Academic Year 2016/2017

When angular momentum is conserved in the evolution, the standard picture of gravitational collapse for asymptotically flat spacetimes leads, using heuristic arguments, to an inequality relating the ADM mass and angular momentum. Such geometric inequalities have been rigorously proved for axisymmetric, asymptotically flat maximal initial data for the vacuum Einstein equations. I will discuss recent work on extending this class of inequalities to higher dimensions, where a number of qualitative differences arise (e.g. black holes can have non-spherical horizon topology). In particular, I will discuss how a lower bound for the mass, in terms of a (regularized) harmonic energy functional is obtained. The unique minimizer of this energy is then shown to correspond to extreme black hole initial data with fixed angular momenta. This provides a variational characterization of extreme black holes.

The double copy is a method for constructing scattering amplitudes in gravity from scattering amplitudes in gauge theory, perturbatively around Minkowski space. This method has been very useful in the study of perturbative supergravity at high loop orders, and even has a catchy slogan: Gravity = (Gauge theory)^2. However, it is not known if the double copy is an intrinsic feature of gauge theory and gravity as perturbative QFTs or just a (highly effective) trick that happens to work in Minkowski space. In this talk I will describe how the robustness of the double copy can be tested by considering the simplest scattering amplitudes in the simplest non-flat backgrounds; namely, 3-point amplitudes on plane wave backgrounds. Despite various subtleties introduced by the non-trivial scattering background, it is clear that a notion of double copy does indeed exist.

Three-dimensional gauge theories with eight supercharges (3d N=4) have a rich moduli space of supersymmetric vacua with different low energy physics. This infrared physics is well understood for theories with large enough number of flavours ("good theories"), but less so if the number of flavours is small ("bad theories"). In this talk I will focus on 3d N=4 super-QCD theories with U(N) gauge group and N_f flavours of fundamental hypermultiplets. After reviewing known results and puzzles about such theories, I will discuss their quantum corrected moduli space of vacua, which consists of Higgs, Coulomb and mixed branches, and their low energy physics as N and N_f are varied. As a by-product, I will clarify if and in which sense bad U(N) gauge theories with N_f >= N flavours flow to their "Seiberg dual" good U(N_f-N) theories with N_f flavours (plus free fields) at low energies, as is suggested by localization results.

The character variety Ch_G(M) is a moduli space of G-local systems on a manifold M. These lie at the heart of the mathematical approaches to Dijkgraaf-Witten and Chern-Simons theories. For a surface S and reductive group G, Atiyah and Bott constructed on Ch_G(S) a natural symplectic form; when S bounds a 3-manifold M, Ch_G(M) defines a Lagrangian subvariety of Ch_G(S). In this talk I'll explain joint work with Ben-Zvi, Brochier, and Snyder to functorially quantize the character variety construction, resulting in a 4-dimensional topological field theory related to N=4 d=4 super Yang-Mills theory, following Kapustin-Witten.

Quantum spin chains are quantum many-body systems that have important applications to condensed-matter theory, quantum information theory, quantum field theory, and string theory. The simplest anisotropic spin chains are arguably those that are integrable and have quantum group symmetry. We begin by reviewing the construction of a large class of such models. We then focus on models associated with the twisted affine Lie algebra A^{(2)}_{2n}. We argue that the Hamiltonians corresponding to two choices of integrable boundary conditions have the symmetries U_q(B_n) and U_q(C_n), respectively. The deformation of C_n is novel, with a nonstandard coproduct. We argue that the degeneracies and multiplicities implied by the quantum group symmetries are completely described by the Bethe ansatz.

In this talk I will first review what are magnetic monopoles, and why singular monopoles are an interesting subject of study. In the core of the talk, based on https://arxiv.org/abs/1603.09575, I will explain a method to construct a large class of singular hyperbolic monopoles by dimensional reduction of smooth instantons defined on a curved 4-manifold. The monopoles so obtained can be described in terms of solutions of a modified Helmholtz equation, and their dependence on boundary data becomes particularly explicit.

A number of recent advances have overcome a many of the obstructions to the construction of superstring field theory - an attempt at a second-quantised description of string theory. In separate developments, the great progress made in the study and simplification of scattering amplitudes in field theory have led to novel `string-like' descriptions of supergravity. I will give a brief overview of some of these developments and show how they lead towards a string field description for supergravity. No prior knowledge of string field theory will be needed and only a passing familiarity with conventional string theory will be assumed.

I shall present a proof of the horizon conjecture which explains the emergence of superconformal symmetry near black hole horizons. This requires the use of global techniques like the index theorem, the Hopf maximum principle and the proof of new Lichnerowicz type of theorems. Then I shall apply similar methods to prove non-existence theorems for smooth warped AdS backgrounds and clarify some puzzles that emerge in the investigation of such systems.

