My research interests revolve around the field of non-Lorentzian string theory and the exotic geometric structures occurring in this context. Naturally, they involve subjects as string theory, supergravity, supersymmetric field theories, and dualities.

Non-Lorentzian string theory was pioneered by Jaume Gomis and Hirosi Ooguri in [1]. This theory is unitary, UV complete, has non-relativistic dispersion rela- tions, and can thus be studied as an independent quantum theory with a non-trivial spectrum of winding modes. What is more, it describes the gravitational interac- tion between these states. In this sense, the Gomis-Ooguri string theory can be regarded as a UV completion of non-relativistic gravity—and thus as an example of non-relativistic quantum gravity. I firmly believe that studying these theories is fascinating both in its own right and from the point of view of relativistic quan- tum gravity. The idea is to isolate non-trivial properties of quantum gravity that are independent of Lorentzian spacetime symmetries. This is best illustrated by the Bronstein cube:

To describe the gravitational sector of the Gomis-Ooguri theory, it is essential to understand the structure of the background geometry to which the string cou- ples. Remarkably, this non-Riemannian geometry has only been understood rather recently in [2]. It is an example of so-called generalized Newton-Cartan structures that have proven to be of interest in studying strong gravity corrections to Newtonian gravity, transport properties of condensed matter systems, non-Lorentzian hologra- phy, and many more. A characteristic feature of Newton-Cartan structures is the occurrence of foliated tangent bundles.

Some of the elegance of (relativistic) string theory comes from the intricate in- terplay of worldsheet and target space dynamics. Cancellation of the worldsheet Weyl anomaly leads to target space beta function constraints. At the same time, these equations can be seen as the Euler-Lagrange equations of a low-energy effec- tive supergravity action. This effective description captures many of the surprising features of the underlying UV description. Moreover, it allows us to access certain non-perturbative aspects of string theory.

In the last years, several research groups [3, 4] have studied the analogous Weyl anomaly occurring in non-Lorentzian string theory. Cancellation of that anomaly imposes differential constraints on the background geometry. These beta functions can be interpreted as the equations of motion of the gravity sector of the theory. In our work [5], we have established a connection of these beta function constraints to an effective NS-NS gravity description.

What is more, we have extended the bosonic effective gravity description to include target-space supersymmetry—in other words, we have established the first non-Lorentzian supergravity multiplet in ten dimensions [6]. This theory is conjec- tured to capture the target space dynamics of non-Lorentzian superstring theory. The effective description shares some features with relativistic supergravity. How- ever, there are two substantial differences: firstly, the smallest multiplet is smaller than the analogous relativistic one. Secondly, the theory requires a certain bosonic a priori constraint on the internal torsion of the geometry for consistency. Our work is only a first step in a larger effort to understand the role of supersymmetry in non- Lorentzian string theories. However, it paves the way for several natural and exciting applications and extensions that I wish to develop in the future. I will describe some of these in the following.

Firstly, it is very tempting to shed light on the space of possible non-Lorentzian superstring theories and supergravities in ten dimensions. Relatedly it will be fasci- nating to study and explore dualities between these theories. The most feasible of which is T-duality. It is known [2] that the notion of T-duality is richer in the context of non-Lorentzian theories. Since the tangent space is naturally foliated, one can un- ambiguously distinguish between T-duality longitudinal and transverse to the string. The latter relates different non-Lorentzian string theories to each other. The former, however, is more unusual in that it relates non-Lorentzian string theory to the DLCQ of relativistic string theory. In fact, this means one can use the Gomis-Ooguri string theory to define the DLCQ of relativistic string theory from first principles. In going forward, I hope to improve the understanding of non-perturbative S- and U-duality. For this, it is essential to build on the results I have established to get a complete understanding of the spectrum of non-Lorentzian supergravities first.

Apart from the above extensions, one can also go further and look for applications in non-Lorentzian holography. There is a large body of work on Lifshitz holography [7] where Newton-Cartan geometries appear naturally in the boundary geometry. It would be fascinating to see whether an alternative version of non-Lorentzian holog- raphy exists that has no a priori relation to relativistic physics. That is, a duality between non-Lorentzian string theory and a non-Lorentzian superconformal CFT. Given my recent work, it seems within reach to go through a decoupling argument analogous to Maldacena’s original derivation of the AdS/CFT correspondence.

Relatedly, I am convinced that it is vital to improve the understanding of dy- namical and non-perturbative aspects of non-Lorentzian field theories. After all, we expect that non-Lorentzian holography establishes a correspondence between non- Lorentzian gravity and the strong-coupling regime of some SCFT. In the original setup, people have managed to test the AdS/CFT correspondence quantitatively by comparing to results from supersymmetric localization [8]. The spectacular results obtained by this technique have motivated a body of work trying to classify manifolds that allow for supersymmetry. In [9], we have adapted well-established techniques for Lorentzian and euclidean spaces [10] to Newton-Cartan structures in three di- mensions. In effect, we have thereby also clarified the structure of non-Lorentzian supercurrent multiplets. Our results open the door for studies of non-perturbative aspects of non-Lorentzian field theories to be applied—among other things—in holography.

To conclude, I want to mention that beyond the topics mentioned above, I am also interested in topics in pure mathematics such as vector bundles, symplectic geometry, category theory, and knot theory. Furthermore, during my undergraduate studies in Vienna, I had the opportunity to follow excellent courses on quantum information, quantum computation, and quantum optics. I maintain an interest in these topics and follow current developments and relations to holography and string theory.

References

[1] J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181].

[2] E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-Duality, JHEP 11 (2018) 133 [1806.06071].

[3] J. Gomis, J. Oh and Z. Yan, Nonrelativistic String Theory in Background Fields, JHEP 10 (2019) 101 [1905.07315].

[4] A. D. Gallegos, U. Gürsoy and N. Zinnato, Torsional Newton Cartan gravity from non-relativistic strings, JHEP 09 (2020) 172 [1906.01607].

[5] E. A. Bergshoeff, J. Lahnsteiner, L. Romano, J. Rosseel and C. Şimşek, A non-relativistic limit of NS-NS gravity, JHEP 06 (2021) 021 [2102.06974].

[6] E. A. Bergshoeff, J. Lahnsteiner, L. Romano, J. Rosseel and C. Simsek, Non-Relativistic Ten-Dimensional Minimal Supergravity, 2107.14636.

[7] M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [1512.03554].

[8] P. Benetti Genolini, Precision holography and supersymmetric theories on curved

spaces, Ph.D. thesis, Oxford U., Inst. Math., 2018.

[9] E. Bergshoeff, A. Chatzistavrakidis, J. Lahnsteiner, L. Romano and J. Rosseel, Non-Relativistic Supersymmetry on Curved Three-Manifolds, JHEP 07 (2020) 175 [2005.09001].

[10] G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [1105.0689].

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