Quantum Fields and Strings

The subject of quantum field theory in mixed states of quantum matter is an old and rich one. The natural setting to discuss field theory in a mixed state is the Schwinger-Keldysh formalism. The subject of this talk is the set of peculiar symmetries that arise in Schwinger-Keldysh theories, and how they may be accounted for in effective field theory. In particular, when the mixed state is thermal, the effective description is constrained by two BRST-like supercharges which, at low energies, generate an algebra akin to minimal supersymmetric quantum mechanics. If time allows, I will also discuss a sort of Schwinger-Keldysh bootstrap for effective actions on more complicated closed-time-contours, which describe the out-of-time-ordered correlation functions that diagnose early-time chaotic growth in quantum systems.

The conformal bootstrap aims to calculate scaling dimensions and correlation functions in various theories, starting from general principles such as unitarity and crossing symmetry.  I will explain that local operators are not independent of each other but organize into analytic functions of spin, and I will present a formula which quantifies the consequences of this fact.  This will include a controlled approximation to the operator spectrum at large spin, as well as new bounds over the strength of bulk higher-derivative interactions in large-N theories with a sparse spectrum. These bounds, previously conjectured to exist using gauge-gravity duality, encode causality of the bulk theory.  Based on 1703.00278.

We study the eigenstate thermalization hypothesis in chaotic conformal field theories (CFTs) of arbitrary dimensions by computing the reduced density matrices of small size in energy eigenstates. We show that in the infinite volume limit this operator is well-approximated by a “universal” density matrix which is its projection to the primary operators that have nonzero thermal one-point functions. These operators in all two-dimensional CFTs and holographic higher-dimensional CFTs are the polynomials of stress tensor. We compute the time-dependent two-point correlators and Renyi entropies of the universal density martix, and compare the results to the thermal Gibbs state and black holes in various dimensions. We demonstrate that the two-dimensional universal density matrix is close in trace distance to the reduced Gibbs state. We put forward the truncation of density matrix to the stress tensor sector as a “hydrodynamic” method to study the out-of-equilibrium dynamics of strong-coupled field theories

I will discuss a new class of supersymmetric Wilson loop operators in pure N=2 Yang-Mills-Chern-Simons theory. These Wilson loops preserve one supercharge on-shell and wrap arbitrary Legendrian knots in the standard contact R^3.  I will also explain a relation, motivated by a global picture of contact three-manifolds, between these loop operators and chiral current algebra in two dimensions.  This talk is directly related to, but independent of, my preceding Friday talk in the Mathematical Physics seminar.

In this talk I will discuss a new use for conformal field theory crossing equation in the context of AdS/CFT:  the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. I will revisit this problem and the dual 1/N expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in 1/N^2, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. The second approach involves Mellin space. As an example, I’ll show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of \phi^4 theory in AdS, the crossing solution reproduces a previous computation of the one-loop bubble diagram. I will end with a discussion on open problems and new developments.

In the first part of the talk, I will explain how the lengths of non-minimal geodesics in AdS3 conical defect backgrounds can be interpreted as the entanglement entropy of certain subalgebras in the dual CFT. This part will be based on 1608.02040. In the second part of the talk, I will discuss how the Ryu-Takayanagi area term seems to be the analog of a certain edge term for EE in a gauge theory, and how one might try to test this. This part is more speculative.

The moduli space of vacua of 3d N=4 gauge theories splits essentially into two ``branches”: the Higgs branch parametrised by the vevs of hypermultiplet scalars and the Coulomb branch parametrised by the vevs of dressed monopole operators. They can be described as complex algebraic varieties. I will present a new approach to the analysis of the Higgs and Coulomb branches based on the type IIB brane realisation of the gauge theory. Focusing on abelian theories, I will show how the addition of new ingredients in the brane setup can be linked to the path integral insertions of Higgs or Coulomb operators. I will explain how this construction can be used to derive the ring relations and study mirror symmetry.

1. The notion of wall-crossing structure (as defined by Maxim Kontsevich and myself in arXiv: 1303.3253)
provides the  universal framework for description of different types of wall-crossing formulas (e.g. Cecotti-Vafa in 2d or KSWCF in 4d). It also gives
a language and tools for proving algebraicity and analyticity of arising generating series (e.g. for BPS invariants).

