Quantum Fields and Strings

It has become a platitute to say that black holes are fascinating objects—but they really are, in part because they challenge our understanding of the fundamental reversibility of physical processes.

In the first part of the talk, I will review some of the classical ways black holes behave as dissipative systems, such as the "hair loss” phenomenon and the monotonic growth of horizon area. In the second part, I will explain how quantum mechanics (more precisely, the coupling of black holes to the quantum vacuum) affects the classical picture at late times, notably through particle creation and evaporation. I will argue that techniques from two-dimensional field theory can help bring clarity to the associated “information loss”

problem, and perhaps also point to new, unexpected predictions. My approach will be as model-independent as possible; that is, rather than investigating a particular scenario for black hole evaporation, I will aim to derive generic consequences from basic assumptions regarding the reversibility of black hole evaporation.

The gauge/gravity enables us to learn about quantum gravity by solving gauge theory. This is not an easy task, of course, and hence numerical techniques should play important roles. So far, properties of super Yang-Mills theories with Euclidean signature, such as the thermodynamic properties, have been studied by using Monte Carlo methods, and good agreement with the dual gravity prediction has been observed, including stringy corrections, both alpha prime and  and g_s. Still, the real-time properties are not well understood.

As a modest first step for the real-time study, we consider classical dynamics of the Banks-Fischler-Shenker-Susskind (BFSS) matrix model, which is expected to describe a highly stringy black hole in type IIA superstring theory. It turns out that this classical model has rather rich structure -- qualitative features of the thermalization of a black hole, the fast scrambling proposed by Sekino and Susskind, and a symptom of the evaporation. By taking into account a part of the quantum effect, we give a classical matrix model which can mimic the formation and evaporation of a black hole. We also argue that a similar calculation could be done for classical Yang-Mills theories with nonzero spatial dimension, without suffering from the UV catastrophe.

This talk is based on collaborations with S. Aoki, N. Iizuka (hep-th) and with E. Berkowitz, G. Gur-Ari, J. Maltz, S. Shenker (in progress).

We will discuss techniques for computing 4-point functions of local operators in 2D conformal field theories, and their implications for semiclassical 3D quantum gravity. For generic 4-point functions, we present new closed-form expansions of the Virasoro conformal blocks. Specializing to correlators of holomorphic operators, these can be efficiently and exactly determined using an analytic implementation of the conformal bootstrap. Applying this method to the sewing construction of 2D CFT partition functions, we compute, via AdS/CFT, the semiclassical expansion of pure 3D quantum gravity around saddle point geometries of genus two; unlike at genus one, this expansion does not truncate at finite loop order.  

In this talk I will propose a general correspondence which associates a non-perturbative quantum mechanical operator to a toric Calabi-Yau manifold, and I will propose a conjectural expression for its spectral determinant. As a consequence of these results, I will derive an exact quantization condition for the operator spectrum. I will give a concrete illustration of this conjecture by focusing on the example of local P2. This approach also provides a non-perturbative Fermi gas picture of topological strings on toric background and suggests the existence of an underlying theory of M2 branes behind this formulation.

Using holography, I will describe an approach for understanding the physics of a big bang singularity by translating the problem into the language of the dual quantum field theory. Certain two-point correlators in the dual field theory are sensitive to near-singularity physics in a dramatic way, and this provides an avenue for investigating how strong quantum gravity effects in string theory might modify the classical description of the big bang.

The talk will be based on a work in progress with Stefano Kovacs
(Dublin IAS) and Yuki Sato (Wits University). In a previous work
(arxiv:1310.0016) we have shown that,
in the M-theory regime (large N with the Chern-Simon level k fixed)
of the duality between ABJM theory and M-theory on AdS4 x S7/Zk,
certain monopole operators with large R charges on the gauge theory side
correspond to spherical membranes
(which is in general in non-BPS excited states) in the pp-wave matrix
model on the dual side.

