# 'Quantum integrability: old and new tools': summary of second Volin's lecture

Summary of second Volin's lecture 'Quantum integrability: old and new tools' (tomorrow Friday May 20th at 10:00). Knowledge of the material from the previous lecture is desirable but not compulsory.

Part IA: Notion of a quantum spectral curve (QSC) is introduced and it is explained in what sense it is a quantisation of a classical curve. Discussion includes the historically first example: relation between Schrodinger and Hamilton-Jacobi equations. On the next step, we introduce a geometric description of QSC for systems with GL(N) symmetry as a special-type section of a holomorphic bundle with fiber being the maximal flag of C^N. This construction naturally arises whenever we solve order-N differential or finite-difference equations. Symmetries behind the geometric constructions are discussed.

Part IB: We discuss the ring of supersymmetric polynomials and demonstrate its isomorphism to the ring of symmetric polynomials, in the case N=infty, M=infty (superduality). Inspired by these observation, we translate these ideas to relation between GL(N|M) and GL(N+M) QSC's thus effectively formulating QSC for GL(N|M) case. Part I is finalised by a practical example: new efficient way to solve Bethe equations for rational spin chains (including the Heisenberg XXX).

Part IIA (After coffee break): Baxter equation has two solutions. We discuss the role of the dominant and subdominant solutions for the case of compact SU(2) and non-compact SL(2) spin chains. Non-compact case is rich with Stoke's phenomena; it also brings us the need to perform Borel re-summation, prescription for the latter naturally introduces the notion of QSC analytic in the upper or lower half-planes.

Part IIB: We motivate and formulate QSC that encodes the spectrum of planar N=4 SYM.