We review modular forms, zeta functions, and the gamma and Euler beta function

After a general discussion of string amplitudes we compute the 4-scalar amplitude and discuss properties of the 4-graviton amplitude, such as resonances and Regge trajectories. Then we introduce background fields and briefly discuss world-sheet beta functions and marginal deformations. We explain how effective actions can be obtained from either scattering amplitudes or beta functions. The string frame and Einstein frame are introduced and the universal part of the string effective action for graviton, Kalb–Ramond field, and dilaton is given. We also discuss background independence.

We review several systems of natural units, and the relation between the gravitational or Planck scale and the string scale.

We review classical relativistic particles, and discuss how to formulate an action principle which is invariant under world-line reparametrizations and space-time Poincaré transformations. The role of constraints is analysed.

We review the key facts about Lie algebras, Lie groups and symmetric spaces used in this book.

We review Young tableux to the extent needed in this book.

We give a mini-review of representations of the Poincaré group, emphasizing the role of irreducible unitary representations in identifying ‘elementary particles’. Young tableaux are used to identify irreducible representations of the little group and thus the particle content of the excitation spectrum of a string.

We return to the topic of partition functions, this time from a space-time perspective. Open string partition functions are introduced as generalisations of particle partition functions. Closed string partition functions arise as a further generalisations, or alternatively by using previous results on CFT partition functions. We observe that through modular invariance theories of closed strings have a ‘built-in UV cut-off’.

We use the the supersymmetric harmonic oscillator to introduce supersymmetry, and to motivate how world-sheet supersymmetry leads to space-time fermions. After a short review of Poincaré Lie superalgebras, we introduce the RNS-model, both as an example of a supersymmetric field theory and as a stepping stone to type-II superstrings. We motivate the GSO-projection through modular invariance and show how the massless spectra as well as the D-branes of type-II string theories arise. Type-I and heterotic strings are briefly introduced, and we conclude with an overview of the dualities that relate the five modular invariant supersymmetric string theories to one another, and to eleven-dimensional M-theory.

We review finite and infinite-dimensional Gaussian integrals, and the integral exponential function.

We continue to carry out the programme of covariant quantisation, this time with emphasis on the massless closed string states: the graviton, the Kalb–Ramond or B-field, and the dilaton. The graviton is related to the linearised Einstein equations, the Kalb–Ramond field is used to discuss the properties and the dualisation of higher rank gauge fields, the dilaton serves as a first example of a ‘moduli field’ and its relation to the string coupling is explained.

We use circle and orbicircle compactifications to illustrate generic features of string compactifications, including non-abelian gauge symmetry enhancement, Kac–Moody algebras and the Higgs effect. T-duality is introduced for both closed and open strings. Toroidal compactification are discussed in some detail, and two-tori are used as a proxi for Calabi–Yau compactifications and mirror symmetry. We discuss the vacuum selection problem and introduce the concepts of ‘landscape’ and ‘swampland’.

We introduce light-cone quantisation, first for particles, then for strings, and show how this leads one to standard representatives for physical states. The shift in the ground state energy of the bosonic string is derived using both zeta-function regularisation and Lorentz covariance. The derivation of the critical dimension D=26 of the bosonic string using Lorentz covariance is provided through a guided exercise.