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Relatively low crossover temperature suggests that chiral symmetry restoration in Quantum Chromodynamics may well be described within the low-energy effective theory. The shape of the pseudocritical line in the T-mu plane is estimated within this assumption. No critical endpoint is found for physical values of quark masses.

We clarify the relationship between probe analysis of the supergravity dual and the large-N solution of the localization matrix model for the planar N=2 super-Yang-Mills theory. A formalism inspired by supergravity allows us to systematically solve the matrix model at strong coupling. Quite surprisingly, we find that quantum phase transitions, known to occur in the N=2 theory, start to be visible at the third order of the strong-coupling expansion and thus constitute a perturbative phenomenon on the string worldsheet.

By considering a Gaussian truncation of N = 4 super Yang-Mills, we derive a set of Dyson equations that account for the ladder diagram contribution to connected correlators of circular Wilson loops. We consider different numbers of loops, with different relative orientations. We show that the Dyson equations admit a spectral representation in terms of eigenfunctions of a Schrodinger problem, whose classical limit describes the strong coupling limit of the ladder resummation. We also verify that in supersymmetric cases the exact solution to the Dyson equations reproduces known matrix model results.

Previous attempts to match the exact N = 4 super Yang-Mills expression for the expectation value of the 1/2-BPS circular Wilson loop with the semiclassical AdS(5) x S-5 string theory prediction were not successful at the first subleading order. There was a missing prefactor similar to lambda(-3/4) which could be attributed to the unknown normalization of the string path integral measure. Here we resolve this problem by computing the ratio of the string partition functions corresponding to the circular Wilson loop and the special 1/4-supersymmetric latitude Wilson loop. The fact that the latter has a trivial expectation value in the gauge theory allows us to relate the prefactor to the contribution of the three zero modes of the "transverse" fluctuation operator in the 5-sphere directions.

Large-N phase transitions occurring in massive N = 2 theories can be probed by Wilson loops in large antisymmetric representations. The logarithm of the Wilson loop is effectively described by the free energy of a Fermi distribution and exhibits second-order phase transitions (discontinuities in the second derivatives) as the size of representation varies. We illustrate the general features of antisymmetric Wilson loops on a number of examples where the phase transitions are known to occur: N = 2 SQCD with various mass arrangements and N = 2* theory. As a byproduct, we solve planar N = 2 SQCD with three independent mass parameters. This model has two effective mass scales and undergoes two phase transitions.

We study the null dipole deformation of N = 4 super Yang-Mills theory, which is an example of a potentially solvable 'dipole CFT': a theory that is non-local along a null direction, has non-relativistic conformal invariance along the remaining ones, and is holographically dual to a Schrodinger space-time. We initiate the field-theoretical study of the spectrum in this model by using integrability inherited from the parent theory. The dipole deformation corresponds to a nondiagonal Drinfeld-Reshetikhin twist in the spin chain picture, which renders the traditional Bethe ansatz inapplicable from the very beginning. We use instead the Baxter equation supplemented with nontrivial asymptotics, which gives the full 1-loop spectrum in the sl(2) sector. We show that anomalous dimensions of long gauge theory operators perfectly match the string theory prediction, providing a quantitative test of Schrodinger holography.

We consider level crossing in a matrix family H = H-0 + lambda V where H-0 is a fixed N x N matrix and V belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of lambda, for which we obtain a number of exact, asymptotic and approximate formulas.

An interplay between localization and holography is reviewed with the emphasis on the AdS5/CFT4 correspondence.

This is the foreword to the special issue on localization techniques in quantum field theory. The summary of individual chapters is given and their interrelation is discussed.