2.1.1

We start by providing background on vertex algebras. We recall their usual definition in §2.1.3, and then state a convenient alternative definition in §2.1.4. We also discuss a notion of weak vertex algebras, which will arise in our constructions below.

2.1.2 Formal power series

Let K be a field. For a finite-dimensional K-vector space $\Lambda $ , let $K {[\![} \Lambda {]\!]} $ be the K-algebra of formal power series on $\Lambda $ with coefficients in K, defined as

$$ \begin{align*} K {[\![} \Lambda {]\!]} = \prod_{n = 0}^\infty {\mathrm{Sym}}^n (\Lambda^\vee) . \end{align*} $$

Elements of $K {[\![} \Lambda {]\!]} $ are often written as $f (z)$ , with a variable $z \in \Lambda $ . Let $K (\!(\Lambda )\!)$ be the fraction field of $K {[\![} \Lambda {]\!]} $ . For a K-vector space V, define

$$ \begin{align*} V {[\![} \Lambda {]\!]} = \prod_{n = 0}^\infty V \underset{K}{\otimes} \mathrm{Sym}^n (\Lambda^\vee) , \qquad V (\!(\Lambda)\!) = V {[\![} \Lambda {]\!]} \underset{K {[\![} \Lambda {]\!]}}{\otimes} K (\!(\Lambda)\!) . \end{align*} $$

For finite-dimensional K-vector spaces $\Lambda _1, \dotsc , \Lambda _n$ , writing $\Lambda = \Lambda _1 \oplus \cdots \oplus \Lambda _n$ , we have a natural isomorphism $K {[\![} \Lambda {]\!]} \simeq K {[\![} \Lambda _1 {]\!]} \cdots {[\![} \Lambda _n {]\!]} $ , and an injective $K {[\![} \Lambda {]\!]} $ -module homomorphism

(2.1.2.1) $$ \begin{align} \iota_{\Lambda_1, \dotsc, \Lambda_n} \colon K (\!(\Lambda)\!) {\hookrightarrow} K (\!(\Lambda_1)\!) \cdots (\!(\Lambda_n)\!) , \end{align} $$

given by power series expansion in the given order. We also denote this map by $\iota _{z_1, \dotsc , z_n}$ if $z_i$ is a set of coordinates of $\Lambda _i$ . For example, we have

(2.1.2.2) $$ \begin{align} \iota_{z, w} \Big( \frac{1}{z + w} \Big) = \sum_{n = 0}^\infty {} (-1)^n \cdot \frac{w^n}{z^{n + 1}} , \end{align} $$

where the notation means that we take $\Lambda _1 \simeq \Lambda _2 \simeq K$ , and z, w are coordinates on $\Lambda _1$ , $\Lambda _2$ .

This also induces a map $\iota _{\Lambda _1, \dotsc , \Lambda _n} \colon V (\!(\Lambda )\!) \to V (\!(\Lambda _1)\!) \cdots (\!( \Lambda _n)\!)$ for any K-vector space V.

2.1.3 Vertex algebras

We recall the usual definition of a vertex algebra, taken from Frenkel and Ben-Zvi [Reference Frenkel and Ben-Zvi18: Definition 1.3.1].

A $\mathbb {Z}$ -graded vertex algebra over K is a quadruple $(V, 1, D, Y)$ , consisting of a $\mathbb {Z}$ -graded K-vector space V, an element $1 \in V$ of degree $0$ , a linear map $D \colon V \to V$ called the translation operator, and a linear map $Y (-, z) (-) \colon V \otimes V \to V (\!(z)\!)$ , satisfying the following axioms:

1. (i) (Unit) For all $a \in V$ , we have $Y (a, z) (1) \in a + z V {[\![} z {]\!]} $ , and $Y (1, z) (a) = a$ .

2. (ii) (Translation) For any $a, b \in V$ , we have

   (2.1.3.1) $$ \begin{align} [D, Y (a, z)] (b) = \frac{\partial}{\partial z} \, Y (a, z) (b), \end{align} $$
   where $[-,-]$ denotes the commutator, and we have $D (1) = 0$ .

3. (iii) (Locality) For homogeneous elements $a, b, c \in V$ , the elements

   $$ \begin{align*} Y (a, z) \circ Y (b, w) (c) & \in V (\!(z)\!) (\!(w)\!) , \\ (-1)^{|a| \, |b|} \cdot Y (b, w) \circ Y (a, z) (c) & \in V (\!(w)\!) (\!(z)\!) \end{align*} $$
   are expansions of the same element in $V {[\![} z, w {]\!]} \, [z^{-1}, w^{-1}, (z - w)^{-1}]$ , where $|{-}|$ denotes the $\mathbb {Z}$ -grading.

2.1.4 An equivalent definition

We introduce an alternative definition of a vertex algebra, following ideas from Kim [Reference Kim28]. This definition is perhaps more naturally motivated than the standard one, and will be important for generalizing Joyce vertex algebras to other situations below. See also Theorem 4.1.7 below for a functorial description of this approach.

A $\mathbb {Z}$ -graded vertex algebra over K is equivalently the data $(V, (X_n)_{n \geq 0})$ , consisting of a $\mathbb {Z}$ -graded K-vector space V and K-linear multiplication maps

$$ \begin{align*} X_n \colon V^{\otimes n} & \longrightarrow V {[\![} z_1, \dotsc, z_n {]\!]} \, [ (z_i - z_j)^{-1} ] , \\ a_1 \otimes \cdots \otimes a_n & \longmapsto X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) , \end{align*} $$

preserving grading, where $\deg z_i = -2$ , and we invert $z_i - z_j$ for $i \neq j$ . In particular, this includes a map $X_0 \colon K \to V$ , thought of as the unit. They should satisfy the following properties:

1. (i) (Unit) For any $a \in V$ , we have

   (2.1.4.1) $$ \begin{align} X_1 (a; 0) = a . \end{align} $$

   More precisely, we have $X_1 (a; z) \in a + z V {[\![} z {]\!]} \subset V {[\![} z {]\!]} $ .

2. (ii) (Commutativity) For any homogeneous elements $a_1, \dotsc , a_n \in V$ , and any permutation $\sigma \in \mathfrak {S}_n$ , we have

   (2.1.4.2) $$ \begin{align} X_n (a_{\sigma (1)}, \dotsc, a_{\sigma (n)}; z_{\sigma (1)}, \dotsc, z_{\sigma (n)}) = \pm X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) , \end{align} $$

   where the sign is determined by the Koszul sign rule: it is ‘ $-$ ’ if and only if $\sigma $ restricts to an odd permutation on the odd-graded elements.

3. (iii) (Associativity) For integers $m, n \geq 0$ and elements $b_1, \dotsc , b_m, a_1, \dotsc , a_n \in V$ , we have

   (2.1.4.3) $$ \begin{align} X_{n+1}& \Big( X_m (b_1, \dotsc, b_m; w_1, \dotsc, w_m), a_1, \dotsc, a_n; \ z_0, \dotsc, z_n \Big) \nonumber\\ &= \iota_{ \{ z_i \}, \{ w_j \} } \, X_{m+n} \big( b_1, \dotsc, b_m, a_1, \dotsc, a_n; z_0 + w_1, \dotsc, z_0 + w_m, z_1, \dotsc, z_n \big) , \end{align} $$
   where $\iota _{ \{ z_i \}, \{ w_j \} }$ is the map defined in (2.1.2.1).

As is common in the literature, we adopt the convenient notation

(2.1.4.4) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) = X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) , \end{align} $$

but the reader should be aware that this is not defined using the individual terms $a_i (z_i) = X_1 (a_i; z_i)$ .

2.1.5 Proof of the equivalence of definitions

Given a graded vertex algebra V as in §2.1.3, define the maps $X_n$ in §2.1.4 by

(2.1.5.1) $$ \begin{align} X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) = \iota _{z_1, \dotsc, z_n} ^{-1} [ Y (a_1, z_1) \circ \cdots \circ Y (a_n, z_n) (1) ] , \end{align} $$

where we take the unique preimage under the embedding

$$\begin{align*}\iota_{z_1, \dotsc, z_n} \colon V {[\![} z_1, \dotsc, z_n {]\!]} \, [(z_i - z_j)^{-1}] {\hookrightarrow} V (\!(z_1)\!) \cdots (\!(z_n)\!) . \end{align*}$$

By Kim [Reference Kim28: Theorem 3.14], this is well-defined, and satisfies the property §2.1.4 (iii). The properties §2.1.4 (i)–(ii) follow from §2.1.3 (i), (iii).

Conversely, given the data $(V, (X_n)_{n \geq 0})$ as in §2.1.4, we may define a $\mathbb {Z}$ -graded vertex algebra structure on V by setting

$$ \begin{align*} 1 = X_0 (1) , \qquad D (a) = \frac{\partial}{\partial z} \, X_1 (a; z) \Big|_{z = 0} , \qquad Y (a, z) (b) = X_2 (a, b; z, 0) \end{align*} $$

for all $a, b \in V$ . Verifying the axioms is straightforward except for (2.1.3.1), which we prove as follows. Applying the associativity relation (2.1.4.3) multiple times, we have

$$ \begin{align*} [D, Y (a, z)] (b) & = \frac{\partial}{\partial w} \Big( X_1 \big( X_2 (a, b; z, 0), w \big) - X_2 (a, X_1 (b, w); z, 0)\! \Big) \Big|_{w = 0} \\ & = \frac{\partial}{\partial w} \big( X_2 (a, b; z + w, w) - X_2 (a, b; z, w)\! \big) \Big|_{w = 0} \\ & = \frac{\partial}{\partial z} \, X_2 (a, b; z, 0) = \frac{\partial}{\partial z} \, Y (a, z) (b) , \end{align*} $$

as desired.

2.1.6 Remark

In fact, in §2.1.4, it is enough to require the operators $X_n$ for $n = 0, 1, 2, 3$ , and only require commutativity and associativity up to $3$ terms, since these are enough for converting to the usual definition, and we can then convert it back to obtain all the higher $X_n$ .

2.1.7 Weak vertex algebras

We introduce a generalized version of vertex algebras, where for the product $a_1 (z_1) \cdots a_n (z_n)$ , in addition to the usual poles at $z_i = z_j$ for $i \neq j$ , we also allow arbitrary poles along any divisor. As we will see, this type of structure arises naturally from geometry, in the set-up of §2.2.

A $\mathbb {Z}$ -graded weak vertex algebra over K is the data $(V, (X_n)_{n \geq 0})$ , consisting of a $\mathbb {Z}$ -graded K-vector space V and K-linear multiplication maps

$$ \begin{align*} X_n \colon V^{\otimes n} \longrightarrow V (\!(z_1, \dotsc, z_n)\!) , \end{align*} $$

preserving grading, where $\deg z_i = -2$ , satisfying the properties §2.1.4 (i)–(iii).

Therefore, such a weak vertex algebra is a vertex algebra if and only if the image of $X_n$ lies in the subspace $V {[\![} z_1, \dotsc , z_n {]\!]} \, [(z_i - z_j)^{-1}] \subset V (\!(z_1, \dotsc , z_n)\!)$ .

2.2.1

As we mentioned in the introduction, Joyce vertex algebras [Reference Joyce23–Reference Joyce25] are vertex algebra structures defined on the homology of certain moduli spaces. Here, we formulate a set of axioms for such moduli spaces.

We use the following terminology and notations:

• ○ A torus is a Lie group T isomorphic to $\mathrm {U} (1)^n$ for some $n \geq 0$ .

• ○ For a torus T, let $\Lambda ^T$ , $\Lambda _T$ be the weight and coweight lattices, $\Lambda ^T = \mathrm {Hom} (T, \mathrm {U} (1)\!)$ and $\Lambda _T = \mathrm {Hom} (\mathrm {U} (1), T)$ , both isomorphic to $\mathbb {Z}^{\dim T}$ .

• ○ $\mathsf {hCW}$ is the category whose objects are topological spaces that are homotopy equivalent to CW complexes, and morphisms are homotopy classes of continuous maps.

• ○ $K (X)$ is the topological K-theory of a space $X \in \mathsf {hCW}$ , defined as the abelian group $K (X) = \mathsf {hCW} (X, \mathrm {BU} \times \mathbb {Z})$ .

2.2.2 The category $\mathcal {T}$

For convenience of presentation, we introduce a category $\mathcal {T}$ of spaces with $\mathrm {B} T$ -actions for tori T, defined as follows:

• ○ Its objects are triples $(T, X, \odot )$ , where $T \simeq \mathrm {U} (1)^n$ is a torus, $X \in \mathsf {hCW}$ is a space, and

  $$ \begin{align*} \odot \colon \mathrm{B} T \times X \longrightarrow X \end{align*} $$
  is a map, such that it defines a $\mathrm {B} T$ -action on X in $\mathsf {hCW}$ .

• ○ A morphism $f \colon (T_1, X_1, \odot _1) \to (T_2, X_2, \odot _2)$ consists of a Lie group homomorphism $f^{\smash {\sharp }} \colon T_2 \to T_1$ , together with a $\mathrm {B} T_2$ -equivariant map $f \colon X_1 \to X_2$ in $\mathsf {hCW}$ .

We often abbreviate the triple $(T, X, \odot )$ as X, and call $\dim T$ the rank of such an object.

The category $\mathcal {T}$ admits finite products, giving a symmetric monoidal structure on $\mathcal {T}$ .

2.2.3 Weights in K-theory

Let $(T, X, \odot )$ be an object of $\mathcal {T}$ . For each weight $\lambda \in \Lambda ^T$ , let $K (X)_\lambda \subset K (X)$ be the subgroup of classes E of weight $\lambda $ , meaning that $\odot ^* (E) = L_\lambda \boxtimes E$ , where $L_\lambda \to \mathrm {B} T$ is the line bundle classified by the map $\lambda \colon \mathrm {B} T \to \mathrm {BU} (1)$ . Let

$$ \begin{align*} K^\circ (X) = \bigoplus_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}}} K (X)_\lambda \subset K (X) \end{align*} $$

be the subgroup of classes which admit finite weight decompositions.

2.2.4 The setting

We assume given the data $(X, \odot , \oplus , 0, \mathbb {T}_X)$ , where

• ○ $X \in \mathsf {hCW}$ is a space.

• ○ $\odot \colon \mathrm {BU} (1) \times X \to X$ is an action in $\mathsf {hCW}$ , giving an object $(\mathrm {U} (1), X, \odot )$ of $\mathcal {T}$ of rank $1$ .

• ○ $\oplus \colon X \times X \to X$ and $0 \colon {*} \to X$ are morphisms in $\mathcal {T}$ , with $\oplus ^\sharp \colon \mathrm {U} (1) \to \mathrm {U} (1)^2$ the diagonal map, defining a commutative monoid structure on X in $\mathcal {T}$ .

• ○ $\mathbb {T}_X \in K (X)_0 \subset K (X)$ is a class of weight $0$ , called the obstruction theory.

In this case, the function $\operatorname {vdim} = \operatorname {rank} (\mathbb {T}_X) \colon \unicode{x3c0} _0 (X) \to \mathbb {Z}$ is called the virtual dimension of X.

For each $n \geq 0$ , let $\oplus _{(n)} \colon X^n \to X$ be the n-fold product using $\oplus $ , and let $\mathbb {T}_{X^n} = \sum _i \mathrm {pr}_i^* (\mathbb {T}_X) \in K (X^n)$ , where $\mathrm {pr}_i \colon X^n \to X$ is the i-th projection for $i = 1, \dotsc , n$ . Define the (virtual) normal bundle of $\oplus _{(n)}$ as the class

(2.2.4.1) $$ \begin{align} {\mathbb{\nu}}_{(n)} = \oplus_{\smash{(n)}}^* (\mathbb{T}_X) - \mathbb{T}_{X^n} \quad {\in} \quad K (X^n) . \end{align} $$

We further assume the following condition:

• ○ For any integer $n \geq 0$ , we have

  (2.2.4.2) $$ \begin{align} {\mathbb{\nu}}_{(n)} \in K^\circ (X^n) \quad \text{and} \quad ({\mathbb{\nu}}_{(n)})_0 = 0 , \end{align} $$
  where $(-)_0$ denotes the part of $\mathrm {U} (1)^n$ -weight $0$ .

