Andrei Okounkov
In this expository paper we summarize the connection between random partitions and 4-dimensional supersymmetric gauge theories discovered in hep-th/0306238. There is also a brief section about how this extends to 3-dimensional partitions which is best read together with math-ph/0412008.
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We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa.
We use the first author's idea on "constructing virtual fundamental cycles without gluing ' to define the quantum correction term between the gauged linear sigma model and the nonlinear sigma model.
We propose, following the AGT correspondence, how the $\mathcal{W}^{\, para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) minimal model characters are obtained from the $U(N)$ instanton counting on ${\mathbb {C}}^2/{\mathbb {Z}}_n$ with $Ω$-deformation by imposing specific conditions which remove the minimal model null states.
We give a mathematically rigorous proof of Nekrasov's conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$ gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of $\mathbb R^4$, we derive a differential equation for the Nekrasov's partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.
We present closed-form expressions of unrefined instanton partition functions for gauge groups of type $BCD$ as sums over Young diagrams. For $\mathrm{SO}(n)$ gauge groups, we provide a fivebrane web picture of our formula based on the vertex-operator formalism of the topological vertex with a new type called O-vertex for an O5-plane.
We introduce the notion of the (instanton part of the) Seiberg-Witten prepotential for general Schrodinger operators with periodic potential. In the case when the operator in question is integrable we show how to compute the prepotential in terms of period integrals; this implies that in the integrable case our definition of the prepotential coincides with the one that has been extensively studied in both mathematical and physical literature. As an application we give a proof of Nekrasov's conjecture connecting certain "instanton counting" partition function for an arbitrary simple group G wit…
Let $M^K_n$ be the moduli space of framed $K$-instantons over $S^4$ with instanton number $n$ when $K$ is a compact simple Lie group of classical type. Let $U^{K}_{n}$ be the Uhlenbeck partial compactification of $M^{K}_{n}$. A scheme structure on $U^{K}_{n}$ is endowed by Donaldson as an algebro-geometric Hamiltonian reduction of ADHM data. In this paper, for $K=SO(N,R)$, $N\ge5$, we prove that $U^{K}_{n}$ is an irreducible normal variety with smooth locus $M^{K}_{n}$. Hence, together with the author's previous result, the K-theoretic Nekrasov partition function for any simple classical group…
We show that Nekrasov instanton partition function for SU(N) gauge theories satisfies recursion relations in the form of U(1)+Virasoro constraints when β = 1. The constraints give a direct support for AGT conjecture for general quiver gauge theories.
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…
Instanton Counting and Donaldson-Thomas Theory on Toric Calabi-Yau Four-Orbifolds
本文通过非交换crepant解消上的瞬子计数研究四维环面卡拉比-丘轨形上的唐纳森-托马斯理论，利用环面局部化计算配分函数并猜想其闭无穷乘积公式。
These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vecτ)$ of N=2 SUSY Yang-Mills theories, which are generating functions of the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$. The second is review of geometry of the Seiberg-Witten curves and the geometric engineering of the gauge theory, which are physical backgrounds of Nekrasov's partition functions. The first part is continuation of math.AG/0306198, where we identified the Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0…
We give explicit expressions for the finite frequency greybody factor, quasinormal modes and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein …
The non-perturbative behavior of the N=2 supersymmetric Yang-Mills theories is both highly non-trivial and tractable. In the last three years the valuable progress was achieved in the instanton counting, the direct evaluation of the low-energy effective Wilsonian action of the theory. The localization technique together with the Lorentz deformation of the action provides an elegant way to reduce functional integrals, representing the effective action, to some finite dimensional contour integrals. These integrals, in their turn, can be converted into some difference equations which define the S…
We consider the problem of computing (irregular) conformal blocks in 2d CFTs whose chiral symmetry algebra is the N=2 superconformal algebra. Our construction uses two ingredients: (i) the relation between the representation theories of the N=2 superconformal algebra and the affine sl(2) algebra, extended to the level of the conformal blocks, and (ii) the relation between affine sl(2) conformal blocks and instanton partition functions in the 4d N=2 SU(2) gauge theory with a surface defect. By combining these two facts we derive combinatorial expressions for the N=2 superconformal blocks in the…
The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $Ω$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical so…
The aim of this memoir for "Habilitation à Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometri…
Wall-crossing of Instantons on the Blow-up
研究四维N=2超对称规范理论在胀点上的瞬子计数，通过构造瞬子模空间为有向图品种，并引入稳定性参数进行正则化，利用Jeffrey-Kirwan残差公式计算瞬子分区函数，并通过超分割分类瞬子贡献。
We prove that the K-theoretic Nekrasov instanton partition functions have a positive radius of convergence in the instanton counting parameter and are holomorphic functions of the Coulomb parameters in a suitable domain. We discuss the implications for the AGT correspondence and the analyticity of the norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on random matrix techniques and relies on an integral representation of the partition function, due to Nekrasov, which we also prove.
