Academic Year 2015/2016

Higher gauge theory is a generalization of ordinary gauge theory. This requires categorifying the underlying mathematical structures in ordinary gauge theory. In this talk we will discuss the case with String 2-groups by defining smooth 2-group bundles as internal functors in the bicategory of finite dimensional Lie groupoids, right-principal bibundles and smooth bibundle maps.

We present the vector sine-Gordon (vSG) equation together with some of its integrability properties such as Lax representaion, Darboux and Backlund transformations. We use the Darboux transformation to construct a related vector Yang-Baxter map and an integrable vector differential-difference equation on the sphere. We will briefly discuss the dressing method and the construction of soliton solutions for the vSG. This is a joint work with Dr J.P. Wang (Kent) and Prof A.V. Mikhailov (Leeds)

Preseminar for PhD Students

If a noncommutative operator algebra models the observables of a quantum system, its empirical content given by classical measurement outcomes lies within the commutative subalgebras. I will survey a programme showing that the operator algebra is determined to a great extent by its partially ordered set of commutative subalgebras, and how adding dynamics leads to a noncommutative notion of configuration space.

This is a joint Topology and EMPG seminar. For details see: http://www.maths.ed.ac.uk/~aar/bismut.pdf

Originally used in the study of Dirac structures, Courant algebroids are now of central intrest in string theory. In typical cases, an underlying Lie bialgebroid exists and an elegant characterization of a Courant algebroid is given by the existence of a homological vector field on the cotangent bundle of the parity reversed version of the starting Lie algebroid. After reviewing this result, we show how the canonical Poisson structure on the total space of this graded manifold can be used to understand the gauge structure of "double field theory", a T-duality invariant description of closed strings. This is more than a beautiful coincidence: First it turns out that the inclusion of string fluxes can be understood as fibre-translations (originally described by Roytenberg). And second, by applying a Moyal-Weyl deformation to the Poisson bracket, the resulting corrections to the Courant bracket coincide with those found recently in heterotic string theory.

I will explain the concept of consistent reductions and uplifts for gravitational theories and discuss recent progress in understanding consistent reductions in supergravity by exploiting duality symmetries.

We extend the theory of bundle gerbes along the lines of higher geometric prequantisation. The 2-category of bundle gerbes as introduced by Waldorf turns out to admit a monoidal structure on its morphism categories. Further, we construct an internal hom functor on these morphism categories, allowing us to obtain a 2-Hilbert space of sections of a bundle gerbe. Finally, we provide an extension of the transgression functor to non-invertible morphisms which is compatible with the above additional structures, and comment on relations with earlier works on higher geometric quantisation.

The Skyrme model is one of the most realistic soliton models for application to particle physics. Skyrmions have a topological charge which is identified with baryon number. The nucleons (proton and neutron) are different quantum states of the basic Skyrmion. Although the Skyrme equations are not integrable, much is known about the static and dynamic solutions, and we can study properties of nuclei like Carbon-12 using Skyrmions. One gains a quite different perspective from this field-theoretic model than from conventional point-nucleon models.

In quantum algebra, the notion of a spherical fusion category plays an important role, and there are two interesting constructions which take such a category C as input. The first is "string nets", an elegant graphical description of the 3-dimensional Turaev-Viro TQFT in terms of string diagrams drawn on surfaces. The second is the "tube algebra" of C, a certain algebra whose category of representations is equivalent to the Drinfeld centre of C. I will describe the relationship between these two constructions.

Preseminar for PhD Students

I will report on ongoing work with Paul de Medeiros and Andrea Santi concerning the algebraic structure of the Lie superalgebra generated by the Killing spinors of a supergravity background, as well as its applications to the classification of supersymmetric supergravity backgrounds and the determination of geometries admitting rigidly supersymmetric field theories.

We study a free scalar field subject to boundary conditions involving second order derivatives, i.e., of generalised Wentzell type. For the classical system, we establish well-posedness of the Cauchy problem and causal propagation. We quantise the system canonically and discuss the relation between the bulk and the boundary field. Based on arXiv:1512.05512.

Preseminar for PhD Students

Abelian S-duality is a generalization of electric-magnetic duality of Maxwell's equations that takes into account important gauge-theoretic aspects such as magnetic monopoles, the Dirac charge quantization condition and Aharonov-Bohm phases. Using a combination of techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology, I show that Abelian S-dualities can be realized as natural isomorphisms between certain Abelian quantum gauge theories. As a spin-off of our formalism, I will also show how to quantize the self-duality equation F = * F for the field strength of a higher Abelian gauge field. This talk is based on joint work with C. Becker, M. Benini and R. J. Szabo ( arXiv:1511.00316 [hep-th] and arXiv:1511.00324 [math.DG] ).

