I work on supersymmetric and topological field and string theories. The main emphasis of my research is the quest and development of non-perturbative methods for quantum theories.
Some of the themes of my research activity have been: supersymmetric indices, string compactifications, topological gravity, matrix models and their applications to string theory, the holomorphic anomaly of topological strings, holographic conformal anomalies, the Wilson-Polchinski renormalization group equation and its large N limit, non-commutative field theories, open string field theory and tachyon condensation, the coupling of topological Yang-Mills and Chern-Simons theories to topological gravity, localization of supersymmetric gauge theories.
More recently I have been exploring a connection between supergravity and topological gravity. I identified emergent topological sectors of supergravity and I showed how they are relevant to the characterization of the moduli space of supersymmetric configurations of supergravity.
Quantum field theories with boundary
One of the first studies on the consequences of the presence of a boundary in field theory was done by Casimir where the pressure between two parallel conducting plates was computed, as an example of modification of the vacuum state of the electromagnetic field. Later, Symanzik faced the general problem of introducing and studying the effect of a boundary in a Quantum Field Theory. Particularly interesting is the case of Topological Quantum Field Theories, which do not possess any local observable, the only cohomologically nontrivial objects being globally defined, like for instance Wilson loops, or surfaces, or geometrical knots. Local physical observables of topological field theories are confined on the boundary, if present. For instance, introducing a boundary in 3D Chern-Simons theory serves to classify all rational conformal field theories, and chiral conserved currents are derived on its edge, which form a Kac-Moody algebra with central charge related to the Chern-Simons coupling constant. This nontrivial boundary structure has a physical interpretation in the Fractional Quantum Hall Effect. Similarly, on the boundary of topological BF theory, which can be defined in any spacetime dimensions, Kac-Moody algebras satisfied by conserved chiral currents are found, thus justifying the claim according to which topological BF theories are the effective field theories of topological insulators.
What are known as “fractons” represent new phases of quantum matter characterized by excitations that exhibit restricted mobility, being either immobile, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. Fractons can be connected to generalized gauge theories, and may be obtained from more familiar theories, like tensor gauge theory. The main properties of the fracton quasiparticles can be obtained from a generalized covariant Maxwell-like action.
The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton Literature, and the field equations of motion, which have the same form as the covariant Maxwell equations, can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Ampère laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the “Bianchi identity” for a rank-three fracton strength tensor, as in Maxwell theory. The equations describing the fracton limited mobility, namely the charge and dipole conservation, in our formalism are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, the known analogies between Linearized Gravity and fracton theory can be increased by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.