Academic Year 2011/2012

After a brief introduction on the theory of framed sheaves and their moduli spaces, I will give a description of a modified Atiyah class of a family of framed sheaves, which allows to define a framed version of the Kodaira-Spencer map. Moreover, I will construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.

This is a report of joint work with T. Koeppe, P. Majumdar and K. Ray. We define new partition functions for theories with targets on toric singularities via products of old partition functions on crepant resolutions.

Subject of this talk is the twistor geometry discussion of a five-dimensional generalisation of the instanton equation known as the contact instanton equation. This equation was recently introduced in the context of (twisted) supersymmetric Yang-Mills theory on the five-dimensional sphere. If time permits, certain extensions to higher dimensions and supersymmetric generalisations will also be presented.

I will talk about a recent construction for an action for non-abelian 2-form in 6-dimensions. The action consists of a non-abelian generalization of the abelian action of Perry and Schwarz for a single M5-brane. It admits a self-duality equation on the field strength as the equation of motion. It has a modified 6d Lorentz symmetry. On dimensional reduction on a circle, our action gives the 5d Yang-Mills action. Based on these properties, we propose that our theory describes the gauge sector of multiple M5-branes in flat space.

I will discuss quantum integrability in a family of 2D sigma-models on superspaces. In particular I will show how to derive perturbatively the Hirota equation that encodes the exact spectrum of these models. As a particular case we obtain a first-principles proof of the Y-system that has been proposed as a solution to the spectrum problem in N=4 SYM.

In 2001 Razumov and Stroganov conjectured that the (properly normalized) components of the ground state of the dense O(1) loop model on a semi-infinite cylinder enumerate fully-packed loop (FPL) configurations on the square, with alternating boundary conditions, refined according to the link pattern for the boundary points. This conjecture has arisen a lot of interest both in the physics and in the mathematics community. In this talk, after reviewing the main background, I shall discuss a stronger (but easier to prove!) version of the correspondence: on the `dense O(1) side', the ground state of Hamiltonian H of the loop model is replaced by the one of the Scattering Matrix S(t); on the `FPL side', one considers the refinement according to the position of the unique straight tile on the last row.

In this talk I consider spin chains and vertex models with dynamical supersymmetry. The corresponding supercharges act non-locally and change the number of sites. The simplest example is the spin-1/2 XXZ chain/six-vertex model with periodic boundary conditions at the combinatorial point, which possesses an N=(2,2) supersymmetry on the lattice. I will explain the construction of the representations of this algebra for arbitrary spin, leading to fused vertex models. Furthermore I will discuss elliptic models (which do not enjoy spin/particle number conservation) with supersymmetry, in particular the spin-1/2 XYZ chain along a special line of couplings, and the elliptic extension of the Fateev-Zamolodchikov spin chain with special emphasis on the properties of the ground states (supersymmetry singlets), which display a variety of connections to combinatorics and classically integrable equations.

In this talk I will discuss the construction of braneworld black holes in the context of the Randall-Sundrum infinite braneworld scenario (RS2). As I shall explain, the existence of large braneworld black holes can be understood as perturbations of the a solution in AdS corresponding to the gravitational dual of N=4 super-Yang-Mills on the background of the 4d Schwarzschild solution.

A noncommutative (NC) gauge theory is typically described in terms of connections on finite projective modules over NC algebras. While the notion of curvature is standard in this setting, there is not yet a general theory of parallel transports. In my talk I will propose a definition of parallel transport on modules over NC algebras. Instead of transporting a vector along a curve, which is the usual LOCAL concept of parallel transport in commutative geometry, we transport an element of the module (a "section" of a NC vector bundle) along a one-parameter group of algebra automorphisms, which is a GLOBAL concept as required by NC geometry. I study in detail parallel transports on finite projective modules over fuzzy spaces and show that they naturally give rise to a set of gauge invariant observables. For modules equipped with a hermitian structure I prove that any hermitian connection can be reconstructed up to gauge equivalence from this set of observables. This is not possible using observables based on the curvature, indicating the importance of module parallel transports in the formulation of (quantum) gauge theories on NC spaces.