An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. It was conjectured by Mills, Robbins and Rumsey in 1982 that the number of ASMs of fixed size is given by a certain simple product formula. A relatively short proof of this conjecture was obtained by Kuperberg in 1996, using the Izergin-Korepin determinant formula for the partition function of the six-vertex model on a square grid with domain-wall boundary conditions, together with a bijection between ASMs and configurations of that model. It was also conjectured by Robbins in the mid 1980's that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula. This conjecture has only recently been proved, in my joint work with Ilse Fischer and Matjaz Konvalinka (see arXiv:1512.06030). Our proof again uses connections with a particular case of the six-vertex model. In the first part of this talk, I'll introduce ASMs, and review Kuperberg's proof. In the second part, I'll outline the proof of the conjecture for the enumeration of diagonally and antidiagonally symmetric ASMs, and present some new results for the associated case of the six-vertex model.

Preseminar for PhD Students

I report on recent progress identifying a classical version of six-dimensional superconformal field theory, the ``(2,0)-theory,'' with higher gauge theory. I start with a review of higher gauge theory and how it overcomes naive no-go-theorems. I then explain that many classical Lagrangians studied in the context of M-theory are in fact higher gauge theories in disguise. Interesting non-trivial solutions to higher gauge theory can be found, and an underlying twistor description reduces the search for the classical (2,0)-theory to a search for the appropriate gauge structure. Finally, I show how to extend conventional higher gauge theory in various ways to allow for more interesting solutions.

After introducing the quantum inverse scattering method I will review the construction of Baxter Q-operators for closed rational spin chains. Subsequently, I will combine the ideas of the Q-operator construction with Sklyanin's formulation of the quantum inverse scattering method for systems with boundaries to construct Q-operators for the open Heisenberg spin chains with diagonal boundary conditions.

Preseminar for PhD Students

We use results from 3D gravity to explore deformations of relativistic wave equations for massive particles. In particular we present a method for deriving standard wave equations using both the representation theory of the Poincare group and a covariance procedure leading to momentum space wavefunctions satisfying mass and spin constraints. Following Fourier transform these constraints become well known wave equations. This procedure is presented for both half integer spin and more generally for anyonic particles, by working with appropriate covering groups. We then review the emergence of deformed symmetries in 3D gravity where the role of the Quantum Double is briefly mentioned. We repeat this procedure now with gravity in the picture, using the representation theory of the Double, to show how this leads to deformed momentum space constraints. Following a choice of momentum space co-ordinates and a formal Group Fourier transform we see the emergence of a deformed spacetime together with modified wave equations.

Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space (arXiv:1502.05383). Lisa Glaser and I have investigated the properties of simple cases of this statistical system using Monte Carlo methods (arXiv:1510.01377). Preliminary results indicate that some of the models have a phase transition, with interesting behaviour near the transition.

Preseminar for PhD Students

The Taub-NUT manifold coupled to an abelian gauge field with self-dual curvature, is considered as a geometric model for the electron in recently proposed Geometric Models of Matter. In this model the spin degrees of freedom are proposed to be given by the zero-modes of the Dirac operator. We compute the zero-modes and show that the gauge field allows the existence of classical bounded orbits and quantum bound states.

Two interesting properties of static curved space QFTs are Casimir Energies, and the Energy Gaps of fluctuations. We investigate what AdS/CFT has to say about these properties by examining holographic CFTs defined on curved but static spatially closed spacetimes. Being holographic, these CFTs have a dual gravitational description under Gauge/Gravity duality, and these properties of the CFT are reflected in the geometry of the dual bulk. We can turn this on its head and ask, what does the existence of the gravitational bulk dual imply about these properties of the CFTs? In this talk we will consider holographic CFTs where the dual vacuum state is described by pure Einstein gravity with negative cosmological constant. We will argue using the bulk geometry first, that if the CFT spacetime's spatial scalar curvature is positive there is a lower bound on the gap for scalar fluctuations, controlled by the minimum value of the boundary Ricci scalar. In fact, we will show that it is precisely the same bound as is satisfied by free scalar CFTs, suggesting that this bound might be something that applies more generally than just in a Holographic context. We will then show, in the case of 2+1 dimensional CFTs, that the Casimir energy is non-positive, and is in fact negative unless the CFT's scalar curvature is constant. In this case, there is no restriction on the boundary scalar curvature, and we can even allow singularities in the bulk, so long as they are 'good' singularities. If time permits, we will also describe some new results about the Hawking-Page transition in this context.

Preseminar for PhD Students

Based on the example of the non-linear Schrodinger hierarchy we define and explore the notion of dual classical integrable hierarchies. The role of Lagrangian formulation is emphasized in this framework, to be taken into account in addition to the more familiar Hamiltonian approach to classically integrable systems. Explicit dual hiererchies of Hamiltonians are built, and a multidimensional procedure is evoked.

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 kinematic space--the space of geodesics on the spatial slice of AdS3.

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by purely Lie algebraic means, our point of departure being the supertranslation ideal. Along the way we compute some cohomological groups and recover exactly the zeroth order terms of the supercovariant connection associated with the four-form flux. This is a joint work with José Figueroa-O’Farrill.

