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.
In this paper, we study 5d $\mathcal{N}=1$ $Sp(N)$ gauge theory with $N_f ( \leq 2N + 3 )$ flavors based on 5-brane web diagram with $O5$-plane. On the one hand, we discuss Seiberg-Witten curve based on the dual graph of the 5-brane web with $O5$-plane. On the other hand, we compute the Nekrasov partition function based on the topological vertex formalism with $O5$-plane. Rewriting it in terms of profile functions, we obtain the saddle point equation for the profile function after taking thermodynamic limit. By introducing the resolvent, we derive the Seiberg-Witten curve and its boundary cond…
Let $K$ be the compact Lie group $USp(N/2)$ or $SO(N, R)$. Let $M^K_n$ be the moduli space of framed K-instantons over $S^4$ with the instanton number $n$. By Donaldson (1984), $M^K_n$ is endowed with a natural scheme structure. It is a Zariski open subset of a GIT quotient of $μ^{-1}(0)$, where $μ$ is a holomorphic moment map such that $μ^{-1}(0)$ consists of the ADHM data. The purpose of the paper is to study the geometric properties of $μ^{-1}(0)$ and its GIT quotient, such as complete intersection, irreducibility, reducedness and normality. If $K=USp(N/2)$ then $μ$ is flat and $μ^{-1}(0)$ …
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…
The Spherical Hecke central (SHc) algebra has been shown to act on the Nekrasov instanton partition functions of $\mathcal{N}=2$ gauge theories. Its presence accounts for both integrability and AGT correspondence. On the other hand, a specific limit of the Omega background, introduced by Nekrasov and Shatashvili (NS), leads to the appearance of TBA and Bethe like equations. To unify these two points of view, we study the NS limit of the SHc algebra. We provide an expression of the instanton partition function in terms of Bethe roots, and define a set of operators that generates infinitesimal v…
We obtain bilinear relations on Nekrasov partition functions, arising from study of tau functions of quantum $q$-Painlevé equations, from Nakajima-Yoshioka blowup relations by an elementary algebraic approach. Additionaly, using this approach, we prove certain relations on Nekrasov partition functions, modified by Chern-Simons term.
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conform…
We prove a Macdonald polynomial analogue of the celebrated Nekrasov-Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of Hausel and Rodriguez-Villegas from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces.
We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (…
We define an elliptic deformation of the Virasoro algebra. We argue that the $\mathbb{R}^4\times \mathbb{T}^2$ Nekrasov partition function reproduces the chiral blocks of this algebra. We support this proposal by showing that at special points in the moduli space the 6d Nekrasov partition function reduces to the partition function of a 4d vortex theory supported on $\mathbb{R}^2\times \mathbb{T}^2$, which is in turn captured by a free field correlator of vertex operators and screening charges of the elliptic Virasoro algebra.
We discuss the implications of studies of partition function zeros and equimodular curves for the analytic properties of the Ising model on a square lattice in a magnetic field. In particular we consider the dense set of singularities in the susceptibility of the Ising model at $H=0$ found by Nickel and its relation to the analyticity of the field theory computations of Fonseca and Zamolodchikov.
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks ha…
We study properties of the full partition function for the $U(1)$ 5D $\mathcal{N}=2^*$ gauge theory with adjoint hypermultiplet of mass $M$. This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function $G_2^C$ associated with a certain moment map cone $C$. The answer exhibits a curious $SL(4,\mathbb{Z})$ modular property. Finally, we propose a s…
It is shown that the free energy associated to a finite dimensional Airy structure is an analytic function at each finite order of the $\hbar$ expansion. Semiclassical series itself is in general divergent. Calculations are facilitated by putting the topological recursion equations into a form exhibiting more explicitly the semiclassical geometry. This formulation involves certain differential operators on the characteristic variety, which are found to satisfy a Lie algebra cocycle condition. It is proven that this cocycle is a coboundary. Developed formalism is applied in specific examples. I…
We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a defi…
A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter $\hbar$. Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in…
We derive the first $ε_2$-correction to the instanton partition functions of $\mathcal{N}=2$ Super Yang-Mills (SYM) in four dimensions in the Nekrasov-Shatashvili limit $ε_2\rightarrow 0$. In the latter we recall the emergence of the famous Thermodynamic Bethe Ansatz-like equation which has been found by Mayer expansion techniques. Here we combine efficiently these to field theory arguments. In a nutshell, we find natural and resolutive the introduction of a new operator $\nabla$ that distinguishes the singularities within and outside the integration contour of the partition function.
We apply exact WKB methods to the study of the partition function of pure N=2 epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of epsilon_1 when passing from the Nekrasov form of the partition function to the topological string t…
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…
We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of arXiv:1709.01928. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO($N$) gauge theories and the pure $G_2$ gauge theory. The obtained results agree with the results known in the literature. We also compute …
本文对D1-D5膜系统的规范线性西格玛模型在一个解析的A_1奇点上进行了精确计算。这是通过超对称局部化爆破的两个球。我们证明了在C^2/Z_2的吹落极限下，配分函数约化为Nekrasov配分函数来计算瞬子模空间的等变体积.对于有限的半径，我们得到了一个世界片瞬子修正塔，我们确定与ADHM模空间的等变Gromov-Witten不变量。我们表明，这些修正可以编码在变形的Seiberg-Witten prepotential。从数学的观点来看，所研究的D1-D5系统表现出双重性质：D1-膜的观点捕获了Givental形式主义中ADHM瞬子模空间的等变量子上同调，而D5-膜的观点与P^1 x C^2的高阶等变Donaldson-Thomas不变量有关。
We explore a new connection between Seiberg-Witten theory and quantum statistical systems by relating the dual partition function of SU(2) Super Yang-Mills theory in a self-dual Omega-background to the spectral determinant of an ideal Fermi gas. We show that the spectrum of this gas is encoded in the zeroes of the Painleve III tau function. In addition we find that the Nekrasov partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional limit of a recently proposed conjecture relating topological strings and spectral theory. In …
We prove the connection between the Nekrasov partition function of N=2 super-symmetric U(2) gauge theory with adjoint matter and conformal blocks for the Virasoro algebra, as predicted by the Alday-Gaiotto-Tachikawa relations. Mathematically, this is achieved by relating the Carlsson-Okounkov Ext vector bundle on the moduli space of rank 2 sheaves with Liouville vertex operators.Our approach is geometric in nature, and uses a new method for intersection-theoretic computations of the Ext operator.
In the context of 4D/2D dualities, SH$^c$ algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH$^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm1}(z)$ with which the algebra and its epresentations are simplified. The instanton partition functions for arbitrary $\mathcal{N}=2$ super Yang-Mills theories with $A_n$ and $A^{(1)}_n$ type quiver diagrams are compactly expressed as a product of four building blocks: Gaiotto state, dil…
We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x S^1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R^2 x S^1. We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quan…
