## Research

### Summary

My research interests are at the interface of algebra and mathematical physics. More specifically, I am interested in mathematical structures arising in three-dimensional gravity, Chern-Simons theory and quantum geometry. These interest of these research subjects is that they tie together a variety of topics from algebra, geometry and topology:

**Algebra**

- Poisson-Lie groups
- Hopf algebras and quantum groups
- representation theory of Lie groups and quantum groups
- (braided) monoidal categories and higher categories
- braid groups and mapping class groups
- Fuchsian groups

**Geometry**

- Moduli spaces of flat connections
- Poisson and symplectic geometry
- Teichmüller theory
- differential geometry in three dimensions
- hyperbolic geometry in two and three dimensions

**Topology**

- link and knot invariants
- quantum invariants of manifolds contructed from representation theory
- topological quantum field theories

### Mathematical structures 3d gravity and Chern-Simons theory

**Hyperbolic geometry, Teichmüller theory, Fuchsian groups and moduli spaces of flat connections**

The classical geometry of three-dimensional spacetimes involves a wide range of topics from two- and three-dimensional differential and hyperbolic geometry. A large set of spacetimes in Lorentzian 3d gravity and Euclidean 3d gravity with negative cosmological constants are obtained as quotients of regions in 3d Minkowski, de Sitter, anti de Sitter and hyperbolic space by the action of Fuchsian groups.
For each value of the time parameter, the associated spatial surface is a Riemann surface.
The description of these Riemann surfaces and their evolution with the time parameter involves topics from the
theory of grafting and earthquakes, measured geodesic laminations and quadratic differentials.

It can be shown that the phase space of three-dimensional gravity with Lorentzian signature and the Euclidean case with negative cosmological constant are realised as a cotangent bundle of Teichmüller space, which establishes a direct link between three-dimensional gravity and Teichmüller geometry. More generally, the phase spaces arising in Chern-Simons theory are moduli spaces of flat connections on oriented two surfaces, whose geometry and Poisson structure have been investigated extensively in mathematics and physics.

**Conformal geometry and conformal field theories**

Another important feature of classical and quantised 3d gravity and Chern-Simons theory is their link with conformal geometry and conformal field theory. It has been shown that three-dimensional spacetimes with two-dimensional boundaries are naturally associated with conformal structures and conformal field theories such as WNZW models on these boundaries. In the quantisation of the theory, the presence of these boundary conformal field theories is a necessary condition that ensures the correct behaviour of the amplitudes under the gluing of manifolds.

**Braid groups and mapping class groups**

Unlike in higher dimensions, the statistics of particles in three-dimensional spacetimes is not governed by the symmetric group, but by braid groups. 3d gravity thus admits anyons - particles whose spin is not limited to integer or half-integer values but can take any real value. Such particles are present generically in Chern-Simons theory and have an application in the modelling of the fractional quantum Hall effect. More generally, the phase space of 3d gravity and the associated quantum theory have mapping class groups of oriented two surfaces as their symmetry groups. The role of these symmetries is of high relevance in classical and quantum gravity, since it is debated if they should be interpreted as gauge symmetries, which reflect a redundancy in the description of the physical states, or as physical symmetries of the theory.

**Poisson-Lie groups and quantum groups**

3d gravity and Chern-Simons theory are naturally related to the theory of Poisson-Lie groups and their quantum counterparts, quantum groups. It has been shown that the Poisson structure on the phase space of Chern-Simons theory, the moduli space of flat connections on a punctured Riemann surface, can be related to certain Poisson structures from the theory of Poisson-Lie groups: a copy of the dual Poisson-Lie structure associated with each puncture and a copy of the Heisenberg double Poisson structure for each handle.

In the quantisation of the theory, these Poisson-Lie structures give rise to quantum group symmetries, which are present in Hamiltonian quantisation approaches such as combinatorial quantisation, state sum models (spin foams) as well as some other approaches to the quantisation of the theory. In particular, one finds that the implementation of the constraints in the quantum theory can be formulated in terms of the representation theory of these quantum groups. Of particular interest in this context is the representation theory of non-compact quantum groups, which is much less understood than the compact case. It is also of particular relevance to physics since all quantum groups arising in 3d gravity with Lorentzian signature are non-compact.

**Knot and link invariants, manifold invariants and topological quantum field theories**

The relation between Chern-Simons theory and the theory of knot invariants was first established by Witten's discovery of the link between Chern-Simons theory and the Jones polynomial. Since then, it has been found that knot, link and manifold invariants arise canonically in 3d gravity and Chern-Simons theory and play an important role in the quantisation of the theory. State sum models for the quantisation of Chern-Simons theory and 3d gravity such as the Reshitikhin-Turaev invariant and the Turaev-Viro invariant give rise to knot and link invariants and quantum invariants of three-manifolds. They place the quantisation of 3d gravity and Chern-Simons theory in the context of topological quantum field theory.

