Gruppo: Fisica Matematica
I am interested in a variety of topics, spanning from classical and quantum field theory, gravity and string theory, as well as applications to dynamical systems and geometric quantum information theory.
The broad research area I have been working on revolves around a cohomological approach to a variety of problems in classical and quantum field theory on manifolds with boundary (and corners). The method I use goes under the name of BV-BFV formalism (after Batalin, Fradkin and Vilkovisky), a modern mathematical approach to variational problems with local symmetries on manifolds with boundary, and to their quantisation.
Here is a brief outline of the main research lines I am currently interested in.
BV-BFV approach to General Relativity
Cohomological methods for gravitational models on manifolds with boundary.
Within the BV-BFV framework, I have set up and developed a new approach to general relativity (GR) on manifolds with boundary. This program connects to a diverse landscape of research endeavours, and the application to general relativity tackles one of the most long-standing problems in theoretical and mathematical physics.
This analysis led me to find a discrepancy that arises when comparing two models of gravity that are otherwise supposed to be equivalent: the Einstein–Hilbert (EH) and Palatini–Cartan (PC) formulations of GR in dimension four and higher. Specifically, I have shown that certain cohomological data, which are naturally assigned to either model, differ in the presence of boundaries.
The outlook of this extended program is that of addressing quantisation of general relativity (with boundary). Without the observations produced in this preliminary phase, an early attempt at directly quantising gravity would have been thwarted by the mentioned obstructions. A large part of my research is then devoted to understanding and overcoming boundary and corner obstructions such as these, furthering our understanding of general relativity, with great potential impact for the research community working in quantum gravity, and field theory more generally.
Dynamical zeta functions and field theory
A new perspective on Fried’s conjecture.
Recently, we were able to find a surprisingly simple rewriting of a longstanding conjecture due to Fried, which asserts the equivalence of Ruelle’s dynamical zeta function for chaotic (Anosov) flows and the analytic torsion of Ray and Singer. This arises as an unexpected application to dynamical systems of the Batalin–Vilkovisky formalism, which frames Fried’s conjecture as the invariance of a physical quantity on a “nonphysical” choice. Our interpretation immediately offers a compelling explanation of the conjecture itself and a heuristic argument in its favour.
Inspired by classic results by A. Schwarz, who showed how the analytic torsion can be represented by the partition function of a field theory called BF theory, we have shown that the Ruelle zeta can be similarly linked to the same theory. Indeed, BF theory can be endowed with an unusual class of gauge-fixing conditions on manifolds that admit a contact structure, a central example being given by sphere bundles of (hyperbolic) Riemannian manifolds, for whose geoedesic flows the Ruelle zeta function is well-defined as a meromorphic function.
Bulk-boundary correspondences and BV
A cohomological treatment of holographic correspondences.
Cohomological techniques are useful to address a broad variety of bulk-boundary correspondences, which I will collectively label as holography for simplicity. These are central to modern investigations in the fields of condensed matter, high energy physics and gravity, for which they are of key relevance, but they also present rich opportunities for the development of pure mathematics.
In order to understand holography, the main guiding example for our purposes has been a well-studied correspondence between Chern–Simons (CS) theory on a three dimensional manifold, and the (chiral, gauged) Wess–Zumino–Witten (WZW) model supported on its boundary. The quantum data of the two models are related: the space of solutions of certain symmetry relations called ‘conformal Ward identities’, satisfied by the correlation functions for the WZW model, coincides with the space of states associated to CS theory. The CS/WZW example is relevant for gravity, which in three dimensions can be cast as a CS theory, as well as for condensed matter applications, since CS theory describes quantum Hall systems.
Using a framework due to Alexandrov, Kontsevich, Schwarz and Zaboronski we showed how one can recover the WZW model for a group-valued field starting from the “infinitesimal” data given by the BFV structure for Chern–Simons theory. Indeed, this approach also gives a field theoretic interpretation of Lie III theorem, readily generaliseable to problems of integration of more general algebraic structures.