Research

I am interested in mathematical aspects of String Theory and Quantum Field Theory.

Presently my reserch interests include nonperturbative effects in string theory and quantum field theory, enumerative geometry, topological quantum field theory, topological strings and black holes. I am also interested in matrix models, noncommutative geometry and conformal field theory.

Right now I have several ongoing projects. One of these concerns defects in quantum field theory. Defects are fundamental modification of a theory, for example by imposing certain conditions on a submanifold. Sometimes they can also be seen as certain operators, such as Wilson or 't Hooft lines, but this point of view is somewhat limited if there are mutually non local charges. I am interested in understanding the BPS spectrum of quantum field theories with (or without) defects, and to study the associated geometrical structures. It turns out this problem has quite amazing ramifications in math (Donaldson-Thomas theory, cluster algebras, quiver representation theory, algebraic curves and many more) and physics (strong coupling effects, phases of gauge theories, wall-crossing formulae and many more).

I am also trying to understand mathematical aspects of quantum gravity. One of the great successes of string theory is a microscopic explanation of black hole entropy. This is a consequence of the relation between black hole microstates with the enumerative geometry of complex varieties. For example in Calabi-Yau compactifications of string theory, one can construct explicitly quantum microstates from the internal geometry (for example by studying sheaves on holomorphic cycles, or special Lagrangian submanifolds) and count them to reproduce the semiclassical limit in Einstein's gravity. This implies a relation between black holes partition functions and enumerative geometry.

Another project concerns the possible applications of techniques from computational topology to string theory. Persistent homology is a relatively new approach to extract topological informations out of a set of point as a function of the scale. This approach has been proven very successful in the field of topological data analysis, and its various applications. Does these techniques have anything to say about string theory? Can we introduce a notion of topological complexity of a string compactification?

The ancillary files for the paper "BPS spectra, barcodes and walls" can be found here . The ancillary files for the paper "Persistent homology and string vacua" are here .

From time to time I also wonder about the application of topological field theory to condensed matter physics.

Here is a sample of some recent talks:

I talk I gave at the "Hilbert Schemes, Sheaves and Representations (VBAC 2013)" conference in SISSA Defects in cohomological gauge theory and Donaldson-Thomas invariants

Here's a talk I gave at Queen Mary University, on Framed BPS quivers and line defects in N=2 QFT

BPS spectra in quantum field theory and representation theory

June 28, 2014

Quantum field theory has a deep connection with representation theory and the underlying algebraic structures. For example, the very definition of a quantum field theory which admits a weakly coupled lagrangian description is based upon Lie algebras and Lie groups. The lagrangian functional, which describes the interactions between light degrees of freedom at a certain energy scale, is mostly determined by a choice of Lie algebras (the "symmetries" of the theory) and their representations (the "fields" of the theory). Even when the theory does not admit a weakly coupled lagrangian functional, such a choice greatly constrain the spectrum of allowed states. Indeed the pattern of hadronic resonances experimentally observed during the '60s, provided strong motivations to consider Yang-Mills theories based on special unitary structure groups. Another example is given by two dimensional conformal field theories, whose states are often organized according to the representation theory of infinite dimensional algebras.

In certain cases representation theory is a tool powerful enough to obtain exact informations about the exact spectrum of a theory, beyond weak coupling. This can happen when a quantum field theory has enough symmetries to cancel most of the quantum corrections to the parameters which describe the particles in the spectrum (that is, their masses and their charges). An example are four-dimensional supersymmetric quantum field theories with eight supercharges. In this case, and under certain conditions, the spectrum of so-called stable BPS states is captured by the representation theory of an algebraic object called quiver.

In supersymmetric theories, physical states are organized according to the representations of a certain Clifford algebra. Those states on which a certain number of generators act trivially, i.e. on which the Clifford algebra is degenerate, are called BPS. BPS representations are generically "rigid", that is invariant under deformations of the physical parameters. In this sense representation theory protects these states from quantum corrections.

To solve for the spectrum of BPS states is equivalent to provide an answer to the question: which BPS particles with fixed electromagnetic charge exist as a physical state for a certain value of the physical parameters? The question is very similar to the study of hadronic resonances in the theory of strong interactions, but thanks to supersymmetry can be solved exactly in terms of auxiliary algebraic structures. The question if a particle corresponds to a stable state or not, can be addressed by studying its effective dynamics. This is relatively standard in quantum field theory, and it involves an auxiliary quantum mechanics model which is defined on the particle world-line, as the particle propagates in time. In general this problem is very hard. However when the quantum field theory has extended supersymmetry, it often happens that this quantum mechanics can be more easily described in terms of a quiver. In this case we say that the quantum field theory has a BPS quiver.

A quiver is a directed graph, given in terms of nodes and arrows between the nodes. More precisely a quiver Q is given in terms of a quadruple (Q₀ , Q₁ , s , t), where Q₀ is the set of vertices, Q₁ the set of arrows and s , t : Q₁ → Q₀ linear maps which identify the starting and terminating node for each arrow. The path algebra of the quiver CQ is simply the algebra generated by the set of all paths, where the product of two paths is defined as their concatenation, or zero if concatenation is impossible. If the quiver has cycles, that is paths which start and end at the same node, one can introduce a superpotential W defined as a formal linear combination of cycles. Then we can define a path algebra with relations CQ / ∂ W by taking the quotient by the ideal ∂ W obtained by taking cyclic derivatives of the superpotential W by all the arrows. In physics the conditions ∂ W = 0 are also known as F-term relations.

A representation of a quiver (or a module over the path algebra) is obtained by the assignment of a vector space V_i at each node i ∈ Q₀, and of linear maps X_a : V_s(a) → V_t(a) to each arrow a ∈ Q₁. If the quiver has a superpotential W, then the maps X_a of the representation are required to obey the same equations as the arrows, obtained from ∂ W = 0. From the point of view of the quantum mechanics model which describe a BPS particle, the nodes of the quiver represent elementary constituents (whose multiplicities are given by the dimensions dim V_i) while the maps between the nodes represents the interactions which can bound them together. Therefore physical BPS states are described by representations of quivers with potentials (plus an additional mathematical condition, called stability, which coincide with the physical notion of stability and makes sure that the energy of the constituents is not too big).

In this way a difficult physical problem is reduced, thanks to supersymmetry, to a purely algebraic question in representation theory.