Currents of the Quantum Boussinesq Hierarchy [HBNI Th268]

Abstract

A conformal field theory (CFT) possessing the Virasoro algebra as its local symmetry algebra is endowed with an infinite set of mutually commuting local integrals of motion, commonly referred to as quantum Korteweg-de Vries (KdV) charges. This system defines the quantum KdV problem, where in the limit of infinite central charge, these quantum charges reduce to their classical counterparts in the KdV hierarchy. Similarly, a CFT with an extended W3 symmetry algebra is expected to admit an infinite tower of conserved local integrals of motion, known as quantum Boussinesq charges. While explicit expressions for the first four such charges are known, this thesis presents a systematic procedure for constructing the higher charges. The construction relies on two key methodological components: 1. The eigenvalues of these charges on the highest-weight state are computed using the ODE/IM correspondence. 2. The most general ansatz for the current densities is formulated, and the thermal correlators of each composite operator appearing in the ansatz are computed. These correlators are evaluated via the Zhu recursion relations, where for an operator O, the thermal expectation value is defined as: ⟨O⟩ = TrV(O qL0−c/24), q = e−β, with β denoting the inverse temperature and the trace taken over the higher-spin module generated by the action of L−n and W−m on a highest-weight state. By matching the low-temperature limit (q → 0) of these thermal correlators with the computed charge eigenvalues, the undetermined coefficients in the current ansatz are determined. To extend this construction to higher-order currents, additional spectral data are required, obtained by computing the eigenvalues of the higher charges in the first and second excited level states and evaluating thermal two-point functions of currents with composite operators. Through a systematic comparison of the q-expansion coefficients (up to O(q2)) with the spectral calculations, all remaining undetermined constants in the current densities are fixed. This approach enables the explicit construction of three previously unknown current densities of the quantum Boussinesq hierarchy: J8, J9, and J11. The main results can be summarized as follows: a systematic study of the integrable structure of W3-symmetric CFTs is performed through their quantum Boussinesq charges; the eigenvalues of these charges on the low-lying states of the higher-spin module are computed; the thermal one-point functions of the currents are computed up to weight eleven; a systematic procedure is established to construct higher conserved currents by combining the spectral data with thermal correlators; and three new current densities are explicitly derived, all of which correctly reduce to their classical Boussinesq counterparts in the large central charge limit.