Quasimap Vertex Functions

• Quasimap Vertex Functions are generating series that encode local enumerative invariants from quasimaps into GIT quotients and Nakajima quiver varieties, including orbifold contributions.
• They are computed using localization methods on graph spaces, which reduce global invariants to explicit algebraic formulas that underpin mirror symmetry phenomena.
• The framework connects enumerative geometry, representation theory, and mathematical physics through wall-crossing, quantum K-theory, and vertex operator techniques.

Quasimap vertex functions are generating series encoding local enumerative invariants associated with moduli spaces of quasimaps from a domain curve (typically $\mathbb{P}^1$) into a GIT quotient or Nakajima quiver variety, incorporating possible orbifold or stacky structures. These functions generalize classical Gromov–Witten vertex functions and play a central role in the intersection of enumerative geometry, representation theory, and mathematical physics. Computed via localization methods and interpreted through virtual structure sheaves, quasimap vertex functions provide explicit algebraic representatives for the quantum geometry of the target space, encapsulating both untwisted and twisted (orbifold sector) contributions and underpinning mirror symmetry phenomena.

1. Definition and Construction of Quasimap Vertex Functions

Quasimap vertex functions encapsulate contributions from enumerative invariants of quasimaps into quotient targets, such as Deligne–Mumford stacks $\mathcal{X} = [W^{ss}/G]$ or Nakajima quiver varieties. Formally, the genus-zero orbifold quasimap vertex function is represented by a $J$-function or I-function as follows: $$J^\varepsilon(t, q, z) = \sum_{\beta} q^\beta \left\langle \varphi / (z - \psi) \right\rangle^{\varepsilon}_{0,1,\beta}$$ where $\varphi$ ranges over Chen–Ruan cohomology classes (with age shifts) in the cyclotomic inertia stack $I\mathcal{X}$ and $\varepsilon$ is the stability parameter (Cheong et al., 2014). In K-theoretic settings relevant to Nakajima varieties, one writes: $$V^{(\tau)}(\mathbf{z}) = \sum_{d} \, ev_{\infty,*}\left(\hat{O}^{\,d}_{\text{vir}} \otimes ev_0^*(\tau)\right) \,\mathbf{z}^{d}$$ where $d$ runs over the effective curve classes, $\hat{O}^{\,d}_{\text{vir}}$ is the symmetrized virtual structure sheaf, and $\tau$ is a descendant class in $K$-theory (Dinkins et al., 19 Feb 2025).

The evaluation maps for orbifold quasimaps naturally land in $I\mathcal{X}$, ensuring twisted sector contributions (with age shifts) are fully incorporated, thus capturing the orbifold quantum invariants. These functions are computed using residue localization over graph spaces equipped with $\mathbb{C}^*$-actions, allowing explicit calculation in terms of fixed points and combinatorial data (Cheong et al., 2014).

2. Structure, Factorization, and Localization Methods

A central computational tool is the use of localization methods on graph spaces or “stacky loop spaces,” utilizing the fixed loci to reduce global vertex invariants to sums over local data. The quasimap $J$-function admits a factorization: $$J^\varepsilon(t, q, z) = S^\varepsilon(z)\left(P^\varepsilon(t, q, z)\right)$$ where $S^\varepsilon(z)$ is constructed from two-pointed quasimap invariants and $P^\varepsilon(t, q, z)$ is the vertex factor (or P-series), encoding the elementary vertex contributions (Cheong et al., 2014). For I-functions in the toric orbifold case, explicit closed forms are derived via stacky loop space localization: $$I(0, q, z) = 1_\mathcal{X} + \sum_{\beta \neq 0} q^\beta \left(ev_*\left[\frac{\text{Residues}}{e_{\mathbb{C}^*}(\text{Normal Bundle})}\right]\right)$$ This systematic reduction, using residues and Euler classes of virtual normal bundles, makes possible explicit formulas for vertex functions in a wide class of geometric contexts, including the K-theory of torus fixed points in quiver varieties (Dinkins et al., 19 Feb 2025).

