The 2-leg vertex in K-theoretic DT theory

TL;DR

This paper derives a formula for a specific K-theoretic vertex in Donaldson-Thomas theory when two entries are nontrivial, advancing computational techniques in enumerative geometry.

Contribution

It provides a new explicit formula for the 2-leg K-theoretic vertex in DT theory, expanding the computational toolkit for curve counting in toric threefolds.

Findings

Derived a formula for the 2-leg K-theoretic vertex with two nontrivial entries.

Discussed applications of the vertex formula in enumerative geometry.

Enhanced computational methods for K-theoretic DT counts.

Abstract

K-theoretic Donaldson-Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two out of three entries are nontrivial. We also discuss some applications of this result.

Equations

$$X=S\times \mathbb{C},\text{where}S={\text{Blow-up}}_{(0,0)}({\mathbb{P}}^1\times \mathbb{C})$$

$$\mathsf{T}={\mathbb{C}}_x^{\times }\times {\mathbb{C}}_y^{\times }\times {\mathbb{C}}_z^{\times }$$

$$D_i\cong {\mathbb{C}}_y\times {\mathbb{C}}_{xz},i=1,2,$$

$$\{\mathsf{w}\}=\hat{\mathsf{a}}(\mathsf{w})={\mathsf{w}}^{1/2}-{\mathsf{w}}^{-1/2},$$

$${\mathcal{O}}_X\stackrel{s}{\to }\mathcal{F}$$

$$\mathrm{ev}:\mathrm{PT}(X/D)\to \mathrm{Hilb}(D,\text{points})$$

$${\mathcal{O}}_D\to {\mathcal{F}}_D\stackrel{\text{def}}{=}\mathcal{F}\otimes {\mathcal{O}}_D\to 0$$

$${\mathrm{ev}}_i:\mathrm{PT}(X/D_1\bigsqcup D_2)\to \text{Hilb}(D_i).$$

$${\mathcal{T}}_{\text{vir}}={\chi }_{X^{\prime }}(\mathcal{F})+{\chi }_{X^{\prime }}(\mathcal{F},{\mathcal{O}}_{X^{\prime }})-{\chi }_{X^{\prime }}(\mathcal{F},\mathcal{F})$$

$${\mathcal{T}}_{\mathrm{Hilb}(D)}=\chi ({\mathcal{F}}_D)+{\chi }_D({\mathcal{F}}_D,{\mathcal{O}}_D)-{\chi }_D({\mathcal{F}}_D,{\mathcal{F}}_D)$$

$${\chi }_X(\mathcal{F},\mathcal{F})=(1-y){\chi }_S(p_{\ast }\mathcal{F},p_{\ast }\mathcal{F}).$$

$${\mathcal{T}}_{\mathrm{Hilb}(D)}^{1/2}$$

$${\mathcal{T}}_{\text{vir}}^{1/2}$$

$$\mathrm{rk}{\mathcal{T}}_{\mathrm{Hilb}(D)}^{1/2}$$

$$\mathrm{rk}{\mathcal{T}}_{\text{vir}}^{1/2}$$

$${\mathcal{T}}_{\mathrm{Hilb}(D)}$$

$${\mathcal{T}}_{\text{vir}}$$

$${\mathcal{K}}_X=\frac{1}{xyz}{\mathcal{O}}_X(-D),{\mathcal{K}}_S=\frac{1}{xz}{\mathcal{O}}_S(-\partial S).$$

$$\kappa =\sqrt{xyz}\text{and}\sqrt{y}.$$

$${\hat{\mathcal{O}}}_{\text{vir}}=(-q{)}^{\chi (\mathcal{F})}{\kappa }^{\mathrm{rk}{\mathcal{T}}_{\text{vir}}^{1/2}}\det {\left({\mathcal{T}}_{\text{vir}}^{1/2}\right)}^{-1}\otimes (\sqrt{xz}{)}^{-|\mathcal{F}{|}_{D_1}{|}^2-|\mathcal{F}{|}_{D_2}{|}^2}\otimes {\mathcal{O}}_{\text{vir}},$$

$${\hat{\mathcal{O}}}_{\text{vir}}=(-q{)}^{\chi (\mathcal{F})}\otimes {\left(\mathcal{K}\otimes \left(\det {\text{H}}^{\bullet }(\mathcal{F}{|}_D)\right)\right)}^{1/2}\otimes {\mathcal{O}}_{\text{vir}},$$

$${\mathrm{ev}}_{\ast }{\hat{\mathcal{O}}}_{\text{vir}}\in K_{\text{eq}}(\mathrm{Hilb}(D))[[q]].$$

