Let @math be a conformal block, with @math consecutive channels @math , @math , in the conformal field theory @math , where @math is a @math minimal model, generated by chiral fields of spin @math , and labeled by two co-prime integers @math and @math , @math , while @math is a free boson conformal field theory. @math is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra @math , spanned by weight vectors @math . We restrict our attention to conformal blocks with vertex operators whose charge vectors point along @math . The charge vectors that label the initial and final states can point in any direction. Following the @math AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute @math , leads to ill-defined expressions. We show that restricting the states that flow in the channels @math , @math , to states labeled by @math partitions that satisfy conditions that we call @math -Burge partitions, leads to well-defined expressions that we identify with @math . We check our identification by showing that a specific non-trivial conformal block that we compute, using the @math -Burge conditions satisfies the expected differential equation.
3. In @cite_2 , Fucito, Morales and Poghossian show that @math supersymmetric Yang-Mills gauge theories on the squashed @math , with rational deformation parameters, are dual to Virasoro minimal models. Ill-defined expressions are handled using a deformation scheme, akin to that used in @cite_48 , and rested to low orders in the combinatorial expansion of the instanton partition functions.
@cite_48 @cite_2
1976841071 2154483904 2215999167 1009009209
We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system. We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum). We study the computational complexity of several natural problems arising in statistical physics and combinatorics. In particular, we consider the following problems: the mean magnetization and mean energy of the Ising model (both the ferromagnetic and the anti-ferromagnetic settings), the average size of an independent set in the hard core model, and the average size of a matching in the monomer-dimer model. We prove that for all non-trivial values of the underlying model parameters, exactly computing these averages is #P-hard. In contrast to previous results of Sinclair and Srivastava (2013) for the mean magnetization of the ferromagnetic Ising model, our approach does not use any Lee-Yang type theorems about the complex zeros of partition functions. Indeed, it was due to the lack of suitable Lee-Yang theorems for models such as the anti-ferromagnetic Ising model that some of the problems we study here were left open by Sinclair and Srivastava. In this paper, we instead use some relatively simple and well-known ideas from the theory of automatic symbolic integration to complete our hardness reductions. Abstract After a very brief recollection of how my scientific collaboration with Ugo started, in this talk I will present some recent results obtained with localization: the deformed gauge theory partition function Z ( τ → | q ) and the expectation value of circular Wilson loops W on a squashed four-sphere will be computed. The partition function is deformed by turning on τ J tr Φ J interactions with Φ the N = 2 superfield. For the N = 4 theory SUSY gauge theory exact formulae for Z and W in terms of an underlying U ( N ) interacting matrix model can be derived thus replacing the free Gaussian model describing the undeformed N = 4 theory. These results will be then compared with those obtained with the dual CFT according to the AGT correspondence. The interactions introduced previously are in fact related to the insertions of commuting integrals of motion in the four-point CFT correlator and the chiral correlators are expressed as τ -derivatives of the gauge theory partition function on a finite Ω -background.