Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)

We describe the algebra of invariants of the vacuum module associated with the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. We give a formula for its Hilbert--Poincar\'{e} series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the $(1,1)$-hook. Our arguments are based on a super version of the Beilinson--Drinfeld--Ra\"{i}s--Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with $\mathfrak{gl}(1|1)$. We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.Comment: 24 pages, final version; contribution to Rodney Baxter volume, J.Phys.

On the low-temperature anomalies in the thermal conductivity of plastically deformed crystals due to phonon-kink scattering

Previous experimental studies of the thermal conductivity of plastically deformed lead crystals in the superconducting state have shown strong anomalies in the thermal conductivity. Similar effects were also found for the thermal conductivity of bent ${}^4\text{He}$ samples. Until now, a theoretical explanation for these results was missing. In this paper we will introduce the process of phonon-kink scattering and show that it qualitatively explains the anomalies that experiments had found.Comment: 3 pages, follow-up paper to appear soo

Spaces of quasi-exponentials and representations of gl_N

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda, the weight subspace of weight $\lambda$ of the tensor product of k polynomial irreducible gl_N-modules with highest weights \lambda^{(1)},...,\lambda^{(k)}, respectively. The Bethe algebra depends on N complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are distinct, we prove that the image of B_K in the endomorphisms of (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the algebra of functions on the intersection of k suitable Schubert cycles in the Grassmannian of N-dimensional spaces of quasi-exponentials with exponents K. We also prove that the B_K-module (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the coregular representation of that algebra of functions. We present a Bethe ansatz construction identifying the eigenvectors of the Bethe algebra with points of that intersection of Schubert cycles.Comment: Latex, 29 page

Gaudin models for gl(m|n)

Date of Acceptance: 16/04/2015We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.Peer reviewedFinal Accepted Versio

Dicke model semiclassical dynamics in superradiant dipolar phase follows the Euler heavy top

Analytic solution is presented of the nonlinear semiclassical dynamics of superradiant photonic condensate that arises in the Dicke model of two-level atoms dipolar coupled to the electromagnetic field in the microwave cavity. In adiabatic limit with respect to photon degree of freedom the system is approximately integrable and its evolution is expressed via Jacobi elliptic functions of real time. Periodic trajectories of the semiclassical coordinate of photonic condensate either localise around two degenerate minima of the condensate ground state energy or traverse between them over the saddle point. An exact mapping of the semiclassical dynamics of photonic condensate on the motion of unstable Lagrange 'sleeping top' is found. Analytic expression is presented for the frequency dependence of transmission coefficient along a transmission line inductively coupled to the resonant cavity with superradiant condensate. Sharp transmission drops reflect Fourier spectrum of the semiclassical motion of photonic condensate and of 'sleeping top' nodding.Comment: 13 pages, 3 figures, submitted to PR