Quantum integrability of $\mathcal{N}=2$ 4d gauge theories

Abstract

We provide a description of the quantum integrable structure behind the Thermodynamic Bethe Ansatz (TBA)-like equation derived by Nekrasov and Shatashvili (NS) for $\mathcal{N}=2$ 4d Super Yang-Mills (SYM) theories. In this regime of the background, -- we shall show --, the instanton partition function is characterised by the solution of a TQ-equation. Exploiting a symmetry of the contour integrals expressing the partition function, we derive a 'dual' TQ-equation, sharing the same T-polynomial with the former. This fact allows us to evaluate to $1$ the quantum Wronskian of two dual solutions (for $Q$) and, then, to reproduce the NS TBA-like equation. The latter acquires interestingly the deep meaning of a known object in integrability theory, as its two second determinations give the usual non-linear integral equations (nlies) derived from the 'dual' Bethe Ansatz equations.

21 pages

References in corpus (12)

• Thermodynamic Bethe Ansatz for the AdS_5 x S^5 Mirror Model
• Deforming SW curve
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• BPS/CFT correspondence II: Instantons at crossroads, Moduli and Compactness Theorem
• BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters
• (p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver
• Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Seiberg-Witten period relations in Omega background

Cited by in corpus (4)

• A slow review of the AGT correspondence
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• More on the SW-QNM correspondence
• Seiberg-Witten period relations in Omega background