Gauge/Vortex Duality and AGT

165

We study N = 1 theories on Hermitian manifolds of the form M 4 = S 1 × M 3 with M 3 a U(1) fibration over S 2 , and their 3d N = 2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D 2 × T 2 and D 2 × S 1 . We prove that when the 4d and 3d anomalies are cancelled, the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D 2 × T 2 and D 2 × S 1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.

Section: Jhep11(2015)155mentioning

confidence: 99%

Factorisation and holomorphic blocks in 4d

Nieri

Pasquetti

2015

165

“…In particular, it is unclear what vertex operator one would write in D 4 -Toda; indeed, in the little string formalism, the defect is the null weight, which suggests a trivial conformal block with no vertex operator insertion! To investigate this issue more carefully, it is useful to keep m s finite and work in the little string proper; there, a computation in the spirit of [7,50,51], shows that the partition function of T 2d is in fact not a q-conformal block of D 4 Toda, due to subtleties of certain non-cancelling fugacities. In other words, the claim that the partition function of T 2d is a q-conformal block of g-type Toda fails precisely when T 2d is an unpolarized defect, and only for those cases.…”

Section: Jhep05(2017)082mentioning

confidence: 99%

Little string origin of surface defects

Haouzi

Schmid

2017

We derive a large class of codimension-two defects of 4d N = 4 Super YangMills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [1]. Furthermore, we provide a geometric perspective on the nilpotent orbit classification of codimension-two defects, and the connection to ADEtype Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.

“…It was shown in [57][58][59][60], that certain refined topological string amplitudes on toric CY three-folds correspond to conformal blocks of the q-deformed Virasoro or W N -algebras. 5 The horizontal legs of the toric diagram represent the Hilbert space of the CFT, on which the conformal algebra acts, and the intersections with vertical legs give vertex operators or intertwiners of the algebra (see figure 1).…”

Section: Q-deformed Cftmentioning

confidence: 99%

Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

Zenkevich

2017

We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identified with refined topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.