This is a technical condition to ensure that the equivariant Euler class $e_z^{-1} ({\mathbb{\nu}}_{(n)})$ in §2.3.1 is well-defined, and is satisfied in most of our examples. See Examples 2.2.6 and 2.2.7 below.

2.2.5 The topological realization of a stack

To give examples of the data in §2.2.4, we will use the topological realization functor

(2.2.5.1) $$ \begin{align} |{-}| \colon \mathsf{Fun} (\mathsf{Aff}_{\mathbb{C}}^{\smash{\mathrm{op}}}, \mathsf{S}) \longrightarrow \mathsf{S} , \end{align} $$

as in Blanc [Reference Blanc2: §3.1], where $\mathsf {Aff}_{\mathbb {C}}$ is the category of affine $\mathbb {C}$ -schemes, and $\mathsf {S}$ is the $\infty $ -category of spaces. This is defined as the left Kan extension of the functor $(-)^{\mathrm {an}} \colon \mathsf {Aff}_{\mathbb {C}} \to \mathsf {S}$ sending an affine $\mathbb {C}$ -scheme to the topological space of its analytification.

For a derived algebraic stack $\mathcal {X}$ over $\mathbb {C}$ , we denote by $|\mathcal {X}|$ the topological realization of its classical truncation defined as above.

2.2.6 Example

We give a class of examples of the data $(X, \odot , \oplus , 0, \mathbb {T}_X)$ as in §2.2.4.

Let $\mathcal {X}$ be a derived linear moduli stack over $\mathbb {C}$ in the sense of the author et al. [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11: §7.1] and [Reference Bu, Davison, Ibáñez Núñez, Kinjo and Pădurariu10: §2.4.6], so that it is equipped with a monoid structure and an action

$$ \begin{align*} \oplus \colon \mathcal{X} \times \mathcal{X} & \longrightarrow \mathcal{X} , \\ \odot \colon \mathrm{B} \mathbb{G}_{\mathrm{m}} \times \mathcal{X} & \longrightarrow \mathcal{X} , \end{align*} $$

satisfying higher coherence and compatibility conditions. Typical examples include the following:

• ○ The derived moduli stack of finite-dimensional representations of a quiver, possibly with relations or a potential.

• ○ The derived moduli stack of vector bundles or coherent sheaves on a smooth, projective $\mathbb {C}$ -variety.

Let $X = |\mathcal {X}|$ be the topological realization as in §2.2.5. The data $\odot , \oplus , 0$ on X are defined by those on $\mathcal {X}$ . The class $\mathbb {T}_X$ is defined as the class of the tangent complex $\mathbb {T}_{\mathcal {X}}$ of $\mathcal {X}$ .

The condition (2.2.4.2) usually holds in this case, since if $\mathcal {X}$ is a moduli stack of objects in a $\mathbb {C}$ -linear abelian category $\mathcal {A}$ , then we often have $\mathbb {T}_{\mathcal {X}} |_E \simeq \mathrm {Ext}_{\mathcal {A}}^{1+{\bullet }} (E, E)$ at a point $E \in \mathcal {A}$ , so $\oplus _{\smash {(n)}}^* (\mathbb {T}_{\mathcal {X}}) |_{E_1, \dotsc , E_n} \simeq \bigoplus _{i, j} \mathrm {Ext}_{\mathcal {A}}^{1+{\bullet }} (E_i, E_j)$ only has $\mathbb {G}_{\mathrm {m}}^n$ -weights of the form $e_i - e_j$ for $i, j \in \{ 1, \dotsc , n \}$ , and the weight $0$ part $\bigoplus _i \mathrm {Ext}_{\mathcal {A}}^{1+{\bullet }} (E_i, E_i)$ agrees with $\mathbb {T}_{\mathcal {X}^n} |_{E_1, \dotsc , E_n}$ .

2.2.7 Example

Another class of examples of $(X, \odot , \oplus , 0, \mathbb {T}_X)$ as in §2.2.4 are given as follows.

Let $\mathcal {C}$ be a $\mathbb {C}$ -linear dg-category of finite type in the sense of Toën and Vaquié [Reference Toën and Vaquié35: Definition 2.4], and let $\mathcal {X}$ be the derived moduli stack of right proper objects in $\mathcal {C}$ , as in [Reference Toën and Vaquié35: Theorem 3.6]. Examples include the following:

• ○ The derived moduli stack of finite-dimensional complexes of representations of a quiver.

• ○ The derived moduli stack of perfect complexes on a smooth, projective $\mathbb {C}$ -variety.

Let $X = |\mathcal {X}|$ be the topological realization of $\mathcal {X}$ , and define the data $\odot , \oplus , 0, \mathbb {T}_X$ using the scalar product, the direct sum, the zero object, and the tangent complex of $\mathcal {X}$ .

In this case, the condition (2.2.4.2) always holds, by the description of the tangent complex of $\mathcal {X}$ by Brav and Dyckerhoff [Reference Brav and Dyckerhoff6: Proposition 3.9].

For example, if $\mathcal {X} = \mathcal {P}\mkern -2mu\mathit {erf}$ is the classifying stack of perfect complexes over $\mathbb {C}$ , as in Toën and Vaquié [Reference Toën and Vaquié35: Proposition 3.7], then we have an equivalence $|\mathcal {P}\mkern -2mu\mathit {erf}| \simeq \mathrm {BU} \times \mathbb {Z}$ , by Blanc [Reference Blanc2: Theorems 4.5 and 4.21].

2.3.1 The equivariant Euler class

Let $(T, X, \odot ) \in \mathcal {T}$ , and let $E \in K^\circ (X)$ be a class such that the weight $0$ part $E_0$ is the class of a vector bundle on X, possibly of mixed rank. Define

(2.3.1.1) $$ \begin{align} e_z (E) = e (E_0) \cdot \prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}} \setminus \{ 0 \}} \sum_{i = 0}^\infty \lambda (z)^{\operatorname{rank} E_\lambda - i} \cdot c_i (E_\lambda) \quad {\in} \quad \prod_{k = 0}^\infty \mathrm{H}^{2 k} (X; \mathbb{Q}) ( z_1, \dotsc, z_n ) , \end{align} $$

where $z = (z_1, \dotsc , z_n)$ is a set of coordinates on $\Lambda _T$ , and $n = \dim T$ . For each $k \geq 0$ , the part of $e_z (E)$ lying in $\mathrm {H}^{2k}$ is a rational function in $z_1, \dotsc , z_n$ , with possible poles at $\lambda (z) = 0$ for $\lambda \in \Lambda ^T$ such that $E_\lambda $ is not the class of a vector bundle, although there may not be a uniform bound for the orders of poles as k increases.

We have the relation

$$ \begin{align*} e_z (E + F) = e_z (E) \, e_z (F) , \end{align*} $$

and in particular, if $E_0 = 0$ , then $e_z (E) \, e_z (-E) = 1$ , in which case we also write $e_z^{-1} (E) = e_z (-E)$ .

2.3.2 The translation operator

Let $(T, X, \odot ) \in \mathcal {T}$ . Define its translation operator

$$ \begin{align*} \tau (z) = \exp (z_1 D_1 + \cdots + z_n D_n) \colon \mathrm{H}_{{\bullet}} (X; \mathbb{Q}) \longrightarrow \mathrm{H}_{{\bullet}} (X; \mathbb{Q}) {[\![} z_1, \dotsc, z_n {]\!]} , \end{align*} $$

with $\deg z_i = -2$ and $z = (z_1, \dotsc , z_n)$ , where we choose an identification $T \simeq \mathrm {U} (1)^n$ , and define $D_i \colon \mathrm {H}_{\bullet } (X; \mathbb {Q}) \to \mathrm {H}_{{\bullet } + 2} (X; \mathbb {Q})$ by

$$ \begin{align*} D_i (a) = \odot_* (t_i \boxtimes a) , \end{align*} $$

where $t_i \in \mathrm {H}_2 (\mathrm {B} T; \mathbb {Q})$ is the generator of the i-th copy of $\mathrm {H}_2 (\mathrm {BU} (1); \mathbb {Q})$ , dual to the universal first Chern class in $\mathrm {H}^2 (\mathrm {BU} (1); \mathbb {Q})$ .

This construction does not depend on the choice of identification $T \simeq \mathrm {U} (1)^n$ , provided that the coordinates $z_i$ on $\Lambda _T$ are transformed accordingly.

2.3.3 Theorem

Let X be as in §2.2.4. Then the assignment

(2.3.3.1) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) = (\oplus_{(n)})_* \circ \tau (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap e_z^{-1} ({\mathbb{\nu}}_{(n)})\! \Big) \end{align} $$

equips the space $V = \mathrm {H}_{{\bullet } + 2 \operatorname {vdim}} (X; \mathbb {Q})$ with the structure of a $\mathbb {Z}$ -graded weak vertex algebra over $\mathbb {Q}$ , where $z = (z_1, \dotsc , z_n)$ , and $\tau (z)$ is the translation operator of the object $X^n \in \mathcal {T}$ .

The product (2.3.3.1) has poles along $\lambda (z) = 0$ for nonzero weights $\lambda \in \mathbb {Z}^n$ appearing in $\oplus _{(n)}^* (\mathbb {T}_X)$ . In particular, if the class $\oplus ^* (\mathbb {T}_X) \in K^\circ (X^2)$ only has $\mathrm {U} (1)^2$ -weights $(-1, 1),\, (0, 0),$ and $(1, -1),$ then V is a $\mathbb {Z}$ -graded vertex algebra.

Proof. The property (2.1.4.1) holds since $X_1 (a; 0) = \exp (0) (a) = a$ . The relation (2.1.4.2) holds since the definition (2.3.3.1) is symmetric (with a sign rule) in the indices $1, \dotsc , n$ .

We now verify (2.1.4.3). Write $a = a_1 \boxtimes \cdots \boxtimes a_n$ and $b = b_1 \boxtimes \cdots \boxtimes b_m$ for short. Expanding the left-hand side of (2.1.4.3), we obtain

$$ \begin{align*} & {\oplus}_* \circ \exp \big( z_0 D_j + z_i D_{m + i} \big) \Big[ \Big( \exp (w_j D_j) \big( b \cap e_w^{-1} ({\mathbb{\nu}}_{(m)})\! \big) \boxtimes a \Big) \cap e_{z_0, z}^{-1} ({\mathbb{\nu}}_{(n+1)}) \Big] \\[1ex]= {} & {\oplus}_* \circ \exp \big( z_0 D_j + z_i D_{m + i} \big) \Big[ \exp (w_j D_j) \Big(\! (b \boxtimes a) \cap e_w^{-1} \big(\!\oplus_{\smash{(m)}}^*\! (\mathbb{T}_X) - (\mathbb{T}_1 + \cdots + \mathbb{T}_m)\! \big)\! \Big) \\* & \hspace{2em} {} \cap e_{z_0, z}^{-1} \Big(\!\oplus^*\! (\mathbb{T}_X) - \big(\!\oplus_{\smash{(m)}}^*\! (\mathbb{T}_X) + \mathbb{T}_{m+1} + \cdots + \mathbb{T}_{m+n} \big)\! \Big) \Big] \\[1ex]= {} & {\oplus}_* \circ \exp \big( z_0 D_j + z_i D_{m + i} \big) \circ \exp (w_j D_j) \Big[ (b \boxtimes a) \cap e_w^{-1} \big(\!\oplus_{\smash{(m)}}^*\! (\mathbb{T}_X) - (\mathbb{T}_1 + \cdots + \mathbb{T}_m)\! \big) \\* & \hspace{2em} {} \cap e_{z_0 + w, z}^{-1} \Big(\!\oplus^*\! (\mathbb{T}_X) - \big(\!\oplus_{\smash{(m)}}^*\! (\mathbb{T}_X) + \mathbb{T}_{m+1} + \cdots + \mathbb{T}_{m+n} \big)\! \Big) \Big] \\[1ex]= {} & {\oplus}_* \circ \exp \big( z_0 D_j + z_i D_{m + i} \big) \circ \exp (w_j D_j) \Big[ (b \boxtimes a) \cap e_{z_0 + w, z}^{-1} \big(\!\oplus^*\! (\mathbb{T}_X) - ( \mathbb{T}_1 + \cdots + \mathbb{T}_{m+n} )\! \big) \Big] \\[1ex]= {} & {\oplus}_* \circ \exp \big(\! (z_0 + w_j) D_j + z_i D_{m + i} \big) \Big(\! (b \boxtimes a) \cap e_{z_0 + w, z}^{-1} ({\mathbb{\nu}}_{(m+n)})\! \Big) , \end{align*} $$

where $\oplus $ is short for $\oplus _{(m+n)}$ , and z is short for $z_1, \dotsc , z_n$ . The indices $i, j$ are implicitly summed over the ranges $1 \leq i \leq n$ and $1 \leq j \leq m$ , respectively. For $i = 1, \dotsc , m+n$ , we denote by $\mathbb {T}_i$ the pullback of $\mathbb {T}_X$ from the i-th copy of X. The second step uses Lemma 2.3.4 below, and the third step uses the fact that ${\mathbb{\nu}}_{(m)}$ has weight zero with respect to the diagonal torus, associated to the variable $z_0$ . This computation agrees with the right-hand side of (2.1.4.3).

For the final statement on usual vertex algebras, it is enough to show that for each $n \geq 0$ , the class $\oplus _{\smash {(n)}}^* (\mathbb {T}_X)$ can only have $\mathrm {U} (1)^n$ -weights of the form $e_i - e_j$ for $i, j \in \{ 1, \dotsc , n \}$ , where ${e_i = (0, \dotsc , 0, 1, 0, \dotsc , 0)}$ with $1$ at the i-th position. This is because the assumption implies that for any weight $(k_1, \dotsc , k_n)$ appearing in $\oplus _{\smash {(n)}}^* (\mathbb {T}_X)$ and any partition $\{ 1, \dotsc , n \} = I \sqcup J$ , writing $k_I = \sum _{i \in I} k_i$ and $k_J = \sum _{j \in J} k_j$ , we have $(k_I, k_J) \in \{ (-1, 1), (0, 0), (1, -1) \}$ , which implies the desired property.

2.3.4 Lemma

Let $(T, X, \odot ) \in \mathcal {T}$ , and let $E \in K^\circ (X)$ , such that its weight $0$ part $E_0$ is the class of a vector bundle on X. Then for any $a \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ , we have

(2.3.4.1) $$ \begin{align} \tau (w) (a) \cap e_z (E) = \iota_{z, w} \big( \tau (w) (a \cap e_{z + w} (E)\!)\! \big) \quad \in \quad \mathrm{H}_{\bullet} (X; \mathbb{Q}) (\!(z)\!) {[\![} w {]\!]} , \end{align} $$

where $w, z$ are sets of n variables, with $n = \dim T$ , and $\iota _{z, w}$ denotes the expansion map from $\mathrm {H}_{\bullet } (X; \mathbb {Q}) {[\![} w {]\!]} (z + w)$ to $\mathrm {H}_{\bullet } (X; \mathbb {Q}) (\!(z)\!) {[\![} w {]\!]} $ .

Proof. Choose an identification $T \simeq \mathrm {U} (1)^n$ , and let $\delta _i \in \mathrm {H}^2 (\mathrm {B} T; \mathbb {Z})$ be the universal first Chern class in the i-th copy of $\mathrm {BU} (1)$ , where $i = 1, \dotsc , n$ . We have

(2.3.4.2) $$ \begin{align} {\odot}^* \, e_z (E) & = \prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}}} \sum_{i = 0}^\infty \lambda (z)^{\operatorname{rank} E_\lambda - i} \cdot c_i (L_\lambda \boxtimes E_\lambda) \notag \\ & = \prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}}} \sum_{i = 0}^\infty \lambda (z)^{\operatorname{rank} E_\lambda - i} \cdot \sum_{j = 0}^i {} \binom{\operatorname{rank} E_\lambda - j}{i - j} \, \lambda (\delta)^{i - j} \boxtimes c_j (E_\lambda) \notag \\ & = \prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}}} \sum_{j = 0}^\infty {} \lambda (z + \delta)^{\operatorname{rank} E_\lambda - j} \cdot c_j (E_\lambda) = e_{z + \delta} (E) , \end{align} $$

where we write $\delta = (\delta _1, \dotsc , \delta _n)$ , and we expand $e_{z+\delta } (E)$ in nonnegative powers of $\delta _i$ . In the final line, we pull back E along the projection to the second factor, $\mathrm {B} T \times X \to X$ .

