Intertwining operator and integrable hierarchies from topological strings

Abstract

In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

Engineering 3D $\mathcal{N}=2$ theories using the quantum affine $\mathfrak{sl}(2)$ algebra

“…In this letter, we have presented a formalism for the algebraic engineering of 3D N = 2 gauge theories that is arguably simpler than the previous ones based on quantum toroidal algebras. We hope that it will help elucidates certain aspects of this technique, such as the role of the auxiliary Fock space supported by NS5-branes, or the relation with integrable hierarchies recently pointed out in [65]. The technique easily extends to type A quiver gauge theories with U(N) gauge groups via the gluing of intertwiners.…”

Shifted Quiver Quantum Toroidal Algebra and Subcrystal Representations

“…• Studying horizontal representations (C = 1) [10,[16][17][18][19][20][21][22][23][24]69] of shifted QQTA is also one of the studies that must be done. Studying generalized intertwiners [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] with shift parameters as Φ r,r : (vertical) r ⊗ (horizontal) r → (horizontal) r+r is interesting. Actually the original motivation of this work was to study two-dimensional crystal representations that might enter in the vertical representation part of the intertwiner.…”

“…Infinite-dimensional alebra has been one of the most powerful tools to study supersymmetric gauge theories [1][2][3][4][5]. In particular, quantum toroidal algebras [6][7][8][9][10][11][12][13][14][15] and their truncations [10,[16][17][18][19][20][21][22][23][24] have played significant roles in the context of 5d AGT correspondence [25][26][27][28][29][30][31][32] and topological vertex [29,30,[32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]…”

Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

“…Given the observed connections between topological strings and integrable hierarchies [38,[40][41][42], the relation between W and gl(∞) comes as no surprise. One of the motivation for this paper is to extend these connections to the subalgebras W X .…”

“…In fact, this property follows from the intertwining relation obeyed by the vacuum component of the AFS intertwiner (see[42]). Thus, it would seem that we are uncovering different aspects of a bigger picture.…”