Spectrum of quantum KdV hierarchy in the semiclassical limit

Abstract

We employ semiclassical quantization to calculate spectrum of quantum KdV charges in the limit of large central charge c. Classically, KdV charges Q2n−1 generate completely integrable dynamics on the co-adjoint orbit of the Virasoro algebra. They can be expressed in terms of action variables Ik, e.g. as a power series expansion. Quantum-mechanically this series becomes the expansion in 1/c, while action variables become integer-valued quantum numbers ni. Crucially, classical expression, which is homogeneous in Ik, acquires quantum corrections that include terms of subleading powers in nk. At first two non-trivial orders in 1/c expansion these “quantum” terms can be fixed from the analytic form of Q2n−1 acting on the primary states. In this way we find explicit expression for the spectrum of Q2n−1 up to first three orders in 1/c expansion. We apply this result to study thermal expectation values of Q2n−1 and free energy of the KdV Generalized Gibbs Ensemble.

Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations

“…It is also worth noting that there is a deep -and not fully understood -connection between classical and quantum integrability that goes beyond a classical limit [38,39], e.g., the generating function of quantum transfer matrices in spin chains can be identified with tau-functions of classical integrable models [40], besides their equivalence to the partition function of a matrix model formulation of 2D quantum and topological gravity [33][34][35][36]. Recently, the KdV hierarchy has been the central object in a number of papers, such as thermal correlation functions of 2D CFTs [41][42][43][44], eigenstate thermalization hypothesis [45,46], and black holes on AdS 3 [47][48][49][50][51][52][53].…”