Let $t_i \in \mathrm {H}_2 (\mathrm {B} T; \mathbb {Z})$ be the dual of $\delta _i$ as in §2.3.2, and equip $\mathrm {H}_{\bullet } (\mathrm {B} T; \mathbb {Q})$ with a ring structure given by the group structure of $\mathrm {B} T$ in $\mathsf {hCW}$ . Then

$$ \begin{align*} \tau (w) (a) \cap e_z (E) & = \odot_* \Big[ \big( \exp (w_i t_i) \boxtimes a \big) \cap \odot^* \, e_z (E) \Big] \\& = \odot_* \Big[ (a \cap e_{z + \partial / \partial t} (E)\!) \exp (w_i t_i) \Big] \\& = \odot_* \Big[ \iota_{z, w} (a \cap e_{z + w} (E)\!) \exp (w_i t_i) \Big] = \iota_{z, w} \big( \tau (w) ( a \cap e_{z + w} (E)\! )\! \big) , \end{align*} $$

where $z + \partial / \partial t$ denotes the set of variables $(z_1 + \partial / \partial t_1, \dotsc , z_n + \partial / \partial t_n)$ , and we expand $e_{z + \partial / \partial t} (E)$ in nonnegative powers of $\partial / \partial t_i$ . The second step uses (2.3.4.2) and the fact that $\delta _i$ acts as $\partial / \partial t_i$ on $\mathrm {H}_{\bullet } (\mathrm {B} T; \mathbb {Q})$ . This proves the lemma.

2.3.5 Morphisms of Joyce vertex algebras

Finally, we mention a construction of morphisms of Joyce vertex algebras, as in Gross, Joyce, and Tanaka [Reference Gross, Joyce and Tanaka20: §2.5].

Theorem

Let $X, X'$ be two spaces as in §2.2.4, and let $f \colon X \to X'$ be a map in $\mathsf {hCW}$ , respecting the monoid structures and $\mathrm {BU} (1)$ -actions. Assume that the class

$$ \begin{align*} \mathbb{T}_f = \mathbb{T}_X - f^* (\mathbb{T}_{X'}) \end{align*} $$

is the class of a vector bundle on X, possibly of mixed rank. Then the map

$$ \begin{align*} Y_f \colon \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (X; \mathbb{Q}) & \longrightarrow \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (X'; \mathbb{Q}) , \\ a & \longmapsto f_* (a \cap e (\mathbb{T}_f)\!) \end{align*} $$

is a morphism of $\mathbb {Z}$ -graded weak vertex algebras, where e denotes the Euler class.

Moreover, if $f \colon X \to X'$ and $g \colon X' \to X''$ are maps as above, then $Y_{g \circ f} = Y_g \circ Y_f$ .

Proof. Let $a_1, \dotsc , a_n \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ . Then

$$ \begin{align*} & \phantom{{} = {}} Y_f (a_1) (z_1) \cdots Y_f (a_n) (z_n) \\[1ex] & = (\oplus_{(n)})_* \circ \tau (z) \smash{ \Big(\! (f^{\times n})_* \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap e (\mathbb{T}_{\smash{f}}^{\boxplus n})\! \Big) \cap e_z (\mathbb{T}_{\smash{X'}}^{\boxplus n} - \oplus_{\smash{(n)}}^* (\mathbb{T}_{X'})\!)\! \Big) } \\[1ex] & = (\oplus_{(n)})_* \circ \tau (z) \circ (f^{\times n})_* \smash{ \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap e_z \Big( \mathbb{T}_{\smash{X}}^{\boxplus n} - (f^{\times n})^* \circ \oplus_{\smash{(n)}}^* (\mathbb{T}_{X'})\! \Big) \Big) } \\[1ex] & = f_* \circ (\oplus_{(n)})_* \circ \tau (z) \smash{ \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap e_z (-{\mathbb{\nu}}_{(n)} + \oplus_{\smash{(n)}}^* (\mathbb{T}_f)\!)\! \Big) } \\[1ex] & = f_* \circ (\oplus_{(n)})_* \smash{ \Big( \tau (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap e_z (-{\mathbb{\nu}}_{(n)})\! \Big) \cap e (\oplus_{\smash{(n)}}^* (\mathbb{T}_f)\!)\! \Big) } \\[1ex] & = Y_f \big( a_1 (z_1) \cdots a_n (z_n)\! \big) , \end{align*} $$

where the second last step follows from Lemma 2.3.4 by setting $z = 0$ in the lemma, which is possible since $\mathbb {T}_f$ is a vector bundle, so $e_z (\oplus _{\smash {(n)}}^* (\mathbb {T}_f)\!)$ does not have negative powers of z.

The final statement is elementary.

3.1.1 Modules

We recall the definition of vertex algebra modules from Frenkel and Ben-Zvi [Reference Frenkel and Ben-Zvi18: §5.1].

Let $(V, 1, D, Y)$ be a $\mathbb {Z}$ -graded vertex algebra over K, as in §2.1.3. Then a V-module is a $\mathbb {Z}$ -graded K-vector space M, equipped with an operation $Y^M (-, z) (-) \colon V \otimes M \to M (\!(z)\!)$ , preserving grading with $\deg z = -2$ , satisfying the following conditions:

1. (i) (Unit) For any $m \in M$ , we have $Y^M (1, z) \, m = m$ .

2. (ii) (Associativity) For any $a, b \in V$ and $m \in M$ , the elements

   $$ \begin{align*} Y^M (a, z) \circ Y^M (b, w) (m) & \in M (\!(z)\!) (\!(w)\!) , \\ Y^M (Y (a, z - w) (b), w) (m) & \in M (\!(w)\!) (\!(z - w)\!) \end{align*} $$
   are expansions of the same element in $M {[\![} z, w {]\!]} \, [z^{-1}, w^{-1}, (z - w)^{-1}]$ .

3.1.2 An equivalent definition

We can also give an alternative definition of a vertex algebra module, in the style of §2.1.4.

Let $(V, (X_n)_{n \geq 0})$ be a $\mathbb {Z}$ -graded vertex algebra over K, in the sense of §2.1.4. Then a V-module is a $\mathbb {Z}$ -graded K-vector space M, equipped with operations

$$ \begin{align*} X_n^M \colon V^{\otimes n} \otimes M & \longrightarrow M {[\![} z_1, \dotsc, z_n {]\!]} \, [ z_i^{-1}, (z_i - z_j)^{-1} ] , \\ a_1 \otimes \cdots \otimes a_n \otimes m & \longmapsto X_n^M (a_1, \dotsc, a_n, m; z_1, \dotsc, z_n) , \end{align*} $$

preserving grading, where $\deg z_i = -2$ , and we invert $z_i - z_j$ for $i \neq j$ . They should satisfy the following properties:

1. (i) (Unit) We have $X_0^M = \mathrm {id}_M$ .

2. (ii) (Associativity) For integers $k, n \geq 0$ and elements $a_1, \dotsc , a_n, b_1, \dotsc , b_k \in V$ , we have

   (3.1.2.1) $$ \begin{align} & X_{n+1}^M \Big( X_k (b_1, \dotsc, b_k; w_1, \dotsc, w_k); a_1, \dotsc, a_n, m; z_0, \dotsc, z_n \Big) \notag \\[-.5ex] & \hspace{2em} = \iota_{ \{ z_i \}, \{ w_j \} } \, X_{n+k}^M \big( b_1, \dotsc, b_k, a_1, \dotsc, a_n, m; z_0 + w_1, \dotsc, z_0 + w_k, z_1, \dotsc, z_n \big), \end{align} $$
   (3.1.2.2) $$ \begin{align} & X_{n}^M \Big( a_1, \dotsc, a_n, X_k^M (b_1, \dotsc, b_k, m; w_1, \dotsc, w_k); z_1, \dotsc, z_n \Big) \notag \\[-.5ex] & \hspace{2em} = \iota_{ \{ z_i \}, \{ w_j \} } \, X_{n+k}^M \big( a_1, \dotsc, a_n, b_1, \dotsc, b_k, m; z_1, \dotsc, z_n, w_1, \dotsc, w_k \big) , \end{align} $$
   where the maps $\iota _{ \{ z_i \}, \{ w_j \} }$ are defined in (2.1.2.1).

Again, this definition may look more complicated than §3.1.1, but it will be more convenient for our purposes, such as for defining weaker notions of modules below.

We adopt the convenient notation

(3.1.2.3) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) \, m = X^M_n (a_1, \dotsc, a_n, m; z_1, \dotsc, z_n) , \end{align} $$

with the same caveats as those below (2.1.4.4) for vertex algebras.

3.1.3 Proof of the equivalence of definitions

We essentially follow the proof of Kim [Reference Kim28: Theorem 3.14], adapting it to the case of modules.

Given a V-module M in the sense of §3.1.1, define the maps $X_n^M$ by

(3.1.3.1) $$ \begin{align} X_n^M (a_1, \dotsc, a_n, m; z_1, \dotsc, z_n) = \iota_{z_1, \dotsc, z_n} ^{-1} \big( Y^M (a_1, z_1) \circ \cdots \circ Y^M (a_n, z_n) (m)\! \big) , \end{align} $$

where we take the unique preimage under the embedding

(3.1.3.2) $$ \begin{align} \iota_{z_1, \dotsc, z_n} \colon M {[\![} z_1, \dotsc, z_n {]\!]} \, [z_i^{-1}, (z_i - z_j)^{-1}] {\hookrightarrow} M (\!(z_1)\!) \cdots (\!(z_n)\!) . \end{align} $$

To see that such a preimage exists, note that for fixed elements $a_1, \dotsc , a_n, m$ as above, there exists $N> 0$ such that in the expression

(3.1.3.3) $$ \begin{align} \bigg( \prod_{1 \leq i < j \leq n} {} (z_i - z_j) \bigg)^N \cdot Y^M (a_1, z_1) \circ \cdots \circ Y^M (a_n, z_n) (m) , \end{align} $$

the order of the operators $Y^M (a_i, z_i)$ can be permuted freely without changing the result, up to an appropriate sign. Indeed, this is true when $n = 2$ , a proof of which can be found in Frenkel and Ben-Zvi [Reference Frenkel and Ben-Zvi18: Remark 5.1.4], and the general case follows from swapping two adjacent operators at a time.

This property implies that the expression (3.1.3.3) must lie in the intersection of the spaces $M (\!(z_{\sigma (1)})\!) \cdots (\!(z_{\sigma (n)})\!) \subset M {[\![} z_i^{\pm 1} {]\!]} $ for all $\sigma \in \mathfrak {S}_n$ , which is $M {[\![} z_i {]\!]} \, [z_i^{-1}]$ . Consequently, the element $Y^M (a_1, z_1) \circ \cdots \circ Y^M (a_n, z_n) (m)$ lies in the image of (3.1.3.2).

From here, verifying the axioms in §3.1.2 is straightforward.

Conversely, given the data $(M, (X_n^M)_{n \geq 0})$ , define

(3.1.3.4) $$ \begin{align} Y^M (a, z) (m) = X^M_1 (a, m; z) \end{align} $$

for all $a \in V$ and $m \in M$ . Then this defines a vertex algebra module structure on M, where the property §3.1.1 (i) holds since $Y^M (1, z) (m) = X^M_1 (X_0 (1), m; z) = X^M_0 (m) = m$ .

3.1.4 Remark

In §3.1.2, it is enough to require the operators $X_n^M$ for $n = 0, 1, 2$ , and the higher ones can be recovered from them, in a similar way to §2.1.6.

3.1.5 Weak modules

Similarly to weak vertex algebras introduced in §2.1.7, we define a weak module over a $\mathbb {Z}$ -graded weak vertex algebra $(V, (X_n)_{n \geq 0})$ to be a $\mathbb {Z}$ -graded vector space M, equipped with multiplication maps

$$ \begin{align*} X^M_n \colon V^{\otimes n} \otimes M \longrightarrow M (\!(z_1, \dotsc, z_n)\!) , \end{align*} $$

preserving grading, where $\deg z_i = -2$ , satisfying the properties §3.1.2 (i)–(ii).

For example, every weak vertex algebra is a weak module over itself.

Note that if V is a vertex algebra, then a weak module is a usual module if and only if the image of $X^M_n$ lies in the subspace $M {[\![} z_1, \dotsc , z_n {]\!]} \, [z_i^{-1}, (z_i - z_j)^{-1}] \subset M (\!(z_1, \dotsc , z_n)\!)$ .

3.1.6 Twisted modules

We introduce twisted modules for vertex algebras equipped with involutions, which are a special class of weak modules. They will play an important role in the theory of wall-crossing for orthosymplectic enumerative invariants. Note that these are different from the notion of twisted modules in, for example, Frenkel and Ben-Zvi [Reference Frenkel and Ben-Zvi18: §5.6].

Let V be a $\mathbb {Z}$ -graded vertex algebra over K. Suppose that V is equipped with a twisted involution $(-)^\vee \colon V \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} V^{\mathrm {op}}$ , where $V^{\mathrm {op}}$ is the vertex algebra with the same underlying graded vector space as V, with multiplication

(3.1.6.1) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) \text{ in } V^{\mathrm{op}} \quad {=} \quad a_1 (-z_1) \cdots a_n (-z_n) \text{ in } V . \end{align} $$

We require that $(-)^\vee $ is an isomorphism of $\mathbb {Z}$ -graded vertex algebras, and $(-)^{\vee \vee } = \mathrm {id}_V$ .

Define a twisted module for V as a weak module $(M, (X_n^M)_{n \geq 0})$ , such that the image of each $X^M_n$ lies in the subspace

$$ \begin{align*} M {[\![} z_1, \dotsc, z_n {]\!]} \, [z_i^{-1}, (z_i \pm z_j)^{-1}] \subset M (\!(z_1, \dotsc, z_n)\!) , \end{align*} $$

where we invert $z_i \pm z_j$ for $i \neq j$ , and such that

(3.1.6.2) $$ \begin{align} a^\vee (z) \, m = a (-z) \, m \end{align} $$

for $a \in V$ and $m \in M$ .

We will construct examples of twisted modules in §3.2 below.

3.1.7 Residues

Let V be a $\mathbb {Z}$ -graded vertex algebra. Recall that the quotient $L = V / D (V)$ , where D is the translation operator in §2.1.3, admits a $\mathbb {Z}$ -graded Lie algebra structure given by

(3.1.7.1) $$ \begin{align} [a, b] = \mathrm{res}_{z = w} \, a (z) \, b (w) , \end{align} $$

with sign rules implemented for odd elements in the axioms of a Lie algebra. A usual module for V gives rise to a module for L, defined by $a \cdot m = \mathrm {res}_{z = 0} \, a (z) \, m$ .

