Periodic Reporting for period 1 - STROMINGER (The Strominger system of differential equations)
Reporting period: 2015-07-01 to 2017-06-30
In addition, in collaboration with Álvarez-Cónsul (PI, ICMAT), García-Prada (ICMAT), and Pingali (Bangalore), the Fellow has studied the gravitating vortex equations, in relation to the uniformization problem for pairs (X,D) consisting of a Riemann surface X and an effective divisor D. These interesting equations comprise as a particular case the Einstein-Bogomol'nyi Equations in the theory of cosmic strings in physics, which describe a special class of solutions of the abelian Higgs model coupled with gravity in four dimensions. The outcome of this research is a complete characterization of the existence of solutions of the Einstein-Bogomol'nyi Equations in terms of a stability condition in Geometric Invariant Theory (GIT), and a uniqueness and existence result for gravitating vortices in genus > 1. In particular, the main result solves a conjecture by Yisong Yang about the non-existence of cosmic strings superimposed at a single point. To our knowledge, this is the first instance of a non-trivial relation between the equations of a unified field theory which involves gravity and a GIT stability condition.
For the interdisciplinary transfer of knowledge, the Fellow and the PI have organised a working seminar on Vertex algebras and mirror symmetry, involving a combination of mathematicians from the ICMAT and physicists from the IFT in Madrid. The researcher has also attended and given lectures at interdisciplinary conferences, with a mixed audience of physicists and mathematicians (Benasque, Stony Brook, (0,2) in Paris, StringPheno 2017).
For the communication and public engagement of the action, the Fellow has given lectures at conferences and seminars (GEOQUANT, NS50, Hitchin70, IHP, StringPheno 2017), has tought courses at the Yau Mathematical Sciences Center (Tsinghua University), the GEOQUANT Conference (ICMAT), the JAE School of Mathematics at ICMAT, and the Universidad de Zaragoza. In addition the Fellow has undertaken several activities in close collaboration with the ICMAT Mathematics Culture Unit and the Servicio de Programas Europeos del CSIC (general public lectures and workshops).