Academic Year 2017/2018

I will introduce basic facts about G2-manifolds (G2-structures, Hitchin’s volume functional) in the special case where the G2-metric admits an isometric action of a compact Lie group with generic orbits of codimension 1. I will then discuss the classical cohomogeneity one examples of complete G2-manifolds found by Bryant–Salamon in 1989. As a warm-up for the seminar talk, I will also review the construction of 4-dimensional ALE and ALF hyperkähler metrics via the Gibbons-Hawking Ansatz.

In the early 2000’s Atiyah–Maldacena–Vafa and Acharya proposed an explanation of the large N duality between Type IIA theory with D6 branes/RR fluxes on the deformed/resolved conifold as a geometric transition in G2-geometry, the so-called G2-flop. In recent joint work with Mark Haskins and Johannes Nordström, we constructed many complete and incomplete G2-metrics with interesting prescribed asymptotic geometry and/or singular behaviour. In particular, we produced infinitely many new asymptotically conical G2-metrics (only three such metrics were previously known) and the first known example of a G2-metric that has an isolated conical singularity, but is otherwise complete (and is not a cone). Our examples provide a metric version of the G2-flop and infinitely many new geometric transitions in G2-geometry.

The classification of black hole solutions is a problem of central importance in general relativity. While in four dimensional Einstein-Maxwell theory this question has been answered under some assumptions (any asymptotically flat stationary spacetime must be a Kerr-Newman solution), for higher-dimensional general relativity an analogous classification of equilibrium states constitutes a major open problem. In this talk I will present some recent work in which we determine all asymptotically flat, supersymmetric and biaxisymmetric soliton and black hole solutions to five dimensional minimal supergravity by combining local constraints from supersymmetry with global constraints for stationary and biaxisymmetric spacetimes. We find that horizon topologies must be one of S^3, S^1×S^2 or a lens space L(p,1), with a large moduli space of black hole spacetimes for each of these allowed horizon topologies. In the absence of a black hole we obtain a classification of the known “bubbling” soliton spacetimes.

One of the great success of string theory is the microscopical explanation of the entropy of a class of asymptotically flat black holes.Much less is known about asymptotically AdS black holes in four dimensions or higher. In this talk I explain how to derive the entropy of a class of asymptotically AdS supersymmetric black holes in four dimensions using holography and localisation, a technique to evaluate exact quantities in supersymmetric QFT. I also discuss what is known in five and higher dimensions.

We present a state-sum construction of TFTs on r-spin surfaces which uses a combinatorial model of r-spin structures. We give an example of such a TFT which computes the Arf invariant for r even. We use the combinatorial model and this TFT to calculate diffeomorphism classes of r-spin surfaces with parametrized boundary.

String theory has new symmetries known as dualities that are rather different from the familiar geometric symmetries of field theories. These dualities can be used to glue together different patches of a solution to construct what have been called ‘non-geometric spaces’; these can be good solutions of string theory even though they would not be allowed in supergravity. In this talk, some recent work will be described that constructs non-geometric analogues of Calabi-Yau manifolds, which preserve the same amount of supersymmetry as Calabi-Yau spaces, but typically have far fewer light moduli and so lead to models with far fewer light particles.

Four-dimensional gauge theories with N=2 supersymmetry have a long history of being labeled by two-dimensional Riemann surfaces. The most recent incarnation of this connection, the Alday-Gaiotto-Tachikawa correspondence, relates partition functions of such theories to correlators of Toda theory, a two-dimensional conformal field theory. This correspondence, which was initially discovered through exact computations, can be understood constructively using compactifications of six-dimensional (2,0) theories. I will review this derivation, which relies on a connection to Chern-Simons theory and Drinfeld-Sokolov reduction through the 3d-3d correspondence. Based on recent work with Sam van Leuven, I will explore several features of this derivation and argue how it can be extended to include generalized Toda theories.

I will give an informal introduction to local aspects of graded symplectic manifolds, homological vector fields and derived bracket constructions. If needed, I will review basic notions of double field theory.

Graded manifolds are locally ringed spaces whose local model is an open subset of R^n together with the tensor product of smooth functions on this subset with the graded symmetric algebra of a sum of vector spaces. A toy example is the exterior algebra of a smooth manifold, where the vector space is the restriction of the cotangent bundle to a chart on the manifold. If the graded manifold in addition carries a vector field whose graded commutator with itself vanishes, it is called a Q-manifold . A crucial observation by D. Roytenberg is that a Courant algebroid can be identified with a specific Z_2-graded Q-manifold. In particular the Dorfman bracket can be recovered as a derived bracket. I will review this result in the Z-graded setting and show how it allows for a unified description of Lie algebroids, Courant algebroids and heterotic Courant algebroids. Furthermore, as a new result I will present how this language can be used to understand the gauge algebra of the closed string inspired "double field theory" as a two-term strongly homotopy Lie algebra. If time permits, I will comment on the notion of torsion and curvature in this setting.