$${{\mathcal{T}}_{\mathrm{Hilb}(D)}|}_{{\mathcal{E}}_d}={{\mathcal{T}}_{S^d{\mathbb{A}}^1}|}_{{\mathcal{E}}_d}+xyz{{\mathcal{T}}_{S^d{\mathbb{A}}^1}^{\vee }|}_{{\mathcal{E}}_d}=\sum _{i=1}^d\left(y^i+\frac{xyz}{y^i}\right),$$

$${{\mathcal{T}}_{\text{vir}}|}_{{\mathcal{E}}_d}=\sum _{i=1}^d{y}^i.$$

$${{\mathcal{T}}_{\mathrm{Hilb}(D)}^{1/2}|}_{{\mathcal{E}}_d}=\frac{1-y^{-d}}{1-y^{-1}}\left(1-\frac{1-y^d}{1-y}(1-xz),\right)$$

$${{\mathcal{T}}_{\text{vir}}^{1/2}|}_{{\mathcal{E}}_d}=(-xz)\frac{\{y^d{\}}^2}{\{y{\}}^2},$$

$${\mathrm{rk}{\mathcal{T}}_{\text{vir}}^{1/2}|}_{{\mathcal{E}}_d}=-d^2.$$

$${{\hat{\mathcal{O}}}_{\text{vir}}|}_{{\mathcal{E}}_d}=y^{-d^2/2}$$

$$(\mathcal{K}\otimes \mathcal{O}(1{)}_{\mathrm{Hilb}(D_1)}{)}^{1/2}.$$

$${{\mathcal{T}}_{\text{vir}}|}_{{\mathcal{E}}_{\lambda }}=\sum _{\square \in \lambda }\left((xz{)}^{-a(\square )}{y}^{l(\square )+1}+(xz{)}^{a(\square )+1}{y}^{-l(\square )}\right),$$

$${{\mathcal{T}}_{\mathrm{Hilb}(D)}|}_{{\mathcal{E}}_{\lambda }}=2\cdot \sum _{\square \in \lambda }\left((xz{)}^{-a(\square )}{y}^{l(\square )+1}+(xz{)}^{a(\square )+1}{y}^{-l(\square )}\right).$$

$${{\mathcal{T}}_{\text{vir}}^{1/2}|}_{{\mathcal{E}}_{\lambda }}=-\frac{1}{2}\cdot {\mathcal{T}}_{\mathrm{Hilb}(D)}^{1/2}({\mathcal{E}}_{\lambda }),$$

Full text

1 Introduction

1.1

Donaldson-Thomas (DT) theories, broadly interpreted, are enumerative theories of objects that look like coherent sheaves on a algebraic threefold. In this very broad spectrum of possibilities, DT counts of curves in an algebraic threefold $X$ stand out due to the intrinsic richness of the subject, of the generality in which such counts may be defined and studies, and also because of the range of connections with other branches of mathematics and mathematical physics. See, for example, [15] for a set of introductory lectures, and also [8] for an early discussion of the meaning of DT counts in theoretical physics. From the perspective of both mathematics and physics, it is particularly natural to study DT counts in equivariant K-theory, which is the setting of this paper.

In contrast to counts defined only with assumptions like $c_{1}(X)=0$, DT counts of curves in general 3-folds $X$ are much more flexible. The degeneration and localization properties of these counts (see [15] for an introduction), make the theory resemble the computation of Chern-Simons (CS) counts for real 3-folds by cutting and gluing. Similarly to how CS counts may be reduced to a few basic tensors (described in terms of quantum groups), there are some basic tensors for DT counts of curves, of which the 3-valent K-theoretic vertex is the most important one.

1.2

The 3-valent vertex is defined as the equivariant count of curves in the coordinate space $X=\mathbb{C}^{3}$. This can be defined as either straight equivariant localization counts for suitable moduli spaces of one-dimensional sheaves on $\mathbb{C}^{3}$, or with relative boundary condition along divisors $D_{1},D_{2},D_{3}$ that compactify $X$ in some ambient geometry like $(\mathbb{P}^{1})^{3}$. In either case, the vertex takes 3 partitions or, more canonically, a triple of elements of $K_{\textup{eq}}(\operatorname{Hilb}(\mathbb{C}^{2},\textup{points}))$ as its argument. Variations in boundary condition result in gauge transformations of the vertex that are understood, albeit complicated. In this paper, we find a particular gauge, that is, a particular relative geometry that makes the 2-leg vertex simple.

Further technical variations of vertices in DT theory come from the possibility to vary stability conditions for DT moduli spaces. While early papers used the Hilbert scheme of curves in $\mathbb{C}^{3}$ to define the vertex, there are many technical advantages to using the Pandharipande-Thomas (PT) moduli spaces instead [18]. Wall-crossing between different stability chambers have not really been explored in fully equivariant K-theory. However, there is little doubt that Hilbert scheme and the PT counts in any $X$ differ by an overall factor that comes from counting [math]-dimensional subschemes in $X$, see [14] for discussion of the latter count. In this paper, we work with the PT counts. Their only disadvantage is that they are harder to visualize, which is why Figure 1 shows examples of torus-fixed points in the Hilbert scheme of curves in $\mathbb{C}^{3}$.