A subclass of (1+1)-dimensional dilaton-Maxwell gravity is studied in the Dirac approach to constrained dynamics. By exploiting a newly found duality with Liouville Field theory, it is shown that the requirement of a quantum realization of the conformal algebra for physical quantum states of the fields naturally constrains the cosmological constant to take values in a well determined and mostly discrete spectrum. Furthermore the contribution of the quantum fluctuations of the single dynamical degree of freedom in the gravitational sector, namely the conformal mode, to the cosmological constant is negative, in contrast to the positive contributions of the quantum fluctuations of the matter fields, possibly opening an avenue towards addressing the cosmological constant problem in a more general context.

I will show how the equations of motion of double field theory can also be derived as the one-loop background field equations of a doubled sigma model, both being the vanishing of the so called 'generalised Ricci tensor'. This raises the possibility of deriving higher order corrections to double field theory from a sigma model approach.

TBA

The bosonic field equations of type IIB supergravity are reduced on a locally (Cartan) symmetric space and through the complete classification of symmetric spaces, we identify all symmetric backgrounds up to local isometry.

This talk is based on joint work with Michael Atiyah and Nick Manton, described in arXiv:1108.5151. The paper contains a proposal to describe particles in terms of Riemannian 4-manifolds. This description is radically different from established models of particle physics. I will review the proposal, and illustrate it with geometric models for the electron and the proton. I will end by sketching some recent ideas for describing the neutron and beta decay geometrically.

In this talk, we discuss the boundary coupling of a Poisson sigma model associated to a Lie algebroid E over a smooth manifold X. The discussion is done within the context of the Batalin–Vilkovisky formalism in the AKSZ geometrical formulation. The main result is that a boundary condition corresponding to a Lie subalgebroid F of E over a submanifold Y of X can be coupled to a representation up to homotopy of F in a way that is consistent with the BRST symmetry. If time permits, I will explain how these considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.

Interpreting the cosmological constant as a pressure, black-hole thermodynamic potentials can be used to define a black-hole volume as the thermodynamically conjugate variable to the pressure. The properties of the thermodynamic volume are analysed and its physical consequences explored. The Euclidean free energy is identified as the Gibbs free energy, beloved of chemists, and the resulting PdV term in the first law modifies the amount of emergy that can be extracted in a Penrose process --- the maximum efficiency of a charged rotating black-hole is shown to be 75% when the PdV term is included, as opposed to the currently accepted value of 50%. The limit of zero angular-momentum is rather subtle and must be trated with care.

I shall start with an idea (somewhat heuristic) I call `Thermal Holography' and use that to probe the thermal behaviour of quantum horizons, i.e., without using any classical geometry, but using ordinary statistical mechanics with Gaussian fluctuations. This approach leads to a criterion for thermal stability for thermally active horizons with an Isolated horizon as an equilibrium configuration, whose (microcanonical) entropy has been computed using Loop Quantum Gravity (I shall outline this computation). As fiducial checks, we briefly look at some very well-known classical black hole metrics for their thermal stability and recover known results. Finally, I shall speculate about a possible link between our stability criterion and the Chandrasekhar upper bound for the mass of stable neutron stars.

TBA

A novel map between two even self-dual lattices of signature (3,19) and (24,0) will be presented. This map is important in identifying a symmetry group larger than the largest finite symplectic automorphisms group of K3 surfaces by orders of magnitude. The significance of this overarching symmetry will be stressed in the context of Mathieu 24 Moonshine.

Instanton equations in dimensions 5 and higher were introduced nearly 30 years ago. They appear naturally in Yang-Mills theory and supergravity, and also in recent attempts to generalise the Donaldson invariants. I will discuss some new examples of Yang-Mills instantons on spheres and Euclidean planes, and more generally, on manifolds with real Killing spinors and their cones.

We present a large family of junction-type representations of the Temperley-Lieb algebra. After introducing the representation, we discuss its associated symmetries. Apart from certain manifest ones, we are able to extract some non-trivial quantum symmetries of the representation. We present the extension to the boundary Temperley-Lieb algebra and we briefly discuss the corresponding spin chain.

The modern approach to integrability proceeds via a Riemann surface, the spectral curve. In many applications this curve is specified by transcendental constraints in terms of periods. In some cases, including the construction of magnetic monopoles, physical symmetries are inherited by the spectral curve and there may be consequent simplifications of the transcendental constraints. We shall look at this interplay of ideas.

