an appendix by C. J. Bushnell and G. Henniart.CrossRefGoogle Scholar
Kim, H. H., and Shahidi, F. 2004. On simplicity of poles of automorphic L-functions. J. Ramanujan Math. Soc., 19(4), 267280.Google Scholar
Kim, M. 2015. Arithmetic Chern-Simons theory I. arXiv:1510.05818 [math.NT].Google Scholar
Kirillov, A. A. 1995. Reduction of additional dimensions in nonuniform quantum Kaluza-Klein cosmological models. JETP Lett., 62, 8994. [Pisma Zh. Eksp. Teor. Fiz.62,81(1995)].Google Scholar
Kirillov, A. A. 1999. Merits and demerits of the orbit method. Bull. Am. Math. Soc., New Ser., 36(4), 433488.CrossRefGoogle Scholar
Kirillov, A. N., and Berenstein, A. D. 1995. Groups generated by involutions, Gel’fand-Tsetlin patterns, and combinatorics of Young tableaux. Algebra i Analiz, 7(1), 92152.Google Scholar
Kiritsis, E. 1998. Introduction to Superstring Theory. Leuven Notes in Mathematical and Theoretical Physics, vol. B9. Leuven University Press, Leuven. arXiv:hep-th/9709062 [hep-th].Google Scholar
Kiritsis, E., and Pioline, B. 1997. On R**4 threshold corrections in IIb string theory and (p, q) string instantons. Nucl. Phys., B508, 509534.CrossRefGoogle Scholar
Kleinschmidt, A., Koehn, M., and Nicolai, H. 2009. Supersymmetric quantum cosmological billiards. Phys. Rev., D80, 061701.Google Scholar
Kleinschmidt, A., and Nicolai, H. 2009. Cosmological quantum billiards. Pages 106124 of Murugan, J., Weltman, A., and Ellis, G. F. R. (eds), Proceedings, Foundations of Space and Time: Reflections on Quantum Gravity. Cambridge University Press, Cambridge.Google Scholar
Kleinschmidt, A., Nicolai, H., and Palmkvist, J. 2012. Modular realizations of hyperbolic Weyl groups. Adv. Theor. Math. Phys., 16(1), 97148.CrossRefGoogle Scholar
Kleinschmidt, A., and Verschinin, V. 2017. Tetrahedral modular graph functions. JHEP, 09, 155.CrossRefGoogle Scholar
Knapp, A. W. 1997. Introduction to the Langlands program. Pages 245302 of Baitey, T. N., and Knapp, A. W. (eds), Representation Theory and Automorphic Forms. Proceedings in Symposia in Pure Mathematics, vol. 61, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Knapp, A. W. 2009a. First steps with the Langlands program. Pages 1020 of Ji, L., Liu, K., Yau, S.-T., and Zheng, Z.-J. (eds), Automorphic Forms and the Langlands Program. Advanced Lectures in Mathematics, vol. 9. International Press of Boston, Somerville, MA.Google Scholar
Knapp, A. W. 2009b. Prerequisites for the Langlands program. Pages 19 of Ji, L., Liu, K., Yau, S.-T., and Zheng, Z.-J. (eds), Automorphic Forms and the Langlands Program. Advanced Lectures in Mathematics vol. 9. International Press of Boston, Somerville, MA.Google Scholar
Kobayashi, T., and Savin, G. 2015. Global uniqueness of small representations. Math. Z., 281(1–2), 215239.CrossRefGoogle Scholar
Koblitz, N. 1984. p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer, New York.CrossRefGoogle Scholar
Koblitz, N. 1993. Introduction to Elliptic Curves and Modular Forms. Second edn. Graduate Texts in Mathematics, vol. 97. Springer, New York.CrossRefGoogle Scholar
Kolyvagin, V. A. 1988. Finiteness of E(Q) and Ш(E, Q) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat., 52(3), 522540, 670–671.Google Scholar
Kontsevich, M. 1995. Homological algebra of mirror symmetry. Pages 120– 139 of Chatterji, S. D. (ed.), Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994). Birkhäuser, Basel.Google Scholar
Kontsevich, M., and Soibelman, Y. 2008. Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. arXiv:0811.2435 [math.AG].Google Scholar
Kostant, B. 1963. Lie group representations on polynomial rings. Amer. J. Math., 85, 327404.CrossRefGoogle Scholar
Kostant, B. 1978. On Whittaker vectors and representation theory. Invent. Math., 48(2), 101184.CrossRefGoogle Scholar
Kostant, B., and Rallis, S. 1971. Orbits and representations associated with symmetric spaces. Amer. J. Math., 93, 753809.CrossRefGoogle Scholar
Kostov, I. K., and Vanhove, P. 1998. Matrix string partition functions. Phys. Lett., B444, 196203.CrossRefGoogle Scholar
Krutelevich, S. 2007. Jordan algebras, exceptional groups, and Bhargava composition. J. Algebra, 314(2), 924977.CrossRefGoogle Scholar
Kubota, T. 1969. On Automorphic Functions and the Reciprocity Law in a Number Field. Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2. Kinokuniya Book-Store Co., Tokyo.Google Scholar