1.3

While there is an in principle understanding of the vertices in terms of the Fock space representations of quantum double loop groups (see [15] for an introduction), having a better handle on them would lead to a significant theoretical and computational progress. The goal of this paper is to provide a direct and explicit description of the vertex with 2 nontrivial legs (as in Figure 1 on the right) in a specific gauge. This result is stated as Theorem 1 below.

Given the complexity of the problem, we find the existence of such an explicit formula quite remarkable. We also think it is unlikely that a comparably direct formula exists for the full 3-valent vertex.

1.4

The shape of our formula definitely suggests an interpretation in terms of counting M2-branes of the M-theory, along the lines explored in [13]. Very visibly, (27) is made up of the contributions of the three basic curves in the geometry: the two coordinate axes and their union.

We note, however, that formula (27) refers to relative DT counts and those currently fall outside of the scope of the conjectural correspondence with membrane counts proposed in [13]. Thus, the framework of [13] needs to be expanded and we hope to return to this question in a future paper.

1.5

As an application of our result, we compute the operator corresponding to the parallel legs in the resolved conifold. It proves that any matrix element of the operator divided by the vacuum matrix element is polynomial in the Kähler parameter.

1.6

We thank the Simons Foundation for financial support within the Simons Investigator program. The results of Section 3 were obtained under support of the Russian Science Foundation under grant 19-11-00275.

We thank N. Nekrasov and A. Smirnov for valuable discussions.

2 The setup and the formula

2.1 The basic geometry

2.1.1

The equivariant vertex with 2 legs can be captured by relative counts in the following threefold

[TABLE]

The toric diagram of $S$ is drawn on the left in Figure 2. The torus

[TABLE]

acts on $X$ with weights as in Figure 2. We denote by

[TABLE]

the two divisors shaded in Figure 2.

In computations we use the following notation: for any weight $\mathsf{w}:\mathsf{T}\to\mathbb{C}^{\times}$ we define

[TABLE]

and extend it multiplicatively to linear combinations.

2.1.2

The Pandharipande-Thomas moduli spaces of stable pairs parametrize complexes of the form

[TABLE]

in which $\mathscr{F}$ is a pure $1$-dimensional sheaf on $X$ and the cokernel of the section $s$ is a [math]-dimensional sheaf.

If a smooth divisor $D\subset X$ is given, there is a very useful relative modification $\operatorname{PT}(X/D)$ of this moduli space. It parametrizes complexes of the form (1) on semistable degenerations $X^{\prime}$ that allow $X$ to bubble off copies of $\mathbb{P}(\mathscr{O}_{D_{i}}\oplus N_{X/D_{i}})$, where $D_{i}$ is a component of $D$. One get such degenerations in families by blowing up $D_{i}\times\{b\}\subset X\times B$ in a trivial family with base $B\owns b$. See e.g. [14] for a hands-on introduction. By allowing degenerations of $X$, one can achieve that $\operatorname{Coker}(s)$ is supported away from $D$. Therefore, there is a well-defined map

[TABLE]

that takes a complex of the form (1) to its restriction

[TABLE]

to the divisor $D$. This map records the intersection of curves in $X$ with the divisor $D$.

In our case, $D=D_{1}\sqcup D_{2}$ has two components. We denote the components of the evaluation map (2) by

[TABLE]

2.1.3

The general formalism of perfect obstruction theories gives the PT moduli spaces their virtual structure sheaves $\mathscr{O}_{\textup{vir}}$. A small, but important detail in setting up the K-theoretic DT counts is to use a certain symmetrized virtual structure sheaf $\widehat{\mathscr{O}}_{\textup{vir}}$. The main difference between $\widehat{\mathscr{O}}_{\textup{vir}}$ and $\mathscr{O}_{\textup{vir}}$ is a twist by a square root of the virtual canonical bundle $\mathscr{K}_{\textup{vir}}$, in parallel to how a Dirac operator on a Kähler manifold $M$ is obtained from the $\overline{\partial}$-operator in $\Omega^{0,\bullet}(M)$ using a twist by $\mathscr{K}_{M}^{1/2}$.

2.1.4

The deformation theory of sheaves on a fixed semistable degeneration $X^{\prime}$ gives $\operatorname{PT}(X/D)$ a relative obstruction theory over the stack of degenerations of $X$. We denote by

[TABLE]

and $\mathscr{K}_{\textup{vir}}=\det(\mathscr{T}_{\textup{vir}})^{-1}$ the virtual tangent bundle and the virtual canonical line bundle of this relative obstruction theory.