Now, suppose that V is equipped with an involution $(-)^\vee $ as in §3.1.6, and let M be a twisted module for V. Then the Lie algebra $L = V / D (V)$ admits an induced involution $(-)^\vee \colon L \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} L^{\mathrm {op}}$ , where $L^{\mathrm {op}}$ is the Lie algebra with the opposite Lie bracket $[a, b]^{\mathrm {op}} = -[a, b]$ . Moreover, the assignment

(3.1.7.2) $$ \begin{align} a \mathbin{\heartsuit} m = \mathrm{res}_{z = 0} \, a (z) \, m \end{align} $$

for $a \in V$ and $m \in M$ establishes M as a twisted module for L, in that we have

(3.1.7.3) $$ \begin{align} \qquad\qquad\qquad a^\vee \mathbin{\heartsuit} m & = -a \mathbin{\heartsuit} m , \end{align} $$

(3.1.7.4) $$ \begin{align} a \mathbin{\heartsuit} (b \mathbin{\heartsuit} m) - (-1)^{|a| \cdot |b|} \cdot b \mathbin{\heartsuit} (a \mathbin{\heartsuit} m) & = [a, b] \mathbin{\heartsuit} m - [a^\vee, b] \mathbin{\heartsuit} m \end{align} $$

for homogeneous elements $a, b \in L$ and $m \in M$ , where $|{-}|$ denotes the degree. The identity (3.1.7.4) can be seen by applying the residue theorem to $a (z) \, b (w) \, m$ , giving

(3.1.7.5) $$ \begin{align} \mathrm{res}_{z = 0} \, \mathrm{res}_{w = 0} \, a (z) \, b (w) \, m = \mathrm{res}_{w = 0} \, \big( \mathrm{res}_{z = 0} + \mathrm{res}_{z = w} + \mathrm{res}_{z = -w} \big) \, a (z) \, b (w) \, m , \end{align} $$

where each of the four terms corresponds to a term in (3.1.7.4), and we use the fact that $a (z) \, b (w) \, m = a^\vee (-z) \, b (w) \, m$ .

Note that this structure of a twisted module for a Lie algebra also appears in motivic Donaldson–Thomas theory for orthosymplectic structure groups, described in the author [Reference Bu9: §5.2.2]. Also, as noted there, when the coefficient ring contains $1/2$ , a twisted module for L is equivalent to a usual module for the subalgebra of L consisting of elements $a \in L$ with $a = -a^\vee $ , with the action $a \cdot m = (1/2) (a \mathbin {\heartsuit } m)$ .

3.2.1 The setting

We assume given the data $(X, \odot , \oplus , 0, \mathbb {T}_X)$ as in §2.2.4, We also assume given the extra data $(Y, \diamond , \mathbb {T}_Y)$ , where

• ○ $Y \in \mathsf {hCW}$ is a space, regarded as an object of $\mathcal {T}$ of rank $0$ .

• ○ $\diamond \colon X \times Y \to Y$ is a map, exhibiting Y as an X-module in $\mathsf {hCW}$ , or equivalently, in $\mathcal {T}$ .

• ○ $\mathbb {T}_Y \in K (Y)$ is a class, called the obstruction theory.

For each integer $n \geq 0$ , let

(3.2.1.1) $$ \begin{align} \diamond_{(n)} \colon X^n \times Y \longrightarrow Y \end{align} $$

denote the n-fold module multiplication map. Define the normal bundle of $\diamond _{(n)}$ by

(3.2.1.2) $$ \begin{align} {\mathbb{\nu}}_{\diamond, \smash{(n)}} = \diamond_{\smash{(n)}}^* (\mathbb{T}_{Y}) - \mathbb{T}_{X^n \times Y} \in K (X^n \times Y) , \end{align} $$

where $\mathbb {T}_{X^n \times Y} = \mathrm {pr}_{\smash {X^n}}^* (\mathbb {T}_{X^n}) + \mathrm {pr}_Y^* (\mathbb {T}_{Y})$ , where $\mathrm {pr}_{X^n}$ , $\mathrm {pr}_Y$ are the projections to $X^n$ and Y. We further assume the following condition:

• ○ For any integer $n \geq 0$ , we have

  (3.2.1.3) $$ \begin{align} {\mathbb{\nu}}_{\diamond, \smash{(n)}} \in K^\circ (X^n \times Y) \quad \text{and} \quad ({\mathbb{\nu}}_{\diamond, \smash{(n)}})_0 = 0 , \end{align} $$
  where $(-)_0$ denotes the part of $\mathrm {U} (1)^n$ -weight $0$ .

3.2.2 Theorem

Let $X, Y$ be as in §3.2.1. Then the assignment

(3.2.2.1) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) \, m = (\diamond_{(n)})_* \circ \exp (z_1 D_1 + \cdots + z_n D_n) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n \boxtimes m) \cap e_z^{-1} ({\mathbb{\nu}}_{\diamond, \smash{(n)}})\! \Big) \end{align} $$

defines a weak module structure on $M = \mathrm {H}_{{\bullet } + 2 \operatorname {vdim}} (Y; \mathbb {Q})$ for the weak vertex algebra $V = \mathrm {H}_{{\bullet } + 2 \operatorname {vdim}} (X; \mathbb {Q})$ in Theorem 2.3.3.

The product (3.2.2.1) has poles along $\lambda (z) = 0$ for nonzero weights $\lambda \in \mathbb {Z}^n$ appearing in $\diamond _{(n)}^* (\mathbb {T}_Y)$ . In particular, if the class $\oplus ^* (\mathbb {T}_X) \in K^\circ (X^2)$ only has $\mathrm {U} (1)^2$ -weights $(-k, k)$ with $|k| \leq 1$ , so that V is a vertex algebra, and if the class $\diamond _{\smash {(2)}}^* (\mathbb {T}_Y) \in K^\circ (X^2 \times Y)$ only has $\mathrm {U} (1)^2$ -weights $(k, \ell )$ with $|k| + |\ell | \leq 1$ , then M is a module for V.

3.2.3 An involution-twisted version

We now discuss a version that often arises from moduli spaces involving the orthogonal and symplectic groups, which will be important in studying enumerative invariants for such moduli spaces.

Let $(X, Y)$ be as in §3.2.1. We further assume given the following data:

• ○ An involution $(-)^\vee \colon X \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} X$ , such that $(-)^{\vee \vee } = \mathrm {id}$ in $\mathsf {hCW}$ .

It should satisfy the following conditions:

• ○ $(-)^\vee $ preserves the monoid structure $\oplus $ .

• ○ $(-)^\vee $ reverses the $\mathrm {BU} (1)$ -action $\odot $ , in that it is equivariant with respect to the map $(-)^{-1} \colon \mathrm {BU} (1) \to \mathrm {BU} (1)$ .

• ○ $(-)^\vee $ preserves the module action $\diamond $ , in that ${\diamond } \circ (\! (-)^\vee \times \mathrm {id}_Y ) = {\diamond }$ .

Again, these are conditions in $\mathsf {hCW}$ , and do not require higher coherence.

3.2.4 Example

Our main source of examples of the situation of §3.2.3 is orthosymplectic enumerative geometry, as in the author [Reference Bu9].

Let $\mathcal {X}$ be a self-dual derived linear moduli stack over $\mathbb {C}$ , that is a derived linear moduli stack $\mathcal {X}$ as in Example 2.2.6, equipped with an involution $(-)^\vee \colon \mathcal {X} \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathcal {X}$ , satisfying conditions similar to those in §3.2.3, as in [Reference Bu9: §2.2.5]. Typical examples include the following:

• ○ The derived moduli stack of representations of a self-dual quiver Q, possibly with potential, where being self-dual means that we are given a contravariant involution $(-)^\vee \colon Q \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} Q^{\mathrm {op}}$ of Q, which induces an involution of the stack. See §6.3 below for details, and see also [Reference Bu9: §6.1].

• ○ The derived moduli stack of objects in a quasi-abelian subcategory $\mathcal {A} \subset \mathsf {Perf} (Z)$ , where Z is a smooth projective $\mathbb {C}$ -variety, such that $\mathcal {A}$ is stable under the dual functor $(-)^\vee = \mathbb {R} {\mathcal {H}\mkern -3mu\mathit {om}} (-, L) [s] \colon \mathsf {Perf} (Z) \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathsf {Perf} (Z)^{\mathrm {op}}$ , where we choose a line bundle $L \to Z$ and an integer s. See also [Reference Bu9: §6.2] for this set-up.

In this case, the derived fixed locus of the involution $\mathcal {X}^{\mathrm {sd}} = \mathcal {X}^{\mathbb {Z}_2}$ is the moduli stack of self-dual objects, whose points correspond to pairs $(x, \phi )$ with $x \in \mathcal {X}$ and $\phi \colon x \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} x^\vee $ , such that $\phi = \phi ^\vee $ . Such objects can often be interpreted as orthogonal or symplectic objects in a linear category, where the difference between being orthogonal and symplectic is encoded in the higher coherence data of the involution on $\mathcal {X}$ as a $\mathbb {Z}_2$ -action. Again, see [Reference Bu9] for details.

We have an action $\diamond \colon \mathcal {X} \times \mathcal {X}^{\mathrm {sd}} \to \mathcal {X}^{\mathrm {sd}}$ given by $(x, y) \mapsto x \oplus y \oplus x^\vee $ , where $x \oplus x^\vee $ is equipped with the obvious self-dual structure.

We now set $X = |\mathcal {X}|$ and $Y = |\mathcal {X}^{\mathrm {sd}}|$ to be the topological realizations, and with the involution on X and the X-action on Y given by the involution on $\mathcal {X}$ and its action on $\mathcal {X}^{\mathrm {sd}}$ . Let $\mathbb {T}_X$ , $\mathbb {T}_Y$ be the classes of the tangent complexes $\mathbb {T}_{\mathcal {X}}$ , $\mathbb {T}_{\smash {\mathcal {X}^{\mathrm {sd}}}}$ , where we note that $\mathbb {T}_{\smash {\mathcal {X}^{\mathrm {sd}}}}$ is given by the $\mathbb {Z}_2$ -invariant part of the pullback of $\mathbb {T}_{\mathcal {X}}$ to $\mathcal {X}^{\mathrm {sd}}$ .

Then, assuming that X satisfies the conditions in §2.2.4, all the extra conditions in §3.2.1 and 3.2.3 will be automatically satisfied, where the weight condition (3.2.1.3) follows from the similar condition (2.2.4.2) for $\mathbb {T}_{\mathcal {X}}$ and the above description of $\mathbb {T}_{\smash {\mathcal {X}^{\mathrm {sd}}}}$ .

3.2.5 Example

We also consider a derived version of Example 3.2.4, similar to Example 2.2.7.

Namely, let $\mathcal {C}$ be a $\mathbb {C}$ -linear dg-category as in Example 2.2.7, and suppose we are given a contravariant involution $(-)^\vee \colon \mathcal {C} \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathcal {C}^{\mathrm {op}}$ , inducing a $\mathbb {Z}_2$ -action on the derived moduli stack, $(-)^\vee \colon \mathcal {X} \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathcal {X}$ . Examples of $\mathcal {C}$ include the following:

• ○ The derived category of representations of a self-dual quiver, with the involution induced by the self-dual structure of the quiver.

• ○ The category of perfect complexes on a smooth projective $\mathbb {C}$ -variety Z, with the involution $(-)^\vee = \mathbb {R} {\mathcal {H}\mkern -3mu\mathit {om}} (-, L) [s]$ as in Example 3.2.4.

Again, the derived fixed locus $\mathcal {X}^{\mathrm {sd}} = \mathcal {X}^{\mathbb {Z}_2}$ is the moduli stack of self-dual objects, and we have an action $\diamond \colon \mathcal {X} \times \mathcal {X}^{\mathrm {sd}} \to \mathcal {X}^{\mathrm {sd}}$ given by $(x, y) \mapsto x \oplus y \oplus x^\vee $ .

Set $X = |\mathcal {X}|$ and $Y = |\mathcal {X}^{\mathrm {sd}}|$ , and let $\mathbb {T}_X$ , $\mathbb {T}_Y$ be the classes of the tangent complexes $\mathbb {T}_{\mathcal {X}}$ , $\mathbb {T}_{\smash {\mathcal {X}^{\mathrm {sd}}}}$ . Then all the conditions in §2.2.4, §3.2.1, and §3.2.3 are automatically satisfied.

3.2.6 Theorem

Let $X, Y$ be as in §3.2.3. Assume the class $\oplus ^* (\mathbb {T}_X) \in K^\circ (X^2)$ only has $\mathrm {U} (1)^2$ -weights $(-k, k)$ with $|k| \leq 1$ , so that $V = \mathrm {H}_{{\bullet } + 2 \operatorname {vdim}} (X; \mathbb {Q})$ is a vertex algebra by Theorem 2.3.3, and the class $\diamond _{\smash {(2)}}^* (\mathbb {T}_Y) \in K^\circ (X^2 \times Y)$ only has $\mathrm {U} (1)^2$ -weights $(k, \ell )$ with $|k| + |\ell | \leq 2$ .

Then the product (3.2.2.1) establishes $M = \mathrm {H}_{{\bullet } + 2 \operatorname {vdim}} (Y; \mathbb {Q})$ as a twisted module for the vertex algebra V.

3.2.7 Morphisms of modules

Finally, we state a result analogous to Theorem 2.3.5, which gives a construction of morphisms between modules for Joyce vertex algebras.

Theorem. Let $(X, Y)$ and $(X', Y')$ be as in §3.2.1, $f \colon X \to X'$ be a map as in Theorem 2.3.5, and $g \colon Y \to Y'$ a map compatible with the actions of X and $X'$ , such that the class

$$ \begin{align*} \mathbb{T}_g = \mathbb{T}_Y - g^* (\mathbb{T}_{Y'}) \end{align*} $$

is the class of a vector bundle on Y, possibly of mixed rank. Then the map

$$ \begin{align*} Y_g \colon \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (Y; \mathbb{Q}) & \longrightarrow \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (Y'; \mathbb{Q}) , \\ a & \longmapsto g_* (a \cap e (\mathbb{T}_g)\!) \end{align*} $$

is compatible with the weak vertex algebra module structures, in the sense that

$$ \begin{align*} Y_g \big( a_1 (z_1) \cdots a_n (z_n) \, m \big) = Y_f (a_1) (z_1) \cdots Y_f (a_n) (z_n) \, Y_g (m) \end{align*} $$

for any $a_1, \dotsc , a_n \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ and $m \in \mathrm {H}_{\bullet } (Y; \mathbb {Q})$ . In particular, if $X = X'$ and $f = \mathrm {id}_X$ , then g is a homomorphism of weak modules.

The proof is very similar to that of Theorem 2.3.5, and we omit it here. See Theorem 4.2.3 below for a proof of a more general result.

4.1.1 Paracategories

A paracategory is, roughly speaking, a ‘category with partially defined composition’. The following definition is adapted from Hermida and Mateus [Reference Hermida and Mateus22: §2]. Compare the notion of a relaxed multilinear category in Borcherds [Reference Borcherds and Kashiwara4: §4].

A paracategory $\mathcal {C}$ is a directed graph $s, t \colon \mathcal {C}_1 \rightrightarrows \mathcal {C}_0$ , together with a partially defined function $\circ _n \colon \mathcal {C}_n \dashrightarrow \mathcal {C}_1$ for all $n \geq 0$ , meaning a function from a subset of the n-fold fibre product set $\mathcal {C}_n = \mathcal {C}_1 \times _{s, \mathcal {C}_0, t} \cdots \times _{s, \mathcal {C}_0, t} \mathcal {C}_1$ to $\mathcal {C}_1$ , satisfying the following axioms:

1. (i) The map $\circ _0 \colon \mathcal {C}_0 \to \mathcal {C}_1$ is fully defined. That is, identity morphisms always exist.

2. (ii) The map $\circ _1 \colon \mathcal {C}_1 \to \mathcal {C}_1$ is fully defined, and is the identity map.

3. (iii) For $(x_1, \dotsc , x_n) \in \mathcal {C}_{k_1} \times _{\mathcal {C}_0} \cdots \times _{\mathcal {C}_0} \mathcal {C}_{k_n}$ , we have

   (4.1.1.1) $$ \begin{align} \circ_{k_1 + \cdots + k_n} (x_1, \dotsc, x_n) = \circ_n (\circ_{k_1} x_1, \dotsc, \circ_{k_n} x_n) , \end{align} $$
   in the sense that whenever one side of the equality is defined, so is the other, and both sides are equal.

Any (small) category can be seen as a paracategory in the obvious way.

