Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

Abstract

We consider Dotsenko-Fateev matrix models associated with compactified Calabi-Yau threefolds. They can be constructed with the help of explicit expressions for refined topological vertex, i.e. are directly related to the corresponding topological string amplitudes. We describe a correspondence between these amplitudes, elliptic and affine type Selberg integrals and gauge theories in five and six dimensions with various matter content. We show that the theories of this type are connected by spectral dualities, which can be also seen at the level of elliptic Seiberg-Witten integrable systems. The most interesting are the spectral duality between the XYZ spin chain and the Ruijsenaars system, which is further lifted to self-duality of the double elliptic system.

Modular properties of 6d (DELL) systems

“…From the viewpoint of integrable systems, Γ corresponds to the spectral curve Γ CM of the elliptic Calogero-Moser model [4,61,62] known to have the same geometrical description [63] (generalization to the case of more than two NS5 branes leads to the spin Calogero model, see [17]). To avoid uncertainties in the notation, from now on, we denote the curve Γ of the 4d theory under consideration by Γ CM .…”

“…One of the research directions here is the interpretation of the corresponding Nekrasov functions in terms of the representation theory of DIM algebras [20,21] and network models [18,22], which generalize the Dotsenko-Fateev (conformal matrix model [23][24][25][26][27][28]) realization of conformal blocks, manifest an explicit spectral duality [16,17,[29][30][31][32][33][34] and satisfy the Virasoro/W-constraints in the form of the qq-character equations [18,21,[35][36][37]. Another direction is study of the underlying integrable systems, where the main unknown ingredient is the double-elliptic (DELL) generalization [38][39][40][41][42][43] of the Calogero-Ruijsenaars model [44][45][46][47][48][49][50][51].…”

Toric Calabi-Yau threefolds as quantum integrable systems. ℛ $$ \mathrm{\mathcal{R}} $$ -matrix and ℛ T T $$ \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} $$ relations

“…The sum over intermediate Young diagrams in the computation of any amplitude can also be interpreted as a "network"-type matrix model [41,[98][99][100]. For certain "balanced" toric diagrams, the corresponding matrix model can be identified with the Dotsenko-Fateev (DF) representation for the multipoint conformal blocks of the q-deformed W N algebra [41].…”

“…The spectral dual interpretation of the resulting amplitude is the partition function of a 6d linear quiver gauge theory compactified on a two-dimensional torus. The AGT relations in this case [121][122][123][124][125][126][127][128][129][130][131][132][133][134] give the conformal block of the q-deformed W -algebras on torus, or the spherical conformal block of the affine W -algebra [98,99].…”

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

“…The second famous example comes from the gauge theory: the equivariant cohomology of the instanton moduli spaces (captured by Nakajima quiver varieties [12][13][14] and the corresponding Nekrasov partition functions [15][16][17][18][19][20][21][22]) is acted on by a certain vertex operator algebra, which turns out to be the W N -algebra of two dimensional conformal field theory. This correspondence between the geometric (moduli space) and algebraic (W N -algebra) objects is known as the AGT relation [23][24][25] and has many known implications and generalizations [26][27][28][29][30][31][32][33][34][35][36]. These two examples are in fact directly related to each other and their relation can be understood on both sides of the algebro-geometric correspondence.…”