L_infinity algebras, the Batalin-Vilkovisky quantisation, and gauge theory
Lead Research Organisation:
University of Surrey
Department Name: Mathematics
Abstract
The project deals with the underlying mathematical structures
of gauge theory. In particular, it has been realised that any Lagrangian
field theory has an underlying L_infinity algebra governing its
dynamics. L_infinity algebras, also known as strong homotopy Lie algebras,
are higher (in the sense of higher category theory) generalisations of
Lie algebras. For instance, for L_infinity algebras the Jacobi identity
is allowed to hold only up to higher coherent homotopy. The origin of
these underlying L_infinity algebras is most easily understood in the
Batalin-Vilkovisky quantisation which, in turn, is a generalisation of
the Becchi-Rouet-Stora-Tyutin quantisation. Within the context of such
algebras it is seen that physically equivalent theories are related by
L_infinity quasi-isomorphisms between their underlying L_infinity
algebras. The project deals with studies of gauge theory scattering
amplitudes, integrability, and renormalisation within this frame work of
higher category theory in the context of supersymmetric Yang-Mills theories.
of gauge theory. In particular, it has been realised that any Lagrangian
field theory has an underlying L_infinity algebra governing its
dynamics. L_infinity algebras, also known as strong homotopy Lie algebras,
are higher (in the sense of higher category theory) generalisations of
Lie algebras. For instance, for L_infinity algebras the Jacobi identity
is allowed to hold only up to higher coherent homotopy. The origin of
these underlying L_infinity algebras is most easily understood in the
Batalin-Vilkovisky quantisation which, in turn, is a generalisation of
the Becchi-Rouet-Stora-Tyutin quantisation. Within the context of such
algebras it is seen that physically equivalent theories are related by
L_infinity quasi-isomorphisms between their underlying L_infinity
algebras. The project deals with studies of gauge theory scattering
amplitudes, integrability, and renormalisation within this frame work of
higher category theory in the context of supersymmetric Yang-Mills theories.
Organisations
People |
ORCID iD |
| Martin Wolf (Primary Supervisor) | |
| Tommaso Macrelli (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509772/1 | 01/10/2016 | 30/09/2021 | |||
| 2120152 | Studentship | EP/N509772/1 | 01/10/2018 | 30/09/2021 | Tommaso Macrelli |
| Description | At the current stage of my research, my most significant achievements are: 1) A tree and loop level generalisation of Berends-Giele recursion relations, valid for for every Lagrangian field theory. Scattering amplitudes are important objects in physics, since they link the theoretical description of a field theory to the experimental results. Understanding the mathematical structure of scattering amplitudes and computing them in an efficient way are among the most important goals of many research directions in theoretical physics. We have showed that for every Lagrangian field theory the associated L-infinity algebra encodes a generalization of Berends-Giele recursion relations for tree-level scattering amplitudes. We have further generalized our approach to off-shell recursion relations, including the quantum case and obtaining formulas to recursively compute arbitrary loop amplitudes. 2) The proof of the all-loop equivalence between the double-copy of Yang-Mills theory and N=0 supergravity. Yang-Mills gauge theory and gravity provide us our best current description of the known fundamental interactions of Nature. Homotopy algebras are an adequate formalism for the investigation of the dualities between gauge theory and gravity. In particular, we have proposed an off-shell Lagrangian approach to the double-copy prescription. Double-copy is an all-loop statement that permits to construct gravity amplitudes from Yang-Mills ones. Its validity relies on BCJ color-kinematic duality, which is true at tree-level and conjectured for loop-level. Following our off-shell approach, we were able to circumvent the difficulties of directly validating color-kinematic duality at loop level, and finally prove (for arbitrary loops) that the double-copy of Yang-Mills theory is equivalent to N=0 supergravity. Understanding the double-copy construction of loop integrands in supergravity is (conceptually and technically) a foundamental goal, and our result is an answer to an important open problem in this field. |
| Exploitation Route | Our work on the duality between gauge theory and gravity opens the door to many possible research directions. This homotopy algebra framework can be useful to give an unified description of double-copy at the level of scattering amplitudes and actions, and can be extended to other field theories. In particular, our off-shell approach can be applied to double-copy supersymmetric gauge theories, establishing quantum equivalences between those theories and a wide family of supergravities. Another natural research direction would be the study of the dualities between open and closed strings: homotopy algebra is the language of string field theory, and the double copy can be considered the offspring of a more foundamental relation at the level of string theory. It will be also interesting to extend our approach to scattering amplitudes recursion relations to on-shell recursion relations (such as BCFW) and in general to obtain new explicit recursion relations. Finding connection to other algebraic and geometric structures that emerge in scattering amplitudes, such as the amplituhedron, would be a compelling task to follow. Our results will be helpful to various people involved in high energy theoretical physics and mathematical physics, from scattering amplitudes experts to homotopy algebra experts. |
| Sectors | Other |