Note that the axioms imply that composition with an isomorphism is always defined, since for example, if $x \in \mathcal {C}_1$ is an isomorphism, and $y \in \mathcal {C}_1$ is such that $s (y) = t (x)$ , then $y = y \circ (x \circ x^{-1}) = (y \circ x) \circ x^{-1}$ , showing that $y \circ x$ is defined.

A functor of paracategories is a map of directed graphs $f \colon \mathcal {C} \to \mathcal {D}$ , such that for any $x \in \mathcal {C}_k$ , if $\circ _k x$ is defined, then $\circ _k f (x)$ is defined and is equal to $f (\circ _k x)$ . Note that such functors can always be composed.

A natural isomorphism of functors $f, g \colon \mathcal {C} \to \mathcal {D}$ consists of an isomorphism $\eta _x \colon f (x) \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} g (x)$ for any $x \in \mathcal {C}_0$ , such that the naturality squares commute. Natural isomorphisms can always be composed.

4.1.2 Monoidal paracategories

A (symmetric) monoidal structure on a paracategory $\mathcal {C}$ consists of an object $1 \in \mathcal {C}_0$ and a functor $\otimes \colon \mathcal {C} \times \mathcal {C} \to \mathcal {C}$ , equipped with natural isomorphisms satisfying the usual axioms.

One can also define (symmetric) monoidal functors between (symmetric) monoidal paracategories in the usual way.

For a (coloured) operad $\mathcal {O}$ and a symmetric monoidal paracategory $\mathcal {C}$ , an $\mathcal {O}$ -algebra in $\mathcal {C}$ is a symmetric monoidal functor $\mathcal {O}^\otimes \to \mathcal {C}$ , where $\mathcal {O}^\otimes $ is the underlying symmetric monoidal category of $\mathcal {O}$ , as in Lurie [Reference Lurie31: Remark 2.2.4.6].

For example, an associative algebra in $\mathcal {C}$ can be described as an object $x \in \mathcal {C}$ , with n-ary multiplication maps $m_n \colon x^{\otimes n} \to x$ for all $n \geq 0$ , satisfying the usual axioms. A commutative algebra is an associative algebra such that the maps $m_n$ are equivariant under permutations.

4.1.3 Vertex spaces

We now define a symmetric monoidal paracategory of vertex spaces, where vertex algebras are algebra objects.

Define a vertex space over a field K as a triple $(\Lambda , V, \tau )$ , consisting of the following:

• ○ A finite-dimensional K-vector space $\Lambda $ , whose dimension is called the rank of the vertex space.

• ○ A $\mathbb {Z}$ -graded K-vector space V, thought of as a space of fields on $\Lambda $ .

• ○ A K-linear map $\tau (z) \colon V \to V {[\![} \Lambda {]\!]} $ , called the translation operator, where z is a set of coordinates for $\Lambda $ , and $\deg \Lambda ^\vee = -2$ , thought of as translating fields along z. It should satisfy

  (4.1.3.1) $$ \begin{align} \tau (0) & = \mathrm{id}_V ,\end{align} $$
  (4.1.3.2) $$ \begin{align} \tau (z) \circ \tau (w) & = \tau (z + w) . \end{align} $$

  In other words, $\tau $ is a representation of the formal group $\hat {\mathbb {G}}_{\mathrm {a}}^r$ of the additive group of $\Lambda $ .

We often write V for the vertex space, and write $\Lambda _V$ and $\tau _V$ for $\Lambda $ and $\tau $ .

4.1.4 Maps of vertex spaces

A map of vertex spaces $f \colon V \to V'$ consists of a K-linear map $f^{\smash {\sharp }} \colon \Lambda _{V'} \to \Lambda _V$ , and a K-linear map

$$ \begin{align*} f (z) \colon V \longrightarrow V' (\!(\Lambda_V)\!) , \end{align*} $$

where $\deg \Lambda _V^\vee = -2$ , such that

(4.1.4.1) $$ \begin{align} f (z) \circ \tau_V (w) & = \iota_{z, w} \, f (z + w) ,\end{align} $$

(4.1.4.2) $$ \begin{align} \tau_{V'} (z') \circ f (z) & = \iota_{z', z} \, f (f^{\smash{\sharp}} (z') + z) , \end{align} $$

for $z, w \in \Lambda _V$ and $z' \in \Lambda _{V'}$ . Here, $f (z)$ can be thought of as translating fields along the vector z, then pulling them back to $\Lambda _{V'}$ along $f^\sharp $ .

The identity map $\mathrm {id} \colon V \to V$ is given by $\mathrm {id}^\sharp = \mathrm {id}_{\Lambda _V}$ and $\mathrm {id} (z) = \tau _V (z)$ .

Note that the relation (4.1.4.1) might tempt one think that $f (z)$ is determined by $f (0)$ via ${f (z) = f (0) \circ \tau _V (z)}$ . However, this is not the case, as $f (z)$ is often singular at $z = 0$ , in which case $f (0)$ is undefined.

4.1.5 Composition

Maps of vertex spaces cannot always be composed. Given a diagram

(4.1.5.1)

of maps of vertex spaces, we say that h is the composition of f and g, written $h = g \circ f$ , if $h^\sharp = f^\sharp \circ g^\sharp $ , and we have

(4.1.5.2) $$ \begin{align} g (z') \circ f (z) = \iota_{z', z} \, h (f^\sharp (z') + z) \end{align} $$

for $z \in \Lambda _V$ and $z' \in \Lambda _{V'}$ . Note that h is unique if it exists, and always exists if f or g is the identity. Composition is associative and unital whenever all involved compositions are defined.

Moreover, this generalizes to define n-fold compositions of maps of vertex spaces for any $n \geq 0$ . This defines the structure of a paracategory.

We denote by $\mathsf {VS}_K$ the paracategory of vertex spaces over K.

4.1.6 The monoidal structure

Given vertex spaces V and $V'$ , we define a vertex space structure on $V \otimes V'$ by

(4.1.6.1) $$ \begin{align} \Lambda_{V \otimes V'} & = \Lambda_V \oplus \Lambda_{V'} ,\end{align} $$

(4.1.6.2) $$ \begin{align} \tau_{V \otimes V'} (z, z') & = \tau_V (z) \otimes \tau_{V'} (z') . \end{align} $$

The tensor product defines a symmetric monoidal structure on $\mathsf {VS}_K$ , and has a unit given by the vertex space K, with $\Lambda _K = 0$ and $\tau _K = \mathrm {id}$ . We include a sign rule when identifying $V \otimes V'$ with $V' \otimes V$ on the odd graded pieces.

4.1.7 Theorem

A $\mathbb {Z}$ -graded weak vertex algebra over a field K, in the sense of §2.1.7, is equivalently a commutative algebra object in $\mathsf {VS}_K$ of rank $1$ . Moreover, a $\mathbb {Z}$ -graded weak module for such an algebra, in the sense of §3.1.5, is equivalently a module object in $\mathsf {VS}_K$ of rank $0$ .

4.2.1

We now introduce the vertex induction, which is a simultaneous generalization of the constructions of Joyce vertex algebras and modules in §§2–3.

As mentioned in the introduction, we expect this construction to be useful for enumerative geometry, especially for formulating wall-crossing formulae for enumerative invariants of nonlinear moduli stacks.

4.2.2 The paracategory $\mathcal {T}'$

To help with stating our main construction in Theorem 4.2.3, we define an auxiliary paracategory $\mathcal {T}'$ as follows. Recall the category $\mathcal {T}$ from §2.2.2.

• ○ An object of $\mathcal {T}'$ is a quadruple $(T_X, X, \odot _X, \mathbb {T}_X)$ , where $(T_X, X, \odot _X) \in \mathcal {T}$ , and $\mathbb {T}_X \in K^\circ (X)$ is a class of weight $0$ .

• ○ A morphism $f \colon (T_X, X, \odot _X, \mathbb {T}_X) \to (T_Y, Y, \odot _Y, \mathbb {T}_Y)$ in $\mathcal {T}'$ is a morphism $f \colon (T_X, X, \odot _X) \to (T_Y, Y, \odot _Y)$ in $\mathcal {T}$ , such that the normal bundle

  $$ \begin{align*} {\mathbb{\nu}}_f = f^* (\mathbb{T}_Y) - \mathbb{T}_X \end{align*} $$
  satisfies the following conditions:

  1. (a) We have ${\mathbb{\nu}}_f \in K^\circ (X)$ . Equivalently, we have $f^* (\mathbb {T}_Y) \in K^\circ (X)$ .

  2. (b) The weight $0$ part $(-{\mathbb{\nu}}_f)_0$ is the class of a vector bundle on X.

Note that composition in $\mathcal {T}'$ is not fully defined, since these conditions are not preserved by composition in general: For example, consider the composition $X \to Y \to Z$ , where $X = Y = Z = \mathrm {BU} (1)$ , the maps are identity maps, $T_X = \mathrm {U} (1)$ with the tautological action of $\mathrm {B} T_X$ on X, $T_Y = T_Z = \{ 1 \}$ , $\mathbb {T}_X = \mathbb {T}_Y = a$ , and $\mathbb {T}_Z = 0$ for some $a \in K (X) \setminus K^\circ (X)$ .

4.2.3 Theorem

There is a functor of paracategories

$$ \begin{align*} V \colon \mathcal{T}' & \longrightarrow \mathsf{VS}_{\mathbb{Q}} , \\ X & \longmapsto (\Lambda_X = \Lambda_{T_X} , \ V_X = \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (X; \mathbb{Q}) , \ \tau_X ) , \\ \smash{(X \overset{f}{\to} Y)} & \longmapsto f_* \circ \tau_X (z) \circ \big(\! (-) \cap e_z^{-1} ({\mathbb{\nu}}_f)\! \big) , \end{align*} $$

where the translation operator $\tau _X$ is defined in §2.3.2.

Proof. The proof is very similar to that of Theorem 2.3.3, and the key is in proving that V preserves composition. Let $X \overset {\smash [t]{f}}{\to } Y \overset {\smash [t]{g}}{\to } Z$ be composable morphisms in $\mathcal {T}'$ . Then for any $a \in V_X$ , we have

$$ \begin{align*} & g_* \circ \tau_Y (z) \Big[ f_* \circ \tau_X (w) \Big( a \cap e_w^{-1} ({\mathbb{\nu}}_f)\! \Big) \cap e_z^{-1} ({\mathbb{\nu}}_g) \Big] \\ = {} & g_* \circ f_* \circ \tau_X (f^\sharp (z)\!) \Big[ \tau_X (w) \Big( a \cap e_w^{-1} ({\mathbb{\nu}}_f)\! \Big) \cap e_z^{-1} (f^* ({\mathbb{\nu}}_g)\!) \Big] \\ = {} & g_* \circ f_* \circ \tau_X (f^\sharp (z)\!) \circ \tau_X (w) \Big( a \cap e_w^{-1} ({\mathbb{\nu}}_f) \cap e_{f^{\smash{\sharp}} (z) + w}^{-1} (f^* ({\mathbb{\nu}}_g)\!)\! \Big) \\ = {} & (g \circ f)_* \circ \tau_X (f^\sharp (z) + w) \Big( a \cap e_{f^{\smash{\sharp}} (z) + w}^{-1} ({\mathbb{\nu}}_{g \circ f})\! \Big) , \end{align*} $$

where the third line uses Lemma 2.3.4, and the last step uses the fact that ${\mathbb{\nu}}_f$ has weight $0$ with respect to $T_Y$ , and the fact that ${\mathbb{\nu}}_{g \circ f} = {\mathbb{\nu}}_f + f^* ({\mathbb{\nu}}_g)$ .

This generalizes the main constructions in §§2–3, in the sense that the associativity and commutativity axioms of vertex algebras and modules are now encoded as the functoriality of the functor V, in view of Theorem 4.1.7. See also §4.3.8 below for more motivations of this construction from enumerative geometry.

4.3.1

We now discuss the vertex induction applied to algebraic stacks over $\mathbb {C}$ , which is the setting we expect to be useful for enumerative geometry.

Throughout, let $\mathcal {X}$ be a derived algebraic stack over $\mathbb {C}$ , locally finitely presented, such that its classical truncation $\mathcal {X}_{\mathrm {cl}}$ is a classical algebraic $1$ -stack over $\mathbb {C}$ , quasi-separated, with affine stabilizers and separated inertia.

4.3.2

We use the formalism of the component lattice of an algebraic stack, following the author et al. [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11]. The component lattice is a combinatorial structure attached to any algebraic stack, and is key to generalizing existing results in enumerative geometry for linear moduli stacks to nonlinear moduli stacks. It provides the indexing set for decomposition-type theorems, or for terms in wall-crossing formulae, etc. See [Reference Bu, Davison, Ibáñez Núñez, Kinjo and Pădurariu10; Reference Bu, Ibáñez Núñez and Kinjo12; Reference Bu, Núñez and Kinjo13] for these applications.

4.3.3 The component lattice

For a finite rank free $\mathbb {Z}$ -module $\Lambda $ , let $T_\Lambda = \operatorname {Spec} (\mathbb {C} \Lambda ^\vee ) \simeq \mathbb {G}_{\mathrm {m}}^{\operatorname {rank} \Lambda }$ be the torus with coweight lattice $\Lambda $ , where $\mathbb {C} \Lambda ^\vee $ is the group algebra of $\Lambda ^\vee $ . Define the stack of $\Lambda ^\vee $ -graded points of $\mathcal {X}$ as the derived mapping stack

$$ \begin{align*} \mathrm{Grad}^\Lambda (\mathcal{X}) = \mathcal{M}\mathit{ap}\, (* / T_\Lambda, \mathcal{X}) . \end{align*} $$

It is also a derived algebraic stack satisfying the conditions in §4.3.1, and this construction is contravariant in $\Lambda $ . We also write $\mathrm {Grad} (\mathcal {X}) = \mathrm {Grad}^{\mathbb {Z}} (\mathcal {X})$ .

The component lattice of $\mathcal {X}$ is the functor

$$ \begin{align*} \mathrm{CL} (\mathcal{X}) \colon \mathsf{Lat} (\mathbb{Z})^{\mathrm{op}} & \longrightarrow \mathsf{Set} , \\ \Lambda & \longmapsto \unicode{x3c0}_0 ( \mathrm{Grad}^\Lambda (\mathcal{X})\! ) , \end{align*} $$

where $\mathsf {Lat} (\mathbb {Z})$ is the category of finite rank free $\mathbb {Z}$ -modules, and $\unicode{x3c0} _0 (-)$ denotes the set of connected components. This does not depend on the derived structure, in that we have $\mathrm {CL} (\mathcal {X}) \simeq \mathrm {CL} (\mathcal {X}_{\mathrm {cl}})$ .

A presheaf on $\mathsf {Lat} (\mathbb {Z})$ is also called a formal lattice, so that $\mathrm {CL} (\mathcal {X})$ is a formal lattice. Its underlying set is the set $| \mathrm {CL} (\mathcal {X}) | = \mathrm {CL} (\mathcal {X}) (\mathbb {Z}) \simeq \unicode{x3c0} _0 ( \mathrm {Grad} (\mathcal {X})\! )$ .

For example, if $\mathcal {X} = V / G$ is a quotient stack, where G is a linear algebraic group over $\mathbb {C}$ and V is a G-representation, then the component lattice $\mathrm {CL} (\mathcal {X})$ is the quotient formal lattice $\Lambda _T / W$ , where $\Lambda _T = \mathrm {Hom} (\mathbb {G}_{\mathrm {m}}, T)$ is the coweight lattice of the maximal torus $T \subset G$ , and $W = \mathrm {N}_G (T) / \mathrm {Z}_G (T)$ is the Weyl group. In general, $\mathrm {CL} (\mathcal {X})$ is usually glued from copies of $\mathbb {Z}^n$ for various n, along their automorphisms and maps between them. See [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11] for details and more examples.

