The unity of scattering amplitudes: gauge theory, gravity, strings and number theory

Projektinformationen

UNISCAMP

ID Finanzhilfevereinbarung: 804286

DOI

10.3030/804286

Projekt abgeschlossen

EK-Unterschriftsdatum 29 November 2018

Startdatum 1 Januar 2019

Enddatum 31 Dezember 2023

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 425 000,00

EU-Beitrag

€ 1 425 000,00

Koordiniert durch

UPPSALA UNIVERSITET
Sweden

Periodic Reporting for period 4 - UNISCAMP (The unity of scattering amplitudes: gauge theory, gravity, strings and number theory)

Berichtszeitraum: 2023-07-01 bis 2023-12-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

The central theme of the project UNISCAMP are scattering amplitudes -- central observables in quantum field theory that provide essential information about the quantum consistency of perturbative gravity. Precise control of the physical and mathematical properties of scattering amplitudes holds the key to long-standing questions on fundamental interactions and the structure of space and time. As a concrete leap in this direction, UNISCAMP addresses predictions in gauge theories, gravity, effective theories and in particular string theory through

- the efficient computation and compact representation of scattering amplitudes and,

- decoding their hidden structures & symmetries and their rich web of connections.

String-theory methods complement conventional approaches to scattering
amplitudes, and UNISCAMP combines the insights from

- the point-particle limit of superstrings & heterotic strings and,

- ambitwistor strings which directly compute field-theory amplitudes.

Both of them naturally incorporate the double-copy relation between gauge-theory & gravity amplitudes and extend the framework to effective field theories including higher-derivative interactions. It is a primary goal of UNISCAMP to pinpoint the unifying principles connecting a wide range of field and string theories.

Moreover, field- and string-theory amplitudes exhibit an intriguing mathematical structure: Their Feynman- and moduli-space integrals yield special functions such as polylogarithms which became a vibrant common theme of high-energy physics, number theory and algebraic geometry. An interdisciplinary goal of UNISCAMP is to

- investigate the low-energy expansion of multiloop string amplitudes and,

- extract an organizing scheme for iterated integrals and modular forms on higher-genus Riemann surfaces.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

During the 60-month funding period of UNISCAMP, up to 5 simultaneous team members produced a total of 45 preprints (apart from 3 white papers, all of them published or to be published in leading journals) and delivered a total of 35 presentations at international conferences and workshops. The scientific results roughly fall into three different directions:

1) understanding and exposing the double-copy structure of gravitational interactions through a fusion of string-theory, field-theory and Lie-algebra methods: The publications of UNISCAMP not only advanced the scattered ideas on the gravitational double copy in the earlier literature but also provided the first coherent approaches to exploit several of them at the same time. For instance, it turned out to be particularly fruitful to combine the complementary input from so-called unitarity methods in the quantum-field-theory literature with the point-particle limit of string-theory interactions. Apart from conference talks of the team members and their collaborators, the white paper arXiv:2204.06547 co-authored by the PI was a valuable channel of dissemination.

2) new synergies between conventional string theories and the more recent ambitwistor and chiral models, informing string interactions of infinitely many modes from field-theory methods: The innovative approach of UNISCAMP overcame two long-standing barriers -- one in unravelling the multiparticle interactions of massive string resonances and another one in systematically investigating the analytic structure of string loop amplitudes in a low-energy expansion. This research line led to a publication in the prestigious journal PRL.

3) particularly rapid and far-reaching progress on the mathematical structure of field- and string-theory amplitudes: A new way of harnessing the integrals on a torus in one-loop string amplitudes was found and applied to find new connections between (a) open- and closed-string interactions and (b) between iterated integrals and modular forms. Moreover, integration on higher-genus surfaces was successfully studied, starting from concrete computations of multi-loop string amplitudes and culminating in a proposal for higher-genus polylogarithms (i.e. a function space on arbitrary two-dimensional surfaces that closes under integration). This line of results led to one PRL publication, two white papers and the largest number of conference talks.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

All of the results mentioned in 1), 2) and 3) go considerably beyond the state of the art. In case of 1), the novelties concern both improved methods to derive known results in more elegant ways and new computations which were previously out of reach (explicit representations of gravitational scattering amplitudes using their double-copy structure). The main results of 2) overcome bottlenecks on massive string tree-level amplitudes and the analytic structure of one-loop amplitudes which were could only be discussed at a qualitative level in the earlier literature.

The work on 3) led to multiple paradigm shifts in the mathematical investigation of string loop amplitudes. A first series of paper derived and applied differential equations for integrals over genus-one surfaces that apply to all orders in the low-energy expansion (going beyond earlier work where differential equations were limited to fixed orders). These differential equations neatly organize the (infinite space of) non-holomorphic modular forms in closed-string low-energy expansions which are know as modular graph forms and received a lot of attention in the mathematics literature. One of UNISCAMP's research highlights is to identify and spell out the connection of modular graph forms with mathematical work on so-called equivariant iterated Eisenstein integrals. This achievement paved the way for two additional papers on the number theory and algebraic geometry of modular graph forms to be finalized in the next months.

Another series of papers on 3) took the synergies between string amplitudes and integration on higher-genus surfaces to the next level. We presented the first explicit result for two-loop five-point amplitudes of massless superstring states that is valid at all energies and verified the agreement of different formalisms. The low-energy expansion of our five-point results then guided the development of novel computational techniques on higher-genus surfaces related to non-holomorphic modular tensors. Towards the end of UNISCAMP's lifetime, our pioneering work on modular tensors matured to the extent that we could present a proposal for the long-sought for higher-genus polylogarithms. This construction will not only reshape string-theory computations but also have drastic impact on both particle physics and pure mathematics. Two or more papers on higher-genus polylogarithms are planned for 2024.

A group photo from the conference ``Elliptics 2021'' hosted from the funds of UNISCAMP