# Higher Structures, Gravity and Fields

Europe/Berlin
02.430 (Mainz Institute for Theoretical Physics, Johannes Gutenberg University)

### 02.430

#### Mainz Institute for Theoretical Physics, Johannes Gutenberg University

Staudingerweg 9 / 2nd floor, 55128 Mainz
Description

Many physical models possess symmetries that cannot be described by Lie algebras, but require some kind of higher structure. This applies already to p-form gauge fields, described by gerbes rather than fibre bundles. An example under active research is generalised or extended geometry — models capturing duality symmetries of gravity or supergravity, often inherited from string theory, and described in terms of differential geometry of L∞-algebras, Courant and Leibniz algebroids, QP manifolds, differential graded algebras, tensor hierarchy algebras, etc.

We organise a 3-week programme bringing together theoretical physicists and mathematicians working in the areas of gravity, dualities, extended geometry, field theory, higher homotopy structures, and other infinite-dimensional superalgebras. The overarching goal is to make progress in constructing “exotic” field theories, where conventional notions of spacetime break down, and thus traditional methods fail. These include exotic forms of gravity, extended (double and exceptional) geometry, string/M-theory and its dualities, and higher spin theories.

The common feature of these theories is the appearance of higher symmetry algebras, generalising the notion of Lie algebras. In this programme, we will bring physicists and mathematicians together working on these “higher structures” in order to better understand the fundamental structures of physical theories and develop a bootstrap based on higher structures.

Keywords:

· gravity, including non-standard versions
· string theory / M-theory
· dualities
· extended and graded geometry
· gauge theory
· higher spin theory
· higher homotopy structures
· infinite-dimensional superalgebras

Contact @ MITP : Dominika Kosler
• Monday, January 9
• 9:00 AM 10:00 AM
registration 1h
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Geometric actions in gravity 1h
Speaker: Glenn Barnich
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Asymptotic structure of gravity with Ehlers symmetry 1h
Speaker: Sucheta Majumdar
• 6:00 PM 7:00 PM
Reception: drinks and snacks at the institute 1h
• Tuesday, January 10
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Higher-spins from higher dualisations 1h
Speaker: Nicolas Boulanger
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Brane wrapping in AKSZ topological field theories 1h
Speaker: David Tennyson
• Wednesday, January 11
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Quest for Background Independence 1h
Speaker: Ivo Sachs
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Higher-derivative corrections and duality invariance 1h
Speaker: Camille Eloy
• 7:00 PM 9:00 PM
workshop dinner 2h
• Thursday, January 12
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Double Field Theory as the Double Copy of Yang-Mills Theory 1h
Speaker: Olaf Hohm
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Supergravity and Supergeometry 1h
Speaker: Pietro Grassi
• Friday, January 13
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles 1h
Speaker: Maxim Grigoriev
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Consistent Truncations and Dualities 1h

A major application of extended geometries in supergravity is
the construction of consistent truncations. Remarkably, all this
construction's ingredients also appear in the context of generalised T-
and U-dualities. After a quick review of both concepts, I will discuss a
conjectured correspondence between consistent truncations and
generalised dualities in string and M-theory. We will prove that all
known generalised dualities give rise to consistent truncations.
Moreover, we will see evidence that after incorporating higher
derivative correction, all consistent truncations might be based on
generalised dualities.

Speaker: Falk Hassler
• Monday, January 16
• 9:00 AM 10:00 AM
registration 1h
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Spinors and geometric structures 1h

My aim will be to explain some relatively well-known (and then less-known) geometric constructions linking structures of the type appearing in double field theory to spinors. The well-known example is that of a split signature metric on a vector space together with a choice of a maximal totally null subspace as encoded by a real pure spinor. But I will also describe less-known examples when the metric together with (a unit) spinor is encoded by a collection of certain differential forms. I will describe how going up in dimension necessitates considering impure spinors, and how these bring with themselves even more interesting geometry. My plan is to end with an example in 14 dimensions, with a split signature metric, where a generic real spinor can be shown to encode the second (dynamical) metric of double field theory.

I hope to be able to make this talk interesting to anyone familiar with generalised geometry and/or double field theory.

Speaker: Kirill Krasnov
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
On the backgrounds of spinning particles 1h

Spinning particles are models for point particles with spin degrees of freedom. They automatically feature N-supersymmetry on the worldline, with fixed albeit arbitrary N. After first quantization one obtains spin-N/2 states in the Hilbert space. Consistency conditions for second quantization, realized in the form of BRST quantization, can severely restrict the background fields in target space, putting them on-shell. In the N=2 case the target space can host a Yang-Mills gauge theory, while in N=4 it is possible to have Einstein's gravity and the NS-NS sector of Supergravity.
After explaining these passages, I will present new results involving the Ramond-Ramond fluxes (based on arXiv:2206.03243 with I. Sachs). I wish to conclude this analysis with some brief comments on almost complex structures (on-going work with O. Hulik and I. Sachs).

