Coherent states in quantum $\mathcal{W}_{1+\infty}$ algebra and qq-character for 5d super Yang–Mills

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

“…In this appendix, which is based on the spectral duality, we present conjectures for explicit expressions of the action of x + ±1 on the generalized Macdonald polynomials, which are defined to be eigenfunctions of the Hamiltonian X (1) 0 . We also conjecture the eigenvalues of higher Hamiltonians acting on the generalized Macdonald polynomials from those of the spectral dual generators provided in [142]. Our conjectures mean that the generalized Macdonald polynomials explicitly realize the spectral dual basis to | u, λ in [142].…”

“…We also conjecture the eigenvalues of higher Hamiltonians acting on the generalized Macdonald polynomials from those of the spectral dual generators provided in [142]. Our conjectures mean that the generalized Macdonald polynomials explicitly realize the spectral dual basis to | u, λ in [142]. …”

“…One can consider a representation of the DIM algebra which is called rank N representation and can be realized in terms of a basis | u, λ called AFLT basis, [142]. This representation is given by the N -fold tensor product of the level (0,1) representations (i.e.…”

“…It is expected that the basis M λ corresponds to the AFLT basis 5 in [142] and realizes the rank N representation through the spectral duality S. That is to say, for any generator a in the DIM algebra, that the action of ρ (N ) u 1 ,...,u N • S(a) on the integral forms M λ are the same as the action of ρ rankN (a) on the basis | u, λ [142]. Indeed, one can check that the action of x + ±1 on the generalized Macdonald polynomials is given by the following conjecture.…”

Modular properties of 6d (DELL) systems

“…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].…”