While the general discussion of [13] about the existence of square roots may be adopted to the setting at hand, there is a more direct argument that applies whenever the pair $(X,D)$ is a line bunde over a surface $(S,\partial S)$, where $\partial S=S\cap D$.

The required twist will also involve the pull-back

[TABLE]

of the tangent bundle to $\operatorname{Hilb}(D)$ under the evaluation map (2). Since $D$ is the anticanonical divisor of $X$, the virtual dimension of the source in (2) is half of the dimension of the target, and in fact the image is a virtual Lagrangian. Reflecting this, we will see a polarization, that is, a certain half of the tangent bundle (5) in the sense of (11) in the formulas.

2.1.5

For simplicity of notation, assume that $X^{\prime}=X$ and let $p:X\to S$ denote the projection. A sheaf $\mathscr{F}$ on $X$ is the same as its pushforward $p_{*}\mathscr{F}$ together with an endomorphism of $p_{*}\mathscr{F}$ given by the multiplication by the 3rd coordinate, thus

[TABLE]

We define

[TABLE]

These are virtual bundles of ranks

[TABLE]

where $(\,\cdot\,,\,\cdot\,)_{S}$ denotes the intersection form on curve classes in the surface $S$.

The first half of the following proposition shows that (7) is a polarization of $\operatorname{Hilb}(D)$, that is, an equivariant half of the tangent bundle.

Proposition 2.1**.**

[TABLE]

Proof.

Follows from (6), Serre duality, and the equivariant identifications

[TABLE]

∎

2.1.6

To have the required twist defined in equivariant K-theory, we pass to the cover $\widetilde{\mathsf{T}}$ of the torus $\mathsf{T}$ with characters

[TABLE]

With this we define

[TABLE]

in other terms

[TABLE]

where $q$ is the boxcounting variable in the DT theory.

Our main object of study is the correspondence in defined by

[TABLE]

2.1.7 Example

Consider sheaves in the class $d\mathscr{O}_{C_{1}}$ with the smallest possible euler characteristic $\chi=d$. The moduli space is isomorphic to $S^{d}\mathbb{A}^{1}=\mathbb{A}^{d}$. There is one fixed point $\mathscr{E}_{d}$ for each $d$, and we can compute the characters of the tangent space to the Hilbert scheme:

[TABLE]

The normal bundle to $\mathbb{P}^{1}\subset X$ with support $C_{1}$ is $\mathscr{O}\oplus\mathscr{O}(-1)$, and the second bundle is acyclic, that is why

[TABLE]

By (7) and (8),

[TABLE]

[TABLE]

[TABLE]

For the stalk of the symmetrized structure sheaf at $\mathscr{E}_{d}$ we get

[TABLE]

in full agreement with

[TABLE]

2.1.8 Example

Consider sheaves in the class $d\mathscr{O}_{C1\cup C2}$ with the smallest possible $\chi=d$. These sheaves are obtained as pullbacks of sheaves on $\mathbb{P}^{1}\times\mathbb{C}^{2}$, where we consider the component of degree $d$ and $\chi=d$. Hence, the moduli space in this case is isomorphic to the diagonal in the product of two Hilbert scheme of $d$ points in $\mathbb{C}^{2}$. The fixed points can be identified with Young diagrams $\lambda=(\lambda_{0}\geq\lambda_{1}\geq...)$ which we denote by by $\mathscr{E}_{\lambda}$. Characters of the tangent space is given by the well-known arms-legs formula

[TABLE]

[TABLE]

Here we have a factor of 2 because of 2 copies of $\mathbb{C}^{2}$. By (7) and (8),

[TABLE]

[TABLE]

The stalk of the symmetrized structure sheaf is equal to

[TABLE]

2.2 Symmetric functions

2.2.1

Theorems of Bridgland-King-Reid [3] and Haiman [6] give an equivalence

[TABLE]

and hence a natural identification of equivariant $K$-theories. The Fourier-Mukai kernel of this identification is $\mathscr{O}_{\mathscr{Z}}$, where $\mathscr{Z}$ is the universal subscheme

[TABLE]

provided by Haiman’s identification of $\operatorname{Hilb}(\mathbb{C}^{2},n)$ with the Hilbert scheme of regular orbits for the diagonal action of $S(n)$ on $\mathbb{C}^{2n}$.

2.2.2

We use the identification

[TABLE]

given by the global sections and extend it to the suitable localization of the right-hand side for all sheaves. Here $\Lambda_{n}$ denotes symmetric functions of degree $n$.

The identification of $K_{S(n)}(\operatorname{pt})$ with symmetric functions sends a module $W$ to the symmetric function $f_{W}$ such that

[TABLE]

where

[TABLE]

the functions $p_{\mu}=\prod p_{\mu_{i}}$ are the power-sum symmetric functions, and the inner product is the standard inner product on $\Lambda_{n}$. In particular, the sheaf

[TABLE]

where $W^{\lambda}$ is an irreducible $S(n)$-module labeled by a diagram $\lambda$ of size $n$, corresponds to the Schur function $s_{\lambda}\in\Lambda_{n}$.

