## Final Report Summary - GAUGE-STRING (Gauge theory - String theory duality: maximally symmetric case and beyond)

The project was aimed at the study of important correspondence between gauge theory and string theory. An example of gauge theory is the theory of strong interactions, but it has been difficult to use it due to large value of its coupling at low energies. Understanding of gauge theory dynamics at large values of coupling is a major problem of physics of strong interactions preventing analytic computation of the hadron spectrum. The research in the last 20 years gave strong evidence that connection of gauge theories to string theory is a key to the solution of this strong coupling problem. Study of string theory in curved spaces should also have important implications for quantum gravity and physics of black holes.

The gauge-string duality and, in particular, Anti deSitter / conformal field theory (AdS/CFT) correspondence was the central theme of this project. The first part of the project studied the most symmetric case - duality between maximally supersymmetric gauge theory in flat 4 dimensions and superstring theory in curved 10-dimensional AdS5 x S5 space. We used perturbative string methods (semiclassical expansion, scattering S-matrix for string excitations) and hidden symmetries (integrability) to establish and cross-check the general set of equations ("Bethe ansatz") describing the string energy spectrum and thus the spectrum of dimensions of the corresponding gauge-theory operators. We also studied an important class of observables (Wilson loops) that can be computed perturbatively on the string side and sometimes exactly on the gauge theory side and as a result fine details of AdS/CFT duality can be checked. In particular, we proposed an interesting relation between gauge theory correlators along a circular Wilson loop and an effective 1-dimensional conformal model dual to induced AdS2 geometry on the string world sheet.

In the second part of the project we applied similar methods to lower dimensional or less symmetric cases. We suggested that integrability-based approach can be used to solve string models in AdS3 x S3 space which represents a near-horizon limit of certain extremal black holes. We made important steps towards the complete Bethe ansatz solution for the spectrum of this model by computing perturbative S-matrix and the dispersion relation for relevant fundamental string excitations and generalizing them to all orders in string tension. It was found, however, that the remarkable integrability property is rather rare for strings in curved spaces and thus the identification of integrable cases with potentially interesting applications is very important. We performed an extensive study of a distinguished class of deformations of AdS string models where integrability is built in and that should be leading to new "non-commutative" examples of gauge-string duality with reduced supersymmetry. As a by-product, we discovered an unexpected generalization of the supergravity equations that describe scale invariance conditions of the superstring model. Another novel direction that have grown out of the work on the project was the study of quantum theories of massless higher spin fields. The spectrum of string theory in AdS in a special tensionless limit should contain an infinite set of massless higher spin fields. Massless higher spin theories in AdS are dual to simple conformal theories at the boundary providing simple example of AdS/CFT duality. We initiated the study of quantum properties of higher spin theories and discovered that the partition function and scattering S-matrix of such theories in flat and AdS space have very simple structure, reflecting their large amount of symmetry. This may have implications for the study of special limits of gauge-string duality and more generally for the investigation of new consistent models of quantum gravity.

The gauge-string duality and, in particular, Anti deSitter / conformal field theory (AdS/CFT) correspondence was the central theme of this project. The first part of the project studied the most symmetric case - duality between maximally supersymmetric gauge theory in flat 4 dimensions and superstring theory in curved 10-dimensional AdS5 x S5 space. We used perturbative string methods (semiclassical expansion, scattering S-matrix for string excitations) and hidden symmetries (integrability) to establish and cross-check the general set of equations ("Bethe ansatz") describing the string energy spectrum and thus the spectrum of dimensions of the corresponding gauge-theory operators. We also studied an important class of observables (Wilson loops) that can be computed perturbatively on the string side and sometimes exactly on the gauge theory side and as a result fine details of AdS/CFT duality can be checked. In particular, we proposed an interesting relation between gauge theory correlators along a circular Wilson loop and an effective 1-dimensional conformal model dual to induced AdS2 geometry on the string world sheet.

In the second part of the project we applied similar methods to lower dimensional or less symmetric cases. We suggested that integrability-based approach can be used to solve string models in AdS3 x S3 space which represents a near-horizon limit of certain extremal black holes. We made important steps towards the complete Bethe ansatz solution for the spectrum of this model by computing perturbative S-matrix and the dispersion relation for relevant fundamental string excitations and generalizing them to all orders in string tension. It was found, however, that the remarkable integrability property is rather rare for strings in curved spaces and thus the identification of integrable cases with potentially interesting applications is very important. We performed an extensive study of a distinguished class of deformations of AdS string models where integrability is built in and that should be leading to new "non-commutative" examples of gauge-string duality with reduced supersymmetry. As a by-product, we discovered an unexpected generalization of the supergravity equations that describe scale invariance conditions of the superstring model. Another novel direction that have grown out of the work on the project was the study of quantum theories of massless higher spin fields. The spectrum of string theory in AdS in a special tensionless limit should contain an infinite set of massless higher spin fields. Massless higher spin theories in AdS are dual to simple conformal theories at the boundary providing simple example of AdS/CFT duality. We initiated the study of quantum properties of higher spin theories and discovered that the partition function and scattering S-matrix of such theories in flat and AdS space have very simple structure, reflecting their large amount of symmetry. This may have implications for the study of special limits of gauge-string duality and more generally for the investigation of new consistent models of quantum gravity.