Abstract
We propose an efficient grassmannian formalism for solution of bi-linear finite-difference Hirota equation (T-system) on T-shaped lattices related to the space of highest weight representations of $gl(K_1,K_2|M)$ superalgebra. The formalism is inspired by the quantum fusion procedure known from the integrable spin chains and is based on exterior forms of Baxter-like Q-functions. We find a few new interesting relations among the exterior forms of Q-functions and reproduce, using our new formalism, the Wronskian determinant solutions of Hirota equations known in the literature. Then we generalize this construction to the twisted Q-functions and demonstrate the subtleties of untwisting procedure on the examples of rational quantum spin chains with twisted boundary conditions. Using these observations, we generalize the recently discovered, in our paper with N. Gromov, AdS/CFT Quantum Spectral Curve for exact planar spectrum of AdS/CFT duality to the case of arbitrary Cartan twisting of AdS$_5\times$S$^5$ string sigma model. Finally, we successfully probe this formalism by reproducing the energy of gamma-twisted BMN vacuum at single-wrapping orders of weak coupling expansion.
Highlights
In 1931, Hans Bethe analysed the very first example of a quantum integrable model — Heisenberg SU(2) XXX spin chain — and showed that it can be reduced to algebraic equations which bear his name [1]
If we focus on the QQrelation (3.14a), we see that if the Q-functions are polynomial, their degrees obey deg Q[At∓;∅1;S] 3 (QA),a|I + deg QA,b|I =
We gave a general description of grassmannian structure emerging from fusion relations in integrable rational Heisenberg super-spin chains
Summary
In 1931, Hans Bethe analysed the very first example of a quantum integrable model — Heisenberg SU(2) XXX spin chain — and showed that it can be reduced to algebraic equations which bear his name [1]. This is a quite well established topic in the literature, in particular its geometric interpretation can be spotted from discussion in [4]. It happens in a very natural way: one should gauge the global rotational GL(N ) symmetry w.r.t. the space of spectral parameter, making it local and introducing a new object: a holomorphic connection A. A potential advantage of our method is the possibility to find the corrections to this state on a regular basis, by the methods similar to [22, 23] as well as application of the efficient numerical procedure of [24], but this is beyond the scope of the current paper
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