The motivic coaction of multiple zeta values and multiple polylogarithms encodes both structural insights on and computational methods for scattering amplitudes in a variety of quantum field theories and in string theory. In this work, we propose coaction formulae for iterated integrals over holomorphic Eisenstein series that arise from configuration-space integrals at genus one. Our proposal is motivated by formal similarities between the motivic coaction and the single-valued map of multiple polylogarithms at genus zero that are exposed in their recent reformulations via zeta generators. The genus-one coaction of this work is then proposed by analogies with the construction of single-valued iterated Eisenstein integrals via zeta generators at genus one. We show that our proposal exhibits the expected properties of a coaction and deduce $\textit{f}$-alphabet decompositions of the multiple modular values obtained from regularized limits.

Germany

Belgium

Axel Kleinschmidt

Sweden

Franziska Porkert

Oliver Schlotterer

Towards motivic coactions at genus one from zeta generators

We propose a novel approach to study hyperbolic Kac-Moody algebras, and more specifically, the Feingold-Frenkel algebra 𝔉, which is based on considering the tensor algebra of level-one states before descending to the Lie algebra by converting tensor products into multiple commutators. This method enables us to exploit the presence of mutually commuting coset Virasoro algebras, whose number grows without bound with increasing affine level. We present the complete decomposition of the tensor algebra under the affine and coset Virasoro symmetries for all levels $\textit{ℓ}$ ≤ 5, as well as the maximal tensor ground states from which all elements of 𝔉 up to level five can be (redundantly) generated by the joint action of the affine and coset Virasoro generators, and subsequent conversion to multi-commutators, which are then expressed in terms of transversal and longitudinal DDF states. We comment on the deep relations between the algebra 𝔉 and Einstein gravity in four space-time dimensions, and outline novel directions for future work.

Switzerland

Hannes Malcha

Hermann Nicolai

From tensor algebras to hyperbolic Kac-Moody algebras

We revisit six-dimensional (1, 0) supergravity coupled to tensor multiplets and Yang-Mills fields for > 1 for which no covariant action exists. We construct the action in the Henneaux-Teitelboim approach and in the presence of a gauge anomaly. We moreover obtain the supersymmetric Green-Schwarz counterterm for the gravitational anomaly for arbitrary matter content.

France

Guillaume Bossard

Austria

United States

Ergin Sezgin

Higher derivative couplings with multi-tensor multiplets in 6D supergravity, action and anomalies

We consider the dimensional reduction of $\textit{N}$ = (2, 0) conformal supergravity in six dimensions on a two-torus to $\textit{N}$ = 4 conformal supergravity in four dimensions. At the level of kinematics, the six-dimensional Weyl multiplet is shown to reduce to a mixture of the $\textit{N}$ = 4 Weyl and vector multiplets, which can be reinterpreted as a new off-shell multiplet of $\textit{N}$ = 4 conformal supergravity. Similar multiplets have been constructed in other settings and are referred to as dilaton Weyl multiplets. We derive it here for the first time in a maximally supersymmetric context in four dimensions. Furthermore, we present the non-linear relations between all the six- and four-dimensional bosonic and fermionic fields, that are obtained by comparing the off-shell supersymmetry transformation rules.

Franz Ciceri

India

United Kingdom

Subrabalan Murugesan

Bindusar Sahoo

Torus reduction of maximal conformal supergravity

We prove the existence of a family of integrable deformations of $\textit{ℤ}$ -coset models in two dimensions. Our approach uses and generalises the method of auxiliary fields that was recently introduced for the principal chiral model by Ferko and Smith.

Mattia Cesàro

Poland

David Osten

Integrable auxiliary field deformations of coset models

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Daniele Dorigoni

Mehregan Doroudiani

Joshua Drewitt

Martijn Hidding

Leila Schneps

Bram Verbeek

Non-holomorphic modular forms from zeta generators

Eleven-dimensional supergravity on S 8 × S 1 is conjectured to be dual to the M-theory matrix model. We prove that the dynamics of a subset of fluctuations around this background is consistently described by D = 2 SO(9) gauged maximal supergravity. We provide the full non-linear uplift formulæ for all bosonic fields. We also present a further truncation to the SO(3)×SO(6) invariant sector and discuss its relation to the BMN matrix model at finite temperature. The construction relies on the framework of generalised Scherk-Schwarz reductions, established for E9 exceptional field theory in a companion paper. As a by-product, we severely constrain the most general gauge deformations in D = 2 admitting an uplift to higher dimensions.

Italy

Gianluca Inverso

Consistent truncation of eleven-dimensional supergravity on S 8 × S 1

We develop in detail the general framework of consistent Kaluza-Klein truncations from D=11 and type II supergravities to gauged maximal supergravities in two dimensions. In particular, we unveil the complete bosonic dynamics of all gauged maximal supergravities that admit a geometric uplift. Our construction relies on generalised Scherk-Schwarz reductions of E9 exceptional field theory. The application to the reduction of D=11 supergravity on S 8 × S 1 to SO(9) gauged supergravity is presented in a companion paper.

Maximal D = 2 supergravities from higher dimensions

We consider non-Lorentzian expansions, Galilean and Carrollian, of the Lorentz force equation in which both the particle position and the electro-magnetic field are expanded. There are two well-known limits in the case of a constant field, called electric and magnetic, that are studied separately. We show that the resulting equations of motion follow equivalently from considering a non-linear realisation of a certain infinite-dimensional algebras.

Spain

José Cerdeira

Joaquim Gomis

Non-Lorentzian expansions of the Lorentz force and kinematical algebras

We provide, through the framework of extended geometry, a geometrisation of the duality symmetries appearing in magical supergravities. A new ingredient is the general formulation of extended geometry with structure group of non-split real form. A simple diagrammatic rule for solving the section constraint by inspection of the Satake diagram is derived.

Martin Cederwall

Jakob Palmkvist

Linus Sundberg

Extended geometry of magical supergravities