Academic Year 2014/2015

Due to the nonlinear and coupled nature of the Einstein field equation, exact black hole solutions are difficult to find in general without sufficient symmetry assumptions. Any spacetime admitting a degenerate Killing horizon (e.g. an extreme black hole) has a well defined near-horizon geometry (NHG). While keeping all the important information about the horizon, NHGs are much easier to solve. However given a NHG, there may or may not exist a corresponding black hole solution, let alone uniqueness. In this talk we investigate black hole candidates which possess certain NHGs in $4$ and $5D$ by linearising the Einstein equation about the NHG. In $4D$ we uncover a \textit{local} uniqueness theorem for extreme Kerr black hole without asymptotic flatness. In $5D$ we show that there is room for new black hole solutions which share the same NHG as the extreme self-dual Myers-Perry black hole.

The Yang-Baxter equation (YBE) is central in the theory of quantum integrable systems. For decades, together with its companion for problems with boundaries (the quantum reflection equation), it has been studied and used in the quantum realm. But it was suggested by Drinfeld in 1990 that the general study of the so-called "set-theoretical YBE" is also important. It turns out that classical integrable field theories provide a means to construct solutions to this equation, called Yang-Baxter maps, by looking at soliton collisions. Using the vector nonlinear Schrödinger (NLS) equation as the main example, we will review this notion of "classical solutions of the quantum YBE". Then, we will show how the new concept of set-theoretical reflection equation naturally emerges by studying the vector NLS on the half-line. Solutions to this equation, which we call reflection maps, arise from the reflection of solitons on the boundary. In the present context, factorization of interactions is the unifying principle behind integrability. This is of course well-know for quantum field theories but is essentially unexplored classically. Time allowing, I will mention applications of these new ideas to fully discrete integrable systems on quad-graphs."

I will compute mutual information between finite intervals in two non-compact 2d CFTs in the thermofield double (TFD) formulation after one of them has been locally perturbed by a primary operator in the large $c$ limit. We will also look at holographic dual: falling particle in eternal BTZ black hole. Time scale, called the scrambling time, at which the mutual information vanishes and the original entanglement between the thermofield double gets destroyed by the perturbation will be computed.

The worldline approach to quantum field theory is a reformulation where the fundamental objects are point particles described by simple quantum mechanics. This formalism can offer significant computational advantages over conventional field theory. I will describe how this quantum theory can be modified to introduce interactions between the particle worldlines and how it can be related to classical electrostatics and the well-known problem of confinement.I'll then introduce some further structure into the theory, generalising it to include non-Abelian interactions of chiral fields. The extra degrees of freedom will be shown to produce the group-theoretic information required in the field theory. The result will be a novel method of generating the representations and chiralities present in SU(5) unified theory and I will also comment on field theories with an underlying symmetry based on the groups of SU(6) and SO(10). There is a theory of interacting spinning strings underlying the model which I will refer to throughout.

We present fundamental notions regarding classical and quantum integrable systems. More precisely, we present the Hamiltonian formulation in classical integrable systems together with the linear auxiliary problem. Using these ideas we extract the local integrals of motion, which are by construction in involution. The non-linear Schrodinger model is used as a paradigm. We then briefly review the algebraic Bethe ansatz formulation in the context of quantum integrable lattice models. We consider the anisotropic Heisenberg model as a prototype to illustrate the identification of the spectrum and the associated Bethe ansatz equations. The issue of classical and quantum integrable impurities is also addressed in this context.

I review the recent construction of exceptional field theories — the manifestly duality covariant formulation of maximal supergravity in which D=11 supergravity and IIB supergravity can be embedded. As an application I discuss consistent IIB truncations, including the maximal AdS5 x S5.

We briefly review recent developments which suggest that non-geometric string backgrounds experience a nonassociative deformation of spacetime geometry. A geometrisation of this frame leads to a sigma-model for closed strings propagating in an effective target space that is the phase space of the original compactification manifold. Quantization of the sigma-model produces an explicit nonassociative star-product algebra on functions on phase space, which is related to the quantization of Nambu-Poisson structures. We use this formalism to develop a phase space formulation of nonassociative quantum mechanics, and demonstrate that, against all odds, a consistent formulation seems indeed possible. Our approach is completely quantitative and adds to previous qualitative discussions of nonassociativity in quantum mechanics, and it moreover avoids previous no-go theorems.

