Quantum Fields and Strings

I will first summarize recent exact localization computations of supersymmetric gauge theories, and then discuss curious connections between SUSY/non-SUSY theories coming from 6d (2,0) theories. I particular, I will focus on our recent proposal relating 3d N=2 theories and 3d SL(2,R) Chern-Simons theories (or more mathematically, geometry of 3-manifolds).

The Polchinski equations for the Wilsonian renormalization group in the $D$--dimensional matrix scalar field theory can be written at large $N$ in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the subsector of $Tr\phi^n$ (for all $n$) operators. We show that at low energies independently of the dimensionality $D$ the Hamiltonian system in question reduces to the {\it integrable} effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger--Hopf) equation with an external potential and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.

The gauge/gravity duality may give a nonperturbative formulation of superstring/M theory, and hence, if one can study the nonperturbative dynamics of the gauge theory, it would be useful to understand the nonperturbative aspects of superstring theory. Although researches in this direction were not successful for long time because of the notorious difficulties in lattice SUSY, however, recent progress made it possible; nonperturbative formulations free from the parameter fine-tuning were proposed, some of them are confirmed to work numerically, and nontrivial evidence for the validity of the gauge/gravity duality has been obtained. In these talks I review the state of the art in this field. I start with reviewing basics of the Monte-Carlo. Then I explain how to put supersymmetric theories on computer and show actual numerical results. 1st talk : basics of Monte-Carlo simulation. 2nd talk : 1d SYM (matrix quantum mechanics). 3rd talk : how to put 2d, 3d and 4d SYM on computer. In the talks I concentrate on basic ideas and omit technical details (e.g. algorithms to accelerate simulations). They will be explained after the talks if people are interested in. References: 1st talk : standard textbooks e.g. Heinz J. Rothe, "Lattice Gauge Theories: An Introduction", Third Edition, World Scientific. 2nd talk : 0706.1647 [hep-lat], 0707.4454 [hep-th], 0811.2081 [hep-th], 0811.3102 [hep-th], 0911.1623 [hep-th], 1012.2913 [hep-th]. 3rd talk : hep-lat/0302017, hep-lat/0311021, 1010.2948 [hep-lat] (2d SYM); hep-th/0211139 (3d SYM); 1004.5513 [hep-lat], 1009.0901 [hep-lat] (4d SYM)

The gauge/gravity duality may give a nonperturbative formulation of superstring/M theory, and hence, if one can study the nonperturbative dynamics of the gauge theory, it would be useful to understand the nonperturbative aspects of superstring theory. Although researches in this direction were not successful for long time because of the notorious difficulties in lattice SUSY, however, recent progress made it possible; nonperturbative formulations free from the parameter fine-tuning were proposed, some of them are confirmed to work numerically, and nontrivial evidence for the validity of the gauge/gravity duality has been obtained. In these talks I review the state of the art in this field. I start with reviewing basics of the Monte-Carlo. Then I explain how to put supersymmetric theories on computer and show actual numerical results. 1st talk : basics of Monte-Carlo simulation. 2nd talk : 1d SYM (matrix quantum mechanics). 3rd talk : how to put 2d, 3d and 4d SYM on computer. In the talks I concentrate on basic ideas and omit technical details (e.g. algorithms to accelerate simulations). They will be explained after the talks if people are interested in. References: 1st talk : standard textbooks e.g. Heinz J. Rothe, "Lattice Gauge Theories: An Introduction", Third Edition, World Scientific. 2nd talk : 0706.1647 [hep-lat], 0707.4454 [hep-th], 0811.2081 [hep-th], 0811.3102 [hep-th], 0911.1623 [hep-th], 1012.2913 [hep-th]. 3rd talk : hep-lat/0302017, hep-lat/0311021, 1010.2948 [hep-lat] (2d SYM); hep-th/0211139 (3d SYM); 1004.5513 [hep-lat], 1009.0901 [hep-lat] (4d SYM)