Having in mind application to
the study of three point functions of the monopole operators
from the dual side, we study the BPS instanton equation
of the pp-wave matrix model. The instanton equation describes, for example,
a process in which a single spherical membrane splits into
two spherical membranes. Under a certain
approximation which is valid when the matrix size is large,
the instanton equation can be recast
into a three dimensional Laplace equation;
a time snapshot of the membrane configuration corresponds
to an equipotential surface of the solution of the Laplace equation.
In order to study the above mentioned splitting process,
we found that one has to introduce a special boundary condition
of the Laplace equation: one prepares two copies of
three dimensional space which are connected in a manner analogous
to Riemann surfaces. We will discuss an exact solution
of the Laplace equation under this boundary condition, and 
the corresponding instanton solution, in which a membrane splits into two.

The analogy between Multi-scale Entanglement Renormalization
Ansatz (MERA) and the spatial slice of three-dimensional anti-de
Sitter space (AdS3) has motivated a great interest in tensor networks
among holographers. I discuss a way to promote this analogy to a
rigorous, quantitative, and constructive relation. A key quantitative
ingredient is the way the strong subadditivity of entanglement entropy
is encoded in MERA and in a holographic spacetime. The upshot is that
the map between MERA and the spatial slice of AdS3 is mediated through
an additional integral transform. Interpreted directly, MERA is a
discretization not of the spatial slice of AdS3, but of the space of
geodesics on the spatial slice of AdS3.

As an alternative to the Holographic Renormalization procedure, we

introduce a regularization scheme for AdS gravity based on the addition

boundary terms which are a given polynomial of the extrinsic and 

intrinsic curvatures (Kounterterms).

Since these terms are closely related to either topological invariants 

or Chern-Simons densities in the corresponding dimension, they can be 

easily generalized to other gravity theories (Einstein-Gauss-Bonnet, 

Lovelock, etc.).

Finally, a general prescription on how to obtained the standard 

counterterm series in AdS gravity is given.​ 

Via the AdS/CFT correspondence, fundamental constraints on the entanglement structure of quantum systems translate to constraints on spacetime geometries that must be satisfied in any consistent theory of quantum gravity. In this talk, we describe some of the constraints arising from strong subadditivity and from the positivity and monotonicity of relative entropy. Our results may be interpreted as a set of energy conditions restricting the possible form of the stress-energy tensor in consistent theories of Einstein gravity coupled to matter."

The moduli space of k G instantons on C^2, where G is a classical gauge group, has a well known HyperKahler quotient formulation known as the ADHM construction. The extension to exceptional groups is an open problem.
In string theory this is realized using a system of branes, and the moduli space of instantons is identified with the Higgs branch of a particular supersymmetric gauge theory with 8 supercharges.
A less known, and less studied aspect of moduli spaces of instantons is that they can be realized as the Coulomb branch of a supersymmetric gauge theory in 2+1 dimensions.
Recent developments on the understanding of the Coulomb branch gives us a nice solution to the problem where G is an exceptional group, thus allowing a systematic study of these moduli spaces which was unavailable so far.
I will discuss these developments, and present the corresponding quivers, and the Coulomb branch Hilbert Series - the main tool which lead to the recent progress.

Recent research has suggested deep connections between geometry and entropy. This connection was first seen in black hole thermodynamics, but has been more fully realized in the Ryu-Takayanagi proposal for calculating entanglement entropies in AdS/CFT. We suggest that this connection is even broader: entropy, and in particular compression, are the fundamental building blocks of emergent geometry. We demonstrate how spatial geometry can be derived from the properties of a recursive compression algorithm for the boundary CFT. We propose a general algorithm for constructing MERA-like tensor networks and elucidate connections to the mathematical field of integral geometry.

The tt* equations define a flat connection on the moduli spaces of 2d, N=2 quantum field theories. For conformal theories with c=3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. I will show that the non-holomorphic content of the tt* equations in the cases d=1,2,3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the anti-holomorphic derivatives of tt* and provides these with a finer and algebraic meaning. The generators of the differential rings of special functions are given by quasi-modular forms for d=1 and their generalizations in d=2,3. For d=3, this can be used to provide a purely Lie algebraic formulation of the higher genus topological string theory amplitudes and of the BCOV holomorphic anomaly equations which govern them.