From March 2023 to February 2025, my work at Wrocław University was supported by a Polonez Bis grant, project No. 2022/45/P/ST2/03995 co-funded by the Polish National Science Centre and the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 945339.

This project investigated 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 lay people

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. 

Duality and reductionism

Physics is the science of mathematical descriptions of physical phenomena. Progress happens when multiple phenomena can be described by a common mathematical description, typically called theory, and this predicts new phenomena that can be observed. The paradigm, on which this progress is based, is sometimes called reductionism — more and more phenomena are described by a reduced amount of theory (mathematical description). A popular example for this paradigm is the quest for a grand unified theory that describes both phenomena of particle physics (handled by quantum theory) and gravity (handled by the theory of general relativity). There have been simpler examples before in history of science: the unification of electric and magnetic phenomena into electromagnetism, for example. Intuitively, the progress of fundamental physics is such that theories have a certain range of validity (for example up to a certain length scale) — a more fundamental theory is then one that can describe physics at an even smaller length scale.

Many approaches to quantum theory of gravity lead to a problem with that intuition: space and time do not make sense up to arbitrarily small length scales. Depending on the model, space and time might be discrete or there is some other mechanism, such that only at big distances it seems smooth and continuous as we know it from day-to-day experience. 

When employing string theory as a candidate for theory of quantum gravity, this problem can be understood due to the existence of dualities. A duality is the observation that a phenomenon can be understood similarly well in more than one way. An easy example is electromagnetism. In the vacuum electric and magnetic field are interchangeable — the physical phenomena, that can be described by electromagnetism, stay the same when the fields switch roles, they are simply described in another way. The so-called T-duality in string theory is mathematically very similar to the electric-magnetic duality but seems physically very different: it roughly states that string theory in a space with length scale R is equivalent to string theory in a space with length scale 1/R. So, a string theory with a small length scale (a very fundamental one in the nomenclature from above) is equivalent to a string theory with a big length scale (a not so fundamental one). The possible interpretation is that string theory implies a natural minimal distance. Such dualities challenge the paradigm, discussed at the beginning: there might be multiple equivalent mathematical descriptions and the quest for a more fundamental theory (reductionism) might make no sense anymore. Hence, dualities hint at the fact, that the correct map between objects in the physical world and their mathematical representation has not been found yet.

Publications under support of the POLONEZ BIS program

• Mattia Cesaro, David Osten: Integrable deformations of dimensionally reduced gravity (JHEP 06 (2025) 064, arXiv:2502.01750)

• Falk Hassler, David Osten & Yuho Sakatani: Duality covariant curvatures for the heterotic string (arXiv:2412.17893, accepted in JHEP)

• Mattia Cesaro, Axel Kleinschmidt, David Osten: Integrable Auxiliary Field Deformations of Coset Sigma Models (JHEP 11 (2024) 028, arXiv:2409.04523)

• Falk Hassler, Ondrej Hulik & David Osten: Current Algebra and Generalised Cartan Geometry (Phys.Rev.D 110 (2024) 12, 126022, arXiv:2409.00176)

• Eric Lescano & David Osten: Non-Relativistic Limits of Bosonic and Heterotic Double Field Theory, (JHEP 07 (2024) 286, arXiv:2405.10362)

• David Osten: On the universal exceptional structure of world-volume theories in string and M-theory (Phys.Lett.B 855 (2024) 138814, arXiv:2402.10269)

• David Osten: A heterotic integrable deformation of the principal chiral model (Phys.Rev.D 109 (2024) 10, 106021, arXiv:2312.10149)

• David Osten: On exceptional QP-manifolds, (JHEP 01 (2024) 028, arXiv:2306.11093)

Activities and Events during the POLONEZ BIS project

Research talks

• • • Nov 20th 2024: Research seminar (invited): 'The universal exceptional structure of brane sigma models in string and M-theory', NYUAD, UAE

    • Oct 17th 2024: Research seminar (invited): 'Current algebra and Generalised Cartan Geometry', Charles U., Prague, Czechia

    • Sep 21st 2024: Research talk (invited): 'Exceptional generalised geometry as a symmetry principle for sigma models' at Workshop on Noncommutative and Generalized Geometry in String theory, Gauge theory and Related Physical Models, Corfu, Greece

    • Sep 3rd 2024: Research talk (contributed): 'On classical integrabilty of the Heterotic String' at 'Eurostrings 2024 meets Fundamental Physics UK', Southampton, UK

    • Aug 29th 2024: Research seminar: (invited) 'Exceptional generalised geometry as symmetry principle for sigma models' at AEI, Potsdam, Germany

    • Jul 16th 2024: Research talk (contributed): An Integrable Deformation of the Heterotic String, 'Integrability, Duality and Deformations' conference, Swansea, UK

    • Sep 23rd 2023: Research talk (invited) 'Exceptional QP-manifolds' at Workshop on Non-commutative and generalised Geometry, Corfu, Greece


    • Sep 15th 2023: Research talk (invited) 'Tensor hierarchy and QP-manifolds' at Workshop on New Developments in Quantum Gravity, Strings and the Swampland', Corfu, Greece


    • Jun 28th 2023: Research talk (invited) 'New classically integrable sigma models based on Z(N)-symmetric homogeneous spaces',
 ETH Zurich, Switzerland

    • Mar 13th 2023: Overview talk (invited) 'Duality symmetries as a fundamental principle for sigma models', IFT Wroclaw, Poland

• From January 15th to March 15th 2024, in the context of an intersectoral secondmend, I took take part in the program Science meets Politics' of a private German Science Foundation (Wilhelm & Else Heraeus Foundation), that funds research, education and science communication in physics. This was a new program aimed at physicists holding a PhD in order to improve the interaction between science and politics. Each of participants worked in the office of a member of the German parliament, taking part in day-to-day work in the office and getting insights into the internal dynamics of the parliamentary processes. I worked in the Bundestag (German Parliament) in the office of MoP Michael Mueller (SPD)
  
  

Outreach

• • • Aug 14th - 29th 2023, together with Dr. Robert Jonsson (Nordita, Stockholm)  I have taught at Deutsche Schülerakademie (two week summer schools for high school students):
      Introduction to Quantum Information Theory

    • Public lectures:

      • 'String Theory and the Limits of Phyiscs'
        Science fair, State School St. Afra, Meissen, Germany, November 4th 2023

      • 'General Relativity: From Einstein's Ideas to Black Holes and the Expansion of the Universe'
        Saturday physics lectures, University of Wroclaw, Poland, May 18th 2024

Guests

The following guest were invited in context of this grant:

• • • Fiona Seibold (Imperial College London, UK), May 14th - 20th 2023

    • Sibylle Driezen (ETH Zurich, Switzerland), October 23rd - 27th 2023

    • Axel Kleinschmidt (Max-Planck Institute, Potsdam, Germany), June 16th - 22nd 2024

    • Sylvain Laxroix (ETH Zurich, Switzerland), June 23rd - 28th 2024

    • Mattia Cesaro (Max-Planck Institute, Potsdam, Germany), November 25th - 29th 2024

    • Anne Spiering (Humboldt University Berlin), Germany, January 24th - 28th 2025