Gauge theory in four dimensions is closely tied to the geometry of Riemann surfaces. For instance, its spectrum is encoded in a so-called spectral network, which is a collection of lines on a Riemann surface defined by a tuple of k-differentials. I will show that spectral networks generate interesting sets of Darboux coordinates on the moduli space of flat connections on a Riemann surface, such as (higher rank generalizations of) Fenchel-Nielsen coordinates. These coordinates in turn teach us about the strongly coupled regime of the gauge theory.

From an algebraic point of view anyonic theories can be described by braided fusion categories. In this talk I will discuss the so(2p+1)_2 fusion categories and a family of one-dimensional anyonic models arising from these categories. This includes how gauge and monoidal equivalences of catogeries leads to symmtries or mappings between models. Finally, certain integral points within these models will be discussed along with Bethe equations.

This talk is devoted to the theory of multivariate σ-functions developed by V. Buchstaber, D. Leykin, V. Enolski, C. Eilbeck and his group, and C. Athorne. The theory is based on series expansions, and has the advantage to be effective and easy for computation. The first part of talk describes a construction of the series expansion of σ-function associated with a so called (n, s)-curve. As a by-product of the construction we obtain the basis of second kind differentials associated to the standard first kind differentials. The general scheme is illustrated by the examples of small genera. Further we discuss on some applications and open problems related so called polylinear relations, namely, bilinear Hirota relations, which can be alternatively obtained from Klein’s bidifferential formula; and trilinear relations, which produce addition formulas. We focus on the problem of regularization of the second kind integrals, which appears nontrivial in non-hyperelliptic case.

Preseminar for PhD students

One dimensional Heisenberg spin chains can be used to model certain quasi one dimensional materials. Using the vertex operator approach due to Jimbo and Miwa, it is possible to compute exact results for correlation functions of the spin 1/2 XXZ chain and so calculate the dynamic structure factors of these materials - objects measurable in inelastic neutron scattering experiments. I hope to give an overview of the techniques involved and also talk about our application of this approach in the spin 1 case, the goal of which being to compute exact form factors.

We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model originally used to describe the formation of ice. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory. This is joint work with Vassily Gorbounov, Aberdeen.

Half BPS states (operators) in N=4 SYM are famously described by free fermions both at weak and strong coupling. I describe a set of conjectures for a preferred class of states in more general conformal field theories that can be tested in supergravity for when such a free fermion description might arise and some motivation for it applying generally. The states in question belong to the chiral ring of a supersymmetric conformal field theory that extremize an additional U(1) charge for fixed dimension and can be reduced to multi-traces of a composite matrix field, which is equivalent to using Young tableaux (Schur polynomials) as a basis. The main conjecture asserts that if the Young tableaux are orthogonal, then the set of extremal three point functions of traces to order 1/N are determined up to a single constant. The conjecture is extended further by providing an exact norm for the Schur basis and this norm arises from a set of free fermions for a generalized oscillator algebra.

N=4 super Yang-Mills can be formulated in twistor space via a twistor action. Feynman diagrams for this formulation in twistor space in an axial gauge are vastly simpler than their standard counterpart. They naturally express the underlying superconformal invariance and have very few terms for a given amplitude. They give a proof of the planar amplitude/Wilson loop correspondence diagram by diagram. They relate simply to recent concepts such as the amplituhedron and naturally lead to the loop integrand in dlog form and can be evaluated to give the expected polylogs without Feynman parameters.

There has been an important progress, over the last 20 years, in describing the correlation functions-expectation values of local operators- for various instances of quantum integrable models, the most prominent example being the XXZ spin-1/2 chain with L sites . The form factors in this model, \textit{viz}. the expectation values of local operators taken between two eigenstates of XXZ Hamiltonian, can be expressed in terms of determinants of matrices whose size grows with L. Usually, one is interested in describing the correlators and form factors in the thermodynamic limit L->\infty. This leads to numerous questions related to extracting the large-L behaviour of the aforementioned determinants. After presenting the XXZ model and reviewing some of the progress that has been made in the field, I shall discuss the large-L behaviour of determinants arising in the description of the massive regime of the chain where the excitation spectrum above the ground state has a gap. I shall then comment on how, in the thermodynamic limit , these large-L asymptotics allow one to describe the large-distance behaviour of two-point functions. The answer I obtained with my collaborators is structurally different from the one issuing from the vertex operator approach, hence leading to highly non-trivial identities between elliptic functions. This is a joint work with M. Dugave, F. Göhmann (Bergische Universität Wuppertal, Germany) and J. Suzuki (Shizuoka University, Japan).