I will give a short tour of discrete integrable systems related to quantum spin chains, and their relation to cluster algebras. I will show how to use integrability to solve these systems and use this to prove positivity conjectures in cluster algebras algebras. The connection to cluster algebras allows us to prove the completeness of Bethe ansatz states of the spin chain. I will also explain how this point of view can be generalized to non-commutative integrable systems.Cluster algebras are a rather general construction in algebraic combinatorics introduced by Fomin and Zelevinsky about a decade ago. Originally inspired by positivity questions in canonical basis theory of quantum groups, they have found applications in Donaldson-Thomas invariant theory, the proof of the periodicity conjectures of Zamolodchikov in TBA, representation theory, etc. I will describe these algebras without assuming any previous familiarity with the subject.

We consider the 6-vertex model on a 2D geometry with an integrable fracture running from the origin to spatial infinity. In the language of the associated XXZ quantum spin chain, the picture is that of a spin chain separated into two separate domains at time t≤0 that are joined together for t>0. Such a scenario is called a local quantum quench. This model is very natural to consider in the language of Baxter's corner transfer matrix. Exploiting this fact, we use the machinery of the vertex operator approach to integrable systems in order to compute exact expressions for correlation functions. Specialising to the case of the magnetization in the vicinity of the tip of the fracture, we obtain a simple exact formula that is analogous to Baxter's formula for the spontaneous magnetisation of the usual XXZ chain.

I will discuss the bosonic excitations present in certain strongly-interacting holographic field theory states with a large density of matter. Particular attention will be paid to the properties of the sound modes of these theories and how they compare with the properties of the sound mode of a Landau-Fermi liquid.

Compactifications of the heterotic string on a Calabi-Yau manifold are a relatively unexplored area of string theory despite their promise in cultivating novel mathematics as well as promising phenomenology. In this talk I will summarise recent developments in understanding the worldsheet descriptions of heterotic vacua, its implications for computing physical observables (e.g. Yukawa couplings) and mathematics (e.g. mirror symmetry).

The main topic of this talk is the correspondence between D-branes in a certain type of rational conformal field theories, namely the Grassmannian Kazama-Suzuki models SU(3)_k/U(2), and matrix factorizations of the associated Landau-Ginzburg theories. Such a correspondence has been found on the level of bulk theories already around 1989, followed by considerable progress in the case of A-type boundary conditions, for which a correspondence between D-branes and solitons of the LG theories with perturbed superpotentials could be established. A correspondence for B-type boundary conditions, which are described by means of matrix factorizations in the LG theory, remained elusive except for certain explicit examples of RCFTs including the minimal models of CFT. The results obtained in collaboration with S. Fredenhagen suggest that this has a beautiful mathematical reason: it appears that the structure of matrix factorizations alone is not enough to find the "dictionary" between D-branes and matrix factorzations. As I will present, at least in the sufficiently generic case of the SU(3)_k/U(2) KS models, a successful strategy involves incorporating defects into the picture, or rather defect functors, a concept that seems to be new and thus will be defined in the seminar. In effect, the symmetry structure of the superpotential is sufficient to construct a defect functor semi-ring that generates all MFs corresponding to Cardy branes in the RCFT from a small set of simple solutions. In addition, these defect functors are the exact analogues of the CFT defects. Finally, an outlook to potential applications of these findings both in mathematics and in physics will be given.

I will summarise the search for candidate dual geometries to 3-dimensional N-extended superconformal field theories, with N>4. This includes a classification of eleven-dimensional homogeneous lorentzian manifolds with a transitive isometric action of SO(3,2) x SO(N). We find no (new) backgrounds with N>5, but several new backgrounds with N=5, at least one of which is supersymmetric.

In physics T-duality is a phenomenon which relates certain types of string theories to one another. From a topological point of view, one can view T-duality as a duality between line bundles carrying a degree three cohomology class (the H-flux). Bundle gerbes provide a geometric realisation of the degree three cohomology of a manifold. In this talk we will use bundle gerbes to give a geometric realisation of the H-flux and explain how to construct the T-dual of a line bundle together with its T-dual bundle gerbe.

TBA

We consider bounds on the entropy of certain infrared-stable conformal field theories at the self-dual temperature which are imposed by modular invariance.

In a non-unitary two-dimensional conformal quantum field theory there may be conserved currents which do not split into a holomorphic and an antiholomorphic components. We construct such models via a current-current perturbation of WZW models on supergroups. Such models emerge in string theory and condensed matter physics. We present some perturbative and non-perturbative results that probe the structure of the current OPE algebra in such models as well as the correlation functions and equal time commutators.