Boundary TBA, trees and loops

Abstract

We derive a graph expansion for the thermal partition function of solvable two-dimensional models with boundaries. This expansion of the integration measure over the virtual particles winding around the time cycle is obtained with the help of the matrix-tree theorem. The free energy is a sum over all connected graphs, which can be either trees or trees with one loop. The generating function for the connected trees satisfies a non-linear integral equation, which is equivalent to the TBA equation. The sum over connected graphs gives the bulk free energy as well as the exact g-functions for the two boundaries. We reproduced the integral formula conjectured by Dorey, Fioravanti, Rim and Tateo, and proved subsequently by Pozsgay. The method is easily extended to the case of non-diagonal bulk scattering and diagonal reflection matrices. Our method can be extended to the case of non-diagonal bulk scattering and diagonal reflection matrices with proper regularization.

arXiv comment26 pages, 7 figures version 2: comments about non-diagonal scattering added version 3: comments on excited state g-function added, goal of section 6 changed

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• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
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• Boundary state bootstrap and asymptotic overlaps in AdS/dCFT
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• Crosscap States in Integrable Field Theories and Spin Chains
• Giant Wilson Loops and AdS${}_2$/dCFT${}_1$
• Duality Relations for Overlaps of Integrable Boundary States in AdS/dCFT
• Structure constants of short operators in planar $\mathcal{N}=4$ SYM theory
• Functional Equations and Separation of Variables for Exact g-Function
• Effective Quantum Field Theory for the Thermodynamical Bethe Ansatz
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• Cumulants of conserved charges in GGE and cumulants of total transport in GHD: exact summation of matrix elements?
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