Flux compactifications in string theory are a promising candidate for an accurate model of the nature of the quantum vacuum. Curiously, closed strings in flux compactifications probe a non-commutative and non-associative geometry. In this talk I will present to you how one might understand the concept of a non-commutative and non-associative spacetime, making use of the capable language of category theory wherein differential geometry is internal to a certain quasi-Hopf representation category and quantisation is achieved by the application of the twist deformation quantisation functor.

TBA

We will review some arguments relating quantum entanglement with geometry.

As is well known, the extremal 5D Reissner-Nordstrom black hole can be embedded in type IIB string theory by using the D-brane construction of Strominger and Vafa. I will discuss a new family of 1/8-BPS type IIB supergravity solutions that represent microstates for this black hole. I will also discuss the dual interpretation, in the AdS/CFT sense, of these geometries as semiclassical states in the D1-D5 CFT.

Preseminar for PhD students

CFTs at large central charge display some universal features which can be inferred from holography. Using these as a guide one can obtain some necessary conditions for a given CFT to admit a classical string dual. I will describe attempts to construct a large class of CFTs satisfying these conditions exploiting some technology of permutation orbifolds.

Inspired by the relations with matrix models and gauge theories, we introduce equations in 2d conformal field theory, which are the analogs of the loop equations of matrix models. Solving a conformal field theory then amounts to solving these equations. We solve them for two values of the central charge: c=1 and c=infinity. In the case c=1, the solution is built from a Lax matrix, which is a matrix square root of the energy-momentum tensor. In the case c=infinity, the solution is built from a non-commutative spectral curve, which (via topological recursion) encodes the perturbative expansion of correlation functions. In both cases, we recover the appropriate limit of the Liouville three-point function.

Preseminar for PhD students

My main purpose will be to discuss what insight can be gained in general relativity by taking seriously Klein's Erlangen Program and Cartan's extension of it to differential geometry. The talk will be divided into three main parts. First, I will introduce Cartan geometry in a gravitational context by describing the geometric role of "breaking" SO(4,1) symmetry to SO(3,1), as in the MacDowell-Mansouri-Stelle-West formulation of gravity. Second, I will describe how a "field of observers" breaks the symmetry further to SO(3), yielding a system of evolving spatial Cartan geometries, or "Cartan geometrodynamics". This gives a Lorentz-covariant version of the Ashtekar-Barbero formulation, which is the classical starting point for loop quantum gravity. Third, I will combine the first two parts, and consider breaking SO(4,1) to SO(3) directly. This describes the geometry of "observer space", the bundle of future-timelike unit vectors in spacetime, giving a perspective with key features of both the spacetime and geometrodynamic approaches. I will also discuss how observer space geometry provides a unified geometric setting for relating various alternative theories of gravity.

This talk is about an exact solution of the 4+1 dimensional vacuum Einstein equations which can be interpreted as an adiabatic evolution of a Euclidean Taub-NUT space from Euclidean 4-space. This solution is not new little studied. Remarkably, the Maxwell and Dirac equations on this background also allow for exact adiabatic solutions. I will interpret them in the framework of a geometric model of the electron and discuss tentative links with Dirac's large number hypothesis.

Preseminar for PhD students

The spin-statistics connection is one of the major structural features of quantum field theory in Minkowski space, and its proof is tied closely to the properties of the Poincare group of spacetime symmetries, which enters into the definition of what spin is, among other things. General curved spacetimes have trivial symmetry group, so the Minkowski space proof cannot be transferred directly to this setting; it is not even clear why spin continues to be a reasonable concept. In a landmark paper [Commun. Math. Phys. 223 (2001) 261--288], Rainer Verch showed how a spin-statistics connection can be proved for locally covariant theories: his proof applies to fields that are sections of vector bundles associated to a spin bundle. In this talk, I describe a new and more algebraic approach to the spin-statistics connection for locally covariant theories. In particular, the concept of spin arises naturally without being inserted by hand, and no explicit reference to fields need be made.

