A & B Model Approaches to Surface Operators and Toda Theories

Journal of High Energy Physics

We study half-BPS surface operators in 5d N = 1 gauge theories compactified on a circle. Using localization methods and the twisted chiral ring relations of coupled 3d/5d quiver gauge theories, we calculate the twisted chiral superpotential that governs the infrared properties of these surface operators. We make a detailed analysis of the localization integrand, and by comparing with the results from the twisted chiral ring equations, we obtain constraints on the 3d and 5d Chern-Simons levels so that the instanton partition function does not depend on the choice of integration contour. For these values of the Chern-Simons couplings, we comment on how the distinct quiver theories that realize the same surface operator are related to each other by Aharony-Seiberg dualities.

Journal of High Energy Physics, 2011

We present a new duality between the F-terms of supersymmetric field theories defined in two-and four-dimensions respectively. The duality relates N = 2 supersymmetric gauge theories in four dimensions, deformed by an Ω-background in one plane, to N = (2, 2) gauged linear σ-models in two dimensions. On the four dimensional side, our main example is N = 2 SQCD with gauge group G = SU(L) and N F = 2L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.

Journal of High Energy Physics, 2014

A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P ,Q] = −i between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and βensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of the modular transformation is then defined as the intertwiner of the two monodromies, and can be obtained straightforwardly, even when the eigenfunction interpretation of the blocks themselves is technically tedious. In this way, we provide an elementary derivation of the old expression for the modular kernel for the one-point toric conformal block.

Journal of High Energy Physics, 2014

We decompose sphere partition functions and indices of three-dimensional N = 2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2, 0) theory on a threemanifold M , the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.

Journal of High Energy Physics - J HIGH ENERGY PHYS, 2011

The AGT relation identifies the Nekrasov functions for various $$ \mathcal{N} = 2 $$ SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (β-ensemble) representations the latter being polylinear combinations of Selberg integrals. The “pure gauge” limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.

Journal of High Energy Physics, 2019

We use entanglement entropy to define a central charge associated to a twodimensional defect or boundary in a conformal field theory (CFT). We present holographic calculations of this central charge for several maximally supersymmetric CFTs dual to eleven-dimensional supergravity in Anti-de Sitter space, namely the M5-brane theory with a Wilson surface defect and three-dimensional CFTs related to the M2-brane theory with a boundary. Our results for the central charge depend on a partition of N M2-branes ending on M M5-branes. For the Wilson surface, the partition specifies a representation of the gauge algebra, and we write our result for the central charge in a compact form in terms of the algebra’s Weyl vector and the representation’s highest weight vector. We explore how the central charge scales with N and M for some examples of partitions. In general the central charge does not scale as M 3 or N 3/2, the number of degrees of freedom of the M5- or M2-brane theory at large M or N...

Journal of High Energy Physics, 2018

We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe qq-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally found in this paper that the presence of the preferred direction brings a nontrivial effect. We provide the modified formula involving this point. We then apply our prescription of the refined geometric transition to proposing the stringy description of doubly quantized Seiberg-Witten curves called qq-characters in certain cases.

Journal of High Energy Physics, 2011

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the N f = 4, SU(2) twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the N f = 4, SU(2) supersymmetric gauge theory from the classical block.

Journal of High Energy Physics, 2016

In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce β-deformed version of those models, and derive differential equations for associated α/β-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.

Journal of High Energy Physics, 2018

As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that hav...

Journal of High Energy Physics, 2013

We continue to investigate the relationship between the infrared physics of N = 2 supersymmetric gauge theories in four dimensions and various integrable models such as Gaudin, Calogero-Moser and quantum spin chains. We prove interesting dualities among some of these integrable systems by performing different, albeit equivalent, quantizations of the Seiberg-Witten curve of the four dimensional theory. We also discuss conformal field theories related to N = 2 4d gauge theories by the Alday-Gaiotto-Tachikawa (AGT) duality and the role of conformal blocks of those CFTs in the integrable systems. As a consequence, the equivalence of conformal blocks of rank two Toda and Novikov-Wess-Zumino-Witten (WZNW) theories on the torus with punctures is found.

Journal of High Energy Physics, 2012

In this article we explore the duality between the low energy effective theory of fivedimensional N = 1 SU (N) M −1 and SU (M) N −1 linear quiver gauge theories compactified on S 1. The theories we study are the five-dimensional uplifts of four-dimensional superconformal linear quivers. We study this duality by comparing the Seiberg-Witten curves and the Nekrasov partition functions of the two dual theories. The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. The result of our study is a map between the gauge theory parameters, i.e., Coulomb moduli, masses and UV coupling constants, of the two dual theories. Apart from the obvious physical interest, this duality also leads to compelling mathematical identities. Through the AGTW conjecture these five-dimentional gauge theories are related to q-deformed Liouville and Toda SCFTs in two-dimensions. The duality we study implies the relations between Liouville and Toda correlation functions through the map we derive.

Journal of High Energy Physics, 2011

We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N = 2 SU(2) n gauge theory with circular quiver diagram by the AGT relation. We first verify that the generalized matrix model is obtained from the perturbative calculation of the Liouville correlation function. We then derive the Seiberg-Witten curve for N = 2 gauge theory as a spectral curve of the generalized matrix model.

Journal of High Energy Physics, 2021

We show how to map Grothendieck’s dessins d’enfants to algebraic curves as Seiberg-Witten curves, then use the mirror map and the AGT map to obtain the corresponding 4d$$ \mathcal{N} $$N= 2 supersymmetric instanton partition functions and 2d Virasoro conformal blocks. We explicitly demonstrate the 6 trivalent dessins with 4 punctures on the sphere. We find that the parametrizations obtained from a dessin should be related by certain duality for gauge theories. Then we will discuss that some dessins could correspond to conformal blocks satisfying certain rules in different minimal models.

Journal of High Energy Physics, 2014

In this article we use 5-brane junctions to study the 5D T N SCFTs corresponding to the 5D N = 1 uplift of the 4D N = 2 strongly coupled gauge theories, which are obtained by compactifying N M5 branes on a sphere with three full punctures. Even though these theories have no Lagrangian description, by using the 5-brane junctions proposed by Benini, Benvenuti and Tachikawa, we are able to derive their Seiberg-Witten curves and Nekrasov partition functions. We cross-check our results with the 5D superconformal index proposed by Kim, Kim and Lee. Through the AGTW correspondence, we discuss the relations between 5D superconformal indices and n-point functions of the q-deformed W N Toda theories.