Speaker: Eugenia Boffo
• 6:00 PM 7:00 PM
Reception: drinks and snacks at the institute 1h
• Tuesday, January 17
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Conformal (higher spin) gravity and Deformation quantization 1h
Speaker: Evgeny Skvortsov
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Post-Newtonian Test of Double Field Theory 1h
Speaker: Jeong-Hyuck Park
• Wednesday, January 18
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
From Lie algebra crossed modules to tensor hierarchies, and beyond 1h

Gauging procedures in supergravity theories depart from classical gauge theories as in the former, the gauge fields take values in the fundamental representation V of the Lie algebra g of global symmetries of the system. The consistency of the theory relies on a pairing V-->g called the embedding tensor, turning V into a Leibniz algebra. As is usually met in higher gauge theories, if the gauge algebra is not Lie, it is replaced by some higher form of Lie algebras. Here, such a higher structure is materialized by a differential graded Lie algebra on a chain complex of g-modules, called the tensor hierarchy. In the present talk we explain how tensor hierarchies are genetically related to Lie algebra crossed modules.
Indeed, two such algebras V and g, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. The canonical assignment (functor) associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra can be extended to the category of Lie-Leibniz triples, giving their associated tensor hierarchies. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of 'lie-ization' of the former. The "oidization" of such Lie-Leibniz triples then conjecturally opens the possibility to define the tensor hierarchies associated to Courant algebroids and G-algebroids.

Speaker: Sylvain Lavau
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Carrollian and Galilean conformal spin-3 theories in 3d 1h
Speaker: Iva Lovrekovic
• Thursday, January 19
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
D=2 supergravity and affine exceptional field theory 1h
Speaker: Franz Ciceri
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
2d gauged supergravity and consistent truncations 1h
Speaker: Gianluca Inverso
• 7:00 PM 10:00 PM
workshop dinner 3h
• Friday, January 20
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
The most complicated way of writing D=11 supergravity 1h
Speaker: Axel Kleinschmidt
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Arborescent Koszul-Tate resolutions and BFV for singular coisotropic reductions 1h

Let $I$ be an ideal in some commutative (associative) algebra $O$. Starting from resolution of $O/I$ as an $O$-module, we construct a Koszul-Tate resolution for this quotient, i.e.\ a graded symmetric algebra over $O$ with a differential which provides simultaneously a resolution as an $O$-module. This algebra resolution has a beautiful structure of a forest of decorated trees and is related to an $A_\infty$ algebra on the original module resolution.

Considering $O$ to be a Poisson algebra and $I$ a finitely generated Poisson subalgebra, we use the above construction to obtain the corresponding BFV formulation. Its cohomology at degree zero is proven to coincide with the reduced Poisson algebra $N(I)/I$, where $N(I)$ is the normaliser of $I$ inside $O$, thus generalising ordinary coisotropic reduction to the singular setting. As an illustration we use the example where $O$ consists of functions on $T^*(\R^3)$ and $I$ is the ideal generated by angular momenta.

This is joint work with Aliaksandr Hancharuk and, in part, with Camille Laurent-Gengoux.

Speaker: Thomas Strobl
• Monday, January 23
• 9:00 AM 10:00 AM
registration 1h
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Extended geometry and restricted associativity 1h
Speaker: Jakob Palmqvist
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Algebraic structures in the pure spinor superfield formalism 1h
Speaker: Ingmar Saberi
• 6:00 PM 7:00 PM
Reception: drinks and snacks at the institute 1h
• Tuesday, January 24
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
A new consistent limit of 11D supergravity 1h
Speaker: Eric Bergshoeff
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Systematics of consistent truncations 1h
Speaker: Michela Petrini
• Wednesday, January 25
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Counting moduli of flux backgrounds 1h
Speaker: Daniel Waldram
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
Quantum space-time, torsion and gravity from quantum effects in the IIB matrix model 1h
Speaker: Harold Steinacker
• 8:30 PM 11:30 PM
workshop dinner 3h
• Thursday, January 26
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Gauge theory, higher structures and compatibility conditions 1h
Speaker: Athanasios Chatzistavrakidis
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
G-algebroids, embedding tensors, and Poisson–Lie duality 1h

I will describe a class of structures, known as G-algebroids, which arises as a natural generalisation of the ordinary, generalised, and exceptional tangent bundles from ordinary, generalised, and exceptional geometry, respectively. I will discuss a classification result in the exceptional case and show how to obtain the possible fluxes/twists of the bracket. In the M-theory and IIB setups, the twists organise themselves naturally into connections and covariantly constant differential forms, while in the IIA case one in particular recovers both the Romans mass and the deformation of Howe–Lambert–West. Finally, I will show how to use these algebroids to answer a question about the realisability of embedding tensors (providing a new perspective on the result of Inverso '17) and to give a joint description of the Poisson–Lie T- and U-duality. This is a joint work with M. Bugden, O. Hulik, and D. Waldram.

Speaker: Fridrich Valach
• Friday, January 27
• 10:00 AM 10:30 AM
Coffee break 30m
• 10:30 AM 11:30 AM
Quantization of braided noncommutative field theories 1h

In this talk we will shortly review the construction of noncommutative field theories via the (braided) L-infinity algebra. Then we will discuss quantization of these theories on two examples: braided scalar field theory and braided U(1) gauge theory coupled with fermions, braided electrodynamics. In the case of scalar field theory we show the absence of UV/IR mixing at one loop order, while in the braided electrodynamics the UV/IR mixing remains present at one loop order.

Speaker: Marija Dimitrijević Ćirić
• 3:00 PM 3:30 PM
Coffee break 30m
• 3:30 PM 4:30 PM
D-branes and doubled geometry 1h
Speaker: Richard Szabo