Research Overview
Emilio Franco
My interest in mathematical physics started quite early, and this interest took me to study both degrees simultaneously at
UAM (Madrid, Spain). Years later, my main motivation for doing research is still that of giving mathematical answers to questions arising from Physics.
Attracted by the natural connection between gauge theory and algebraic geometry that offers the theory of Higgs bundles, I started my thesis project in this subject under the supervision of
O. García-Prada at
ICMAT and the advise of
P. E. Newstead. A great level of explicitness in the description of moduli spaces is usually rare, but in the case studied in
my PhD, I could achieved a concrete description of the moduli space of Higgs bundles over elliptic curves. Therefore, I gained an explicit understanding of a moduli space with an extremely rich geometry and this circumstance has allowed me to jump easily to other areas of research, working during my postdoc with
M. Jardim at
Campinas State University (Brazil) with moduli spaces of a variety of geometrical objects such as quiver representations, instantons or sheaves on symplectic surfaces. Before going to Brazil, I spent 3 months in Berlin, working with
A. Schmitt at
Free University of Berlin.
Being in Brazil was a great experience, both personal and professional. There I had the great pleasure of working M. Jardim and a variety of young researchers such as
P. Tortella,
S. Marchesi and
G. Menet. Besides, within my Brazilian fellowship, I had the opportunity of spending up to a year visiting a foreign research centre. I made the most of it by going to
Imperial College London, where I visited
Richard Thomas. My research was boosted thanks to all I learned there, specially about Mirror Symmetry. This was crucial to complete a paper with a promising young researcher,
A. Peón-Nieto which constitutes the first example of Fourier-Mukai transform of branes in the critical locus of the Hitchin fibration.
At the beginning of 2017, I started working with
P. Gothen and
A. Oliveira at
U. Porto (Portugal). I was offered to continue one more year at Porto when my contract ended (in January 2018) but I declined to dedicate myself to the care of my daughter
Inés, born on November 2017. I did so until February 2019, when I started a research visit at
ICMAT.
In June 2019 I started my current position at
CAMGSD (
Instituto Superior Técnico, U. Lisboa)}, where I enjoy an
Investigador FCT fellowship for the next 6 years. This FCT program aims to attract independent researchers so they can consolidate their careers in Portugal.
Brief description of my papers
Higgs bundles over elliptic curves, with Oscar Garcia-Prada and Peter Newstead.
In this paper we study \(G\)-Higgs bundles over an elliptic curve when the structure group \(G\) is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable \(G\)-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable \(G\)-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of \(G\)-Higgs bundles and we study the generic and non-generic fibres.
Branes in the moduli space of framed instantons, with Marcos Jardim and Simone Marchesi.
In the physicist's language, a brane in a hyperkähler manifold is a submanifold which is either complex or lagrangian with respect to three Kähler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.
Moduli spaces of Lambda-modules on abelian varieties, with Pietro Tortella.
We study the moduli space \(\mathrm{M}_X(\Lambda, n)\) of semistable \(\Lambda\)-modules of vanishing Chern classes over an abelian variety \(X\), where \(\Lambda\) belongs to a certain subclass of \(D\)-algebras. In particular, for \(\Lambda = \mathcal{D}_X\) (resp.\(\Lambda= \mathrm{Sym}^\bullet \mathcal{T} X\)) we obtain a description of the moduli spaces of flat connections (resp. Higgs bundles).
We give a description of \(\mathrm{M}_X(\Lambda, n)\) in terms of a symmetric product of a certain fibre bundle over the dual abelian variety \(\hat{X}\). We also give a moduli interpretation to the associated Hilbert scheme as the classifying space of \(\Lambda\)-modules with extra structure. Finally, we study the non-abelian Hodge theory associated to these new moduli spaces.
Higgs bundles over elliptic curves for complex reductive Lie groups, with Oscar Garcia-Prada and Peter Newstead.
We study Higgs bundles over an elliptic curve with complex reductive structure group, describing the (normalization of) its moduli spaces and the associated Hitchin fibration. The case of trivial degree is covered by the work of Thaddeus in 2001. Our arguments are different from those of Thaddeus and cover arbitrary degree.
