Vertex Functions in Enumerative Geometry

• Vertex functions are K-theoretic generating series defined by counting equivariant quasimaps from P¹ to holomorphic symplectic varieties.
• They satisfy explicit q-difference equations and localization formulas that connect Macdonald theory with 3d mirror symmetry and gauge theory dualities.
• Applications of vertex functions extend to the study of flag varieties, Nakajima quiver varieties, and vertex operator algebras, highlighting their broad impact in enumerative geometry and mathematical physics.

Vertex functions are generating series attached to geometric, algebraic, or statistical data, but in contemporary enumerative geometry the term most often denotes the $K$-theoretic analog of an $I$-function, defined by counting equivariant quasimaps from $\mathbb P^1$ to a holomorphic symplectic variety. For finite type $A$ bow varieties, they are formal series in Kähler variables whose coefficients lie in localized equivariant $K$-theory, and they are controlled by explicit localization formulas, $q$-difference equations, and 3d mirror symmetry. In that setting, vertex functions furnish a bridge among quasimap enumerative geometry, Macdonald theory, elliptic stable envelopes, and the Higgs/Coulomb branch duality of 3d $N=4$ gauge theories (Botta et al., 17 Jul 2025).

1. Geometric definition and parameter space

A finite type $A$ bow variety $X$ is a smooth holomorphic symplectic variety constructed from a brane diagram $D$ consisting of NS5 and D5 branes arranged along a line, separated by D3 segments. It carries an action of a torus

$I$0

with $I$1 generated by the $I$2 factors for each D5 brane and $I$3 scaling the symplectic form with weight $I$4. Vertex functions are $I$5-theoretic analogs of $I$6-functions, defined by counting equivariant quasimaps from $I$7 to $I$8. They depend on Kähler or Novikov variables $I$9, equivariant or flavor parameters $\mathbb P^1$0, the symplectic weight $\mathbb P^1$1, and the parameter $\mathbb P^1$2 coming from the maximal torus $\mathbb P^1$3 acting on the domain. The Kähler variables can be written as $\mathbb P^1$4, where $\mathbb P^1$5 are coordinates on the Kähler torus $\mathbb P^1$6 (Botta et al., 17 Jul 2025).

Bow varieties $\mathbb P^1$7 are realized as GIT quotients $\mathbb P^1$8 from suitable linear data associated to the brane diagram, with stability conditions denoted $\mathbb P^1$9–$A$0. For separated or co-separated diagrams, $A$1 admits a polarization $A$2 such that

$A$3

The dual variety $A$4 is obtained from the diagram $A$5 by swapping NS5 and D5 branes, and the fixed points of $A$6 and $A$7 are in bijection. Important examples arise from cotangent bundles of flag varieties: when all D5 branes have weight $A$8, $A$9 is isomorphic to $K$0, and when all NS5 branes also have weight $K$1, $K$2 (Botta et al., 17 Jul 2025).

2. Quasimaps, virtual structure, and localization

Let $K$3 be a bow variety. The relevant moduli stack is the stack of quasimaps

$K$4

with finite base points, together with the open substack $K$5 of quasimaps nonsingular at a fixed point $K$6. The perfect obstruction theory produces a virtual structure sheaf, and after choosing a polarization one defines the symmetrized virtual structure sheaf $K$7. The vertex function is then

$K$8

Restriction to a fixed point $K$9 gives

$q$0

Because $q$1 is proper in localized $q$2-theory, the pushforward is well defined (Botta et al., 17 Jul 2025).

The fixed-quasimap combinatorics is explicit. Fixed quasimaps are classified as stable reverse plane partitions of the butterfly diagram attached to the fixed point, and Proposition 3.5 gives a localization formula in which $q$3 is a sum over $q$4, weighted by powers of $q$5, $q$6, and products of $q$7-Pochhammer symbols

$q$8

For flag cases with all D5 weights equal to $q$9, there is a refined series formula in terms of difference variables $N=4$0, and this formula matches the “Macdonald function/non-stationary Ruijsenaars function.” To streamline mirror-symmetry statements, the vertex is normalized by the rescaling

$N=4$1

with a corresponding convention for $N=4$2 (Botta et al., 17 Jul 2025).