Kudla, S. S., and Rallis, S. 1988a. On the Weil-Siegel formula. J. Reine Angew. Math., 387, 168.Google Scholar
Kudla, S. S., and Rallis, S. 1988b. On the Weil-Siegel formula, II: The isotropic convergent case. J. Reine Angew. Math., 391, 6584.Google Scholar
Kudla, S. S., and Rallis, S. 1994. A regularized Siegel-Weil formula: The first term identity. Ann. of Math. (2), 140(1), 180.CrossRefGoogle Scholar
Kumar, S. 2002. Kac-Moody Groups, Their Flag Varieties and Representation Theory. Progress in Mathematics, vol. 204. Birkhäuser Boston, Boston, MA.CrossRefGoogle Scholar
Lafforgue, L. 2002. Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math., 147(1), 1241.CrossRefGoogle Scholar
Lafforgue, V., and Lysenko, S. 2009. Geometric Weil representation: Local field case. Compos. Math., 145(1), 5688.CrossRefGoogle Scholar
Lafforgue, V., and Lysenko, S. 2011. Compatibility of the theta correspondence with the Whittaker functors. Bull. Soc. Math. France, 139(1), 7588.CrossRefGoogle Scholar
Lafforgue, V., and Lysenko, S. 2013. Geometrizing the minimal representations of even orthogonal groups. Represent. Theory, 17, 263325.CrossRefGoogle Scholar
Lai, K. F. 1974. On the Tamagawa Number of Quasi-split Groups. Ph.D. thesis, Yale University.Google Scholar
Lai, K. F. 1976. On the Tamagawa number of quasi-split groups. Bull. Amer. Math. Soc., 82(2), 300302.CrossRefGoogle Scholar
Lambert, N., and West, P. 2010. Perturbation theory from automorphic forms. JHEP, 1005, 098.CrossRefGoogle Scholar
Lambert, N., and West, P. C. 2007. Duality groups, automorphic forms and higher derivative corrections. Phys. Rev., D75, 066002.Google Scholar
Lang, S. 1975. SL(2, R). Graduate Texts in Mathematics, vol. 105. Springer, New York.Google Scholar
Lang, S. 1994. Algebraic Number Theory. Second edn. Graduate Texts in Mathematics, vol. 110. Springer, New York.CrossRefGoogle Scholar
Lang, S. 2002. Algebra. Third edn. Graduate Texts in Mathematics, vol. 211. Springer, New York.CrossRefGoogle Scholar
Langlands, R. P. 1966. Eisenstein Series. Pages 235252 of Borel, A., and Mostow, G. D. (eds), Algebraic Groups and Discontinuous Subgroups. Proceedings of Symposia in Pure Mathematics, vol. 9. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Langlands, R. 1967a. Euler Products. James K. Whittemore Lectures in Mathematics, Yale Univesity, April 1967. Published with corrections by the author as Yale Mathematical Monographs, vol. 1, Yale Univesity Press, New Haven, CT (1971).Google Scholar
Langlands, R. P. 1967b. Letter to André Weil. http://publications.ias.edu/sites/default/files/ltw_1.pdf.Google Scholar
Langlands, R. P. 1970. Problems in the theory of automorphic forms. Lectures in Modern Analysis and Applications III, Lecture Notes in Mathematics, 170.Google Scholar
Langlands, R. P. 1976. On the Functional Equations Satisfied by Eisenstein Series. Lecture Notes in Mathematics, vol. 544. Springer, Berlin, New York.CrossRefGoogle Scholar
Langlands, R. P. 1978. L-functions and automorphic representations. Pages 165– 175 of Lehto, O. (ed.), Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Academia Scientiarium Fennica, Helsinki (1980).Google Scholar
Laumon, G. 1987. Correspondance de Langlands géométrique pour les corps de fonctions. Duke Math. J., 54(2), 309359.CrossRefGoogle Scholar
Laumon, G. 1990. Faisceaux automorphes liés aux séries d’Eisenstein. Pages 227281 of Clozel, L., and Milne, J. S. (eds), Automorphic Forms, Shimura Varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988). Perspectives in Mathematics, vol. 10. Academic Press, Boston, MA.Google Scholar
Lee, K.-H., and Zhang, Y. 2015. Weyl group multiple Dirichlet series for sym-metrizable Kac-Moody root systems. Trans. Amer. Math. Soc., 367(1), 597625.CrossRefGoogle Scholar
Littelmann, P. 1996. An effective method to classify nilpotent orbits. Pages 255269 of González-Vega, L., and Recio, T. (eds), Algorithms in Algebraic Geometry and Applications (Santander, 1994). Progress in Mathematics, vol. 143. Birkhäuser, Basel.Google Scholar
Littelmann, P. 1998. Cones, crystals, and patterns. Transform. Groups, 3(2), 145179.CrossRefGoogle Scholar
Liu, D. 2015. Eisenstein series on loop groups. Trans. Amer. Math. Soc., 367(3), 20792135.CrossRefGoogle Scholar
Lu, H., Pope, C. N., and Stelle, K. S. 1998. Multiplet structures of BPS solitons. Class. Quant. Grav., 15, 537561.CrossRefGoogle Scholar
Lusztig, G. 1990. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc., 3(2), 447498.CrossRefGoogle Scholar
