T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve

TL;DR

This work develops a Grassmannian, exterior-form formulation of Q-systems to solve the Hirota T-system on T-hooks for ${ m gl}(K_1,K_2|M)$, unifying Strips and L-/T-hooks via Wronskian Q-function determinants. It then extends these algebraic insights to twisted boundary conditions, introducing a twisted QSC framework that accommodates arbitrary Cartan twisting in AdS$_5$/CFT$_4$, including gamma- and beta-deformations, via exponential prefactors or holomorphic connections. The authors analyze untwisting subtleties, Bäcklund flows, and grading, and demonstrate the formalism by reproducing the one-wrapping energy of the gamma-deformed BMN vacuum from the twisted QSC. This approach provides a geometrically transparent, operator-friendly route to exact planar spectra under twists and promises systematic extensions to other sigma models and higher-loop regimes. The results offer a structurally rich, computationally efficient toolkit for AdS/CFT integrability with deformations and non-compact representations.

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.

Figures (14)

• Figure 1: Q-functions define a fibration of grassmannians over the Riemann surface of the spectral parameter $u$. Relation between grassmannians of different rank is restricted by \ref{['eq:81D']}.
• Figure 2: Deformation of the fibration by introducing a connection. This connection "rotates" the spaces $V_{(n)}$ via the parallel transport from point $u\pm\frac{\mathbbm i}{2}$ to point $u$ where the equation (\ref{['eq:81D']}) can be used.
• Figure 3: The Young diagrams of compact representations of ${\mathsf{gl}}(N)$ group are confined to a half-strip, depicted on fig.(a), of width $N$ on infinite representational $(a,s)$-lattice. The vertices within this strip are in one-to-one correspondence with rectangular Young tableux of size $a\times s$, as depicted on fig.(b), as well as with corresponding characters or T-functions.
• Figure 4: Fat hook and T-hook, for supersymmetric symmetry groups.
• Figure 5: Hasse diagram for ${\mathsf{gl}}(3)$ Q-functions.