Project description
Pushing the boundaries of rough analysis to solve quantum gauge theories
The ERC-funded SQGT project seeks to develop mathematical solutions for several open problems of quantum gauge theories (QGTs), including that of the challenging Yang-Mills ‘existence and mass gap’ Millennium Prize Problem. It aims to elucidate constructions in finite volume of 2D and 3D non-exactly solvable QGTs, potentially leading to the physical case of 4D constructions as well. To do so, it will devise new methods in the field of rough analysis, notably singular stochastic partial differential equations (SPDEs), which have seen groundbreaking progress in recent years. This new approach aims at extending the limits of rough analysis, primarily by investigating discrete approximations of SPDEs, presenting novel theories for geometric solutions and coupling SPDEs with random matrix theory.
Objective
This proposal aims to solve central open problems in the mathematical foundation of quantum gauge theories (QGTs), an important challenge comprising the Yang-Mills (YM) Millennium Prize Problem. A key outcome of the proposal will be the first constructions in finite volume of 2- and 3-dimensional non-exactly solvable QGTs, with a view towards the physical case of 4 dimensions.
The principal tools that will be developed and used to address these problems are in the field of rough analysis, in particular singular stochastic partial differential equations (SPDEs). Singular SPDEs appear widely in the study of dynamics with randomness and have seen revolutionary progress in the past decade. By developing new rough analytic methods applicable to QGTs, the proposal will push the frontiers of rough analysis, in particular studying discrete approximations of SPDEs, introducing novel geometric solution theories, and linking SPDEs with random matrix theory.
My research has shown that the stochastic quantisation equations of YM (SYM) can be renormalised in a geometrically faithful way,
which has already revealed new properties of the exactly solvable 2D YM measure. This is strong evidence that rough analytic techniques can bring new light to the study of QGTs and render their construction in 2D and 3D finally within reach.
The proposal is split into the following three long-term projects.
1. Two-dimensional theories: solve and identify the invariant measure of SYM for non-trivial principal bundles; prove large N convergence of SYM; construct the non-Abelian YM-Higgs measure in finite volume.
2. Three-dimensional theories: give the first construction of the 3-dimensional YM measure in finite volume; prove a discrete version of the BPHZ renormalisation theorem in regularity structures.
3. Axiomatic quantum gauge theory: formulate and prove the Osterwalder-Schrader reconstruction theorem applicable to QGTs; prove Uhlenbeck’s regularity theorem for distributions.
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Keywords
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2023-STG
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34136 Trieste
Italy
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