Lusztig, G. 2010. Introduction to Quantum Groups. Modern Birkhäuser Classics. Birkhäuser/Springer, New York. Reprint of the 1994 edition.CrossRefGoogle Scholar
Lysenko, S. 2006a. Moduli of metaplectic bundles on curves and theta-sheaves. Ann. Sci. École Norm. Sup. (4), 39(3), 415466.CrossRefGoogle Scholar
Lysenko, S. 2006b. Whittaker and Bessel functors for GSP4. Ann. Inst. Fourier (Grenoble), 56(5), 15051565.CrossRefGoogle Scholar
Lysenko, S. 2011. Geometric theta-lifting for the dual pair. Ann. Sci. École Norm. Sup. (4), 44(3), 427493.CrossRefGoogle Scholar
Lysenko, S. 2017. Twisted Whittaker models for metaplectic groups. Geom. Funct. Anal., 27(2), 289372.CrossRefGoogle Scholar
Mafra, C. R., and Schlotterer, O. 2014. The structure of n-point one-loop open superstring amplitudes. JHEP, 08, 099.CrossRefGoogle Scholar
Mafra, C. R., and Schlotterer, O. 2017. Non-abelian Z-theory: Berends-Giele recursion for the α′-expansion of disk integrals. JHEP, 01, 031.CrossRefGoogle Scholar
Mafra, C. R., Schlotterer, O., and Stieberger, S. 2013a. Complete N-point superstring disk amplitude, I: Pure spinor computation. Nucl. Phys., B873, 419460.CrossRefGoogle Scholar
Mafra, C. R., Schlotterer, O., and Stieberger, S. 2013b. Complete N-point superstring disk amplitude, II: Amplitude and hypergeometric function structure. Nucl. Phys., B873, 461513.CrossRefGoogle Scholar
Magaard, K., and Savin, G. 1997. Exceptional Θ-correspondences, I. Compos. Math., 107(1), 89123.CrossRefGoogle Scholar
Maldacena, J. M., Moore, G. W., and Strominger, A. 1999. Counting BPS black holes in toroidal type II string theory. arXiv:hep-th/9903163 [hep-th].Google Scholar
Manschot, J. 2010. Stability and duality in N=2 supergravity. Commun. Math. Phys., 299, 651676.CrossRefGoogle Scholar
Manschot, J., and Moore, G. W. 2010. A modern farey tail. Commun. Num. Theor. Phys., 4, 103159.CrossRefGoogle Scholar
Matumoto, H. 1987. Whittaker vectors and associated varieties. Invent. Math., 89(1), 219224.CrossRefGoogle Scholar
Mazur, B. 1978. Rational isogenies of prime degree. Invent. Math., 44(2), 129162. With an appendix by D. Goldfeld.CrossRefGoogle Scholar
McNamara, P. J. 2011. Metaplectic Whittaker functions and crystal bases. Duke Math. J., 156(1), 131.CrossRefGoogle Scholar
McNamara, P. J. 2012. Principal series representations of metaplectic groups over local fields. Pages 299327 of Bump, D., Friedberg, S., and Goldfeld, D. (eds), Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol. 300. Birkhäuser/Springer, New York.CrossRefGoogle Scholar
Miller, S. D. 2013. Residual automorphic forms and spherical unitary representations of exceptional groups. Ann. of Math. (2), 177(3), 11691179.CrossRefGoogle Scholar
Miller, S. D., and Moore, G. 2000. Landau-Siegel zeroes and black hole entropy. Asian J. Math., 4(1), 183211.CrossRefGoogle Scholar
Miller, S. D., and Sahi, S. 2012. Fourier coefficients of automorphic forms, character variety orbits, and small representations. J. Number Theory, 132(12), 30703108.CrossRefGoogle Scholar
Miller, S. D., and Schmid, W. 2004a. The highly oscillatory behavior of automorphic distributions for SL(2). Lett. Math. Phys., 69, 265286.CrossRefGoogle Scholar
Miller, S. D., and Schmid, W. 2004b. Summation formulas, from Poisson and Voronoi to the present. Pages 419440 of Delorme, P., and Vergne, M. (eds), Noncommutative Harmonic Analysis. Progress in Mathematics, vol. 220. Birkhäuser Boston, Boston, MA.CrossRefGoogle Scholar
Miller, S. D., and Schmid, W. 2008. The Rankin-Selberg method for automorphic distributions. Pages 111150 of Kobayashi, T., Schmid, W., and Yang, J.-H. (eds), Representation Theory and Automorphic Forms. Progress in Mathematics, vol. 255. Birkhäuser Boston, Boston, MA.CrossRefGoogle Scholar
Milne, J. S. 2015. Algebraic groups (v2.00). www.jmilne.org/math/.Google Scholar
Misner, C. W. 1969. Mixmaster universe. Phys. Rev. Lett., 22, 10711074.CrossRefGoogle Scholar
Mizoguchi, S., and Schroeder, G. 2000. On discrete U duality in M theory. Class. Quant. Grav., 17, 835870.CrossRefGoogle Scholar
Mœglin, C. 1994. Représentations unipotentes et formes automorphes de carré intégrable. Forum Math., 6(6), 651744.CrossRefGoogle Scholar
Mœglin, C. 1996. Front d’onde des représentations des groupes classiques p-adiques. Amer. J. Math., 118(6), 13131346.CrossRefGoogle Scholar