2. Holomorphic Floer theory is the Floer theory of a pair of complex Lagrangian subvarieties  of a complex symplectic manifold (maybe infinite-dimensional).
This geometry underlies several important topics, both in mathematics and physics. Those  include  questions about analytic continuation of exponential integrals
(e.g. Feynman integrals),
deformation quantization of holomorphic symplectic manifolds (and related Riemann-Hilbert correspondences), Geometric Langlands correspondence, etc.

3. It was known for a long time that many a priori divergent series (like e.g. formal WKB series for solutions of equations with a small parameter) become analytic or meromorphic functions after taking their Laplace transform
(Borel resummation). This property was called  the resurgence property of the divergent series. The relation of the resurgence phenomenon to simplest wall-crossing formulas was realized in the early 90's in the work of French mathematicians (Ecale, Voros, Pham , Malgrange and  others).

I plan to discuss how the combination of 1 and 2 can be applied to 3 in a very general situation.
Main idea goes back to our theory of Donaldson-Thomas invariants (arXiv: 0811.2435).
Namely, analyticity of the formal series  follows from existence of a global analytic object which is glued from the local ones by means of the formal series.
This analytification of  an a priori formal variety is a by-product of the growth estimates on the data of the underlying wall-crossing structure.

In case of the Holomorphic Floer theory (which underlies e.g. the Stokes phenomenon for the WKB solutions of PDE or difference equations) one needs an estimate on the number of pseudo-holomorphic discs
with the boundary on the union of our Lagrangian subvarieties. In practice it often appears as an estimate on the number of gradient lines between two critical points
of the action or potential.

The past few years have seen a surge of interest in six-dimensional superconformal field theories (6D SCFTs).  Notably, 6D SCFTs have recently been classified using F-theory, which relates these theories to elliptically-fibered Calabi-Yau manifolds.  Classes of 6D SCFTs have remarkable connections to structures in group theory and therefore provide a physical link between two seemingly-unrelated mathematical objects.  In this talk, we describe this link and speculate on its implications for future studies of 6D SCFTs.

3d N=4 theories on the sphere have interesting supersymmetric sectors described by 1d QFTs and defined as the cohomology of a certain supercharge. One can define such a 1d sector for the Higgs branch or for the Coulomb branch. We study the Higgs branch case, meaning that the 1d QFT captures exact correlation functions of the Higgs branch operators of the 3d theory. The OPE of the 1d theory gives a star-product on the Higgs branch which encodes the data of these correlation functions. When the 3d theory is superconformal, the 1d theory is topological and coincides with the known construction in flat space, where the topological 1d theory lives in the cohomology of Q+S. Our construction thus generalizes it away from the conformal point. We then focus on theories constructed from vector and hypermultiplets. Using supersymmetric localization, we explicitly describe their 1d sector as the gauged topological quantum mechanics, or equivalently a gaussian theory coupled to a matrix model. This provides a very simple technique to compute the Higgs branch correlators.

The Sachdev-Ye-Kitaev model exhibits conformal invariance and a maximal Lyapunov exponent in the large-N and low temperature limit, and thus belongs to the same universality class as a two-dimensional anti-de Sitter black hole. Poles corresponding to a tower of operators that are bilinear in the microscopic Majorana fermions can be found in the four-point function of the fermions. We propose a renormalization theory for UV perturbations to the model, and derive an effective action for the soft mode of the model that results from integrating out the lowest resonance and which is dominant in the IR. We show that a two-dimensional dilaton theory with a quadratic term precisely reproduces this effective action on its boundary.

We will study the entanglement structure of states in Chern-Simons (CS) theory defined on n-copies of a torus. We will focus on states created by performing the path-integral of CS theory on special 3-manifolds, namely link complements of n-component links in S^3. The corresponding entanglement entropies provide new framing independent link-invariants. In U(1)_k CS theory, we will give a general formula for the entanglement entropy across a bi-partition of a generic n-link into sub-links. In the non-Abelian case, we study various interesting 2 & 3-links including the Whitehead link & Borromean rings, both of which have non-trivial entanglement entropies.