We study Penner type matrix models in relation with the Nekrasov partition function of four dimensional \mathcal{N}=2, SU(2) supersymmetric gauge theories with N_F=2,3 and 4. By evaluating the resolvent using the loop equation for general β, we explicitly construct the first half-genus correction to the free energy and demonstrate the result coincides with the corresponding Nekrasov partition function with general Ω-background, including higher instanton contributions after modifying the relation of the Coulomb branch parameter with the filling fraction. Our approach complements the proof using the Selberg integrals directly which is useful to find the contribution in the series of instanton numbers for a given deformation parameter.
The five dimensional AGT correspondence implies the connection between the q-deformed Virasoro block and the 5d Nekrasov partition function. In this paper, we determine a q-deformation of the four-point block in the Coulomb gas representation from the 5d Nekrasov function, and obtain an expression of the q-deformed vertex operator. If we use only one kind of the q-vertex operators, one of the insertion points of them must be modified in order to hold the 2d/5d correspondence.
Nekrasov partition function for the supersymmetric gauge theories with general Lie groups is not so far known in a closed form while there is a definition in terms of the integral. In this paper, as an intermediate step to derive it, we give a recursion formula among partition functions, which can be derived from the integral. We apply the method to a toy model which reflects the basic structure of partition functions for BCD type Lie groups and obtained a closed expression for the factor associated with the generalized Young diagram.
We study $SU(2)$ gauge theories coupled to $(A_1,D_N)$ theories with or without a fundamental hypermultiplet. For even $N$, a formula for the contribution of $(A_1,D_N)$ to the Nekrasov partition function was recently obtained by us with Y.~Sugawara and T.~Uetoko. In this paper, we generalize it to the case of odd $N$ in the classical limit, under the condition that the relevant couplings and vacuum expectation values of Coulomb branch operators of $(A_1,D_N)$ are all turned off. We apply our formula to the $(A_2,A_5)$ theory to find that its prepotential is related to that of the $SU(2)$ gaug…
We discuss the effective twisted superpotentials of 2d $\mathcal{N}=(2,2)$ theories arising upon the reduction of 4d $\mathcal{N}=2$ gauge theories on the $Ω$-deformed cigar-like geometry. We explain field-theoretic origins of the gaps in the spectrum in the corresponding quantum mechanical (QM) systems. We find local 2d descriptions of the physics near these gaps by resumming the non-perturbative part of the twisted superpotential and discuss arising wall-crossing phenomena. The interpretation of the associated phenomena in the classical Liouville theory and in the scattering of two heavy sta…
We reinterpret the OSV formula for the on-shell action/entropy function of asymptotically flat BPS black holes as a fixed point formula that is formally equivalent to a recent gluing proposal for asymptotically AdS$_4$ black holes. This prompts a conjecture that the complete perturbative answer for the most general gravitational building block of 4d $\mathcal{N}=2$ supergravity at a single fixed point takes the form of a Nekrasov-like partition function with equivariant parameters related to the higher-derivative expansion of the prepotential. In turn this leads to a simple localization-like p…
We study the multi-instanton partition functions of the $Ω$-deformed $\mathcal N =2^{*}$ $SU(2) $ gauge theory in the Nekrasov-Shatashvili (NS) limit. They depend on the deformation parameters $ε_{1}$, the scalar field expectation value $a$, and the hypermultiplet mass $m$. At fixed instanton number $k$, they are rational functions of $ε_{1}, a, m$ and we look for possible regularities that admit a parametrical description in the number of instantons. In each instanton sector, the contribution to the deformed Nekrasov prepotential has poles for "large" deformation parameters. To clarify the pr…
We show that the three different looking BPS partition functions, namely the elliptic genus of the 6d $\mathcal{N}=(1,0)$ $Sp(1)$ gauge theory with $10$ flavors and a tensor multiplet, the Nekrasov partition function of the 5d $\mathcal{N}=1$ $Sp(2)$ gauge theory with $10$ flavors, and the Nekrasov partition function of the 5d $\mathcal{N}=1$ $SU(3)$ gauge theory with $10$ flavors, are all equal to each other under specific maps among gauge theory parameters. This result strongly suggests that the three gauge theories have an identical UV fixed point. Type IIB 5-brane web diagrams play an esse…
We prove functional equations of Nekrasov partition functions for $A_{1}$-singularity, suggested by Ito-Maruyoshi-Okuda. Our proof uses the method by Nakajima-Yoshioka based on the theory of wall-crossing formula developed by Mochizuki.
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…
We propose a new prescription for computing the Nekrasov partition functions of five-dimensional theories with eight supercharges realized by gauging non-perturbative flavor symmetries of three five-dimensional superconformal field theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with flavor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various five-dimensional gauge theories such as $\mathrm{SO}(2N)$ gauge theories with…
It is shown that the generating function of $\mathscr{N}=2$ topological strings, in the heterotic weak coupling limit, is identified with the partition function of a six-dimensional Melvin background. This background, which corresponds to an exact CFT, realises in string theory the six-dimensional $\varOmega$-background of Nekrasov, in the case of opposite deformation parameters $ε_1=-ε_2$, thus providing the known perturbative part of the Nekrasov partition function in the field theory limit. The analysis is performed on both heterotic and type I strings and for the cases of ordinary $\mathsc…
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 investigate the quasi-Hopf twist of the quantum toroidal algebra of $\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum…
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.
We exactly compute the partition function for $U(2)_k\times U(2)_{-k}$ ABJM theory on $\mathbb S^3$ deformed by mass $m$ and Fayet-Iliopoulos parameter $ζ$. For $k=1,2$, the partition function has an infinite number of Lee-Yang zeros. For general $k$, in the decompactification limit the theory exhibits a quantum (first-order) phase transition at $m=2ζ$.
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 (…
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 construct the ($β$-deformed) partition function hierarchies with $W$-representations. Based on the $W$-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), $N\times N$ complex matrix model, $β$-deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.
We demonstrate the presence of modular properties in partition functions of $T\bar{T}$ deformed conformal field theories. These properties are verified explicitly for the deformed free boson. The modular features facilitate a derivation of the asymptotic density of states in these theories, which turns out to interpolate between Cardy and Hagedorn behaviours. We also point out a sub-sector of the spectrum that remains undeformed under the $T\bar{T}$ flow. Finally, we comment on the deformation of the CFT vacuum character and its implications for the holographic dual.
We consider $\mathcal N=2$ $SU(2)$ gauge theories in four dimensions (pure or mass deformed) and discuss the properties of the simplest chiral observables in the presence of a generic $Ω$-deformation. We compute them by equivariant localization and analyze the structure of the exact instanton corrections to the classical chiral ring relations. We predict exact relations valid at all instanton number among the traces $\langle\text{Tr}φ^{n}\rangle$, where $φ$ is the scalar field in the gauge multiplet. In the Nekrasov-Shatashvili limit, such relations may be explained in terms of the available q…
We propose $q$-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé $τ$ function. Namely we propose formula for $τ$ function for $q$-difference Painlevé equation corresponding to $A_7^{(1)}{}'$ surface (and $A_1^{(1)}$ symmetry) in Sakai's classification. In this formula $τ$ function equals the series of $q$-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure $SU(2)$ 5d theory).
We compute the partition functions of $\mathcal{N} = 1$ gauge theories on $S^2 \times \mathbb{R}^2_\varepsilon$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of $\mathbb{R}^2_\varepsilon$. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the $\mathcal{N} = 1$ partition functions on the $Ω$-background, and show that the Nekrasov partition functions can be recovered from these building blocks.