4.3.4 Special faces

As in [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11], define a face of $\mathcal {X}$ as a morphism of formal lattices $\alpha \colon \Lambda \to \mathrm {CL} (\mathcal {X})$ for some $\Lambda \in \mathsf {Lat} (\mathbb {Z})$ , regarded as a formal lattice via the Yoneda embedding. Such faces naturally form a category $\mathsf {Face} (\mathcal {X})$ . For a face $\alpha \in \mathsf {Face} (\mathcal {X})$ , write

$$ \begin{align*} \mathcal{X}_\alpha \subset \mathrm{Grad}^\Lambda (\mathcal{X}) \end{align*} $$

for the connected component corresponding to $\alpha $ . For a morphism of faces $f \colon \alpha \to \beta $ , there is an induced morphism $f^\circ \colon \mathcal {X}_\beta \to \mathcal {X}_\alpha $ , giving a functor

(4.3.4.1) $$ \begin{align} \mathcal{X}_{(-)} \colon \mathsf{Face} (\mathcal{X})^{\mathrm{op}} \longrightarrow \mathsf{dSt}_{\mathbb{C}} , \end{align} $$

where $\mathsf {dSt}_{\mathbb {C}}$ is the $\infty $ -category of derived stacks over $\mathbb {C}$ .

A special face of $\mathcal {X}$ is a face $\alpha $ such that one cannot enlarge $\alpha $ without changing $\mathcal {X}_\alpha $ . More precisely, $\alpha $ is special if for any morphism of faces $f \colon \alpha \to \beta $ such that $f^\circ $ is an isomorphism, it admits a retraction $g \colon \beta \to \alpha $ , so that $g \circ f = \mathrm {id}_\alpha $ . (Note that [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11] uses rationalized faces, but the integral version defined here also works, which we use for convenience of presentation, and taking the rationalization gives an equivalence of the two versions.)

For example, if $\mathcal {X} = V / G$ as in §4.3.3, then the special faces of $\mathcal {X}$ are precisely the intersections of hyperplanes in $\Lambda _T$ dual to weights of V and roots of G in $\Lambda ^T = (\Lambda _T)^\vee $ . For such a face $\alpha \colon \Lambda \hookrightarrow \Lambda _T \twoheadrightarrow \Lambda _T / W$ , the stack $\mathcal {X}_\alpha \simeq V^\alpha / L_\alpha $ is the quotient of the fixed locus $V^\alpha \subset V$ of the corresponding subtorus $T_\alpha \subset T$ by the Levi subgroup $L_\alpha = \mathrm {Z}_G (T_\alpha ) \subset G$ .

Let $\mathsf {Face}^{\mathrm {sp}} (\mathcal {X}) \subset \mathsf {Face} (\mathcal {X})$ be the full subcategory of special faces. There is a special closure functor

$$ \begin{align*} (-)^{\mathrm{sp}} \colon \mathsf{Face} (\mathcal{X}) \longrightarrow \mathsf{Face}^{\mathrm{sp}} (\mathcal{X}) , \end{align*} $$

which is left adjoint to the inclusion. The functor (4.3.4.1) factors through this functor, and in particular, we have $\mathcal {X}_{\alpha ^{\mathrm {sp}}} \simeq \mathcal {X}_\alpha $ for any $\alpha \in \mathsf {Face} (\mathcal {X})$ , so $\alpha ^{\mathrm {sp}}$ is a canonical replacement of $\alpha $ without changing the stack $\mathcal {X}_\alpha $ .

By the finiteness theorem [Reference Bu, Halpern-Leistner, Ibáñez Núñez and Kinjo11: Theorem 6.2.3], if $\mathcal {X}$ is quasi-compact and satisfies a very mild condition called quasi-compact graded points, then $\mathcal {X}$ has only finitely many special faces.

4.3.5 Tangent weights

For any face $\alpha \colon \Lambda \to \mathrm {CL} (\mathcal {X})$ , write $\alpha ^\star (\mathbb {T}_{\mathcal {X}}) = \mathrm {tot}_\alpha ^* (\mathbb {T}_{\mathcal {X}}) \in \mathsf {Perf} (\mathcal {X}_\alpha )$ , where $\mathbb {T}_{\mathcal {X}}$ is the tangent complex of $\mathcal {X}$ , and $\mathrm {tot}_\alpha \colon \mathcal {X}_\alpha \to \mathcal {X}$ is the forgetful morphism. The canonical $* / T_\Lambda $ -action on $\mathcal {X}_\alpha $ induces a $\Lambda ^\vee $ -grading of $\alpha ^\star (\mathbb {T}_{\mathcal {X}})$ , and we have

(4.3.5.1) $$ \begin{align} \mathbb{T}_{\mathcal{X}_\alpha} \simeq \alpha^\star (\mathbb{T}_{\mathcal{X}})_0 , \end{align} $$

where $(-)_0$ denotes the degree $0$ part with respect to the $\Lambda ^\vee $ -grading.

Define the set of tangent weights of $\mathcal {X}$ on $\alpha $ by

(4.3.5.2) $$ \begin{align} \mathrm{wt} (\mathcal{X}, \alpha) = \{ \lambda \in \Lambda^\vee \mid \alpha^\star (\mathbb{T}_{\mathcal{X}})_\lambda \neq 0 \} . \end{align} $$

We say that $\mathcal {X}$ has finite tangent weights if this set is finite for all faces $\alpha $ , or equivalently, for all special faces $\alpha $ .

4.3.6 A functor to $\mathcal {T}'$

For any stack $\mathcal {X}$ as in §4.3.1, with finite tangent weights as in §4.3.5, and any face $\alpha \colon \Lambda \to \mathrm {CL} (\mathcal {X})$ , there is an associated object

$$ \begin{align*} (|T_\Lambda|, |\mathcal{X}_\alpha|, \odot, \mathbb{T}_{\mathcal{X}_\alpha}) \in \mathcal{T}' , \end{align*} $$

where $|T_\Lambda | \simeq (\mathbb {C}^\times )^n \simeq \mathrm {U} (1)^n$ in $\mathsf {hCW}$ for $n = \operatorname {rank} \Lambda $ , and $\odot $ is induced by the canonical $* / T_\Lambda $ -action on $\mathcal {X}_\alpha $ . This defines a functor

$$ \begin{align*} |\mathcal{X}_{(-)}| \colon \mathsf{Face} (\mathcal{X})^{\mathrm{op}} \longrightarrow \mathcal{T}' , \end{align*} $$

which factors through the special closure functor $(-)^{\mathrm {sp}} \colon \mathsf {Face} (\mathcal {X}) \to \mathsf {Face}^{\mathrm {sp}} (\mathcal {X})$ .

4.3.7 Theorem

Let $\mathcal {X}$ be a derived algebraic stack over $\mathbb {C}$ as in §4.3.1, and assume that it has finite tangent weights as in §4.3.5. Then there is a functor

$$ \begin{align*} \mathsf{Face}^{\mathrm{sp}} (\mathcal{X})^{\mathrm{op}} & \longrightarrow \mathsf{VS}_{\mathbb{Q}} , \\ \alpha & \longmapsto (\Lambda , \ V_\alpha = \mathrm{H}_{{\bullet} + 2 \operatorname{vdim}} (\mathcal{X}_\alpha; \mathbb{Q}) , \ \tau_\alpha) , \\ \smash{(\alpha \overset{f}{\to} \beta)} & \longmapsto (f^\circ)_* \circ \tau_{\beta} (z) \circ \big(\! (-) \cap e_z^{-1} ({\mathbb{\nu}}_{f^\circ})\! \big) , \end{align*} $$

where $\Lambda $ is the underlying $\mathbb {Z}$ -lattice of $\alpha $ , $\tau _\alpha = \tau _{|\mathcal {X}_\alpha |}$ is the translation operator in §2.3.2, $f^\circ $ is defined in §4.3.4, and ${\mathbb{\nu}}_{f^\circ } = (f^\circ )^* (\mathbb {T}_{\mathcal {X}_\alpha }) - \mathbb {T}_{\mathcal {X}_\beta } \in K^\circ (|\mathcal {X}_\beta |)$ .

4.3.8

We expect that the maps of vertex spaces associated to f in Theorem 4.3.7, or the vertex induction maps, together with their residues along their poles in the z variables, will be useful in studying the structure of enumerative invariants. In particular, we expect that the wall-crossing formulae for a conjectural version of Joyce’s homological invariants, discussed in §1.1.3, should be expressed using these maps, and should have the same form as those for intrinsic Donaldson–Thomas invariants to appear in [Reference Bu, Núñez and Kinjo13].

5.1.1

We introduce a generalization of the vertex induction, where we allow the obstruction theory to be an oriented real K-theory class instead of a complex K-theory class. This will recover the constructions in §§2–4 if the oriented real K-theory class comes from a complex K-theory class.

We expect that this construction will be important in studying DT4 invariants for general $(-2)$ -shifted symplectic stacks, especially in the nonlinear cases, where the stack is not the moduli stack of objects in a linear category. The linear case is discussed in Gross, Joyce, and Tanaka [Reference Gross, Joyce and Tanaka20: §4.4].

5.1.2 Real K-theory

For a space $X \in \mathsf {hCW}$ , define the real topological K-theory of X as the commutative ring

(5.1.2.1) $$ \begin{align} \mathit{KO} (X) = \mathsf{hCW} (X, \mathrm{BO} \times \mathbb{Z}) . \end{align} $$

There is a forgetful map $\mathit {KO} (X) \to K (X)$ given by complexification.

Given a class $E \in \mathit {KO} (X)$ , an orientation of E is a trivialization of the $\mathbb {Z}_2$ -bundle on X classified by the composition $X \to \mathrm {BO} \times \mathbb {Z} \to \mathrm {B} \mathbb {Z}_2$ , with the second map induced by the projection $\det \colon \mathrm {O} \to \mathbb {Z}_2$ on each component. Such an orientation exists if and only if the class $w_1 (E) \in \mathrm {H}^1 (X; \mathbb {Z}_2)$ is zero, in which case the orientations form an $\mathrm {H}^0 (X; \mathbb {Z}_2)$ -torsor.

Recall the category $\mathcal {T}$ from §2.2.2. For an object $(T, X, \odot ) \in \mathcal {T}$ , define a subset

$$ \begin{align*} \mathit{KO}^\circ (X) \subset \mathit{KO} (X) \end{align*} $$

of elements that map to the subgroup $K^\circ (X) \subset K (X)$ defined in §2.2.3.

5.1.3 The square root Euler class

For a space X and an $\mathrm {SO} (n)$ -bundle E on X, recall the square root Euler class

$$ \begin{align*} \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut} (E) \in \mathrm{H}^{n} (X; \mathbb{Z}) , \end{align*} $$

which is the Euler class of the underlying oriented real vector bundle, and is only nonzero if n is even. It satisfies $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut } (E)^2 = (-1)^n \cdot e (E)$ , where $e (E) = c_n (E)$ is the top Chern class of the associated complex vector bundle.

5.1.4 The equivariant square root Euler class

Let $(T, X, \odot ) \in \mathcal {T}$ , with X connected. Let $E \in \mathit {KO}^\circ (X)$ be a class such that the weight $0$ part $E_0$ is the class of an $\mathrm {O} (n)$ -bundle on X. Suppose that E is equipped with an orientation, so in particular, $E_0$ is also an $\mathrm {SO} (n)$ -bundle.

We define an equivariant square root Euler class $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (E)$ , which is a real analogue of the class $e_z (E)$ in §2.3.1.

Consider the hyperplane arrangement on $\Lambda _T \otimes \mathbb {R}$ given by hyperplanes dual to nonzero T-weights appearing in E. For an open chamber $\sigma \subset \Lambda _T \otimes \mathbb {R}$ of this hyperplane arrangement, define

(5.1.4.1) $$ \begin{align} \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E) = \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut} (E_0) \cdot \prod_{\langle \sigma, \lambda \rangle> 0} e_z (E_\lambda) \quad \in \quad \prod_{k = 0}^\infty \mathrm{H}^{2k} (X; \mathbb{Q}) ( z_1, \dotsc, z_n ) , \end{align} $$

where $E_0$ is equipped with the orientation induced by writing $E_0 = E - \sum _{\lambda \neq 0} E_\lambda $ , where the last term has the orientation given by $\sum _{\langle \sigma , \lambda \rangle> 0} E_\lambda $ , that is, factoring $\sum _{\lambda \neq 0} E_\lambda \colon X \to \mathrm {BO} \times \mathbb {Z}$ as $X \to \mathrm {BU} \times \mathbb {Z} \to \mathrm {BSO} \times \mathbb {Z} \to \mathrm {BO} \times \mathbb {Z}$ , with the first map given by $\sum _{\langle \sigma , \lambda \rangle> 0} E_\lambda $ , and the second map given by $E \mapsto E + E^\vee $ .

One can verify that $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (E)$ does not depend on the choice of $\sigma $ , while the orientation of $E_0$ depends on $\sigma $ . The class $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (E)$ is only nonzero if E has even rank, and we have the relations

(5.1.4.2) $$ \begin{align} \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E^{\mathrm{op}}) & = -\mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E) , \end{align} $$

(5.1.4.3) $$ \begin{align} \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E)^2 & = (-1)^{\operatorname{rank} E / 2} \cdot e_z (E) , \end{align} $$

where $E^{\mathrm {op}}$ denotes E with the opposite orientation.

When X is not connected, we define $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (E)$ on each connected component of X as above.

We have the relation

(5.1.4.4) $$ \begin{align} \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E + F) = \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E) \, \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (F) , \end{align} $$

where $E + F$ is equipped with the induced orientation, which does not depend on the order of E and F if they have even rank. In particular, if $E_0 = 0$ , then $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (E) \, \mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (-E) = 1$ , in which case we also write $\mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z^{\,-1} (E) = \mathord {\smash {\mathchoice {\textstyle }{}{}{}\sqrt {e}}\mathstrut }_z (-E)$ .

5.1.5 Orientation of the normal bundle

For a map of spaces $f \colon X \to Y$ and classes $\mathbb {T}_X \in \mathit {KO} (X)$ and $\mathbb {T}_Y \in \mathit {KO} (Y)$ , equipped with orientations, the normal bundle ${\mathbb{\nu}}_f = f^* (\mathbb {T}_Y) - \mathbb {T}_X \in \mathit {KO} (X)$ is equipped with the induced orientation, defined by the property that the sum $f^* (\mathbb {T}_Y) = {\mathbb{\nu}}_f + \mathbb {T}_X$ preserves orientation.

This depends on the order of the sum, in that if we used $\mathbb {T}_X + {\mathbb{\nu}}_f$ instead, the orientation would be changed by a factor of $(-1)^{\operatorname {rank} \mathbb {T}_X \cdot \operatorname {rank} {\mathbb{\nu}}_f}$ . The orientation is reversed whenever the orientation of either $\mathbb {T}_X$ or $\mathbb {T}_Y$ is reversed.

5.1.6 The paracategory $\mathcal {T}''$

Recall the category $\mathcal {T}'$ from §4.2.2. We define a similar paracategory $\mathcal {T}''$ , using real K-theory, as follows:

• ○ An object is a quintuple $(T_X, X, \odot _X, \mathbb {T}_X, o_X)$ , where $(T_X, X, \odot _X) \in \mathcal {T}$ , and $\mathbb {T}_X \in \mathit {KO}^\circ (X)$ is a class of weight $0$ , and $o_X$ is an orientation of $\mathbb {T}_X$ .

• ○ A morphism $f \colon (T_X, X, \odot _X, \mathbb {T}_X) \to (T_Y, Y, \odot _Y, \mathbb {T}_Y)$ is a morphism $f \colon (T_X, X, \odot _X) \to (T_Y, Y, \odot _Y)$ in $\mathcal {T}$ , such that the normal bundle ${\mathbb{\nu}}_f = f^* (\mathbb {T}_Y) - \mathbb {T}_X$ satisfies ${\mathbb{\nu}}_f \in \mathit {KO}^\circ (X)$ , and the part $(-{\mathbb{\nu}}_f)_0$ of weight $0$ is the class of a vector bundle on X.

There is a forgetful functor $\mathcal {T}'' \to \mathcal {T}'$ , although this will not be used below.

