The Hirota equation for string theory in AdS₅ × S⁵ from the fusion of line operators

Abstract

We present a perturbative derivation of the T-system that is believed to encode the exact spectrum of planar N = 4 SYM. The T-system is understood as an operator identity between some special line operators, the quantum transfer matrices. By computing the quantum corrections in the process of fusion of transfer matrices, we show that the Tsystem holds up to first order in a semi-classical expansion. This derivation does not rely on any assumption. We also discuss the extension of the proof to other theories, including models describing string theory on various AdS spaces.

On the quantization of continuous non-ultralocal integrable systems

“…Nonetheless, this problem has once more become a very active and relevant area of research since it was discovered that AdS 5 × S 5 string theory is a classically integrable system of this type (for a review see [8] and references therein). In spite of all the attention devoted recently to this area, because of its importance to quantizing the AdS 5 × S 5 superstring and thus improving our understanding of the AdS/CFT correspondence [9][10][11][12][13][14], there is still no satisfactory general method to resolve all the difficulties involved in the quantization process of non-ultralocal theories.…”

On the equivalence theorem for integrable systems

“…From the point of view of the inverse scattering method [14][15][16][17] the complicated structure of the Dirac brackets makes it hard to analyze the algebra of the Lax operator. In [8,9] it was shown that the classical algebra of Lax operators displays a non-ultralocal behavior [18][19][20][21][22][23][24][25][26][27][28][29]. 1 While for simpler models classical integrability can be analyzed in detail, and the action-angle variables can be found [9,[18][19][20][21][22], for the AAF model the non-linear structure of the Dirac bracket is too complicated, and the corresponding action-angle variables are still unknown.…”

The r-matrix of the Alday-Arutyunov-Frolov model

“…The latter method, being more elegant and physically clear, is, nevertheless, hard to use in practice for more involved models, and, moreover, it still requires putting the system on the lattice. 1 The method due to Maillet, however, does not use any lattice regularization, and although it is not obvious how to quantize such systems, some essential progress in understanding the integrability of such models, e.g., the complex sine-Gordon model and non-linear sigma models, has been made in the classical theory (for more recent applications see [29][30][31][32][33] and the references therein).…”