Spectral duality in integrable systems from AGT conjecture

Abstract

We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture. Namely, we prove the equivalence (spectral duality) between the N -cite Heisenberg spin chain and a reduced gl N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the SeibergWitten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous AHH duality.

Modular properties of 6d (DELL) systems

“…An additional check of (6.30) is provided by the perturbative limit Im τ → +∞: 32) which matches the exact answer for the perturbative part of the prepotential (6.3).…”

“…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].…”

Classical integrability

“…In [27], the Jacobi identity is shown, which proves that the two spans discussed above form Lie subalgebras of g. 16 A generalisation of the loop algebra is the so-called affine Kac-Moody algebraĝ, associated to a finitedimensional Lie algebra g. To obtain such a generalisation, one allows for a non-trivial central extension c. If we denote the generators of the affine Kac-Moody algebra as sa,n ≡ sa ⊗ v n in terms of a formal parameter v, we can then write the defining relations ofĝ as 19) with (, ) the scalar product induced on g by the Killing form. One usually adjoins a derivation d to the algebra: 20) in order to remove a root-degeneracy (see e.g.…”

“…The KdV equation (5.1) admits, as a particular solution 19 , a travelling soliton parameterised by two arbitrary real constants x 0 and υ:…”

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

“…1 S corresponds to rotation of the integer lattice by π 2 clockwise. The action of this element on the algebra realizes the spectral duality [72][73][74][75][76][77][78][79][80] of different representations: in particular, the central charge vector is rotated; the horizontal representations become the vertical ones and vice versa. The action of S is illustrated in figure 1.…”