Descrizione del progetto
Spingersi ai limiti dell’analisi approssimativa per risolvere le teorie di gauge quantistiche
Il progetto SQGT, finanziato dal CER, intende sviluppare soluzioni matematiche per diversi problemi aperti delle teorie di gauge quantistiche (QGT), tra cui quello dell’impegnativo problema per il millennio di Yang-Mills «esistenza e gap di massa». L’obiettivo è quello di chiarire le costruzioni in volume finito di QGT non esattamente risolvibili in 2D e 3D, portando potenzialmente anche al caso fisico di costruzioni in 4D. A tal fine, svilupperà nuovi metodi nel campo dell’analisi approssimativa, in particolare delle equazioni differenziali parziali stocastiche singolari (SPDE), che hanno registrato progressi rivoluzionari negli ultimi anni. Questo nuovo approccio intende estendere i limiti dell’analisi approssimativa, principalmente studiando approssimazioni discrete delle SPDE, presentando nuove teorie per le soluzioni geometriche e accoppiando le SPDE con la teoria delle matrici casuali.
Obiettivo
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 Uhlenbecks regularity theorem for distributions.
Parole chiave
Programma(i)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Argomento(i)
Invito a presentare proposte
(si apre in una nuova finestra) ERC-2023-STG
Vedi altri progetti per questo bandoMeccanismo di finanziamento
HORIZON-ERC -Istituzione ospitante
1010 Wien
Austria