3. $N=4$3-difference equations and Macdonald structure

For $N=4$4 with all D5 weights equal to $N=4$5, the vertex functions satisfy Macdonald difference equations. The first operator acting in the equivariant variables $N=4$6 is

$N=4$7

and in the notation used for bow varieties,

$N=4$8

After introducing a gauge factor $N=4$9 and setting $A$0, Theorem 6.1 states

$A$1

where $A$2 is an explicit linear combination depending on the brane weights and on whether the diagram is separated or co-separated (Botta et al., 17 Jul 2025).

The analytic structure of these solutions is equally rigid. For $A$3 with all weights equal to $A$4, $A$5 admits analytic continuation with simple poles at

$A$6

and the product $A$7 is entire. A uniqueness theorem states that solutions

$A$8

to $A$9 are uniquely determined by their constant term in the equivariant variables. This uniqueness is one of the structural inputs in the mirror-symmetry argument, because it turns the enumerative problem into a rigid $X$0-difference problem with controlled asymptotics (Botta et al., 17 Jul 2025).

4. Elliptic stable envelopes and the mirror-symmetry matrix relation

The elliptic stable envelope $X$1 is a distinguished meromorphic section on $X$2, characterized by diagonal and support axioms. For a finite fixed set $X$3, the stable-envelope restrictions form a matrix

$X$4

which is upper-triangular with respect to the chamber order. There is also an opposite-chamber matrix satisfying

$X$5

For dual bow varieties $X$6 and $X$7, Theorem 4.3 identifies the normalized stable-envelope matrices after swapping Kähler and equivariant parameters and inverting $X$8, thereby matching the expected mirror transformation (Botta et al., 17 Jul 2025).

The central mirror-symmetry statement for vertex functions is the matrix relation

$X$9

where $D$0 sends

$D$1

Here $D$2 are $D$3-gamma factors from the attracting and repelling parts with respect to the standard chamber. In this form, the elliptic stable-envelope matrix is exactly the matrix relating the vertex functions of $D$4 and $D$5, and its entries form a basis of solutions to the joint $D$6-difference system (Botta et al., 17 Jul 2025).

5. Resolution procedures, flag specialization, and degeneration

A large part of the theory is a reduction mechanism. For D5 resolutions, one constructs a closed embedding $D$7 together with a torus inclusion $D$8. Theorem 5.4 states

$D$9

This compatibility is described as behaving like fusion of $I$00-matrices at the enumerative level. For NS5 resolutions, one relates $I$01 to a finer space $I$02 through a Lagrangian correspondence

$I$03

where $I$04 is a Grassmannian fibration and $I$05 is a closed immersion. In flag-variety language, $I$06 is the cotangent bundle of a finer flag (Botta et al., 17 Jul 2025).

The NS5 step is analytically subtler. A specialization $I$07 of Kähler parameters forces the finer-flag vertex function to a boundary point of convergence. The analytic continuation of $I$08 then has a simple pole, canceled by a universal factor $I$09. The resulting specialization identity recovers the vertex functions of the coarser flag from those of the finer flag, with simple poles proved in Corollary 6.7. Combining the D5 and NS5 statements reduces mirror symmetry of vertex functions to the complete-flag case $I$10, where the $I$11-difference equations can be identified explicitly with the Macdonald system (Botta et al., 17 Jul 2025).

