Integrability of the AdS_5 x S^5 superstring and its deformations

Based on the author's PhD thesis, this article reviews the application of integrability to the spectral problem of strings on AdS_5 x S^5 and its deformations. We begin with a pedagogical introduction to integrable field theories culminating in the description of their finite-volume spectra through the thermodynamic Bethe ansatz. Next, we apply these ideas to the AdS_5 x S^5 string and in later chapters discuss how to account for particular integrable deformations. Through the AdS/CFT correspondence this gives an exact description of anomalous scaling dimensions of single trace operators in planar N=4 supersymmetry Yang-Mills theory, its `orbifolds', and beta and gamma-deformed supersymmetric Yang-Mills theory. We also touch upon some subtleties arising in these deformed theories. Furthermore, we consider complex excited states (bound states) in the su(2) sector and give their thermodynamic Bethe ansatz description. Finally we discuss the thermodynamic Bethe ansatz for a quantum deformation of the AdS_5 x S^5 superstring S-matrix, with close relations to among others Pohlmeyer reduced string theory

Quantum field theory through a stringy mirror - Integrability of the AdS5xS5 superstring and its deformations

Describing quantum field theories at finite coupling is a daunting open problem in theoretical physics. One particular theory where we are able to gain valuable insight is planar N=4 supersymmetric Yang-Mills theory (SYM), related to superstring theory on AdS5 x S5 through the AdS/CFT correspondence. Thanks to integrable structures in both theories we can compute the energy spectrum of the string exactly, giving us the spectrum of scaling dimensions in N=4 SYM at finite coupling. In this thesis we firstly briefly review the computation of energy spectra in two-dimensional integrable quantum field theories by means of the (mirror) thermodynamic Bethe ansatz and show how this applies to the AdS5 x S5 superstring. We then fully extend these considerations to various integrable deformations of the superstring and its dual N=4 SYM, breaking some or all of the supersymmetry. We also discuss the application of these methods to a particularly complicated type of excited string state with complex momenta. Finally we consider deformations of the integrable structure of the superstring related to Pohlmeyer reduced string theory and find the corresponding thermodynamic Bethe ansatz equations which exhibit remarkable features

Quantum field theory through a stringy mirror - Integrability of the AdS5xS5 superstring and its deformations

Describing quantum field theories at finite coupling is a daunting open problem in theoretical physics. One particular theory where we are able to gain valuable insight is planar N=4 supersymmetric Yang-Mills theory (SYM), related to superstring theory on AdS5 x S5 through the AdS/CFT correspondence. Thanks to integrable structures in both theories we can compute the energy spectrum of the string exactly, giving us the spectrum of scaling dimensions in N=4 SYM at finite coupling. In this thesis we firstly briefly review the computation of energy spectra in two-dimensional integrable quantum field theories by means of the (mirror) thermodynamic Bethe ansatz and show how this applies to the AdS5 x S5 superstring. We then fully extend these considerations to various integrable deformations of the superstring and its dual N=4 SYM, breaking some or all of the supersymmetry. We also discuss the application of these methods to a particularly complicated type of excited string state with complex momenta. Finally we consider deformations of the integrable structure of the superstring related to Pohlmeyer reduced string theory and find the corresponding thermodynamic Bethe ansatz equations which exhibit remarkable features

Fermionic reductions of the AdS4×CP3 uperstring

We discuss fermionic reductions of type IIA superstrings on AdS4×CP3 in relation to the conjectured AdS4/CFT3 duality. The superstring theory is described by means of a coset model construction, which is classically integrable. We discuss the global light-cone symmetries of the action and related κ-symmetry gauge choices, and also present the complete quartic action in covariant form with respect to these. Further, we study integrable (fermionic) reductions, in particular, a reduction yielding a quadratic action of two complex fermions on the string world-sheet. Interestingly, this model appears to be exactly the same as the corresponding integrable reduction found in the AdS5×S5 case

Bound States in the Mirror TBA

The spectrum of the light-cone AdS_5 \times S^5 superstring contains states composed of particles with complex momenta including in particular those which turn into bound states in the decompactification limit. We propose the mirror TBA description for these states. We focus on a three-particle state which is a finite-size representative of a scattering state of a fundamental particle and a two-particle bound state and dual to an operator from the su(2) sector of N=4 SYM. We find that the analytic behavior of Y-functions differs drastically from the case of states with real momenta. Most importantly, Y_Q-functions exhibit poles in the analyticity strip which leads to the appearance of new terms in the formula for the energy of this state. In addition, the TBA equations are supplied by quantization conditions which involve Y_2. Considering yet another example of a three-particle state, we find that the corresponding quantization conditions do not even involve Y_1. Our treatment can be generalized to a wide class of states with complex moment

The quantum deformed mirror TBA I

We derive the ground state thermodynamic Bethe ansatz equations for the quantum deformation of the AdS5 S5 mirror model, taking the deformation parameter to be a root of unity. By virtue of the deformation, the resulting equations show an interesting structure between a nite number of Y-functions

Twisting the Mirror TBA

We study finite-size corrections to the magnon dispersion relation in three models which differ from string theory on AdS5 x S5 in their boundary conditions. Asymptotically, this is accomplished by twisting the transfer matrix in a way which manifestly preserves integrability. In model I all world-sheet fields are periodic, whereas model II represents a particular orbifold of AdS5 x S5 and model III is a beta-deformed theory. For models I and II we construct the one-particle TBA equations and use them to determine the leading finite-size correction to the asymptotic Bethe equation. We also make some interesting observations concerning the quantization conditions for the momentum. For the same models we compute the leading and for model II the next-to-leading order finite-size corrections to the asymptotic magnon dispersion relation. Furthermore, we apply L\"uscher's formulae to compute the leading finite-size corrections in beta-deformed theory. In addition to reproducing known results, we provide new predictions for two-particle states from the sl(2) sector, to be confronted with explicit field-theoretic calculations in the dual gauge theory. Finally, we prove that the leading finite-size correction to the energy of an sl(2) magnon in orbifold theory is the same as the one for an su(2) magnon in beta-deformed theory for special values of beta. We also speculate that for these values of beta our result for the next-to-leading order correction in the orbifold model might coincide with the corresponding correction to the energy of su(2) magnon in beta-deformed theor