Higgs bundles over elliptic curves for real groups, with Oscar Garcia-Prada and Peter Newstead.
We study topologically trivial \(G\)-Higgs bundles over an elliptic curve \(X\) when the structure group \(G\) is a connected real form of a complex semisimple Lie group \(G^\mathbb{C}\). We achieve a description of their (reduced) moduli space, the associated Hitchin fibration and the finite morphism to the moduli space of \(G^\mathbb{C}\)-Higgs bundles.
Brane involutions on irreducible holomorphic symplectic manifolds, with Marcos Jardim and Gregoire Menet.
In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists' language, i.e. a submanifold which is either complex or lagrangian submanifold with respect to each of the three Kähler structures of the associated hyperkähler structure.
Starting from a brane involution on a \(K3\) or abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier-Mukai transform. Later, we recall the lattice-theoretical approach to Mirror Symmetry. We provide two ways of obtaining a brane involution on the mirror and we study the behaviour of the brane involutions under both mirror transformations, giving examples in the case of a \(K3\) surface and \(K3^{[2]}\)-type manifolds.
Involutions of the moduli spaces of G-Higgs bundles over elliptic curves, with Indranil Biswas, Luis Angel Calvo and Oscar Garcia-Prada.
We present a systematic study of involutions on the moduli space of \(G\)-Higgs bundles over an elliptic curve \(X\), where \(G\) is complex reductive affine algebraic group. The fixed point loci in the moduli space of \(G\)-Higgs bundles on X, and in the moduli space of representations of the fundamental group of \(X\) into \(G\), are described. This leads to an explicit description of the moduli spaces of pseudo-real \(G\)-Higgs bundles over \(X\).
Mirror symmetry for Nahm branes, with Marcos Jardim.
Using the Dirac-Higgs bundle and the morphism given by tensorization, we consider a new class of virtual hyperholomorphic bundles over the moduli space of \(\mathrm{M}\) of degree 0 semistable Higgs bundles. This construction generalizes the Nahm transform of a stable Higgs bundle. In the physicist's language, our class of hyperholomorphic vector bundles can be seen as (virtual) space filling (BBB)-branes on \(\mathrm{M}\). We then use the Fourier-Mukai-Nahm transform to describe the corresponding dual branes restricted to the smooth locus of the Hitchin fibration. The dual branes are checked to be (BAA)-branes supported on a complex Lagrangian multisection of the Hitchin fibration.
The Borel subgroup and branes on the Hitchin system, with Ana Peón-Nieto.
We study mirror symmetry on the singular locus of the Hitchin system at two levels. Firstly, by covering it by (supports of) (BBB)-branes, corresponding to Higgs bundles reducing their structure group to the Levi subgroup of some parabolic subgroup \(\mathrm{P}\), whose conjectural dual (BAA)-branes we identify. Heuristically speaking, the latter are given by Higgs bundles reducing their structure group to the unipotent radical of \(\mathrm{P}\). Secondly, when \(\mathrm{P}\) is a Borel subgroup, we are able to construct a family of hyperholomorphic bundles on the (BBB)-brane, and study the variation of the dual under this choice. We give evidence of both families of branes being dual under mirror symmetry via an ad-hoc Fourier-Mukai integral functor.
Unramified covers and branes on the Hitchin system, with André Oliveira, Peter Gothen and Ana Peón-Nieto.
We study the locus of the moduli space of Higgs bundles on a curve given by those Higgs bundles obtained by pushforward under an unramified cover. We equip these loci with a hyperholomorphic bundle so that they can be viewed as BBB-branes, and we introduce corresponding BAA-branes which can be described via Hecke modifications. We then show how these branes are naturally dual via explicit Fourier-Mukai transform, where we recall that the structure group \(\mathrm{GL}(n,\mathbb {C})\) is Langlands self dual. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration.
As a particular case, our construction describes the behaviour under mirror symmetry of the fixed loci for the action of tensorization by a line bundle of order \(n\). These loci play a key role in the work of Hausel and Thaddeus on topological mirror symmetry for Higgs moduli spaces.
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