The basic example is $I$12. Its two fixed-point vertex functions are basic hypergeometric series,

$I$13

with the second fixed point obtained by swapping $I$14. Mirror symmetry is verified here using Heine’s transformation and Watson’s connection formula. At the other end of the theory, the formal limit $I$15 degenerates the $I$16-theoretic picture to a cohomological one: $I$17 A plausible implication is that the bow-variety statements interpolate between elliptic and $I$18-theoretic mirror symmetry on one side and Givental-type $I$19-function phenomena on the other (Botta et al., 17 Jul 2025).

6. Related developments and alternative usages

Closely related work extends the quasimap meaning of vertex functions beyond finite type $I$20 bow varieties. For type $I$21 Nakajima quiver varieties, an explicit embedding into a quiver variety with all framings at the rightmost vertex preserves vertex functions up to a simple $I$22-shift of Kähler variables,

$I$23

thereby reducing general computations to cotangent bundles of partial flag varieties (Dinkins, 2023). For type $I$24 Nakajima quiver varieties with isolated torus fixed points, the fixed-point coefficients are described through reverse plane partitions on minuscule posets, and for minuscule framings the vertex functions admit product formulas

$I$25

which prove a degeneration of the conjectured 3d mirror symmetry and identify spin vertex functions with partition functions of half-space Macdonald processes (Dinkins et al., 19 Feb 2025). For $I$26, capped descendent vertex functions with exterior-algebra insertions admit explicit Fock-space formulas that provide a one-parameter deformation of the generating function for normalized Macdonald polynomials, and the capped vertex is a rational function of the quantum parameter (Ayers et al., 2024). For $I$27, 3D mirror symmetry yields integral polynomials $I$28 solving the quantum differential equation modulo $I$29, and $I$30 converges in the $I$31-adic norm to the vertex function while satisfying Dwork-type congruences and infinite-product formulas modulo $I$32 (Smirnov et al., 2023).

The phrase “vertex function” is also used in several non-enumerative senses. The following table records representative usages already established in the literature.

Domain	Meaning of “vertex function”	Representative paper
Many-body theory	Three-point or multipoint interaction vertex extracted from correlators	(Wölfle et al., 2015, Lihm et al., 2023)
Stochastic or integrable vertex models	Symmetric rational functions or partition functions of path ensembles	(Borodin et al., 2016, Aggarwal et al., 2021)
Vertex operator algebras	Graded trace functions or one-point theta functions	(Krauel, 2016, Dong et al., 2018, Carnahan et al., 2017)
Graph theory	Partition function of a vertex model as a graph parameter	(Draisma et al., 2011)

In many-body theory, the term can denote the three-point vertex coupling fermionic particle-hole pairs to spin or charge fluctuations at finite momentum, or the multipoint vertices entering parquet, fRG, and nonequilibrium transport; recent work constructs symmetric improved estimators that separate core and asymptotic contributions and avoid unstable leg amputation (Wölfle et al., 2015, Lihm et al., 2023). In solvable lattice models, vertex functions are symmetric rational functions such as $I$33 and $I$34, realized as partition functions of path ensembles and controlled by Yang–Baxter and Cauchy identities, or their colored fermionic analogues related to LLT and Macdonald polynomials (Borodin et al., 2016, Aggarwal et al., 2021). In vertex-operator-algebra theory, the same phrase can denote graded trace functions, McKay–Thompson series, or one-point theta functions with modular transformation laws under $I$35 (Krauel, 2016, Dong et al., 2018, Carnahan et al., 2017). In graph theory, a vertex function is the partition function of a vertex model over an algebraically closed field, characterized by multiplicativity and antisymmetrizer identities (Draisma et al., 2011).

This terminological dispersion suggests that “vertex function” is not a single invariant notion. In current algebraic and enumerative geometry, however, the dominant usage is the quasimap-generated $I$36-theoretic series controlled by localization, $I$37-difference equations, and mirror symmetry, with finite type $I$38 bow varieties providing a fully worked-out model in which the mirror matrix is identified with the elliptic stable-envelope matrix (Botta et al., 17 Jul 2025).