My research investigates the role of dualities and the consequences of their existence at the example of sigma models, a broad class of theories that is used in particle physics, condensed matter physics but also as the fundamental description of objects in string theory. 

Duality Symmetry as a Fundamental Organising Principle for σ-models

σ-models, prominent examples of quantum field theories, appear ubiquitously in many branches of physics — from particle physics, over statistical physics and condensed matter to gravity. They are also the fundamental description of objects in string and M-theory. Latter are subject to an intricate web of dualities, of which some are better understood at the level of the σ-model – like T-duality, that is already realised as a canonical transformation at the classical level – whereas some are not as well understood – like S- or U-dualities, that are non-perturbative mappings and even relate σ-models with different dimensionalities.

The broad applicability of σ-models in such different areas of physics originates in their quite generic definition. They are theories of embeddings of one manifold into another one, the so-called target space. As a consequence, a feature that all σ-model share — sometimes dubbed geometric paradigm — is that physical properties of these models correspond to geometric properties of the target space. The central aim of this program is to put these two topics together and propose new geometric approach to the study of σ-models, based on so-called generalised geometry. For typical examples of the geometric paradigm, standard Riemannian geometry was sufficient. Instead, the central goal of this program is to establish the following answer to these questions: σ-models have a natural Hamiltonian description in terms of generalised geometry – a generalisation of Riemannian geometry that geometrises certain underlying groups, here typically O(d, d) or Ed(d) (the T- or U-duality groups of string or M-theory). Schematically, the mapping between geometric data and the Hamiltonian formulation looks as follows:

Hamiltonian ~ generalised metric
Poisson structure, constraints ~ duality group invariants

There are two key advantages of this ’duality symmetry approach’ to σ-models. It generalises to setups, in which the σ-model couples to higher form gauge fields in addition to a Riemannian metric. In addition, the duality symmetry underlying the description via generalised geometry motivate certain non-trivial connections between different σ-models that are fairly unexplored for world-volumes with more than than two dimensions. Also, rather unexpectedly, it gives a new perspective on the question, which objects are allowed as fundamental objects of string theory — the so-called brane scan. So the duality symmetry, might be an alternative fundamental organising principle to supersymmetry or conformal invariance that have been central in the progress of string theory so far.

Constructions, based on duality symmetries respectively generalised geometry, of classical σ-models in string and M-theory for target spaces with up to six dimensions have been established in the literature, also involving recent work of mine. In the end the aim is to understand how broadly applicable this established approach is, also outside string theory, and to attack canonical quantisation. Open conceptual questions of this unexplored field are plentiful, like generalisation of known constructions to higher target space dimensions or relations to graded symplectic geometry, i.e. realisations of the target spaces as QP-manifolds and topological parts of the σ-models as AKSZ σ-models. Moreover, this program aims to understand, whether and how these string theory motivated approaches are applicable and insightful for σ-models outside of string theory. Examples for σ-models with similar properties are those used in the description of magnetic skyrmions or those in the relatively new field of non-relativistic gravity, that tries to approach quantum gravity via a quantisation of Newtonian gravity first. Besides unraveling conceptual properties of σ-models, it is expected that the generalised geometry approach to σ-models applies to otherwise difficultly accessible situations, as for examples in co-called non-geometric backgrounds, or for which the generalised geometry is particularly feasible due to the unified treatment of all backgrounds fields.

For Laypersons

Dealing with dualities has been central in the history of physics. Before a consistent description of quantum mechanics was found, there was a a long debate about the duality between wave and particle interpretations of photons and electrons. My research investigates the role of dualities and the consequences of their existence at the example of σ-models, a broad class of theories that is used in particle physics, condensed matter physics but also as the fundamental description of objects in string theory.