Mœglin, C. 1998. Correspondance de Howe et front d’onde. Adv. Math., 133(2), 224285.CrossRefGoogle Scholar
Mœglin, C., and Waldspurger, J.-L. 1987. Modèles de Whittaker dégénérés pour des groupes p-adiques. Math. Z., 196(3), 427452.CrossRefGoogle Scholar
Mœglin, C., and Waldspurger, J.-L. 1989. Le spectre résiduel de GL(n). Ann. Sci. École Norm. Sup. (4), 22(4), 605674.CrossRefGoogle Scholar
Mœglin, C., and Waldspurger, J.-L. 1995. Spectral Decomposition and Eisenstein Series. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Moody, R. V., and Pianzola, A. 1995. Lie Algebras with Triangular Decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York.Google Scholar
Moore, G. W. 1998. Arithmetic and attractors. arXiv:hep-th/9807087 [hep-th].Google Scholar
Moore, G. W. 2014. Physical mathematics and the future. www.physics.rutgers.edu/∼gmoore/PhysicalMathematicsAndFuture.pdf.Google Scholar
Moore, G. W. 2016. Desperately seeking moonshine. www.physics.rutgers.edu/∼gmoore/DaveDayFinal.pdf.Google Scholar
Moore, G. W., Nekrasov, N., and Shatashvili, S. 2000. D particle bound states and generalized instantons. Commun. Math. Phys., 209, 7795.CrossRefGoogle Scholar
Mordell, L. J. 1922. On the rational solutions of the indeterminate equations of the third and fourth degree. Proc. Cam. Phil. Soc., 21, 179192.Google Scholar
Myers, R. C. 1999. Dielectric branes. JHEP, 12, 022.CrossRefGoogle Scholar
Narain, K. S. 1986. New heterotic string theories in uncompactified dimensions <10. Phys. Lett., 169B, 4146.CrossRefGoogle Scholar
Narita, H. 2006. Fourier-Jacobi expansion of automorphic forms on Sp(1, q) generating quaternionic discrete series. J. Funct. Anal., 239, 638682.CrossRefGoogle Scholar
Neitzke, A., Pioline, B., and Vandoren, S. 2007. Twistors and black holes. JHEP, 04, 038.CrossRefGoogle Scholar
Nepomechie, R. I. 1985. Magnetic monopoles from antisymmetric tensor gauge fields. Phys. Rev., D31, 1921.Google Scholar
Neukirch, J. 2006. Algebraische Zahlentheorie. Springer, Berlin.Google Scholar
Nilsson, B. E. W., and Tollsten, A. K. 1986. Supersymmetrization of zeta (3) (R μνρσ)**4 in superstring theories. Phys. Lett., B181, 6366.CrossRefGoogle Scholar
Obers, N., and Pioline, B. 1999. U duality andM theory. Phys. Rep., 318, 113225.CrossRefGoogle Scholar
Obers, N. A., and Pioline, B. 2000a. Eisenstein series and string thresholds. Comm. Math. Phys., 209(2), 275324.CrossRefGoogle Scholar
Obers, N. A., and Pioline, B. 2000b. Eisenstein series in string theory. Class. Quant. Grav., 17, 12151224.CrossRefGoogle Scholar
Orloff, T. 1985. Dirichlet series and automorphic forms on unitary groups. Trans. Amer. Math. Soc., 290(2), 431456.CrossRefGoogle Scholar
Paquette, N. M., Persson, D., and Volpato, R. 2016. Monstrous BPS-algebras and the superstring origin of moonshine. Commun. Num. Theor. Phys., 10, 433526.CrossRefGoogle Scholar
Patnaik, M. September, 2017. Automorphic forms on loop groups. Talk at the workshop ‘Automorphic Forms, Mock Modular Forms and String Theory’, Simons Center for Geometry and Physics, Stony Brook, NY, 31 August 2016.Google Scholar
Patnaik, M., and Puskás, A. 2017. On Iwahori-Whittaker functions for metaplectic groups. Adv. Math., 313, 875914.CrossRefGoogle Scholar
Patnaik, M. M. 2017. Unramified Whittaker functions on p-adic loop groups. Amer. J. Math., 139(1), 175213.CrossRefGoogle Scholar
Persson, D. 2010. Arithmetic and Hyperbolic Structures in String Theory. Ph.D. thesis, Free University of Brussels. arXiv:1001.3154 [hep-th].Google Scholar
Persson, D. 2012. Automorphic instanton partition functions on Calabi-Yau threefolds. J. Phys. Conf. Ser., 346.CrossRefGoogle Scholar
Persson, D., and Volpato, R. 2014. Second quantized Mathieu moonshine. Commun. Num. Theor. Phys., 08, 403509.CrossRefGoogle Scholar
Persson, D., and Volpato, R. 2015. Fricke S-duality in CHL models. JHEP, 12, 156.Google Scholar
Peterson, D. H., and Kac, V. G. a. 1983. Infinite flag varieties and conjugacy theorems. Proc. Nat. Acad. Sci. U.S.A., 80(6 i.), 17781782.CrossRefGoogle ScholarPubMed
Petropoulos, P. M., and Vanhove, P. 2012. Gravity, strings, modular and quasimodular forms. Ann. Math. Blaise Pascal, 19(2), 379430.CrossRefGoogle Scholar
Piatetski-Shapiro, I. I. 1979. Multiplicity one theorems. Pages 209212 of Borel, A., and Casselman, W. (eds), Automorphic Forms, Representations and L-functions. Proceedings of Symposia in Pure Mathematics, vol. 33. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Pioline, B. 1998. A note on nonperturbative R**4 couplings. Phys. Lett., B431, 7376.CrossRefGoogle Scholar