A recent development in this subject is a more abstract formulation of these invariants in terms of categories, which generalise the representation categories of the concrete quantum groups arising in these models. A particular advantage of this approach is that it relates the inclusion of particles and matter into these models to the theory of higher categories.

### 3d gravity and Chern-Simons theory in physics: quantum gravity and quantum geometry

**3d gravity as a model in quantum gravity**

Gravity is the only one of the four fundamental interactions for which quantum theory could not be
constructed so far. This is due to the presence of fundamental problems which cannot be addressed
within existing quantisation approaches. This includes the construction of a quantum theory without
relying on a fixed background spacetime - in quantum gravity it is time and space themselves that are
to be quantised - the role of time in the quantum theory, the question if time and space are continuous
or discrete and the emergence of gravity at low energies.

While a full quantum theory of four-dimensional gravity is currently not available, 3d gravity (one time, two space dimensions) serves as a toy model which allows one to address these conceptual questions in a fully quantised theory and to gain insights relevant to higher dimensions. In three dimensions, Einstein's theory of gravity simplifies dramatically. The theory has no local gravitational degrees of freedom: Any vacuum solution of Einstein's equations of motion is locally isometric to three-dimensional Minkowski space, de Sitter space or anti de Sitter space. However, the theory has global degrees of freedom that arise for spacetimes of non-trivial topology or with matter. As the phase space of the theory is finite dimensional, the quantisation of the theory becomes more tractable, and important progress has been made in the quantisation of the theory.

However, despite these simplifications, the theory remains far from trivial. It presents the same conceptual challenges as in higher dimensions as well as other physically interesting features such as black hole solutions, interacting point particles and vacuum spacetimes with a rich geometry. The theory can thus serve as a toy model which allows one to investigate the fundamental challenges of quantum gravity in a mathematically simpler theory and to identify promising quantisation approaches for higher dimensions.

**Quantising 3d gravity**

Since its beginnings, the subject of 3d gravity has undergone rapid development and attracted the
attention of many mathematicians and physicists. As a consequence, a thorough understanding
has been achieved on the classical level. This includes the discovery that the theory can be
formulated as a Chern-Simons gauge theory or BF theory, a (near complete) classification of
vacuum solutions of the three-dimensional Einstein equations as well as the parametrisation of
its phase space and Poisson structure for spacetimes of general genus and with a general number
of massive particles. There is also a large body of work on specific aspects of the theory such
as boundary conditions, black hole solutions and multi-particle systems.

Moreover, important progress has been made in the quantisation of the theory in a multitude of quantisation approaches. Like in higher dimensions, two of the most prominent ones are Hamiltonian quantisation formalisms and state sum or spin foam models, which follow a very different quantisation philosophy. Hamiltonian quantisation formalisms such as the combinatorial quantisation formalism based on the formulation of the theory as a Chern-Simons gauge theory work with a (2+1)-decomposition of a fixed manifold, an explicit description of phase space and Poisson structure and aim to quantise the theory via Dirac's quantisation formalism for constrained systems.

Spin foam or state sum models follow a path integral approach to quantisation based on a discretisation of the underlying manifold usually via triangulations or cellular decompositions. They resemble models from lattice gauge theory and have a direct relation to the theory of manifold invariants and topological quantum field theory. Unlike the Hamiltonian formalisms, they also include the possibility of topology change. The two most prominent and mathematically tractable spin foam models are the Ponzano-Regge model and the Turaev-Viro model for Euclidean 3d gravity with, respectively, vanishing and positive cosmological constant.

Major progress in the classical description of 3d gravity and its quantisation was triggered by the discovery that the theory can be formulated as a Chern-Simons gauge theory. This related the phase space of the theory to moduli spaces of flat connections on two-dimensional surfaces, whose Poisson structure and geometry has been investigated extensively. Moreover, it related 3d gravity to the theory of link, knot and manifold invariants and allowed one to apply gauge theoretical concepts and methods to the quantisation of the theory. An example of this are Reshetikhin-Turaev invariants, which provide a quantisation of Chern-Simons theories with compact gauge groups. Another important quantisation approach based on quantum groups is the combinatorial quantisation formalism for Chern-Simons gauge theory, which has been applied successfully also to Chern-Simons theories with non-compact gauge groups. In the application of 3d gravity, this includes Lorentzian 3d gravity with positive cosmological constant, Euclidean 3d gravity with negative cosmological constant as well as Lorentzian and Euclidean 3d gravity with vanishing cosmological constant.

** Chern-Simons theory and black holes in 4d quantum gravity**

Another important application of Chern-Simons theory is its role in the description of black holes in four-dimensional quantum gravity. It has been shown that the quantum gravitational degrees of freedom on the boundary of a black hole can be modelled in terms of a quantised SU(2)-Chern-Simons theory. This model is used in attempts to derive a formula for the black hole entropy in terms of microscopic quantum gravitational states at its horizon. The aim is to give a quantum gravitational derivation of the famous relation between the area and entropy of a black hole in black hole thermodynamics.