In this talk I will discuss recent work with my supervisor Christian Saemann, where we construct a classical six-dimensional superconformal field theory containing a non-abelian tensor multiplet that satisfies many of the properties of a desired M5-brane model. I will review the underlying mathematical background of higher gauge theory before turning towards the construction of the model itself, reviewing some of its promising properties and elaborating on remaining open problems. Based on arXiv:1712.06623.

I will review the concept of duality in quantum systems from the 2D Ising model to superconformal field theories in higher dimensions. Using some of these latter theories, I will explain how a generalized concept of duality emerges: these are dualities not between full theories but between algebraically well-defined sub-sectors of strikingly different theories.

The Nahm-Schmid equations are an "indefinite" version of the Nahm equations. Their moduli spaces come equipped with a natural hypersymplectic geometry and are related to "monopoles" on the 3-dimensional Minkowski space. In the talk I will concentrate mainly on the hypersymplectic geometry of the moduli spaces and its relation to spectral curves. This is a joint work with Markus Roeser and Nuno Romao.

Symplectic implosion is an abelianisation construction for group actions on symplectic spaces. There is also an analogue in hyperkahler geometry, and both the symplectic and hyperkahler implosions can be viewed as quotients by nonreductive groups. We discuss current work (with Hanany and Kirwan) on identifying symplectic duals of hyperkahler implosions.

We construct a colored operad whose category of algebras is canonically isomorphic to the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and physically very interesting constructions, such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen's universal algebra. Also the operadic approach will turn out to be crucial to give a formulation of algebraic quantum field theory up to coherent homotopy. This is joint work with Marco Benini and Alexander Schenkel.

In its simplest form, AdS/CFT links the field theory on a stack of D3-branes to type IIB supergravity on AdS5 x S5. When the theory on the branes is supersymmetric, it can admit marginal deformations that preserve supersymmetry. Understanding the supergravity solutions that are dual to the deformed theories is a generally difficult. I will begin by defining the 5d analogue of Calabi-Yau geometry for generic D=5, N=1 supergravity backgrounds with flux using the language of generalised geometry. I will then discuss how deformations of this geometric structure reproduce the marginal deformations of the dual field theories. Finally, I will comment on how this formalism might be used to recover the corresponding supergravity solutions.

In the case where the two boundary conditions for a scalar field on AdS_d both lead to unitary CFTs on the AdS boundary, we consider the situation where both CFTs are present and separated by the corresponding renormalization group interface. We compute the scalar two-point correlation functions and the interface free energy at large N in AdS, and perform some checks on the CFT side.

Introduction to euclidean monopoles

The moduli space of (non-abelian, euclidean, SU(2)) monopoles has been of interest to mathematicians and mathematical physicists since the mid-1980s. It was proved around that time that the natural L^2 metric is hyperKaehler and complete; and its role in low-energy dynamics of monopoles was extensively discussed and analyzed. After the advent of S-duality in supersymmetric gauge theories in the 1990s, Sen made a striking conjecture about the spectrum of supersymmetric quantum states on the monopole moduli spaces. From the mathematical point of view, Sen’s conjectures are about the existence of L^2 harmonic forms on monopole moduli spaces and the analysis of this problem requires a good understanding of the monopole metric. I shall describe recent progress on this problem which will at least prove a part of Sen’s conjectures. This is joint work with Karsten Fritzsch and Chris Kottke.

This talk will consist of an account of several generalised notions of classical phase spaces (which I take to be symplectic manifolds of finite dimension) presented in the context of Dirac geometry. Lie algebroids and Courant algebroids will be introduced and the general notion of Dirac structure will be shown to be a good candidate for a generalised phase space. Then the natural notions of morphism and reduction for this structures, which directly generalise those of symplectic manifolds, will be discussed.

We will discuss some constructions of set-theoretical solutions of the Yang-Baxter equation (Yang-Baxter maps) and study the integrable aspects of the discrete systems associated with them. As an application we will consider one-dimensional elastic relativistic collisions as Yang-Baxter maps and we will investigate the integrability of the corresponding transfer maps on lattices which represent particular periodic sequences of particle collisions.

Starting from considering vortex equations in two dimensions, in particular hyperbolic and Popov vortices, we will see how to view the vortex equations as the flatness conditions for a Cartan connection. This enables us to relate the vortex equations to the three dimensional geometry of SU(1,1) and SU(2) respectively. We can then find three dimensional "vortex equations" solutions of which can be used to construct zero-modes for a Dirac operator on the three manifolds and from there magnetic zero-modes on flat three dimensional space. For the case of Popov vortices the zero-modes constructed are related to those studied by Loss and Yau.

Introduction to the (analytic) conformal bootstrap

The holographic principle is believed to be a key element in the search for a fundamental quantum gravitational theory. It found a concrete realization in the AdS/CFT correspondence but in principle, neither AdS spacetimes nor CFTs are necessary for the holographic principle to be true and it is interesting to ask if there are more general lessons for quantum gravity. One promising testing ground for a more general correspondence is three-dimensional (higher spin) gravity which permits a Chern-Simons description. I will show how non-AdS theories can be constructed and why double extensions play a fundamental role. Higher-spin theories will then be reviewed and a classification of kinematical higher-spin theories will be discussed.