Pioline, B. 2005. BPS black hole degeneracies and minimal automorphic representations. JHEP, 08, 071.CrossRefGoogle Scholar
Pioline, B. 2006. Lectures on black holes, topological strings and quantum attractors. Class. Quant. Grav., 23, S981.CrossRefGoogle Scholar
Pioline, B. 2010. R**4 couplings and automorphic unipotent representations. JHEP, 03, 116.CrossRefGoogle Scholar
Pioline, B. 2015. D6R4 amplitudes in various dimensions. JHEP, 04, 057.CrossRefGoogle Scholar
Pioline, B. 2016. A theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces. J. Number Theory, 163, 520541.CrossRefGoogle Scholar
Pioline, B., and Kiritsis, E. 1998. U duality and D-brane combinatorics. Phys. Lett., B418, 6169.CrossRefGoogle Scholar
Pioline, B., Nicolai, H., Plefka, J., and Waldron, A. 2001. R**4 couplings, the fundamental membrane and exceptional theta correspondences. JHEP, 0103, 036.CrossRefGoogle Scholar
Pioline, B., and Persson, D. 2009. The automorphic NS5-brane. Commun. Num. Theor. Phys., 3, 697754.Google Scholar
Pioline, B., and Russo, R. 2015. Infrared divergences and harmonic anomalies in the two-loop superstring effective action. JHEP, 12, 102.Google Scholar
Pioline, B., and Vandoren, S. 2009. Large D-instanton effects in string theory. JHEP, 0907, 008.CrossRefGoogle Scholar
Pioline, B., and Waldron, A. 2003a. Automorphic forms: A physicist’s survey. Pages 277302 of Cartier, P. E., Julia, B., Moussa, P., and Vanhove, P. (eds), Frontiers in Number Theory, Physics, and Geometry 1: On Random Matrices, Zeta Functions and Dynamical Systems. Les Houches, France, March 9–21, 2003. Springer, Berlin.Google Scholar
Pioline, B., and Waldron, A. 2003b. Quantum cosmology and conformal invariance. Phys. Rev. Lett., 90, 031302.CrossRefGoogle ScholarPubMed
Pioline, B., and Waldron, A. 2004. The automorphic membrane. JHEP, 0406, 009.CrossRefGoogle Scholar
Platonov, V., and Rapinchuk, A. 1994. Algebraic Groups and Number Theory. Pure and Applied Mathematics, vol. 139. Academic Press, Boston, MA. Translated from the 1991 Russian original by Rachel Rowen.Google Scholar
Polchinski, J. 1994. Combinatorics of boundaries in string theory. Phys. Rev., D50, 60416045.Google Scholar
Polchinski, J. 1995. Dirichlet branes and Ramond-Ramond charges. Phys. Rev. Lett., 75, 47244727.CrossRefGoogle ScholarPubMed
Polchinski, J. 1996. Tasi lectures on D-branes. 293–356. arXiv:hep-th/9611050 [hep-th].Google Scholar
Polchinski, J. 2007. String Theory. Vol. 1: An Introduction to the Bosonic String; Vol. 2: Superstring Theory and Beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge.Google Scholar
Polchinski, J., and Cai, Y. 1988. Consistency of open superstring theories. Nucl. Phys., B296, 91128.CrossRefGoogle Scholar
Prasad, G. 1977. Strong approximation for semi-simple groups over function fields. Ann. of Math. (2), 105(3), 553572.CrossRefGoogle Scholar
Proskurin, N. V. 1984. Expansion of automorphic functions. J. Sov. Math., 26(3), 19081921.CrossRefGoogle Scholar
Ribet, K. A. 1990. On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math., 100(2), 431476.CrossRefGoogle Scholar
Ribet, K. A. 1995. Galois representations and modular forms. Bull. Amer. Math. Soc. (N.S.), 32(4), 375402.CrossRefGoogle Scholar
Riemann, B. 1859. Über die Anzahl der Primzahlen unter einer gegebenen Größe. Monatsber. Berliner Akademie, 671680.Google Scholar
Robles-Llana, D., Rocek, M., Saueressig, F., Theis, U., and Vandoren, S. 2007. Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry. Phys. Rev. Lett., 98, 211602.CrossRefGoogle Scholar
Rodier, F. 1975. Modèle de Whittaker et caractères de représentations. Pages 151171 of Carmona, J., Dixmier, J., and Vergne, M. (eds), Non-commutative Harmonic Analysis. Lecture Notes in Mathematics, vol. 466. Springer, Berlin, New York.CrossRefGoogle Scholar
Satake, I. 1963. Theory of spherical functions on reductive algebraic groups over p-adic fields. Inst. Hautes Études Sci. Publ. Math., 18, 169.CrossRefGoogle Scholar
Savin, G. K-types of minimal representations (p-adic case). www.math.utah.edu/∼savin/k-tipovi.pdf.Google Scholar
Savin, G., and Woodbury, M. 2007. Structure of internal modules and a formula for the spherical vector of minimal representations. J. Algebra, 312(2), 755772.CrossRefGoogle Scholar
Schimmrigk, R. 2011. Emergent spacetime from modular motives. Commun. Math. Phys., 303, 130.CrossRefGoogle Scholar