2.2.3

Let $\mathsf{T}$ acts on $\mathbb{C}^{2}$ by

[TABLE]

and let

[TABLE]

be the half-dimensional subspace defined by the vanishing of the coordinates of weight $t_{2}$. We have

[TABLE]

and therefore for any $S(n)$-module $W$

[TABLE]

Substitutions of the kind (18) are known in the literature as plethystic substitutions.

2.2.4

Induction and restriction of representations give symmetric functions

[TABLE]

their natural multiplication and comultiplication, see [19].

It is possible to produce correspondences between Hilbert schemes of different size that act as multiplication by a certain symmetric function. Concretely, let $b$ be the coordinate of weight $t_{2}$ and consider the correspondence

[TABLE]

which can be shown to be smooth and Lagrangian. The following Proposition may be deduced e.g. from the computations in [12].

Proposition 2.2**.**

The correspondence $E_{d}$ acts as multiplication by a symmetric function.

2.2.5

To determine the symmetric function in (2.2), it suffices to apply the correspondence to the Hilbert scheme of [math] points, in which case we get structure sheaf $\mathscr{O}_{L}$ as the corresponding element of $K_{S(n)\times\mathsf{T}}(\mathbb{C}^{2n})$. This corresponds to the trivial module $W=\mathbb{C}$ in (18), and thus to the complete homogeneous symmetric function $h_{n}=s_{(n)}$. From the generating function

[TABLE]

we conclude that $E_{d}$ multiplies by the degree $d$ component of

[TABLE]

Here $H_{(d)}$ is the Macdonald polynomial in Haiman’s normalization, it corresponds to the unique fixed points in

[TABLE]

The denominator is given by the tanget weights to (20). The correspondence $E_{d}$ maps to (20) by the class of $I_{2}/I_{1}$ and, in principle, we can pull back the point class instead of the structure sheaf from there. That would give multiplication by $H_{(d)}$.

2.3 Symmetric algebras

2.3.1

Let $V$ be a representation of a group $G$. The symmetric algebra of $V$

[TABLE]

is a representation of $V$ with character

[TABLE]

Because of the relation

[TABLE]

it is enough to check (21) for a $1$-dimensional module of some weight $w$, in which case it gives

[TABLE]

The operation {\mathsf{S}}^{\raisebox{1.42262pt}{\scriptscriptstyle\bullet}} extends naturally to $K_{G}(\operatorname{pt})$, that is, to virtual representations of $G$, by the rule

[TABLE]

2.3.2

Now suppose $V$ is direct sum of representations of $G$ with action of some symmetric group $S(k)$, $k\geq 1$, that is,

[TABLE]

We define

[TABLE]

where the sum is over all multisubsets, that is, subsets with repetitions, $\mathbf{k}$ of $\{1,2,\dots\}$ and the group $\operatorname{Aut}(\mathbf{k})$ permutes equal parts of $\mathbf{k}$. The representation (22) is the only natural representation that can be made out of unordered collections of representations $V_{k_{i}}$.

2.3.3

For any pair of groups $H\subset G$, Frobenius reciprocity implies

[TABLE]

This yields the following compatibility between {\mathsf{S}}^{\raisebox{1.42262pt}{\scriptscriptstyle\bullet}} and {\mathbb{S}}^{\raisebox{1.42262pt}{\scriptscriptstyle\bullet}}

[TABLE]

The bars in the bottom line of the diagram (23) stand for the required completions.

2.3.4

Take $f(g)\in K_{G}(\operatorname{pt})\otimes\Lambda$, where the argument $g$ denotes an element of the group $G$. Define the Adams operations $\Psi_{n}$ by

[TABLE]

Lemma 2.3**.**

[TABLE]

Proof.

The identification (17) of symmetric functions with representation of the symmetric group may be restated as follows. Let $W$ be a representation of $S(k)$ and suppose we want to evaluate $f_{W}$ as a symmetric polynomial of some variables $x_{1},\dots,x_{N}$. This evaluation is given by

[TABLE]

where

[TABLE]

It follows that

[TABLE]

The diagram (23) thus reduces (24) to (21)  . ∎

2.3.5

Given a collection $\mathscr{F}=\{\mathscr{F}_{k}\}$ of $S(k)\times\mathsf{T}$-equivariant sheaves $\mathscr{F}_{k}$ on $\mathbb{C}^{2k}$, the procedure (22) outputs a new collection of sheaves that we denote {\mathbb{S}}^{\raisebox{1.42262pt}{\scriptscriptstyle\bullet}}\mathscr{F}. More precisely,

[TABLE]

where $\boxtimes$ denotes the exterior tensor product over the coordinate rings of $\mathbb{C}^{2k_{i}}$. The class of (25) in K-theory may be computed by the formula (24).