Starting from a given set of fusion rules one-dimensional lattice models of interacting anyons can be constructed. For a system of particles satisfying the fusion rules of $SO(5)_2$ fine-tuning of the coupling constants leads to integrable anyon chains with commuting transfer matrices of 'interactions round the face' (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. We find two critical points which are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra related to the underlying $SO(5)$ symmetry of the anyon chain. The modular partition function and fusion rules of these RCFTs are derived.

A brief introduction to quantum field theory

Non-extremal black holes, which emit thermal Hawking radiation, have two horizons: the event horizon or outer horizon and the Cauchy horizon or inner horizon. Surprisingly, for a broad class of solutions to the Einstein equations, the product of the areas of the inner and outer horizons is the square of the area of the horizon of the zero temperature black hole obtained from taking the smooth extremal limit. We use the attractor mechanism in supergravity to motivate this result. We motivate these results in terms of CFT.

Quantum physics shows nonlocal features, formally connected with the so-called collapse of the wave function and most evidently visible in the violation of Bell's inequalities. It is argued that nevertheless, by an appropriate split between observables and states, the principle of locality is fulfilled on the side of the observables, whereas the non-localities are due to correlations induced by states.

Preseminar for PhD Students

The thermodynamics of a black hole in asymptotically anti-de Sitter space-time can be generalised beyond the usual treatment by including the cosmological constant as a thermodynamic variable with the physical interpretation of pressure. New phases and phase transitions, analagous to those of a van der Waals gas, have been uncovered in this picture. Recent results will be summarised and their application to conformal field theory, via the AdS/CFT correspondence, will be presented. Some consequences for the physics of the quark-gluon plasma and the deconfining phase transition in QCD will be discussed.

Dual superconformal symmetry is a remarkable, hidden feature of N=4 SYM in 4 dimensions. Via AdS/CFT, such a symmetry corresponds to the invariance of the AdS(5) x S(5) superstring under specific combinations of bosonic and fermionic T-dualities. We show that AdS(d) x S(d) x S(d) superstrings with D(2,1;\alpha) isometry supergroup are T-self-dual if additional T-dualities along complexified S(d) directions are performed. This implies a new type of dual superconformal symmetry for the CFTs dual to AdS(d) x S(d) x S(d) x T(10-3d) superstrings.

Preseminar for PhD Students

We give a functional analytic construction of algebraic states for CAR algebras on a globally hyperbolic Lorentzian manifold. We show that in Minkowski space we recover the vacuum state and when we couple the Dirac equation to a time-dependent external potential, which is smooth and decays faster than quadratically for large times, we obtain Hadamard states.

We study a particular class of supersymmetric M-theory eight-dimensional non-geometric compactification backgrounds to three-dimensional Minkowski space-time, proving that the global space of the non-geometric compactification is still a differentiable manifold, although with very different geometric and topological properties with respect to the corresponding standard M-theory compactification background: it is a compact complex manifold admitting a K\"ahler covering with deck transformations acting by holomorphic homotheties with respect to the K\"ahler metric. We show that this class of non-geometric compactifications evade the Maldacena-Nu\~nez no-go theorem by means of a mechanism originally developed by Mario Garc\'ia-Fern\'andez and the author for Heterotic Supergravity, and thus do not require lP-corrections to allow for a non-trivial warp factor or four-form flux. We obtain an explicit compactification background on a complex Hopf four-fold that solves all the equations of motion of the theory. We also show that this class of non-geometric compactification backgrounds is equipped with a holomorphic principal torus fibration over a projective K\"ahler base as well as a codimension-one foliation with nearly-parallel G2-leaves, making thus contact with the work of M. Babalic and C. Lazaroiu on the foliation structure of the most general M-theory supersymmetric compactifications.

Deformation quantization and star products

We show that higher-degree Dirac currents of twistor and Killing spinors correspond to the hidden symmetries of the background. Hidden symmetries are defined as the antisymmetric generalizations of Killing and conformal Killing vectors and are called as Killing-Yano and conformal Killing-Yano forms respectively. In the case of Killing spinors, we find that the equations satisfied by the higher-degree Driac currents are related to the Maxwell-like and Duffin-Kemmer-Petiau equations. We also analyze the supergravity twistor and Killing spinor cases in ten and eleven dimensional supergravity theories and find that although different inner product classes induce different involutions on spinors, the higher-degree Dirac currents still correspond to the hidden symmetries of the background. As a result, we discuss the possibilities of the extension of Killing superalgebras of supergravity backgrounds by adding Killing-Yano forms.

Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires (weak) hermiticity and the Weyl condition. We also propose an iterative procedure that allows to compute the star product up to any desired order.

We discuss a connection between classes of linear ODEs and quantum integrable models, also known as the ODE/IM correspondence. We present recent results concerning the extension of this connection to include certain integrable PDEs, as well as other aspects of the correspondence.

Abstract of the corresponding paper arXiv:1503.01474: Interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) domain walls? Analyzing these questions in a wide variety of contexts -- from counting RG walls to AdS/CFT correspondence -- will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT's and point out connections with the superconformal index and classifying spaces of global symmetry groups.