5.1.7 Example. Shifted symplectic stacks

Let $\mathcal {X}$ be an n-shifted symplectic stack over $\mathbb {C}$ , with $n \in 4 \mathbb {Z} + 2$ . An important special case is when $n = -2$ , such as when $\mathcal {X}$ is a moduli stack of coherent sheaves on a Calabi–Yau fourfold.

In this case, the symplectic structure $\mathbb {T}_{\mathcal {X}} \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathbb {L}_{\mathcal {X}} [n]$ gives rise to an orthogonal structure on the complex $\mathbb {T}_{\mathcal {X}} [n/2]$ , and hence a class $\mathbb {T}_{X} = -[\mathbb {T}_{\mathcal {X}} [n/2]] \in \mathit {KO} (X)$ , where $X = | \mathcal {X} |$ is the topological realization. An orientation of $\mathbb {T}_X$ , as defined in §5.1.2, is equivalent to an isomorphism $\det (\mathbb {T}_{\mathcal {X}}) \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} \mathcal {O}_{\mathcal {X}}$ that squares to the isomorphism $\det (\mathbb {T}_{\mathcal {X}})^{\otimes 2} \simeq \mathcal {O}_{\mathcal {X}}$ given by the shifted symplectic structure, which agrees with Borisov and Joyce [Reference Borisov and Joyce5: Definition 2.12].

5.1.8 Theorem

There is a functor of paracategories

$$ \begin{align*} V \colon \mathcal{T}" & \longrightarrow \mathsf{VS}_{\mathbb{Q}} , \\ X & \longmapsto V_X = \mathrm{H}_{{\bullet} + \operatorname{vdim}} (X; \mathbb{Q}) , \\ \smash{(X \overset{f}{\to} Y)} & \longmapsto f_* \circ \tau_X (z) \circ \big(\! (-) \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z^{\,-1} ({\mathbb{\nu}}_f)\! \big) , \end{align*} $$

where $V_X$ is equipped with the translation operator $\tau _X$ defined in §2.3.2, and ${\mathbb{\nu}}_f$ is equipped with the induced orientation as in §5.1.5.

In particular, we have the following:

1. (i) Let X be as in §2.2.4, but with $K (-)$ , $K^\circ (-)$ replaced by $\mathit {KO} (-)$ , $\mathit {KO}^\circ (-)$ instead, and assume that $\mathbb {T}_X \in \mathit {KO} (X)$ is equipped with an orientation. Then there is a $\mathbb {Z}$ -graded weak vertex algebra structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (X; \mathbb {Q})$ , defined by

   (5.1.8.1) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) &= (-1)^{\sum_{1 \leq i < j \leq n} |a_i| \operatorname{vdim}_j} \cdot {} \nonumber\\&\qquad (\oplus_{(n)})_* \circ \tau (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z^{\,-1} ({\mathbb{\nu}}_{(n)})\! \Big) , \end{align} $$
   where $a_i \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ are homogeneous elements, each supported on a single connected component of X, and $\operatorname {vdim}_i$ denotes the rank of $\mathbb {T}_X$ on the support of $a_i$ .

2. (ii) Let $X, Y$ be as in §3.2.1, but with $K (-)$ , $K^\circ (-)$ replaced by $\mathit {KO} (-)$ , $\mathit {KO}^\circ (-)$ instead, and assume that $\mathbb {T}_X$ , $\mathbb {T}_Y$ are equipped with orientations. Then there is a weak module structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (Y; \mathbb {Q})$ for the weak vertex algebra $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (X; \mathbb {Q})$ in (i), defined by

   (5.1.8.2) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) \, m &= (-1)^{\sum_{1 \leq i < j \leq n + 1} |a_i| \operatorname{vdim}_j} \cdot {} \nonumber \\& \qquad (\diamond_{(n)})_* \circ \tau (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n \boxtimes m) \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z^{\,-1} ({\mathbb{\nu}}_{\diamond, \smash{(n)}})\! \Big) , \end{align} $$
   where $a_i \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ and $m \in \mathrm {H}_{\bullet } (Y; \mathbb {Q})$ are homogeneous elements supported on single components, $\operatorname {vdim}_i$ is as in (i), and $\operatorname {vdim}_{n + 1}$ is the rank of $\mathbb {T}_Y$ on the support of m.

3. (iii) Let $\mathcal {X}$ be as in Theorem 4.3.7, equipped with an n-shifted symplectic structure for $n \in 4 \mathbb {Z} + 2$ , and an orientation. Then there is a functor

   (5.1.8.3) $$ \begin{align} \mathsf{Face}^{\mathrm{sp}} (\mathcal{X})^{\mathrm{op}} & \longrightarrow \mathsf{VS}_{\mathbb{Q}} , \notag \\ \alpha & \longmapsto V_\alpha = \mathrm{H}_{{\bullet} + \operatorname{vdim}} (\mathcal{X}_\alpha; \mathbb{Q}) , \notag \\ \smash{(\alpha \overset{f}{\to} \beta)} & \longmapsto (f^\circ)_* \circ \tau_{\beta} (z) \circ \big(\! (-) \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z^{\,-1} ({\mathbb{\nu}}_{f^\circ})\! \big) . \end{align} $$

Proof. For the main statement, the proof of Theorem 4.2.3 can be adapted to this situation without much change, and we only need to check the real version of Lemma 2.3.4, that is, for $(T, X, \odot ) \in \mathcal {T}$ and $E \in \mathit {KO}^\circ (X)$ , with an orientation, such that $E_0$ is the class of a vector bundle, we have

(5.1.8.4) $$ \begin{align} \tau (w) (a) \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_z (E) = \iota_{z, w} \big( \tau (w) (a \cap \mathord{\smash{\mathchoice{\textstyle}{}{}{}\sqrt{e}}\mathstrut}_{z + w} (E)\!)\! \big) \end{align} $$

for any $a \in \mathrm {H}_{\bullet } (X; \mathbb {Q})$ . But it is straightforward to adapt the proof of Lemma 2.3.4 to this situation.

The statement (i) follows from the main statement, where we precompose the vertex induction map $V_{X^n} \to V_X$ with the map $V_{\smash {X}}^{\otimes n} \to V_{X^n}$ given by

$$ \begin{align*} a_1 \otimes \cdots \otimes a_n \longmapsto (-1)^{\sum_{1 \leq i < j \leq n} |a_i| \operatorname{vdim}_j} \cdot a_1 \boxtimes \cdots \boxtimes a_n , \end{align*} $$

with notations as in the statement of (i), with the sign inserted so that it is equivariant under (signed) permutations of the factors. The sign is needed because in $V_X$ , compared to homology, the parity of elements is reversed on components with odd virtual dimension.

The statement (ii) holds analogously.

For (iii), as in §4.3.6, there is a functor

$$ \begin{align*} |\mathcal{X}_{(-)}| \colon \mathsf{Face} (\mathcal{X})^{\mathrm{op}} \longrightarrow \mathcal{T}" , \end{align*} $$

sending a face $\alpha \colon \Lambda \to \mathrm {CL} (\mathcal {X})$ to the object $(|T_\Lambda |, |\mathcal {X}_\alpha |, \odot , \mathbb {T}_{\mathcal {X}_\alpha }, o_{\mathcal {X}_\alpha }) \in \mathcal {T}''$ , where we choose an arbitrary orientation of $\mathbb {T}_{\mathcal {X}_\alpha }$ for each $\alpha $ , which exists by (4.3.5.1), as the class of $\mathbb {T}_{\mathcal {X}}$ is orientable and $\alpha ^\star (\mathbb {T}_{\mathcal {X}})_\lambda = \alpha ^\star (\mathbb {T}_{\mathcal {X}})_{-\lambda }^\vee $ in $K (|\mathcal {X}_\alpha |)$ .

5.1.9

We can also apply Theorem 5.1.8 (i)–(ii) to shifted symplectic stacks as follows.

Suppose that $\mathcal {X}$ is an oriented n-shifted symplectic stack over $\mathbb {C}$ , with $n \in 4 \mathbb {Z} + 2$ , and that it is also equipped with the structure of a derived linear moduli stack, as in Example 2.2.6. Suppose that the weight condition (2.2.4.2) is satisfied, which is usually the case, as discussed in Example 2.2.6. Then (5.1.8.1) defines a vertex algebra structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (\mathcal {X}; \mathbb {Q})$ . If, moreover, $\mathcal {X}$ is equipped with a self-dual structure as in Example 3.2.4, such that its $\mathbb {Z}_2$ -action preserves the shifted symplectic structure, then the fixed locus $\mathcal {X}^{\mathrm {sd}}$ is also n-shifted symplectic, and if it is also orientable, then (5.1.8.2) defines a twisted module structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (\mathcal {X}^{\mathrm {sd}}; \mathbb {Q})$ .

For example, when $\mathcal {X}$ is an open substack of the moduli stack of perfect complexes on a Calabi–Yau fourfold Z, an orientation is given by Joyce and Upmeier [Reference Joyce and Upmeier26: Theorem 13.7], under a topological assumption on Z. In this case, self-dual structures may be obtained as in Example 3.2.4, but we do not know if orientations exist on $\mathcal {X}^{\mathrm {sd}}$ .

Alternatively, we could also consider derived versions as in Examples 2.2.7 and 3.2.5. Namely, if $\mathcal {C}$ and $\mathcal {X}$ are as in Example 2.2.7, and if $\mathcal {C}$ is equipped with a $(2-n)$ -Calabi–Yau structure, with $n \in 4 \mathbb {Z} + 2$ , then $\mathcal {X}$ has an n-shifted symplectic structure by Brav and Dyckerhoff [Reference Brav and Dyckerhoff6: Theorem 5.6]. If it is orientable, then (5.1.8.1) defines a vertex algebra structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (\mathcal {X}; \mathbb {Q})$ , and if $\mathcal {C}$ is self-dual as in Example 3.2.5, where the involution is compatible with the Calabi–Yau structure, then $\mathcal {X}^{\mathrm {sd}}$ is also n-shifted symplectic, and if it is also orientable, then (5.1.8.2) defines a twisted module structure on $\mathrm {H}_{{\bullet } + \operatorname {vdim}} (\mathcal {X}^{\mathrm {sd}}; \mathbb {Q})$ .

5.2.1

We introduce a K-theory version of vertex induction, generalizing Liu’s [Reference Liu30] construction of a K-theory version of Joyce vertex algebras. We expect this to be useful for generalizing the K-theoretic invariants of Liu [Reference Liu30] to more general quasi-smooth stacks.

An important difference between the K-theory version and the homology version is that the former is multiplicative in its nature, rather than additive. This is reflected in the fact that we obtain multiplicative vertex algebras and vertex spaces, rather than the usual versions.

5.2.2 Multiplicative vertex algebras

Define a multiplicative vertex algebra over K to be the data $(V, (X_n)_{n \geq 0})$ , consisting of a K-vector space V and K-linear multiplication maps

$$ \begin{align*} X_n \colon V^{\otimes n} & \longrightarrow V {[\![} x_1, \dotsc, x_n {]\!]} \, [(x_i - x_j)^{-1}] , \\ a_1 \otimes \cdots \otimes a_n & \longmapsto X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) , \end{align*} $$

where we write $z_i = 1 + x_i$ , satisfying the following properties:

1. (i) (Unit) For any $a \in V$ , we have

   (5.2.2.1) $$ \begin{align} X_1 (a; 1) = a . \end{align} $$

2. (ii) (Commutativity) For any homogeneous elements $a_1, \dotsc , a_n \in V$ , and any permutation $\sigma \in \mathfrak {S}_n$ , we have

   (5.2.2.2) $$ \begin{align} X_n (a_{\sigma (1)}, \dotsc, a_{\sigma (n)}; z_{\sigma (1)}, \dotsc, z_{\sigma (n)}) = X_n (a_1, \dotsc, a_n; z_1, \dotsc, z_n) . \end{align} $$

3. (iii) (Associativity) For integers $m, n \geq 0$ and elements $b_1, \dotsc , b_m, a_1, \dotsc , a_n \in V$ , we have

   (5.2.2.3) $$ \begin{align} X_{n+1} &\Big( X_m (b_1, \dotsc, b_m; w_1, \dotsc, w_m), a_1, \dotsc, a_n; \ z_0, \dotsc, z_n \Big) \nonumber \\ &= \iota_{ \{ z_i - 1 \}, \{ w_j - 1 \} } \, X_{m+n} \big( b_1, \dotsc, b_m, a_1, \dotsc, a_n; z_0 w_1, \dotsc, z_0 w_m, z_1, \dotsc, z_n \big) . \end{align} $$

Similarly, we define weak multiplicative vertex algebras as above, but with the codomain of $X_n$ enlarged to $V (\!(x_1, \dotsc , x_n)\!)$ .

As usual, we abbreviate the product $X_n (a_1, \dotsc , a_n; z_1, \dotsc , z_n)$ as $a_1 (z_1) \cdots a_n (z_n)$ .

5.2.3 Multiplicative modules

We have a similar notion of a module over a multiplicative vertex algebra $(V, (X_n)_{n \geq 0})$ , defined as a K-vector space M equipped with K-linear multiplication maps

$$ \begin{align*} X^M_n \colon V^{\otimes n} \otimes M & \longrightarrow M {[\![} x_1, \dotsc, x_n {]\!]} \, [x_i^{-1}, (x_i - x_j)^{-1}] , \\ a_1 \otimes \cdots \otimes a_n \otimes m & \longmapsto X^M_n (a_1, \dotsc, a_n, m; z_1, \dotsc, z_n) , \end{align*} $$

where we write $z_i = 1 + x_i$ , with the following properties:

1. (i) (Unit) We have $X_0^M = \mathrm {id}_M$ .

2. (ii) (Associativity) For integers $k, n \geq 0$ and elements $a_1, \dotsc , a_n, b_1, \dotsc , b_k \in V$ , we have

   (5.2.3.1) $$ \begin{align} & X_{n+1}^M \Big( X_k (b_1, \dotsc, b_k; w_1, \dotsc, w_k); a_1, \dotsc, a_n, m; z_0, \dotsc, z_n \Big) \notag \\[-.5ex]& \hspace{2em} = \iota_{ \{ z_i - 1 \}, \{ w_j - 1 \} } \, X_{n+k}^M \big( b_1, \dotsc, b_k, a_1, \dotsc, a_n, m; z_0 w_1, \dotsc, z_0 w_k, z_1, \dotsc, z_n \big) , \end{align} $$
   (5.2.3.2) $$ \begin{align} & X_{n}^M \Big( a_1, \dotsc, a_n, X_k^M (b_1, \dotsc, b_k, m; w_1, \dotsc, w_k); z_1, \dotsc, z_n \Big) \notag \\[-.5ex]& \hspace{2em} = \iota_{ \{ z_i - 1 \}, \{ w_j - 1 \} } \, X_{n+k}^M \big( a_1, \dotsc, a_n, b_1, \dotsc, b_k, m; z_1, \dotsc, z_n, w_1, \dotsc, w_k \big) . \end{align} $$

Define a weak module over a weak multiplicative vertex algebra in the same way as above, but with the codomain of $X_n^M$ enlarged to $M (\!(x_1, \dotsc , x_n)\!)$ .

Similarly to §3.1.6, we also have a notion of twisted modules. Namely, if a multiplicative vertex algebra $(V, (X_n)_{n \geq 0})$ is equipped with a twisted involution, meaning an involution $(-)^\vee \colon V \mathrel {\overset {\smash {{} {-.8ex}{{} {\sim }}}\mspace {3mu}}{\to }} V^{\mathrm {op}}$ such that

(5.2.3.3) $$ \begin{align} \big( a_1 (z_1) \cdots a_n (z_n)\! \big)^\vee = a_1^\vee (z_1^{-1}) \cdots a_n^\vee (z_n^{-1}) \end{align} $$

for all $a_1, \dotsc , a_n \in V$ , then we define a twisted module for V as a weak module such that the image of $X^M_n$ lies in the subspace $M {[\![} x_1, \dotsc , x_n {]\!]} \, [x_i^{-1}, (x_i - x_j)^{-1}, (x_i + x_j + x_i x_j)^{-1} : i \neq j]$ , and such that $a^\vee (z) \, m = a (z^{-1}) \, m$ for all $a \in V$ and $m \in M$ . Here, inverting $x_i + x_j + x_i x_j$ means we allow poles also at $z_i z_j = 1$ for $i \neq j$ .

