Quantum Fields and Strings

The 4D rotating black hole described by the Kerr geometry possesses many of what was called by Chandrasekhar "miraculous" properties. Most of them are related to the existence of a fundamental hidden symmetry of a principal conformal Killing-Yano (PCKY) tensor. In my talk I shall demonstrate that hidden symmetry of the PCKY tensor plays exceptional role also in higher dimensions. Namely, I shall present the most general spacetime admitting the PCKY tensor and show that is possesses the following properties: 1) It is of the algebraic type D and admits the Kerr-Schild form 2) It allows a separation of variables for the Hamilton-Jacobi, Klein-Gordon, Dirac, and stationary string equations. 3) When the Einstein equations with the cosmological constant are imposed the metric describes the most general known (spherical) Kerr-NUT-AdS black hole spacetime. I will also discuss the generalization of Killing-Yano symmetries for spacetimes with natural "torsion 3-form", such as the black hole of D=5 minimal supergravity, or the Kerr-Sen solution of heterotic string theory, and comment on connection to special Riemannian manifolds admiting Killing spinors.

ABJM theory is a world-volume theory for an arbitrary number of M2-branes. One of the unique features of ABJM theory is its characteristic scaling behaviour, exhibited for example by the free energy and correlation functions of chiral primary operators. In more detail, ABJM theory has a holographic dual where thermodynamics at strong coupling is determined by a system of black M2-branes. The zero-coupling (black-body radiation) free energy disagrees with the strong coupling result. Even the scaling in the 't Hooft coupling is different (strongly suppressed at strong coupling). It is therefore important to check that the weak and strong coupling results converge as loop corrections are taken into account. The leading order computation indeed confirms that the first correction goes in the right direction.

We describe a classification of 4d N=2 superconformal theories obtained from the compactification of 6d N=(2,0) A_N theories on punctured Riemann surfaces. We show the basic building blocks to construct any such theory and its various S-dual frames. A host of new S-dualities and interacting (non-Lagrangian) superconformal theories are unconvered. We also compute various properties of these interacting superconformal theories.

I will discuss a novel approach to time-dependent background fields of string theory, which allows the identification of configurations which satisfy the conformal invariance conditions to all orders in the Regge slope parameter alpha'. I will present a graviton, dilaton and tachyon configuration which is a resummation of all orders in the Regge slope and is shown to satisfy the conformal invariance conditions order by order, to all orders in alpha', up to field redefinitions which do not change the dependence of the fields on the target space-time coordinates, due to a property termed as "homogeneity" of the Weyl anomaly coefficients. It describes an inflationary universe in the sigma-model frame, while in the Einstein frame it corresponds either to a de Sitter or to a power-law universe, depending on the values of the dilaton and the tachyon amplitudes. Possible ways to connect these different solutions, which may correspond to different epochs of the universe, will also be discussed.

Gauge/gravity duality, a concept which emerged from string theory, holds promise for revealing the secrets of certain strongly interacting real world condensed matter systems. Historically, string theorists presented their subject as a promising framework for a quantum theory of gravity. More recently, the AdS/CFT correspondence and gauge/gravity dualities have emerged as powerful tools for using what we already know about gravity to investigate the properties of strongly interacting field theories. I will cherry pick and discuss a few recent developments where black holes are used to calculate the thermodynamic and transport properties of quantum critical systems, superconductors, and superfluids.

N=8 supergravity in 4 dimensions exhibits a surprisingly favorable UV behavior -- it is known from explicit computations that the 4-point amplitudes in N=8 supergravity are finite up to 4-loop order. I explain how a "matrix-element approach" can be used to study candidate counterterms for UV divergences in this theory. This approach both demystifies the finiteness found in previous computations, and predicts finiteness of arbitrary n-point amplitudes in N=8 supergravity below the 7-loop order. It also points to the 7-loop 4-point amplitude as the first amplitude whose finiteness is not guaranteed by any known symmetry of the theory.

Mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory---our leading attempt to explain the universe to date. The audience will gain a sense of how mathematicians think and approach the world and realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but it is instead crucial to our understanding of the physical world.

Hints for the possibility of two times emerged in M-theory in 1995. If taken seriously this required new concepts that could solve unitarity (ghost) and causality problems so that physics could be described sensibly in a spacetime with two times. The necessary concept turned out to be a gauge symmetry in phase space. This is an unfamiliar concept, but is one that extends Einstein's approach to the formulation of fundamental equations of physics, by removing the perspective of the observer, not only in position space but more generally in phase space. This approach led in 1998 to what is now called 2T-physics, which has been formulated so far in classical and quantum mechanics, field theory and partially in string theory. In this lecture I will explain the fundamental aspects of 2T-physics, and will outline the progress from classical mechanics, through the standard model and gravity, all the way to supergravity in d-space plus 2-time dimensions. I will describe how 2T-physics is consistent with 1T-physics in (d-1)-space plus 1-time dimensions, but also how it goes beyond 1T-physics, by making in principle a vast number of verifiable predictions that are systematically missed in 1T-physics, as well as providing new computational tools for physics.

We show how to extract from conjectured S-dualities the dimensions and flavor symmetry transformation properties of certain Higgs branches (hypermultiplet flat directions) of strongly coupled N=2 d=4 superconformal field theories. This leads to an expansion and refinement of the exact data (conformal dimensions, central charges) that can be computed for N=2 SCFTs.

The properties of a superfluid phase transition with a d-wave order parameter in a strongly interacting field theory with gravity dual are considered. In the context of the AdS/CFT correspondence, this amounts to writing down an action for a charged, massive spin two field on a background, and I will discuss all technical problems. In the second part I will show that coupling bulk fermions to the spin two field and studying the fermionic two-point function, one recovers interesting features of d-wave superconductors, like d-wave gap, Dirac nodes and Fermi arcs.

We derive a holographic dual description of free quantum field theory in arbitrary dimensions, by reinterpreting the exact renormalization group, to obtain a higher spin gravity theory of the general type which had been proposed and studied as a dual theory