PSU(2, 2|4) character of quasiclassical AdS/CFT

Abstract

Abstract:We solve the recently proposed T-and Y-systems (Hirota equation) for the exact spectrum of AdS/CFT in the strong coupling scaling limit for an arbitrary quasiclassical string state. The corresponding T-functions appear to be super-characters of the SU(2, 2|4) group in unitary representations with a highest weight, with the classical AdS 5 × S 5 superstring monodromy matrix as the group element. We propose a concise first Weyl-type formula for these characters and show that they correctly reproduce the results of quasiclassical one-loop quantization in all sectors of the superstring, under some natural assumptions. We also speculate about possible relation between the T-functions and the quantum monodromy matrix.

Solving the AdS/CFT Y-system

“…The AdS 5 /CFT 4 Y-and T-systems with T-hook boundary conditions proposed in [7] and summarized in figure 1 were later shown to be equivalent, with certain analyticity requirements [11][12][13], to the TBA equations [14][15][16]. It was shown in [17,18] (see also [19,20]) that the T-system, and hence the Y-system, in T-hook can be formally solved in terms of Wronskian determinants of a finite number of Q-functions -a generalization of Baxter's Q-function. But the corresponding finite system of nonlinear integral equations (FiNLIE), a remote analogue of Destri-de Vega equations, was still missing.…”

“…A similar scenario may exist in the AdS/CFT case, however the physical interpretation of the AdS/CFT T-functions and their operatorial realization is not yet established. In the strong coupling limit the T-functions were identified with the psup2, 2|4q characters of the monodromy matrix [17]. This classical monodromy matrix was perturbatively quantized, up to two loops on the world sheet, in [29,30] and the Hirota functional equation (1.1) was shown to hold at least in this approximation.…”

“…in the asymptotic large volume limit [7] and in the strong coupling limit [17]. It is very natural to assume that the formulation of analytic properties at the level of T-functions should be also more revealing and simple.…”

A next-to-leading Lüscher formula

“…In the case of the O(4) σ-model, instead, one needs to use 20) where the scalar factor S 0 is the same as that for the Gross-Neveu model (4.2). In more detail, the main formula for the double-wrapping correction (3.7) becomes 21) where T SU (2) a , U and Λ SU (2) are the same quantities involved in the SU(2) chiral GrossNeveu model, while the NLIE for the U(1) sector changes to [41] A…”

“…In particular, the study of the finite-volume corrections [2][3][4][5][6][7][8][9] for the anomalous dimensions/string energies spectrum of AdS 5 /CF T 4 , has culminated in the formulation of the so-called Thermodynamic Bethe Ansatz (TBA) equations and Y-system [10][11][12][13][14][15][16][17][18], which in principle govern the spectrum exactly at any order of the coupling constant and the volume parameter. Very recently, the TBA equations have been reduced first to few non-linear integral (so-called FiNLIE) equations [19] (see [17,18,[20][21][22][23][24][25][26][27] for some previous developments in that direction), then to an impressively simple set of Riemann-Hilbert equations [28].…”