Jun Li, Kefeng Liu, Jian Zhou
We propose to use the identification of topological string partition functions as equivariant indices on framed moduli spaces of instantons to study the Gopakumar-Vafa conjecture for some local Calabi-Yau geometries.
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Equivariant versions of the radial index and of the GSV-index of a vector field or a 1-form on a singular variety with an action of a finite group are defined. They have values in the Burnside ring of the group. Poincaré-Hopf type theorems for them are proven and some of their properties are described.
For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of the group. We show that on a germ of a smooth complex-analytic G-variety these indices coincide. This permits to consider the difference between them as a version of the equivariant Milnor number of a germ a G-variety with an isolated singular point.
If two G-manifolds are G-cobordant then characteristic numbers corresponding to the fixed point sets (submanifolds) of subgroups of G and to normal bundles to these sets coincide. We construct two analogues of these characteristic numbers for singular complex G-varieties where G is a finite group. They are defined as sums of certain indices of collections of 1-forms (with values in the spaces of the irreducible representations of subgroups). These indices are generalizations of the GSV-index (for isolated complete intersection singularities) and the Euler obstruction respectively.
In this paper we identify the problem of equivariant vortex counting in a $(2,2)$ supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the ${\cal I}$ and ${\cal J}$-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show h…
We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A_1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C^2/Z_2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a…
From equivariant volumes to equivariant periods
本文将等变体积推广为 1d、2d、3d 超对称 GLSM 的配分函数，研究其对等变参数的依赖，推广移位方程，并揭示等变参数展开包含亏格零 Gromov-Witten 不变量信息。
Equivariant quasisymmetry and noncrossing partitions
本文引入等变拟对称性定义，构造双Schur与双Schubert多项式的拟对称推广（双基本与双森林多项式），以非交叉划分替代$S_n$，并预示拟对称旗簇将主导其几何。\n
Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex
通过拓扑顶点方法，研究五重叠空间的Gromov-Witten理论和膜指数，证明局部反对角拓扑动作下的猜想。
We compute section class relative equivariant Gromov-Witten invariants of the total space of P^2-bundles of the form P(O+L1+L2)-->C where C is a genus g curve, O is the trivial bundle, and L1 (resp. L2) is an arbitrary line bundle of degree k1 (resp. k2) over C. We prove a gluing formula for the partition functions of these invariants. Using this gluing formula together with localization techniques, we construct three explicit 3x3 matrices G, U1 and U2 with entries in Q((u))(t0,t1,t2), where u is the genus parameter, and t0,t1,t2 are the equivariant parameters. Then we prove that the partition…
We present a statistical mechanical model whose random variables are solid partitions, i.e. Young diagrams built by stacking up four dimensional hypercubes. Equivalently, it can be viewed as the model of random tessellations of ${\bf R}^{3}$ by squashed cubes of four fixed orientations. The model computes the refined index of a system of D0-branes in the presence of $D8$-$\overline{D8}$ system, with a $B$-field strong enough to support the bound states. Mathematically, it is the equivariant K-theoretic version of integration over the Hilbert scheme of points on ${\bf C}^{4}$ and its higher ran…
We explore three-dimensional gravity with negative cosmological constant via canonical quantization. We focus on chiral gravity which is related to a single copy of $\mathrm{PSL}(2,\mathbb{R})$ Chern-Simons theory and is simpler to treat in canonical quantization. Its phase space for an initial value surface $Σ$ is given by the appropriate moduli space of Riemann surfaces. We use geometric quantization to compute partition functions of chiral gravity on three-manifolds of the form $Σ\times \mathrm{S}^1$, where $Σ$ can have asymptotic boundaries. Most of these topologies do not admit a classica…