Shell formulas for instantons and gauge origami
本文提出壳公式框架，统一描述由任意维度Young图分类的分区函数，提供多种物理系统的闭合表达式和递推关系，包括5维纯超Yang-Mills理论的瞬子分区函数及规范折纸配置。
Tetrahedron Instantons on Orbifolds
该研究构造轨形商奇点非交换解上的上同调规范理论，刻画不变四面体瞬子模空间，通过虚拟环面局部化计算配分函数，并联系Donaldson-Thomas理论给出奇点配分函数的闭式表达。
In arXiv:0908.4052, Nekrasov and Shatashvili pointed out that the N=2 instanton partition function in a special limit of the Omega-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. In this work we present an explicit derivation of this fact as well as generalizations to quiver gauge theories. To do so we combine various techniques like the iterated Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation. The TBA equations derived entirely within gauge theory have been proposed to encode t…
Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb {C}}^2$ with $Ω$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N = 2, 3, \ldots$, on ${\mathbb {C}}^2/{\mathbb {Z}}_n$, $n = 2, 3, \ldots$, with $Ω$-deformation and 2D conformal field theories with $\mathcal{W}^{\, para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) symmetry and $\widehat{\mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with $\mathcal{W}^{\, par…
We present an expression of a deformed partition function for N=2 U(1) gauge theory on C^2/Z_k by using plethystic exponentials.
Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, …
The treatment of the number-theoretical problem of integer partitions within the approach of statistical mechanics is discussed. Historical overview is given and known asymptotic results for linear and plane partitions are reproduced. From numerical analysis of restricted plane partitions an asymptotic formula is conjectured for an intermediate number of parts.
Counting partitions by genus. I. Genus 0 to 2
该研究将亏格0分划计数的泛函方程推广到高亏格，结合（半）本原分划重构方法，得到亏格1、2下所有分划类型的生成函数显式结果，为普通与自由累积量展开提供二阶插值。
Probing M-theory with tetrahedron instantons
本文通过考虑相交 D6-branes 系统及其与四面体瞬子的 T-dual 关系，猜想了 K-理论四面体瞬子配分函数的闭式表达式，并揭示了其与 M-theory 指数及 magnificent four 模型的深刻联系。
Large N expansion of mass deformed ABJM matrix model: M2-instanton condensation and beyond
本文为双参数质量变形的ABJ理论配分函数建立了新的双线性关系，并结合对偶性实现了关于N的递推精确求解，进而研究了超临界质量区域的大N渐近行为。
Recently, Kallen and Zabzine computed the partition function of a twisted supersymmetric Yang-Mills theory on the five-dimensional sphere using localisation techniques. Key to their construction is a five-dimensional generalisation of the instanton equation to which they refer as the contact instanton equation. Subject of this article is the twistor construction of this equation when formulated on K-contact manifolds and the discussion of its integrability properties. We also present certain extensions to higher dimensions and supersymmetric generalisations.
We consider some aspects of counting BPS operators which are annihilated by two supercharges, in superconformal field theories. For non-zero coupling, the corresponding multi-variable partition functions can be written in terms of generating functions for vector partitions or their weighted generalisations. We derive asymptotics for the density of states for a wide class of such multi-variable partition functions. We also point out a particular factorisation property of the finite N partition functions. Finally, we discuss the concept of a limit curve arising from the large N partition functio…
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