Via the BKRH identification (16), this operation may be transported to the Hilbert schemes of points.

2.3.6

If the sheaf $\mathscr{F}$ is a representation of a further group $G$ that commutes with $S(k)\times\mathsf{T}$, then so is {\mathbb{S}}^{\raisebox{1.42262pt}{\scriptscriptstyle\bullet}}\mathscr{F} and one should remember to apply the operations $\Psi_{n}$ to the elements of $G$ in (24).

In particular, from the perspective of M-theory, the grading by the Euler characteristic in the DT theory is the grading with respect to a multiplicative group $\mathbb{C}^{\times}\owns q$. Therefore, we set

[TABLE]

2.4 Main result

2.4.1

Theorem 1**.**

[TABLE]

We recall that it is very important to keep in mind that this identification includes the action of $q$ as in (26).

3 Proof of Theorem 1

3.1 The $1$-leg vertex

3.1.1

To set up the strategy of the proof, we prove a special case of the formula first. It will also serve as an auxiliary statement in the proof of the full statement.

This special case concerns curves that do not meet the divisor $D_{2}$. In other words, they are in the homology class of multiples of the curve $C_{1}\subset S$. In terms of the formula (27) this means taking the constant term in the $\bar{p}_{k}$’s, thus the claim to prove is

[TABLE]

3.1.2

We note that there are very few reduced irreducible complete curves in $X$. Indeed, they all must be of the form

[TABLE]

where $C\subset S$ is reduced and irreducible, and thus either a smooth fiber of the blow-down map

[TABLE]

or one of the irreducible components of the special fiber $C_{1}\cup C_{2}\subset S$. The smooth fiber is a smoothing of $C_{1}\cup C_{2}$.

In particular, irreducible curves that do not meet $D_{2}$ are all of the form $C_{1}\times\textup{point}\subset S\times\mathbb{C}$.

3.1.3

Consider the one rank 1 torus that scales the $xz$- and $y$-axes with opposite weights, attracting and repelling respectively. We will pair $\operatorname{ev}_{*}\widehat{\mathscr{O}}_{\textup{vir}}$ with the stable envelopes $\operatorname{Stab}(\lambda)$ of fixed points for this torus action. See Chapter 9 in [14] for an introduction to stable envelopes in equivariant K-theory.

By the discussion of Section 3.1.2 the intersections of supports of $\operatorname{ev}_{*}\widehat{\mathscr{O}}_{\textup{vir}}$ and $\operatorname{Stab}(\lambda)$ is proper, thus the pairing is a series in $q$ with coefficients in nonlocalized K-theory

[TABLE]

In fact, by a judicious choice of the slope and the polarization of stable envelopes, we may achieve a sharper result.

3.1.4

We take stable envelopes $\operatorname{Stab}(\lambda)$ with zero slope and the polarization defined by by the polarization (7).

Lemma 3.1**.**

With this choice of parameters, we have

[TABLE]

Proof.

It suffices to check that $\chi(\operatorname{Stab}(\lambda)\otimes\operatorname{ev}_{*}\widehat{\mathscr{O}}_{\textup{vir}})$ remains bounded as

[TABLE]

By equivariant localization, the result is a sum of contribution of fixed points, each of which is a rational function of $x,y,z,\kappa$.

In this rational functions, terms that correspond to the first and the second summands in (12), after the twist by $\det\left(\mathscr{T}^{1/2}_{\textup{vir}}\right)^{-1}$ in (13), become balanced, in the sense that they stay finite in the limit (30).

The terms corresponding to last summand in (12) become balanced after pairing with $\operatorname{Stab}(\lambda)$ by the weight condition in the definition of stable envelopes.

∎

3.1.5

We can compute (29) by taking any specific limit of the form (30). Note from the above proof that in this computation we won’t need to consider the contribution of fixed points in $\operatorname{Hilb}(D_{1})$ other than the starting point $\lambda$. Indeed, the weights in the restriction of $\operatorname{Stab}(\lambda)$ to other fixed points satisfy strict inequalities and hence their contribution goes to [math] in the limit (30).

3.1.6

Of all possible limits in (30), we choose

[TABLE]

which, in the language of [13] means that make computations in the refined vertex limit, with the $y$-direction preferred.

The details of this computation will be worked out in the full $2$-leg generality below. Here we only state the result

Proposition 3.2**.**

We have

[TABLE]

3.1.7

To finish the proof in the 1-leg case we need the following statements.

Proposition 3.3**.**

Upon identification with symmetric function we have

[TABLE]

Proposition 3.4**.**

Formulas (32) and (28) agree.

3.1.8

Proof of Proposition 3.3.

