## Periodic Reporting for period 1 - STROMINGER (The Strominger system of differential equations)

**Reporting period:**2015-07-01

**to**2017-06-30

## Summary of the context and overall objectives of the project

This project is devoted to the study of a system of partial differential equations of great relevance in modern geometry and theoretical physics. The Strominger system arises in heterotic string theory in physics and has been proposed by S.-T. Yau as one of the fundamental perspectives of complex geometry, in relation to the moduli problem for Calabi-Yau manifolds. The overall objectives are, on the one hand, to make progress on Yau’s conjecture for the Strominger system and, on the other hand, to understand rigorously, in one simple example, certain aspects of a conjectural fundamental symmetry of the underlying physical theory, known as (0,2)-mirror symmetry.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During the lifespan of the Marie Sklodowska-Curie action STROMINGER the Fellow has obtained several results about the moduli problem for the Strominger system (WP1), its relation with generalized geometry as defined by Hitchin, and the construction of a toy model for (0,2)-mirror symmetry (WP2). The outcome of the study of the moduli problem, in collaboration with R. Rubio (IMPA, Weizmann Institute of Science), C. Tipler (Université de Bretagne Occidentale, Brest), A. Clarke (UFRJ) and C. Shahbazi (Leibniz University, Hannover), is the construction of the infinitesimal moduli space for the Strominger system in dimensions 6 and 7 and the partial understanding of its local structure. As a part of this collaboration, the Fellow has also studied the classification of holomorphic Courant algebroids and the deformation theory for holomorphic Courant algebroids and complex manifolds, establishing a comparison with the deformation theory for the Strominger system. As for WP2, the Fellow has achieved a general proof that the solutions of the Strominger system are exchanged under T-duality, going beyond one of the objectives of the project. The general geometric tools introduced for this last result have been used to construct a toy model example of the Strominger system problem in a homogeneous manifold. This research suggests an analogue for the Strominger system of Yau's existence theorem for Calabi-Yau manifolds, and a refinement of Yau's Conjecture.

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).

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).

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The progress made during this Marie Sklodowska-Curie action provides new insight on the moduli problem and suggests an analogue for the Strominger system of Yau's existence theorem for Calabi-Yau manifolds, and a refinement of Yau's Conjecture. The result on T-duality and the Strominger system breaks new ground towards a SYZ version of (0,2)-mirror symmetry. We expect that the general geometric tools introduced for our results have future applications in the theory of vertex algebras in mathematics and string phenomenology in physics.