The gauge/gravity duality may give a nonperturbative formulation of superstring/M theory, and hence, if one can study the nonperturbative dynamics of the gauge theory, it would be useful to understand the nonperturbative aspects of superstring theory. Although researches in this direction were not successful for long time because of the notorious difficulties in lattice SUSY, however, recent progress made it possible; nonperturbative formulations free from the parameter fine-tuning were proposed, some of them are confirmed to work numerically, and nontrivial evidence for the validity of the gauge/gravity duality has been obtained. In these talks I review the state of the art in this field. I start with reviewing basics of the Monte-Carlo. Then I explain how to put supersymmetric theories on computer and show actual numerical results. 1st talk : basics of Monte-Carlo simulation. 2nd talk : 1d SYM (matrix quantum mechanics). 3rd talk : how to put 2d, 3d and 4d SYM on computer. In the talks I concentrate on basic ideas and omit technical details (e.g. algorithms to accelerate simulations). They will be explained after the talks if people are interested in. References: 1st talk : standard textbooks e.g. Heinz J. Rothe, "Lattice Gauge Theories: An Introduction", Third Edition, World Scientific. 2nd talk : 0706.1647 [hep-lat], 0707.4454 [hep-th], 0811.2081 [hep-th], 0811.3102 [hep-th], 0911.1623 [hep-th], 1012.2913 [hep-th]. 3rd talk : hep-lat/0302017, hep-lat/0311021, 1010.2948 [hep-lat] (2d SYM); hep-th/0211139 (3d SYM); 1004.5513 [hep-lat], 1009.0901 [hep-lat] (4d SYM)

We discuss the calculation of higher loop contributions to the anomalous dimensions of operators in N=6 Superconformal Chern-Simons theories. These lead us to conjecture the form of an interpolating function of the 't Hooft coupling that is not determined by integrability.

I will discuss the construction of a holographic dictionary for theories with non-relativistic conformal symmetry, relating the field theory to the dual spacetime. I will focus on the case of Lifshitz spacetimes, giving a definition of asymptotically locally Lifshitz spacetimes and discussing the calculation of field theory observables and holographic renormalization.

What is the gravity dual of a strongly interacting state of matter at zero temperature and finite charge density? The simplest candidates are extremal black holes. The presence of charged matter in the bulk can often mean that extremal black holes are not the ground state. In this talk I will discuss the physics of a class of solutions, essentially charged neutron stars, that can be thermodynamically preferred over extremal black holes.

Mass, a concept familiar to all of us, is also one of the deepest mysteries in nature. Almost all of the mass in the visible universe, you, me and any other stuff that we see around us, emerges from QCD, a theory with a negligible microscopic mass content. How does QCD and the family of gauge theories it belongs to generate a mass? This class of non-perturbative problems remained largely elusive despite much effort over the years. Recently, new ideas based on compactification have been shown useful to address some of these. Two such inter-related ideas are circle compactifications, which avoid phase transitions and large-N volume independence. Through the first one, we realized the existence of a large-class of "topological molecules", e.g. magnetic bions, which generate mass gap in a class of compactified gauge theories. The inception of the second, the idea of large-N volume independence is old. The new progress is the realization of its first working examples. This property allows us to map a four dimensional gauge theory (including pure Yang-Mills) to a quantum mechanics at large-N.

Correlation functions in the gauge-gravity correspondence (AdS/CFT) are dual to scattering amplitudes in anti-de Sitter space (AdS). In this talk, I will describe how techniques that were recently developed to study scattering amplitudes in flat space can be generalized to AdS leading to a new and efficient method of computing correlation functions in AdS/CFT. References: 1) S. Raju, "Generalized Recursion Relations for Correlators in the Gauge Gravity Correspondence", Phys.Rev.Lett. 106 (2011) 091601. http://arxiv.org/abs/arXiv:1011.0780 2) S. Raju, "Recursion Relations for AdS/CFT Correlators", http://arxiv.org/abs/arXiv:1102.4724