We follow the notations of Section 2.2.3. Let $\Bbbk$ be a field and consider the algebra

[TABLE]

This is the rational Cherednik algebra with parameter [math] and thus the simplest quantization of the orbifold $T^{*}\Bbbk^{n}/S(n)$.

Assume that $p=\textup{char}\,\Bbbk\gg 0$, which, in particular, implies that irreducible representations of the symmetric group $S(n)$ are indexed by all partitions $\lambda$ of $n$. Given such representation $W^{\lambda}$, we consider the corresponding Verma module over $\mathscr{A}_{0}$

[TABLE]

where $b_{i}$ act on $W^{\lambda}$ by zero.

The algebra $\mathscr{A}_{0}$ is a free module of rank $p^{2n}$ over its subalgebra

[TABLE]

and as a $\mathscr{A}_{0,p}$-module, the Verma module (34) is $p^{n}$ copies of $W^{\lambda}\otimes\operatorname{Fr}^{*}\mathscr{O}_{L}$, where $L$ is the Lagrangian subvariety

[TABLE]

and $\operatorname{Fr}$ is the Frobenius map. Thus the Bezrukavnikov-Kaledin equivalence [1]

[TABLE]

sends Verma modules to modules of the form $W_{\lambda}\otimes\mathscr{O}_{L}$, the K-theory class of which was discussed earlier in (18). The twist by 1 in the $\left(\mathbb{C}^{2n}\right)^{(1)}$ term in (35) denotes the Frobenius twist.

The spherical subalgebra in $\mathscr{A}_{0}$ is a quantization of $\operatorname{Hilb}(\mathbb{C}^{2},n)$ for zero value of the quantization parameter, thus the Bridgland-King-Reid-Haiman equivalence (16) is also an example of a Bezrukavnikov-Kaledin equivalence at zero slope. It is known [2] that this equivalence sends Verma modules to suitably normalized stable envelopes, in particular

[TABLE]

The prefactor may be found from comparing the leading monomials, that is, the restrictions to the fixed point indexed by $\lambda$. This concludes the proof. ∎

3.1.9

Proof of Proposition 3.4.

[TABLE]

Here we used that

[TABLE]

∎

This concludes the proof of the $1$-leg case.

3.2 Refined vertex limit

Here we study the limit

[TABLE]

of the equivariant PT vertex with 2 nontrivial legs along (nonpreferred) directions $t_{1},t_{2}$. In this section we use the convention that the tangent weights at the origin are $t_{i}^{-1}$. The tangent space at a fix point $T$ decomposes as

[TABLE]

Note that

[TABLE]

so the contribution of a fixed point is equal to

[TABLE]

Proposition 3.5**.**

Refined limit of the PT vertex with 2 legs:

[TABLE]

Proof.

The tangent space to a sheaf

[TABLE]

[TABLE]

Let us denote by $\mathscr{F}^{\prime}\subset\mathscr{F}$ the minimal sheaf with the same outgoing cylinders as $\mathscr{F}$ on $\mathbb{C}^{3}$, and by $\mathscr{E}^{\prime}_{i}\subset\mathscr{E}_{i}$ the same for $\mathscr{E}_{i}$ on $\mathbb{C}^{2}$, and the quotients by $\mathscr{V}_{i}$, so that we have exact sequences of sheaves on $\mathbb{C}^{2}$:

[TABLE]

Writing contributions of $t_{3}$-slices, we get

[TABLE]

where

[TABLE]

If we denote

[TABLE]

where $K_{i}$ are some torsion sheaves (which we identify with their characters) on the Hilbert scheme with tangent weights $t_{1},t_{2}$ (dual Hilbert scheme), we get the following expression

[TABLE]

If we denote ideals

[TABLE]

then we can write the operator as

[TABLE]

which is identical to the Carlson-Okounkov Ext-operator, so it does not have weights $(t_{1}t_{2})^{n}$, and that’s why multiplied by $1-t_{3}^{-1}$ get the same number of positive and negative weights.

So, the index limit of $T_{3}(\mathscr{F})$ is a sum of contributions of each box, and the contribution of box $t_{1}^{i}t_{2}^{j}t_{3}^{k}$ depends on the sign of $i-j$:

[TABLE]

Let us define operators on the Fock space:

[TABLE]

The refined limit of the 2-leg vertex then can be represented as

[TABLE]

and using the properties

[TABLE]

[TABLE]

we get the statement.