(This is joint with NBMPS) Many quantities in mathematical physics are periods in the sense of Kontsevich and Zagier. Francis Brown's theory of motivic periods enriches these mere numbers to algebraic objects and thereby provides an extremely powerful new tool to understand their relations, namely the motivic coaction (dual to an action of a motivic Galois group). I shall explain this structure in the case of multiple zeta values, where Goncharov's formula allows us to compute the coaction explicitly. Then I will focus on the example of Feynman integrals in perturbative phi^4 quantum field theory and briefly review methods of computing such periods in terms of multiple polylogarithms (based on hyperlogarithms and single-valued integration). These results show a remarkable feature: the Feynman periods appear to be closed under the motivic coaction, which explains their highly constrained structure. This is joint work with Oliver Schnetz.

I will begin by discussing the notion of exact quantum entropy for supersymmetric black holes. I will then discuss how to compute it in a highly symmetric example (32 supercharges) using supersymmetric localization. This macroscopic computation can then be used to make statements about the underlying microscopic theories of quantum gravity like string theory. I will discuss this interplay in the case of string theory with 16 and 8 supercharges where there new supersymmetric configuration that enter the microscopic counting function.

In recent years, we have learned that there is a close connection between scattering amplitudes in gauge theory and gravity. Specifically, there is a specific notion of a “double copy” which allows one to determine gravitational amplitudes directly from Yang-Mills amplitudes. This formula is proven in the tree approximation. In this talk, I will describe these developments and build on them to relate certain classical solutions of Yang-Mills theory and general relativity.

It is well-known that many R-matrices, i.e. matrix solutions of the quantum Yang-Baxter equation (YBE), are associated to representations of affine quantum groups (deformed universal enveloping algebras of affine Lie algebras). Restrictions of these representations to certain coideal subalgebras are known to be related to the reflection equation (RE), which involves the aforementioned R-matrix and an additional object called K-matrix. It appeared in the 1980s in work by Cherednik, Sklyanin and others in the study of quantum integrable systems with boundaries. A particularly useful class of coideal subalgebras can be defined in terms of generalized Satake diagrams, following ideas of Letzter and Kolb. In joint work with Vidas Regelskis, the corresponding solutions of the RE in the vector representation of affine quantum groups of classical (A, B, C, D) Lie type have been found by solving a suitable intertwining equation. Hence this work is a boundary version of Jimbo’s 1986 paper in which he obtained explicit formulas for R-matrices using their intertwining property. It appears that all matrix solutions of the RE involving these R-matrices can be obtained by this method. The underlying Satake diagram governs how many nonzero entries a particular solution of the RE may have, how many free parameters it may depend on and how it may be related to another solution through similarity transformations. Furthermore, there are potential applications to finding RTT-presentations and Schur-Weyl dualities for coideal subalgebras.

I present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2- and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T- and U-duality invariant field theories.

Monopoles are solitons that live in Yang-Mills-Higgs theory in three (space) dimensions. Monopole chains are monopoles invariant under a discrete 1-dimensional translation group; equivalently, they are monopoles on the manifold R^2xS^1. A classical problem for solitons is the determination of allowed symmetry groups; in this talk I will discuss the classification of monopole chains with cyclic symmetry. The classification depends on much of the technology that has been developed to analyse monopole chains, including spectral curves, the Nahm transform, Hitchin systems, and Higgs bundles (all of which will be explained in the talk!).

Perturbing a CFT by a relevant operator on a half space and letting the perturbation flow to the far infrared we obtain an RG interface between the UV and IR CFTs. If the IR CFT is trivial we obtain an RG boundary condition. The space of massive perturbations thus breaks up into regions labelled by conformal boundary conditions of the UV fixed point. For the 2D critical Ising model perturbed by a generic relevant operator we find the assignment of RG boundary conditions to all flows. We use some analytic results but mostly rely on TCSA and TFFSA numerical techniques. We investigate real as well as imaginary values of the magnetic field and, in particular, the RG trajectory that ends at the Yang-Lee CFT. We argue that the RG interface in the latter case does not approach a single conformal interface but rather exhibits oscillatory non-convergent behaviour.

In this talk I will give an introduction to exceptional field theory, a framework with manifest E_{d(d)} symmetry that is useful in studying dualities in string and M-theory. I will discuss how exceptional field theory can be used to find 1/2-maximal consistent truncations in seven dimensions, by considering generalised SU(2)$-structure backgrounds, which I will define in the talk. Such truncations lead to seven-dimensional gauged supergravities coupled to an arbitrary number of vector multiplets and hence have global symmetry group SO(3,n). I will also show that this formalism can be used to obtain the 10-dimensional heterotic supergravity from exceptional field theory. Finally, I will show how the duality between M-theory on K3 and the heterotic string on T^3 arises in exceptional field theory.