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large $s$, $s$ being the charge of the operators under rotations in the space transverse to the defect. Our tools include a formula that inverts the bulk to defect OPE and is analytic in $s$, analogous to the Caron-Huot formula for the four-point function. Some important assumptions are made in deriving this result: we comment on them.

Deformation quantization is a formulation of the relationship between classical and quantum mechanics. The idea is that the quantization of a classical system corresponds to a noncommutative deformation of its algebra of observables that, at leading order, recovers the classical Poisson bracket. In 1997, Kontsevich proved a deep result known as the "formality theorem" which gives an explicit formula for the quantization of any classical phase space (Poisson manifold). Remarkably, while the formula is about quantum mechanics, it is can be derived as a perturbative expansion in a certain topological string theory. I will give an introduction to this formula and describe the few cases in which it can actually be computed directly.

I will describe forthcoming joint work with Banks and Panzer, in which we prove that the integrals appearing in Kontsevich's perturbative quantization formula can be expressed in terms of simpler transcendental constants, called multiple zeta values, which generalize the special values of the Riemann zeta function. The proof involves the introduction of a new algebra of "single-valued multiple polylogarithms" on the moduli space of marked disks, extending work of Brown on the moduli space of genus zero curves. It gives an effective algorithm for calculating the integrals, and allows to calculate deformation quantizations on a computer for the first time.

The auxiliary linear problem (ALP) is a widely used construction in classical integrable systems, which consists of a system of overdetermined equations between the Lax pair and some auxiliary vector. For quantum integrable systems, however, only the spatial half is ever considered. Therefore, in this talk I will discuss a method for constructing the temporal half of the ALP, in order to reinforce the connections between classical and quantum integrable systems. Then, in order to actually make use of this novel construction, I will discuss the construction of a quantum Bäcklund transformation, using the q-oscillator model as an example. This is based off of work done in: arXiv:1706.06052.

I will present recent results in the understanding of the theories emerging at the boundary of three dimensional quantum gravity when the induced quantum metric is fixed on the boundary. This is done in a fully non perturbative context, where the gravitational degrees of freedom are fully quantized. According to “how quantum” the boundary metric is, different boundary theories will be seen emerging. I will focus on two extreme cases, those presently best understood: for deeply quantum boundary metrics a duality with an isotropic Heisenberg chain emerges; at the other end of the spectrum, we study a semiclassical state on the torus, and show how the bulk classical geometry is reconstructed from the boundary theory and how the partition function reproduces an appropriately regularized version of the BMS3 character formula.

The monopole formula provides the Hilbert series of the Coulomb branch for a 3-dimensional N = 4 gauge theory. In this talk, I will discuss how the two geometric notions "fan" and "monoid" can be very fruitful for the understanding of the monopole formula. After a brief reminder of the monopole formula, I will introduce the matter fan and reorganise the monopole formula accordingly. I then discuss the resulting benefits such as: (1) Explicit expressions for the Hilbert series for any gauge group. (2) Proof that the order of the pole at t=1 and t → ∞ equals the complex or quaternionic dimension of the Coulomb branch. (3) Identification of a sufficient set of chiral ring generators.

Introduction to generalised geometry in supergravity

Generalised geometry is an extension of (pseudo-)Riemannian geometry which provides an elegant geometrical description of the supergravity theories underlying string theory and M theory. After a brief introduction to these topics, I will describe recent developments showing that general supersymmetric Minkowski flux backgrounds can be described as the analogue of special holonomy manifolds in this new geometry. The Killing superalgebra, which has a neat manifestation in this language, plays a key role in the proof of the result for N > 2 supersymmetry and we are able to fix its form explicitly using this technology. I will also discuss the corresponding picture for supersymmetric AdS solutions.

We will discuss the interplay between the String Theory transformation known as non-Abelian T-duality and the AdS/CFT correspondence. We will review various examples that illustrate that non-Abelian T-duality changes the dual CFT of a given AdS space, as opposed to the action of its Abelian counterpart. This opens a new way for the holographic study of CFTs whose gravity duals were previously unknown.

Introduction to non-Lorentzian Geometry

Although the study of BPS monopoles is now over 30 years old there are still few analytic results known for the Higgs and gauge fields. For su(2) monopoles without spherical or axial symmetry the only known results are for the Higgs field on a coordinate axis for charge 2. By combining integrable systems and twistor constructions we show how the problem becomes algebraic and one may explicitly construct the fields from the spectral curve. In pursuing this programme a number of new results have been found. The approach will be illustrated by presenting the general charge 2 fields and a ‘movie’.

Boundary geometries of spacetimes such as asymptotically flat or Lifshitz spaces are not described by ordinary pseudo-Riemannian geometry. Instead the geometry is what we call a non-Lorentzian geometry. Such geometric structures therefore play an important role in non-AdS holography. A relatively well known example of such a geometry is called Newton-Cartan geometry. This and related geometric structures appear in many other places in theoretical physics as well, e.g. in field theories relevant for condensed matter physics, non-relativistic limits of string theory, Horava-Lifshitz gravity, etc. In this talk I will introduce various types of non-Lorentzian geometries and give an overview of their role in holographic dualities.