As usual, we abbreviate $X^M_n (a_1, \dotsc , a_n, m; z_1, \dotsc , z_n)$ as $a_1 (z_1) \cdots a_n (z_n) \, m$ .

5.2.4 Topological K-homology

We now define the underlying vector space where our multiplicative vertex algebras and vertex spaces will live.

Let $X \in \mathsf {hCW}$ be a space. Define the topological K-homology of X by

(5.2.4.1) $$ \begin{align} K_\circ (X) = \unicode{x3c0}_0 (\mathit{KU} \wedge \Sigma^\infty X_+) , \qquad K_\circ (X; \mathbb{Q}) = K_\circ (X) \otimes \mathbb{Q} , \end{align} $$

where $\mathit {KU}$ is the K-theory spectrum, $X_+ = X \sqcup \{ * \}$ is X with an extra base point, and $\Sigma ^\infty $ denotes taking the suspension spectrum. This is the zeroth homology group in the generalized homology theory represented by $\mathit {KU}$ ; see Adams [Reference Adams1: Part III, §6] for background on this.

For example, we have

(5.2.4.2) $$ \begin{align} K_\circ (\mathrm{BU} (1)\!) \simeq \mathbb{Z} [\ell] , \qquad \ell^k \cap (-) = \frac{1}{k!} \cdot \Big( \frac{\partial}{\partial L} \Big)^k , \end{align} $$

where $\ell ^k \cap (-)$ acts on $K (\mathrm {BU} (1); \mathbb {Z}) \simeq \mathbb {Z} {[\![} L - 1 {]\!]} $ , and $L \to \mathrm {BU} (1)$ is the universal line bundle. See Adams [Reference Adams1: Part II, Lemma 2.14] for this fact; compare Liu [Reference Liu30: Proposition 2.3.5].

The following lemma characterizes a finiteness property of topological K-homology, which will be useful in dealing with convergence issues in §5.2.8 later.

5.2.5 Lemma

Let $X \in \mathsf {hCW}$ be a space. Then for any class $a \in K_\circ (X; \mathbb {Q})$ , there exists $N \in \mathbb {N}$ , such that $a \cap E = 0$ for all $E \in K (X; \mathbb {Q})$ with $\mathrm {ch}_i (E) = 0$ for all $i < N$ .

5.2.6 The multiplicative translation operator

Let $(T, X, \odot ) \in \mathcal {T}$ . Let $z = (z_1, \dotsc , z_n)$ be a set of coordinates on $\Lambda _T$ , and write $x_i = z_i - 1$ .

Following ideas of Liu [Reference Liu30: §3.3.6], define the multiplicative translation operator

$$ \begin{align*} D (z) \colon K_\circ (X; \mathbb{Q}) & \longrightarrow K_\circ (X; \mathbb{Q}) \, {[\![} x_i {]\!]} , \\ a & \longmapsto \odot_* \bigg( {} \sum_{k_1, \dotsc, k_n \geq 0} {} x^k \cdot (\ell^k \boxtimes a) \bigg) , \end{align*} $$

where $x^k = x_1^{\smash {k_1}} \cdots x_n^{\smash {k_n}}$ , $\ell ^k = \ell _1^{\smash {k_1}} \boxtimes \cdots \boxtimes \ell _n^{\smash {k_n}}$ , and $\ell _i^{\smash {k_i}} \in K_\circ (\mathrm {BU} (1)\!)$ is the element in (5.2.4.2).

By the computation in Liu [Reference Liu30: Lemma 3.3.7], we have

(5.2.6.1) $$ \begin{align} D (z) (a) \cap E_\lambda = z^{\lambda} \cdot D (z) (a \cap E_\lambda) \end{align} $$

for $a \in K_\circ (X; \mathbb {Q})$ and a class $E_\lambda \in K^\circ (X)$ of weight $\lambda \in \Lambda ^T$ ,

5.2.7 Multiplicative vertex spaces

Similarly to §4.1.3, we also define multiplicative vertex spaces, which we use to describe multiplicative versions of the constructions in §4.

A multiplicative vertex space over a field K is a triple $(\Lambda , V, D)$ , consisting of a finite-dimensional K-vector space $\Lambda $ , a K-vector space V, and a K-linear map $D (z) \colon V \to V {[\![} \Lambda {]\!]} $ , called the translation operator, where we set $z = (z_1, \dotsc , z_n)$ for $z_i = 1 + x_i$ , for $(x_1, \dotsc , x_n)$ a set of coordinates for $\Lambda $ , satisfying

(5.2.7.1) $$ \begin{align} D (1) & = \mathrm{id}_V , \end{align} $$

(5.2.7.2) $$ \begin{align} D (z) \circ D (w) & = D (z w) . \end{align} $$

In other words, D is a representation of the formal group $\hat {\mathbb {G}}_{\mathrm {m}}^n$ .

We define maps of multiplicative vertex spaces similarly to §4.1.4, and define compositions similarly to §4.1.5. We obtain a paracategory $\mathsf {VS}^{\mathrm {mul}}_K$ of multiplicative vertex spaces over K.

5.2.8 The equivariant exterior power

Let $(T, X, \odot ) \in \mathcal {T}$ , and $E \in K^\circ (X)$ be a class such that $E_0$ is the class of a vector bundle.

Heuristically, we wish to define the equivariant exterior power ${\wedge }_{-z} (E)$ by

(5.2.8.1) $$ \begin{align} {\wedge}_{-z} (E) = \prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}}} {} \bigg( {} \sum_{k \geq 0} {} (-z^\lambda)^{k} \cdot {\wedge}^k (E_\lambda) \bigg) , \end{align} $$

where $z = (z_1, \dotsc , z_n)$ is a set of coordinates on $\Lambda _T$ , and we write $z^\lambda = z_1^{\smash {\lambda _1}} \cdots z_n^{\smash {\lambda _n}}$ , etc. However, the expression (5.2.8.1) does not make sense yet, since the sum can be infinite when $E_\lambda $ is not the class of a vector bundle.

To fix this, we expand the sum in the variables $x_i = z_i - 1$ as

(5.2.8.2) $$ \begin{align} {\wedge}_{-z} (E) =\,& \bigg( {} \sum_{k \geq 0} {} (-1)^k \cdot {\wedge}^k (E_0) \bigg) \cdot {} \nonumber\\ &\prod_{\lambda \in \Lambda ^{{}{-.1ex}{{}{\scriptscriptstyle T}}} \setminus \{ 0 \}} {} \bigg( {} \big( 1 - (1 + x)^\lambda \big)^{\operatorname{rank} E_\lambda} \cdot \sum_{k \geq 0} {} \bigg( \frac{-(1 + x)^\lambda}{1 - (1 + x)^\lambda} \bigg)^k \cdot {\vee}^k (E_\lambda) \bigg) \quad {\in} \quad K^\circ (X) (\!(x_i)\!)^\wedge , \end{align} $$

where $E_0$ is a vector bundle by assumption, $(1 + x)^\lambda = \prod _i {} (1 + x_i)^{\smash {\lambda _i}}$ , $K^\circ (X) (\!(x_i)\!)^\wedge $ denotes the completion of $K^\circ (X) (\!(x_i)\!)$ with respect to $I^N (X) (\!(x_i)\!)$ , where $I^N (X) \subset K^\circ (X)$ is the ideal of classes E with $\mathrm {ch}_i (E) = 0$ for all $i < N$ , and

(5.2.8.3) $$ \begin{align} \vee^k (E_\lambda) = \sum_{i = 0}^k {} (-1)^{k-i} \cdot \binom{\operatorname{rank} E_\lambda - i}{k - i} \cdot {\wedge}^i (E_\lambda) , \end{align} $$

which satisfies $\vee ^k (E_\lambda ) \in I^k (X)$ . For example, if $E_\lambda = L_1 + \cdots + L_r$ is a sum of line bundles, then $\vee ^k (E_\lambda ) = \sum _{1 \leq i_1 < \cdots < i_k \leq r} {} (L_{i_1} - 1) \cdots (L_{i_k} - 1)$ .

We take (5.2.8.2) as the definition of ${\wedge }_{-z} (E)$ from now on. By Lemma 5.2.5, the operation $(-) \cap {\wedge }_{-z} (E)$ is well-defined on $K_\circ (X; \mathbb {Q})$ . We have the relation

(5.2.8.4) $$ \begin{align} {\wedge}_{-z} (E + F) = {\wedge}_{-z} (E) \cdot {\wedge}_{-z} (F) \end{align} $$

whenever the right-hand side is defined.

5.2.9 Theorem

There is a functor of paracategories

$$ \begin{align*} V \colon \mathcal{T}' & \longrightarrow \mathsf{VS}_{\mathbb{Q}}^{\mathrm{mul}} , \\ X & \longmapsto K_\circ (X; \mathbb{Q}) , \\ \smash{(X \overset{f}{\to} Y)} & \longmapsto f_* \circ D_X (z) \circ \big(\! (-) \cap \wedge_{-z} (-{\mathbb{\nu}}_f)^\vee \big) , \end{align*} $$

where $K_\circ (X; \mathbb {Q})$ is equipped with the translation operator $D_X$ defined in §5.2.6.

In particular, we have the following:

1. (i) Let X be as in §2.2.4. Then the assignment

   (5.2.9.1) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) = (\oplus_{(n)})_* \circ D (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n) \cap {\wedge}_{-z} (-{\mathbb{\nu}}_{\smash{(n)}})^\vee \Big) \end{align} $$
   defines a weak multiplicative vertex algebra structure on $K_\circ (X; \mathbb {Q})$ , with poles along the locus $z_1^{\smash {\lambda _1}} \cdots z_n^{\smash {\lambda _n}} = 1$ for nonzero weights $\lambda \in \mathbb {Z}^n$ appearing in $\oplus _{(n)}^* (\mathbb {T}_X)$ . In particular, if the class $\oplus ^* (\mathbb {T}_X) \in K^\circ (X^2)$ only has $\mathrm {U} (1)^2$ -weights $(-1, 1),\, (0, 0),$ and $(1, -1),$ then $K_\circ (X; \mathbb {Q})$ is a multiplicative vertex algebra.

2. (ii) Let $X, Y$ be as in §3.2.1. Then the assignment

   (5.2.9.2) $$ \begin{align} a_1 (z_1) \cdots a_n (z_n) \, m = (\diamond_{(n)})_* \circ D (z) \Big(\! (a_1 \boxtimes \cdots \boxtimes a_n \boxtimes m) \cap {\wedge}_{-z} (-{\mathbb{\nu}}_{\diamond, \smash{(n)}})^\vee \Big) \end{align} $$
   defines a weak module structure on $K_\circ (Y; \mathbb {Q})$ for the weak multiplicative vertex algebra $K_\circ (X; \mathbb {Q})$ , with poles along the locus $z_1^{\smash {\lambda _1}} \cdots z_n^{\smash {\lambda _n}} = 1$ for nonzero weights $\lambda \in \mathbb {Z}^n$ appearing in $\diamond _{(n)}^* (\mathbb {T}_Y)$ .

3. (iii) Let $\mathcal {X}$ be as in Theorem 4.3.7. Then there is a functor

   (5.2.9.3) $$ \begin{align} \mathsf{Face}^{\mathrm{sp}} (\mathcal{X})^{\mathrm{op}} & \longrightarrow \mathsf{VS}^{\mathrm{mul}}_{\mathbb{Q}} , \notag \\ \alpha & \longmapsto K_\circ (\mathcal{X}_\alpha; \mathbb{Q}) , \notag \\ \smash{(\alpha \overset{f}{\to} \beta)} & \longmapsto (f^\circ)_* \circ D_{\mathcal{X}_\beta} (z) \circ \big(\! (-) \cap {\wedge}_{-z} (-{\mathbb{\nu}}_{f^\circ})^\vee \big) . \end{align} $$

Proof. For the main statement, we need to verify that the construction preserves composition. Let $X \overset {\smash [t]{f}}{\to } Y \overset {\smash [t]{g}}{\to } Z$ be composable morphisms in $\mathcal {T}'$ . Then for any $a \in K_\circ (X)$ , we have

$$ \begin{align*} & g_* \circ D_Y (z) \Big[ f_* \circ D_X (w) \Big( a \cap {\wedge}_{-w} (-{\mathbb{\nu}}_f)^\vee \Big) \cap {\wedge}_{-z} (-{\mathbb{\nu}}_g)^\vee \Big] \\ = {} & g_* \circ f_* \circ D_X (f^\sharp (z)\!) \Big[ D_X (w) \Big( a \cap {\wedge}_{-w} (-{\mathbb{\nu}}_f)^\vee \Big) \cap {\wedge}_{-z} (f^* (-{\mathbb{\nu}}_g)\!)^\vee \Big] \\ = {} & g_* \circ f_* \circ D_X (f^\sharp (z)\!) \circ D_X (w) \Big( a \cap {\wedge}_{-w} (-{\mathbb{\nu}}_f)^\vee \cap {\wedge}_{-f^{\smash{\sharp}} (z) - w} (f^* (-{\mathbb{\nu}}_g)\!)^\vee \Big) \\ = {} & (g \circ f)_* \circ D_X (f^\sharp (z) + w) \Big( a \cap {\wedge}_{-f^{\smash{\sharp}} (z) - w} (-{\mathbb{\nu}}_{g \circ f})^\vee \Big) , \end{align*} $$

where the third line uses Lemma 5.2.10 below, and the last step uses the fact that ${\mathbb{\nu}}_f$ has weight $0$ with respect to $T_Y$ .

It is straightforward to check that the other statements follow from the main statement.

5.2.10 Lemma

Let $(T, X, \odot ) \in \mathcal {T}$ , and let $E \in K^\circ (X)$ such that $E_0$ is the class of a vector bundle. Then for any $a \in K_\circ (X)$ , we have

(5.2.10.1) $$ \begin{align} D (w) (a) \cap {\wedge}_{-z} (E) = \iota_{z - 1, w - 1} \big( D (w) (a \cap {\wedge}_{-zw} (E)\!)\! \big) . \end{align} $$

This is a K-theory version of Lemma 2.3.4. See also Liu [Reference Liu30: Lemma 3.3.12].

Proof. As ${\wedge }_{-z} (E)$ is multiplicative in E, it is enough to prove this when E is of pure weight $\lambda $ . If $\lambda = 0$ , then both sides are equal to $a \cap {\wedge }_{-1} (E)$ . If $\lambda \neq 0$ , by (5.2.6.1), it is enough to show that

(5.2.10.2) $$ \begin{align} a \cap ( w^{\mathrm{wt}} \cdot {\wedge}_{-z} (E)\! ) = \iota_{z - 1, w - 1} (a \cap {\wedge}_{-z w} (E)\!) \end{align} $$

for any $a \in K_\circ (X; \mathbb {Q})$ , where $w^{\mathrm {wt}}$ is the operator that acts as $w^\mu $ on the weight $\mu $ component. To prove (5.2.10.2), we may restrict both sides to a finite subcomplex $U \subset X$ where a is supported, as in the proof of Lemma 5.2.5. Both sides are, by definition, expansions of rational functions in z and w valued in $K_\circ (U; \mathbb {C})$ , and these rational functions are equal to the series $\sum _{k \geq 0} {} (-z^\lambda w^\lambda )^k \cdot (a \cap {\wedge }^k (E)\!)$ in the region of $(z, w) \in \mathbb {C}^{2n}$ where the series converges. But ${\wedge }^k (E) |_U$ has polynomial growth in k, which can be seen by writing it in terms of the $\vee ^j (E) |_U$ as in §5.2.8, which vanish for $j \gg 0$ . The series thus converges in the region $|z^\lambda w^\lambda | < 1$ , which implies that the two rational functions are equal.