Schlotterer, O., and Stieberger, S. 2013. Motivic multiple zeta values and superstring amplitudes. J. Phys., A46, 475401.Google Scholar
Schmid, W. 2000. Automorphic distributions for SL(2, R). Pages 345387 of Dito, G., and Sternheimer, D. (eds), Conférence Moshé Flato 1999, Vol. I (Dijon). Mathematical Physics Studies, vol. 21. Kluwer Academic Publishers, Dordrecht.Google Scholar
Schulze-Pillot, R. 1998. Theta liftings: A comparison between classical and representation-theoretic results. Pages 142153 of Sugano, T. (ed.), Automorphic Forms and Number Theory. RIMS Kôkyûroku, vol. 1052. Research Institute for Mathematical Sciences, Kyoto University, Kyoto.Google Scholar
Schwarz, J. H., and Sen, A. 1993. Duality symmetries of 4-D heterotic strings. Phys. Lett., B312, 105114.CrossRefGoogle Scholar
Seiberg, N. 1988. Observations on the moduli space of superconformal field theories. Nucl. Phys., B303, 286304.CrossRefGoogle Scholar
Sekiguchi, J. 1987. Remarks on real nilpotent orbits of a symmetric pair. J. Math. Soc. Japan, 39(1), 127138.CrossRefGoogle Scholar
Sen, A. 1994. Strong–weak coupling duality in four-dimensional string theory. Int. J. Mod. Phys., A9, 37073750.CrossRefGoogle Scholar
Sen, A. 2008. Black hole entropy function, attractors and precision counting of microstates. Gen. Relativity Gravitation, 40, 22492431.CrossRefGoogle Scholar
Sen, A. 2009. Arithmetic of quantum entropy function. JHEP, 08, 068.CrossRefGoogle Scholar
Sen, A. 2010. Discrete information from CHL black holes. JHEP, 11, 138.CrossRefGoogle Scholar
Sen, A. 2014. Microscopic and macroscopic entropy of extremal black holes in string theory. Gen. Relativity Gravitation, 46, 1711.CrossRefGoogle Scholar
Serre, J.-P. 1973. A Course in Arithmetic. Graduate Texts in Mathematics, vol. 7. Springer, New York, Heidelberg.CrossRefGoogle Scholar
Serre, J.-P. 1987. Sur les représentations modulaires de degré 2 de Gal(Q/Q). Duke Math. J., 54(1), 179230.CrossRefGoogle Scholar
Sethi, S., and Stern, M. 1998. D-brane bound states redux. Commun. Math. Phys., 194, 675705.CrossRefGoogle Scholar
Shahidi, F. 1978. Functional equation satisfied by certain L-functions. Compos. Math., 37(2), 171207.Google Scholar
Shahidi, F. 1981. On certain L-functions. Amer. J. Math., 103(2), 297355.CrossRefGoogle Scholar
Shahidi, F. 1985. Local coefficients as Artin factors for real groups. Duke Math. J., 52(4), 9731007.CrossRefGoogle Scholar
Shahidi, F. 1990. A proof of Langlands’ conjecture on Plancherel measures: Complementary series for p-adic groups. Ann. of Math. (2), 132(2), 273330.CrossRefGoogle Scholar
Shahidi, F. 1996. Intertwining Operators, L-functions and Representation Theory. Lecture Notes of the Eleventh KAIST Mathematics Worskshop, vol. 11. Korea Advanced Institute of Science and Technology, Daejeon, South Korea.Google Scholar
Shahidi, F. 2002. Automorphic L-functions and functoriality. Pages 655666 of Tatsien, L. (ed.), Proceedings of the International Congress of Mathematicians, Beijing, 2002. vol. II. Higher Education Press, Beijing.Google Scholar
Shahidi, F. 2005. Infinite dimensional groups and automorphic L-functions. Pure Appl. Math. Q., 1(3, part 2), 683699.CrossRefGoogle Scholar
Shahidi, F. 2010. Eisenstein Series and Automorphic L-functions. American Mathematical Society Colloquium Publications, vol. 58. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Shalika, J. A. 1974. The multiplicity one theorem for GLn. Ann. of Math. (2), 100, 171193.CrossRefGoogle Scholar
Shenker, S. H. 1990. The strength of nonperturbative effects in string theory. Pages 191200 of Alvarez, O., Marinari, E., and Windey, P. (eds), Random Surfaces and Quantum Gravity: Proceedings, NATO Advanced Study Institute, Cargèse, France, May 27-June 2, 1990. NATO ASI Series B, vol. 262. Plenum Press, New York.Google Scholar
Shih, D., Strominger, A., and Yin, X. 2006a. Counting dyons in N=8 string theory. JHEP, 06, 037.CrossRefGoogle Scholar
Shih, D., Strominger, A., and Yin, X. 2006b. Recounting dyons in N=4 string theory. JHEP, 0610, 087.Google Scholar
Shimura, G. 1958. Correspondances modulaires et les fonctions ζ de courbes algébriques. J. Math. Soc. Japan, 10, 128.Google Scholar
Shimura, G. 1971. On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J., 43, 199208.CrossRefGoogle Scholar
Shimura, G. 1989. Yutaka Taniyama and his time: Very personal recollections. Bull. London Math. Soc., 21(2), 186196.CrossRefGoogle Scholar
