New results in N=2 theories from non-perturbative string

*This is a possible type of post*
“New results in N=2 theories from non-perturbative string”

This is a paper that Alba Grassi (ICTP-ST&FI), Alessandro Tanzini (SISSA  – GAST) and myself (SISSA-ST&FI) wrote recently.

It can be downloaded from the arXiv at [ https://arxiv.org/pdf/1704.01517.pdf ]

In this paper we describe the magnetic phase of SU(N) N = 2 Super Yang-Mills theories in the self-dual Ω background in terms of
a new class of multi-cut matrix models. These arise from a non-perturbative completion of topological strings in the dual four
dimensional limit which engineers the gauge theory in the strongly coupled magnetic frame. The corresponding spectral
determinants provide natural candidates for the τ -functions of isomonodromy problems for flat spectral connections associated to
the Seiberg-Witten geometry.

The paper is a continuation of a research project we started in a previous paper, namely “Seiberg–Witten theory as a Fermi gas”
by the same authors [ https://arxiv.org/pdf/1603.01174.pdf ] where we attached the problem of the SU(2) case. In this earlier
paper we explore a new connection between Seiberg–Witten theory and quantum statistical systems by relating the dual partition
function of SU(2) Super Yang-Mills theory in a self–dual Ω–background to the spectral determinant of an ideal Fermi gas. We show
that the spectrum of this gas is encoded in the zeroes of the Painlev´e III3 τ function. In addition we find that the Nekrasov
partition function on this background can be expressed as an O(2) matrix model. Our construction arises as a four-dimensional
limit of a recently proposed conjecture relating topological strings and spectral theory. In this limit, we provide a
mathematical proof of the conjecture for the local P1×P1 geometry.

Our aim is to make use of the topological string/spectral theory correspondence, proposed in “Topological Strings from Quantum
Mechanics” by Alba Grassi, Yasuyuki Hatsuda and Marcos Marino [ https://arxiv.org/pdf/1410.3382.pdf ] and to give at the same
time calculable examples where the proposal can be tested against gauge theory results. (For futher developments about the TS/ST
correspondence, check the inspire record of subsequent papers by the above authors)

A key tool in this perspective is the relation between the dual Nekrasov-Okunkov partition function — as defined in
“Seiberg-Witten theory and random partitions” by Nikita Nekrasov and Andrei Okounkov [ https://arxiv.org/pdf/hep-th/0306238.pdf ]
— and the tau-function of Painleve’ equations, or more general isomonodromic deformation problems, as in the paper “On
Painleve’/gauge theory correspondence” by Giulio Bonelli (SISSA – ST&FI), Oleg Lisovyy (Tours U., CNRS), Kazunobu Maruyoshi
(Seikei U.), Antonio Sciarappa (Korea Inst. Advanced Study, Seoul – former ST&FI) and Alessandro Tanzini (SISSA – GAST) [
https://arxiv.org/pdf/1612.06235.pdf ] (see also the previous papers by Lisovyy and others on the subject).

The open points in my view are mainly

– extension to other gauge groups

– extention to gauge theories with matter multiplets

– extension to quiver gauge theories

– extension to higher dimensional gauge theories

– proof of the TS/ST correspondence from first principles

Notice that in a forecoming version 2 of the paper, an exact S-duality integral transform in obtained (in the spirit of
Aganagic-Bouchard-Klemm) which could be compared with the results obtained some time ago by a Torino-TorVergata collaboration in
the ST&FI IS and others. (We discuss the unrefined version of the SYM theory, while they discuss the refined version of massive deformed conformal gauge theories, so the comparison could be not immediate).