Matrix model conjecture for exact BS periods and Nekrasov functions

Abstract

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-, where ε and β are related to the LNS parameters ǫ1 and ǫ2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies. Today it is clear that a new unification of the fundamental importance emerges, bringing together at a principally new level the CFT, the theory of loop algebras, SW theory, quantization theory, Baxter equations, DV phase of matrix models, loop equations, the theory of hypergeometric functions, symmetric groups, Hurwitz theory, Kontsevich models and modern combinatorics. This unification is capable to resolve a number of long-standing problems in each of the fields. The goal of this paper is to briefly summarize our knowledge about this emerging pattern, which is scattered and expressed in length in a number of fresh [29,30], makes the picture complete, resolves the remaining uncertainties (about the shape of the second deformation) in [56,57] and finalizes the program [3] to reformulate SW theory of [1,2] in terms of the BS integrals and underlying integrable systems. We do not discuss long formulas, checks and even evidence in favor of all these conjectures: all calculations in these fields remain long and tedious, and most statements still need to be checked and proved, however, the entire picture is starting to get relatively clear.

Quantum geometry of resurgent perturbative/nonperturbative relations

“…We adopt the semiclassical path integral approach of Balian and Bloch [34,35] and the geometric Stokes diagram and monodromy picture [6][7][8]36], to express the P/NP relations in terms of all-orders WKB ("exact WKB") actions and dual actions. In so doing, it proves useful to adopt some of the language and ideas from supersymmetric gauge theories, integrability, conformal field theory and wall-crossing [37][38][39][40][41][42][43][44][45][46][47][48], given the close connection of such theories with exact WKB [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. This is similar to the philosophy of [27], where the refined holomorphic anomaly equation of topological string theory is shown to describe these P/NP relations for some 1d quantum oscillator systems.…”

“…For the Mathieu system, which is associated with N = 2 supersymmetric gauge theory [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65], the P/NP relation is:…”

Seiberg-Witten curve via generalized matrix model

“…(See also [42] for 1/N correction.) It has been discussed that direct integral calculation leads to the instanton (q-)expansion of the Nekrasov partition function (and the corresponding expansion of the conformal block) [43][44][45][46][47][48][49].…”

О Калибровочных Теориях Как Матричных Моделях

“…Дифференциал (2.13) удовлетворяет условию (здесь вариация производится при по-стоянной координате z и постоянном масштабном множителе Λ) 15) поскольку условие их непротиворечивости сводится к симметричности матрицы пе-риодов -частному случаю билинейных соотношений Римана.…”

“…Это происходит весьма схожим об-разом с квазиклассикой матричной модели, хотя некоторые подробности (которые будут рассмотрены ниже) отличаются в этих двух случаях весьма существенно. Уже много раз было предложено (см., например, недавние попытки в работах [14], [15]) интерпретировать функции Некрасова в духе матричных моделей и вне рамок ква-зиклассического приближения, по крайней мере, в рамках пертурбативного раз-ложения, с помощью которого вычисляются поправки к препотенциалу. Однако подобное соотношение вовсе не является очевидным.…”