Shintani, T. 1976. On an explicit formula for Class-1 ‘Whittaker Functions’ on GL(n) over p-adic fields. Proc. Japan. Acad., 52, 180182.Google Scholar
Shintani, T. 1979. On automorphic forms on unitary groups of order 3. Unpublished manuscript.Google Scholar
Sinha, A. 2002. The Ĝ4λ16 term in IIB supergravity. JHEP, 08, 017.CrossRefGoogle Scholar
Soulé, C. 2007. An introduction to arithmetic groups. Pages 247276 of Cartier, P., Moussa, P., Julia, B., and Vanhove, P. (eds), Frontiers in Number Theory, Physics, and Geometry II. Springer, Berlin.Google Scholar
Spaltenstein, N. 1982. Classes Unipotentes et Sous-groupes de Borel. Lecture Notes in Mathematics, vol. 946. Springer, Berlin, New York.CrossRefGoogle Scholar
Stade, E. 1990. On explicit integral formulas for GL(n, R)-Whittaker functions. Duke Math. J., 60(2), 313362. With an appendix by D. Bump, S. Friedberg and J. Hoffstein.CrossRefGoogle Scholar
Stanley, R. P. 1986. A baker’s dozen of conjectures concerning plane partitions. Pages 285293 of Labelle, G., and Leroux, P. (eds), Combinatoire Énumérative (Montreal, Que., 1985/Quebec, Que., 1985). Lecture Notes in Mathematics, vol. 1234. Springer, Berlin, New York.Google Scholar
Stelle, K. S. 1996. Lectures on supergravity p-branes. Pages 287339 of Gava, E., Masiero, A., Narain, K. S., Randjbar-Daemi, S., and Shafi, Q. (eds), High Energy Physics and Cosmology. Proceedings, Summer School, Trieste, Italy, June 10-July 26, 1996. World Scientific, Singapore.Google Scholar
Stieberger, S. 2011. Constraints on tree-level higher order gravitational couplings in superstring theory. Phys. Rev. Lett., 106, 111601.CrossRefGoogle ScholarPubMed
Stieberger, S. 2014. Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator. J. Phys., A47, 155401.Google Scholar
Stieberger, S., and Taylor, T. R. 2006a. Amplitude for N-gluon superstring scattering. Phys. Rev. Lett., 97, 211601.CrossRefGoogle ScholarPubMed
Stieberger, S., and Taylor, T. R. 2006b. Multi-gluon scattering in open superstring theory. Phys. Rev., D74, 126007.Google Scholar
Stieberger, S., and Taylor, T. R. 2014. Closed string amplitudes as single-valued open string amplitudes. Nucl. Phys., B881, 269287.CrossRefGoogle Scholar
Strominger, A., and Vafa, C. 1996. Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett., B379, 99104.CrossRefGoogle Scholar
Strominger, A., and Witten, E. 1985. New manifolds for superstring compactification. Commun. Math. Phys., 101, 341.CrossRefGoogle Scholar
Takhtajan, L. A. 1992. A simple example of modular forms as tau-functions for integrable equations. Theor. Math. Phys., 93, 13081317.CrossRefGoogle Scholar
Tamagawa, T. 1963. On the ζ-functions of a division algebra. Ann. of Math. (2), 77, 387405.CrossRefGoogle Scholar
Tate, J. T. 1967. Fourier analysis in number fields, and Hecke’s zeta-functions. Pages 305347 of Cassels, J. W. S., and Fröhlich, A. (eds), Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society, Brighton, September 1–17, 1965. London Mathematical Society, London.Google Scholar
Taylor, R., and Wiles, A. 1995. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3), 553572.CrossRefGoogle Scholar
Teitelboim, C. 1986a. Gauge invariance for extended objects. Phys. Lett., B167, 6368.CrossRefGoogle Scholar
Teitelboim, C. 1986b. Monopoles of higher rank. Phys. Lett., B167, 6972.CrossRefGoogle Scholar
Terras, A. 1985. Harmonic Analysis on Symmetric Spaces and Applications, I. Springer, New York.CrossRefGoogle Scholar
Terras, A. 1988. Harmonic Analysis on Symmetric Spaces and Applications, II. Springer, Berlin.CrossRefGoogle Scholar
Tokuyama, T. 1988. A generating function of strict Gel′fand patterns and some formulas on characters of general linear groups. J. Math. Soc. Japan, 40(4), 671685.CrossRefGoogle Scholar
Tong, D. 2009. String theory. arXiv:0908.0333 [hep-th].Google Scholar
Tseytlin, A. A. 1997. On non-abelian generalisation of the Born-Infeld action in string theory. Nucl. Phys., B501, 4152.CrossRefGoogle Scholar
Unterberger, A. 2011. Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms. Pseudo-Differential Operators: Theory and Applications, vol. 8. Birkhäuser/Springer, Basel.CrossRefGoogle Scholar
Vinberg, È. B. 1975. The classification of nilpotent elements of graded Lie algebras. Dokl. Akad. Nauk SSSR, 225(4), 745748.Google Scholar