Mapping spaces serve as mathematical models for configuration spaces in classical field theory. There are different approaches to add fermionic degrees of freedom to this description. One of them consists in replacing mapping spaces between ordinary manifolds by suitable mapping spaces between supermanifolds which carry themselves the structure of a supermanifold, i.e. contain bosonic as well as fermionic elements. We will give a short introduction to the functorial approach to supergeometry and show that it allows for a natural construction of the required objects in the sense of infinite-dimensional supermanifolds. In addition, the formalism provides a framework for the study of PDEs for anticommuting quantities. We will eventually describe how these mathematical structures may be used as a starting point for the discussion of classical field theories with fermionic degrees of freedom.

A short introduction to Lorentzian manifolds and normally hyperbolic operators

Locally covariant QFT (LCQFT) has been developed by Brunetti, Fredenhagen and Verch as an axiomatic framework for QFT on curved spacetimes. While the simplest models of interest, such as the Klein-Gordon and Dirac field, obey the axioms of LCQFT, recent studies have shown that this is not the case for gauge theories. Using the language of differential cohomology, I will present the construction of the full U(1) Yang-Mills model (including bundles as dynamical degrees of freedom) and clarify which properties of gauge theories lead to a violation of which axioms of LCQFT. As a new result I will show that the additivity property is violated in gauge theories, i.e. that the algebras of local observables in, say, diamond regions do not generate the algebra of observables on the full spacetime. Inspired by the theory of stacks I will propose new axioms for QFTs on curved spacetimes, which replace the usual locality axiom by an additivity axiom being dual to the `gluing conditions' in a stack.

I will talk about work in progress on the numerical computation of entanglement entropy for free quantum fields in Minkowski spacetime relative to an inaccessible region. This extends classic work of Srednicki's (which I will review) in several directions: introduction of a mass parameter and considering dimensions other than 4. The talk will be quite elementary, requiring little else but familiarity with the quantum harmonic oscillator!

Preseminar for PhD students

In recent years, due to the method of supersymmetric localization, many exact results have been achieved in the study of supersymmetric gauge theories on compact spaces of various dimension and topology, leading to the discovery of surpraising structures. An important example is provided by the correspondence introduced by Alday, Gaiotto and Tachikawa, relating the partition functions of a large class of supersymmetric gauge theories on S4 and S2 to correlators in Liouville CFT. In this talk, I will explain how this picture can be lifted to higher dimensional gauge theories via the correspondence of partition functions on S5, S4xS1, S3 and S2xS1 to correlators in theories whose underlying symmetry is given by a quantum deformation of the Virasoro algebra. In particular, I will discuss how 3-point functions can be derived by the bootstrap approach and used to define this novel class of q-deformed CFTs. I will also discuss some aspects related to integrable structures in these models, such as reflection coefficients, as well as possible generalisation.

I shall give a brief introduction to discrete holomorphicity and its role in the analysis of quantum integrable lattice models and their conformal limit. I will then present a general formalism for constructing discretely holomorphic operators in terms of the non-local currents of an underlying quantum group. This method has been successfully applied in a range of lattice models and I shall describe the examples of the O(n) loop and Chiral Potts models.

Preseminar for PhD students

Studying the cohomology of square-integrable harmonic forms on a non-compact manifold is generally a difficult problem. In the case of ALF gravitational instantons I will show that, thanks to results by Hitchin, Hausel et al, the problem simplifies much and it is possible to give an explicit description of the L^2 harmonic cohomology. In particular L^2 harmonic 2-forms have an interesting geometric description as Poincaré duals of minimal area embedded surfaces.

In the picture of the world presented to us by classical physics the concept of a smooth manifold is the central mathematical tool. On one side, there is the space-time in which we live, and on the other side there are the state-spaces of various physical systems. Quantum theory changes the idea of a physical system in a fairly well-understood way: we know how the quantum states are related to the classical state-space. But the main content of classical physics is the account of how the state-spaces are constructed straightforwardly from space-time, and it is the analogue of this step which is problematical and ill-understood in quantum theory. My talk will describe some of the problems and some possible approaches to them. Ideally, one would want space-time to be an 'emergent' concept - but it is far from clear what one hopes will emerge.