Equivariant Chow functions and Wonderful Models in type A and B
研究B型wonderful模型上超八面体群作用的等变庞加莱多项式，通过等变关联代数反演公式将其化为A型级数与B型特征级数，并建立B型复合恒等式及显式公式。
Motivated by work of Gusein-Zade, Luengo, and Melle-Hernández, we study a specific generating series of arm and leg statistics on partitions, which is known to compute the Poincaré polynomials of Z_3-equivariant Hilbert schemes of points in the plane, where Z_3 acts diagonally. This generating series has a conjectural product formula, a proof of which has remained elusive over the last ten years. We introduce a new combinatorial correspondence between partitions of n and {1,2}-compositions of n, which behaves well with respect to the statistic in question. As an application, we use this corres…
Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories
本文通过四维N=1超对称规范理论的涡旋配分函数提出量子椭圆上同调概念，利用等变局域化计算U(1)理论，并探讨反常与Thom层及虚拟结构层的联系。
Quiver BPS Indices from Crystal Profiles
推导出二维含\(\mathcal{N}\geq2\)超荷的椭圆亏格及其一维、零维对应物的新精确公式，结果表示为对晶体几何/组合结构的离散和，推广了Nekrasov配分函数并揭示新组合学。
Equivariant Localization for N=2 Theories with Hypermultiplets
Ω背景下的等变局域化技术被应用于CP²上的N=2超对称场论，计算等变关联函数并与低能场论结果对比，确定了不同规范群配分函数的差异因子，还结合S对偶讨论了相关结果。
Jack combinatorics of the equivariant edge measure
该研究聚焦环面三维流形Donaldson-Thomas不变量局部化计算中隐含的等变边测度，通过组合方法证明其在约定选择下等价于Jack-Plancherel测度，为相关不变量研究提供了组合学视角的新结论。
We provide a contour integral formula for the exact partition function of ${\cal N}=2$ supersymmetric $U(N)$ gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for $U(2)$ ${\cal N}=2$ theory on $\mathbb{P}^2$ for all instanton numbers. In the zero mass case, corresponding to the ${\cal N}=4$ supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the ge…
We introduce a deformed topological vertex and use it to define deformations of the topological string partition functions of some local Calabi-Yau geometries. We also work out some examples for which such deformations satisfy a deformed Gopakumar-Vafa integrality and can be identified with the equivariant indices of some naturally defined bundles on the framed moduli spaces.
We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lie in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We…
Resultant of an equivariant polynomial system with respect to the reflection group $G(r,n)$
本文针对复反射群$G(r,n)$等变齐次多项式系统，利用分拆索引的差商建立结式分解公式，并结合结式-判别式关系将不变齐次多项式的判别式分解为更小子系统结式之积，简化计算并给出实例。
For a semi-simple simply connected algebraic group G we introduce certain parabolic analogues of the Nekrasov partition function (introduced by Nekrasov and studied recently by Nekrasov-Okounkov and Nakajima-Yoshioka for G=SL(n)). These functions count (roughly speaking) principal G-bundles on the projective plane with a trivialization at infinity and with a parabolic structure at the horizontal line. When the above parabolic subgroup is a Borel subgroup we show that the corresponding partition function is basically equal to the Whittaker matrix coefficient in the universal Verma module over c…
We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kahler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspond…
For $G$ a finite group acting linearly on $\mathbb{A}^2$, the equivariant Hilbert scheme $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ is a natural resolution of singularities of $\operatorname{Sym}^r(\mathbb{A}^2/G)$. In this paper we study the topology of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ for abelian $G$ and how it depends on the group $G$. We prove that the topological invariants of $\operatorname{Hilb}^r[\mathbb{A}^2/G]$ are periodic or quasipolynomial in the order of the group $G$ as $G$ varies over certain families of abelian subgroups of $GL_2$. This is done by using the Bialynicki-Biru…
Non-Perturbative Corrections to 3d BPS Indices and Topological Strings
本文通过SL(2,Z)变换Birkhoff连接矩阵捕捉三维GLSM的非微扰修正，并证明已解锥上二维GLSM的三维提升可作为非微扰拓扑弦的世界体积对偶，其微扰指标计算GV配分函数，解析延拓匹配非微扰完备化提议。
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