3. Twisted Sectors, Age Shifts, and Orbifold Features

The orbifold extension of quasimap theory fundamentally requires working with cohomology classes in $H_T^*(I\mathcal{X})$, which are graded according to age shifts. This is necessitated by the fact that twisted sectors (components of $I\mathcal{X}$ associated to nontrivial stabilizers) contribute nontrivially to the enumerative invariants. As a result, quasimap vertex functions explicitly record orbifold data, ensuring the enumeration respects the stacky, gerbe, or twisted structure of the target. This is crucial for applications to mirror symmetry, where matching predictions with orbifold Gromov–Witten invariants demands finer bookkeeping (Cheong et al., 2014).

4. Mirror Theorems and Wall-Crossing Phenomena

Quasimap vertex functions are deeply connected with mirror symmetry. In particular, the orbifold I-function is shown to lie on the overruled Lagrangian cone $\mathcal{L}_{\mathcal{X}}$ of Givental’s formalism, generalizing the classical mirror theorem to the orbifold (and stacky) setting (Cheong et al., 2014). The interplay between varying the stability parameter $\varepsilon$ and the wall-crossing formula, as developed by Ciocan–Fontanine and Kim, is central for relating quasimap invariants in adjacent chambers, especially for elliptic (genus 1) quasimap counts and their translation into elliptic Gromov–Witten invariants (Lho et al., 2016). Wall-crossing corrections, often involving logarithmic terms in the expansion of I-functions, explicitly account for changes in the enumerative invariants as stability conditions cross boundaries.

5. Algebraic and Representation-Theoretic Interpretations

Vertex operator formalism provides an algebraic framework for encoding and manipulating quasimap vertex functions. Operators such as $T_+$ and $T_-$, satisfying plethystic commutation relations, act as creation and annihilation operators on symmetric functions and quasimaps, assembling local data into global generating series (Carlsson et al., 2016, Jing et al., 2016). The operator mechanism underpins gluing formulas of the form: $H_{g+1,k}(u; q, t) = P_u \cdot H_{g,k+2}(u; q, t)$ which reconstruct higher-genus generating functions from basic three-punctured contributions, paralleling the decomposition of moduli spaces in topological quantum field theory (Carlsson et al., 2016). In quantum K-theory for hypertoric and Nakajima varieties, quasimap vertex functions furnish solutions to holonomic $q$-difference systems and generate modules over Coulomb branch algebras, with Whittaker functions in Verma modules describing their localized forms (Zhou, 2021, Smirnov et al., 2020).

6. Explicit Formulas, Computational Applications, and Connections

Quasimap vertex functions enable explicit calculations in a wide range of settings. Key formulas include:

• Genus 1 quasimap potentials for local Calabi-Yau varieties:

$$P_{1,0}(q) = \sum_{d \ge 0} q^d \deg[Q_{1,0}^{0+}(X,d)]^{\rm vir}$$

• Product expressions for type D quiver varieties:

$$V(\mathbf{z}) = \prod_{\alpha \in \Phi'} F\left((q/\hbar)^{a_\alpha} z_\alpha\right), \quad F(z) = \sum_{d\ge0} \frac{(\hbar)_d}{(q)_d} z^d$$

with the vertex function localizing to combinatorial sums over minuscule posets; these product formulas are equivalent to partition functions for half-space Macdonald processes and establish connections with integrable probability (Dinkins et al., 19 Feb 2025).

In cyclic quiver varieties, capped vertex functions reduce to evaluations of wreath Macdonald polynomials in the large framing limit, revealing deep ties to symmetric function theory and quantum toroidal algebras (Ayers et al., 9 Oct 2024).

7. Implications and Unification Across Theories

Quasimap vertex functions serve as a bridge unifying enumerative geometry (Gromov–Witten/quasimap invariants), representation theory (vertex algebra, quantum algebras, Whittaker modules), and mathematical physics (mirror symmetry, q-difference modules, integrable probability). Their formulation incorporates stacky and orbifold features, enables wall-crossing analysis, and supports explicit computations in quantum K-theory and moduli problems. This framework generalizes classical results, reconciles approaches to mirror symmetry, and provides robust computational tools for a wide class of geometric and algebraic objects.

A plausible implication is that the combinatorial and algebraic descriptions of vertex functions, such as those using symmetric functions, Macdonald polynomials, and poset partition functions, will continue to inform and inspire computational advances and theoretical developments in modern enumerative geometry and representation theory.