Then we have obtained the formula up to a prefactor. To figure out the prefactor, note that in the definition of vertex we have to consider

[TABLE]

rather then

[TABLE]

where $C_{i}$ are the 2 cylinders such that

[TABLE]

Their difference is

[TABLE]

Though there is a very easy combinatorial formula for this expression, we need only its limit, and it has already been investigated in the paper [13]. Then the prefactor is equal to a product over boxes in $C_{1}\cap C_{2}$, and

[TABLE]

This has a clear interpretation: the vertex starts when we have ”holes” in the union of 2 cylinders (in the intersection boxes have multiplicities 1 instead of 2), and that’s why the contributions in (41) and (42) are the opposite. ∎

3.3 Factorizable sheaves

3.3.1

We now consider the full two legs geometry with two evaluation maps $\operatorname{ev}_{i}$ to the Hilbert scheme of points of two divisors $D_{1}$ and $D_{2}$. We may view the two leg vertex as an operator acting from $K_{\mathsf{T}}(\operatorname{Hilb}(D_{2}))$ to $K_{\mathsf{T}}(\operatorname{Hilb}(D_{1}))$. In particular, consider

[TABLE]

The already established case of a $1$-leg vertex gives

[TABLE]

Since the multiplication by this symmetric function is invertible, we may define

[TABLE]

where the dot denotes multiplication of symmetric functions.

The sheaves $\mathscr{F}_{\lambda}$ and $\mathscr{G}_{\lambda}$ are related by the action of correspondences from Section 2.2.4.

3.3.2

Let

[TABLE]

denote the full attracting set for the torus action as in Section 3.1.3, that is, the set of point that have a limit as the $(xz)$-coordinate is scaled to [math] while the $y$-coordinate is scaled to $\infty$. We have

[TABLE]

Our next goal is the following

Proposition 3.6**.**

[TABLE]

In other words, $\mathscr{G}$ comes from the first term in the following exact sequence

[TABLE]

3.3.3

The sheaf $\mathscr{F}_{\lambda}$ is factorizable in the following sense. Consider an open subset $U$ of $\operatorname{Hilb}(D_{1})$ formed by subschemes of the form $Z_{1}\cup Z_{2}$ such that

[TABLE]

and $\operatorname{supp}Z_{1}$ does not meet the $(xz)$-axis. Then

[TABLE]

where

[TABLE]

Indeed, the only curves in our geometry that can leave the zero section $S\subset X$ are curves of the class $C_{1}$. Once they leave the zero section, their deformation theory is independent of the rest of the curve and of the partition $\lambda$.

3.3.4

Proof.

By construction,

[TABLE]

Using the factorization (47), we now prove (46) by induction of the number $n$ in $\operatorname{Hilb}(D_{1},n)$. The statement being vacuous for $n=0$, assume that the sheaf

[TABLE]

is supported on the set $\operatorname{Attr}$. Consider

[TABLE]

By factorization

[TABLE]

as was to be shown. ∎

3.4 Conclusion of the proof

Let’s pair $\mathscr{G}_{\mu}$ with a class of the stable envelope correspinding to the opposite chamber. Similarly as in the 1-leg case, we have

Lemma 3.7**.**

[TABLE]

Thus this function can be extracted from the refined limit (31), where $y$-direction is preferred. To get the formulas simpler, we should divide it by the contribution of the gluing operator ${\bf G}:K_{eq}(\operatorname{Hilb}(D_{2}))\to K_{eq}(\operatorname{Hilb}(D_{2}))$, which is known by [16]:

[TABLE]

Lemma 3.8**.**

[TABLE]

Proof.

By localization,

[TABLE]

where capping is the solution to a certain $q$-difference equation as in [14]. In the limit we are considering

[TABLE]

[TABLE]

which basically means that the solution of a $q$-difference equation is trivial in the limit $q\to 0$, but we have to renormalize the basis appropriately, that is why we consider the matrix element between a stable envelope and a torus-fixed point. The prefactors here are monomials and exactly compensate the ones in (39). ∎

Lemma 3.9**.**

Formulas (27) and (49) are equivalent.

Proof.

[TABLE]

∎

4 Operator corresponding to resolved conifold (4-point

function)

In this section we will apply the computation of the two-legged vertex for the following relative geometry $X/D$:

This 3-fold has $H_{2}(X,\mathbb{Z})=\mathbb{Z}$, and we define the virtual structure sheaf

[TABLE]

Let

[TABLE]

be the restriction map. As in the previous section, we identify

[TABLE]

Theorem 2**.**

[TABLE]

Proof.

The result is obtained by gluing two capped 2-leg vertices along one leg, and can be represented pictorially.

The operator is the composition of the following operators:

[TABLE]

[TABLE]

[TABLE]

They act on the following Fock spaces:

[TABLE]

Here we introduce $Q$ into the gluing operator in order to measure degrees of curves. We contract bilinear forms using the standard Hall product on symmetric functions.

Taking the composition of the first vertex operator with the gluing matrix, we obtain,

[TABLE]

Finally, composing with the second vertex operator, we obtain 6 terms from contractions:

[TABLE]

The first term in the plethystic exponential is the partition function of ${\cal O}(-1)\oplus{\cal O}(-1)\to\mathbb{P}^{1}$,

[TABLE]

which factors out completely, concluding the proof.

∎