I will talk about recent work with Jose Figueroa-O'Farrill on the algebraic structure of the Lie superalgebra g=g0+g1 generated by the Killing spinors of an eleven-dimensional supergravity background. I will first explain how any such g can be regarded as an appropriate deformation of a subalgebra of the Poincare superalgebra p=V+S+so(V) and then discuss applications to the classication of highly supersymmetric backgrounds. In particular, we will see that preserving more than half the supersymmetry implies the supergravity field equations.

I will present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. I will illustrate this method through two fundamental models with sl(2) Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. New long-range boundary Hamiltonians are obtained, and I will demonstrate that they exhibit Yangian symmetries, thus ensuring integrability. The method presented provides a "bottom-up" approach for constructing integrable boundaries and can be applied to any spin chain model.

We will describe constraints on the three-point functions of operators with spin which follow from conformal symmetry and unitarity.

We will describe constraints on the three-point functions of operators with spin which follow from conformal symmetry, unitarity, and some extra assumptions natural for CFTs with gravity duals.

For bi-partite quantum systems, the amount of entanglement between the two regions can be expressed by the entanglement entropy. There are other related quantities such as the Renyi entropy or the logarithm negativity, which are also important. In general, these quantities are difficult to evaluate due to the explicit dependence of the reduced density matrix that needs to be diagonalized. In the scaling limit, one can use quantum field theory (QFT) to find exact results and universal behaviour. In particular, in the context of conformal field theory (CFT), the Renyi entropy has two equivalent interpretations. The first one is of geometric type, because it is given by the evaluation of a partition function on a Riemann surface of higher genus. In principle such partition functions are classified by modular invariance, but for general Riemann surfaces only the set of theories that have central charge equal to one are explicitly known. The second interpretation is algebraic. The Renyi entropy can be written as a correlator of twist fields. These fields live in a extended version of the Virasoro algebra, usually referred to as the cyclic orbifold CFT, where they are associated to branch points on the sphere. This second point of view is particularly useful because when one consider theory away from criticality, or in a massive QFT, correlators of twist fields can be calculated using the form factor program.

In M-theory, M5-branes interact by M2-branes extending between them. The one-dimensional intersections are so-called selfdual strings charged under a selfdual 2-form field. Their dynamics is governed by higher gauge theories. We construct higher gauge theories containing 2- and 3-form curvatures in various dimensions making use of the supergeometric QP-manifold method, which induces a BRST-BV formalism. Structurally equivalent higher gauge theories are identified on the gauge algebra level. A method of off-shell covariantization is proposed that achieves to covariantize the gauge transformation of the 3-form curvature while circumventing the so-called fake curvature condition by constraining the auxiliary gauge fields. We explicitly derive such a gauge algebra in 5 dimensions as a nontrivial extension of a differential crossed module within a symplectic Lie 4-algebra and show its off-shell closure.

After a review of holographic features of general relativity in 3 and 4 dimensions, I will show how to derive the transformation laws of the Bondi mass and angular momentum aspects under finite supertranslations, superrotations and complex Weyl rescalings.

A few years ago, Nekrasov and his collaborators proposed a double quantum deformation of Seiberg-Witten curve with omega background parameters which describes 4D and 5D supersymetric gauge theories. It illuminates not only the quantization of geometry but also gives an intimate relation with integrable models. In this talk, we present a purely algebraic derivation of his proposal, based on the quantum deformation of W(infinity) algebra. We introduced the building blocks of Nekrasov partition function, Gaiotto state, intertwiner and so on, and a simple characterization of such states and operators. It gives the qq-deformed SW curve in the form of Ward identities. The connection with the integrable models will be also explained to some detail.

The notion of "soft hair" refers to zero energy excitations in the near horizon region of black holes or cosmologies, advocated by Hawking, Perry and Strominger. I review recent results on soft hair in three spacetime dimensions. In particular, I focus on the near horizon symmetry algebra, which turns out to be surprisingly simple, namely infinite copies of the Heisenberg algebra. The results are universal (in a sense that I shall make precise) and could generalize to higher dimensions.

It is usually hard to compute entanglement entropy and mutual information for conformal field theories (CFT). Ryu-Takayanagi proposals allows us to find the same quantities using calculations in gravity. In this talk I will show how to find holographic entanglement entropy and scrambling time for BTZ black hole perturbed by a heavy (backreacting) particle. Holographic bulk description improves on the schock-wave approximation in 3d bulk dimensions. I will also discuss current attempts to generalize this calculation to the rotating BTZ black hole.