Vinogradov, A., and Takhtadžjan, L. 1978. Theory of the Eisenstein series for the group SL(3, R) and its application to a binary problem, I: Fourier expansion of the highest Eisenstein series. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., 76, 552.Google Scholar
Wakimoto, M. 2001. Lectures on Infinite-Dimensional Lie Algebra. World Scientific Publishing, River Edge, NJ.CrossRefGoogle Scholar
Walcher, J. 2012. On the arithmetic of D-brane superpotentials: Lines and conics on the mirror quintic. Commun. Num. Theor. Phys., 6, 279337.CrossRefGoogle Scholar
Wallach, N. R. 2003. Generalized Whittaker vectors for holomorphic and quaternionic representations. Comment. Math. Helv., 78(2), 266307.CrossRefGoogle Scholar
Wang, Y., and Yin, X. 2015. Supervertices and non-renormalization conditions in maximal supergravity theories. arXiv:1505.05861 [hep-th].Google Scholar
Weil, A. 1929. L’arithmétique sur les courbes algébriques. Acta Math., 52(1), 281315.CrossRefGoogle Scholar
Weil, A. 1949. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55, 497508.CrossRefGoogle Scholar
Weil, A. 1965. Sur la formule de Siegel dans la théorie des groupes classiques. Acta Math., 113, 187.CrossRefGoogle Scholar
Weil, A. 1967. Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann., 168, 149156.CrossRefGoogle Scholar
Weil, A. 1995. Séries de Dirichlet et fonctions automorphes. Pages 547552 of Séminaire Bourbaki, Vol. 10, Exp. No. 346. Société Mathématique de France, Paris.Google Scholar
West, P. C. 2001. E(11) and M theory. Class. Quant. Grav., 18, 44434460.CrossRefGoogle Scholar
Wiles, A. 1995. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3), 443551.CrossRefGoogle Scholar
Wiles, A. 2006. The Birch and Swinnerton-Dyer conjecture. Pages 3141 of Carlson, J., Jaffe, A., and Wiles, A. (eds), The Millennium Prize Problems. Clay Mathematics Institute, Cambridge, MA.Google Scholar
Witten, E. 1982. Constraints on supersymmetry breaking. Nucl. Phys., B202, 253.CrossRefGoogle Scholar
Witten, E. 1988. Quantum field theory, Grassmannians, and algebraic curves. Commun. Math. Phys., 113, 529.CrossRefGoogle Scholar
Witten, E. 1995. String theory dynamics in various dimensions. Nucl. Phys., B443, 85126.CrossRefGoogle Scholar
Witten, E. 1996. Bound states of strings and p-branes. Nucl. Phys., B460, 335350.CrossRefGoogle Scholar
Witten, E. 2000. World sheet corrections via D instantons. JHEP, 02, 030.CrossRefGoogle Scholar
Witten, E. 2007. Gauge theory and wild ramification. arXiv:0710.0631 [hep-th].Google Scholar
Witten, E. 2010. Mirror Symmetry, Hitchin’s Equations, and Langlands Duality. Pages 113128 of Garcia-Prada, O., Bourguignon, J.-P., and Salamon, S. (eds), The Many Facets of Geometry: A Tribute to Nigel Hitchin. Oxford Science Publications. Oxford University Press, Oxford.CrossRefGoogle Scholar
Witten, E. 2015. More on gauge theory and geometric Langlands. arXiv:1506.04293 [hep-th].Google Scholar
Yi, P. 1997. Witten index and threshold bound states of D-branes. Nucl. Phys., B505, 307318.CrossRefGoogle Scholar
Zagier, D. 1984. L-series of elliptic curves, the Birch-Swinnerton-Dyer conjecture, and the class number problem of Gauss. Notices Amer. Math. Soc., 31(7), 739743.Google Scholar
Zagier, D. 1990. The Bloch-Wigner-Ramakrishnan polylogarithm function. Math. Ann., 286(1–3), 613624.CrossRefGoogle Scholar
Zagier, D. 1991. The Birch-Swinnerton-Dyer conjecture from a naive point of view. Pages 377389 of van der Geer, G., Oort, F., and Steenbrink, J. H. M. (eds), Arithmetic Algebraic Geometry (Texel, 1989). Progress in Mathematics, vol. 89. Birkhäuser Boston, Boston, MA.CrossRefGoogle Scholar
Zagier, D. 2008. Elliptic modular forms and their applications. Pages 1103 of Ranestad, K. (ed.), The 1–2–3 of Modular Forms: Lectures from the Summer School on Modular Forms and Their Applications, Held at Nordfjordied, Norway, June 2004. Universitext. Springer, Berlin.Google Scholar
Zelevinsky, A. V. 1980. Induced representations of reductive p-adic groups, II: On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4), 13(2), 165210.CrossRefGoogle Scholar
Zerbini, F. 2016. Single-valued multiple zeta values in genus 1 superstring amplitudes. Commun. Num. Theor. Phys., 10, 703737.CrossRefGoogle Scholar
Zhang, S.-W. 2010. Gross-Schoen cycles and dualising sheaves. Invent. Math., 179(1), 173.CrossRefGoogle Scholar
Zwiebach, B. 2004. A First Course in String Theory. Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×