AGMP 2015-2019 research plan
Being formally established by the end of 2014 only, the team is presenting here its first joint research plan reflecting the new synergy which will encourage interaction, discussions and collaborations between researchers currently working in distinct directions. The research plan is divided according to several subjects. Due to the limited space we are covering here only selected projects.
Mathematical Relativity
Research in mathematical relativity will be mostly performed by V. Pravda, A. Pravdová, M. Ortaggio, T. Málek, students and collaborators.
1.Exact solutions of generalized gravities
1.1 Universal spacetimes
In a 2015 preprint [HMPP15], we presented explicit examples of type II and D universal spacetimes(i.e., metrics that are simultaneous solutions of virtually all gravity theories derived from the Lagrangian) for all composite number dimensions. However, we have no examples for prime number dimensions and we have proved that such spacetimes do not exist in five dimensions. These results hint at the possibility that type II and D universal spacetimes do not exist for all prime number dimensions. Although we believe that a general proof of such statement is beyond the reach of current methods, in collaboration with S. Hervik,we will attempt to study the (non-)existence of type D/II universal spacetimes in seven and, possibly, eleven dimensions. In collaboration with G. Dotti, we will also study stability of type N and III universal spacetimes found in [433091] using the methods recently developed in [D14].
[HMPP15] S. Hervik, T. Málek, V. Pravda, A. Pravdová, arXiv:1503.08448 (2015)
[D14] G. Dotti,Phys. Rev. Lett 112 (2014) 191101
1.2Solutions with massless fields
Corrections to Einstein's equations coming, e.g., from string theory, will in general depend also on other massless fields such as certain p-forms. As an extension of our previous work, it will be of interest to explore the Einstein-Maxwell systems that admit a “universal” property. We already obtained partial results in this direction and we expect to publish them in 2015. A complementary research topic will be a study of “universality” for type N and III aligned Weyl and Ricci tensors and corresponding vacuum solutions to particular theories of gravity (e.g. L(Riemann) gravities).
1.3Robinson-Trautman spacetimes
Since very few explicit vacuum solutions of generalized gravities (such as L(Riemann) gravity) are known (obviously apart from the universal solutions found in [433091]and [HMPP15]) we plan to search for non-universal exact solutions of generalized gravities. Natural candidates for such solutions are the Robinson-Trautman spacetimes.This project already started as a collaboration with J. Podolský and R. Švarc (MFF UK, Prague).
2. Exact solutions of the Einstein and Einstein-Maxwellequations in higher dimensions
2.1We will search for specific algebraically special exact solutions of the Einstein and Einstein-Maxwell equations, building also on our previous results on the Goldberg-Sachs theorem.
2.2We will also study Kerr-Schild and extended Kerr-Schild spacetimes admitting electromagnetic fields (p-forms). Special subcases of this family of solutions will include also certain Robinson-Trautman and Kundt electrovacuum spacetimes. This will be a partial continuation of [441470](published in 2015) and of our ongoing studies of Kundt spacetimes coupled to null Maxwell fields.
3. Asymptotic properties of Einstein spacetimes
We plan to continue our previous works (e.g. [434752]) and employ the asymptotic approach, combined with a study of the characteristic initial value problem, to study asymptotic properties of type II non-degenerate Einstein spacetimes and the “uniqueness” of such spacetimes (as proven recently by Reall et al. in a closed form in five dimensions). Under some regularity assumptions, the same methods will be subsequently applied to general asymptotically flat spacetimes, in which radiation may be present. This will be a development of a recent paper [434752]and will make contact with a previous related work [GR12] based on a different technique. Students may also be involved in these projects.
[GR12] M. Godazgar, H. S. Reall, Phys.Rev. D85 (2012) 084021
4. Further projects in Mathematical Relativity
4.1 Above mentioned Kundt spacetimes also appear in the context of geometrization of the Hamiltonian dynamics [P07]. We plan to study geometrical properties of such “Hamiltonian” Kundt spacetimes and their overlap with VSI, universal, recurrent and other subclasses of Kundt spacetimes.
4.2 A. P. and the Master's student M. Kuchyňka are studying appropriate non-vacuum generalizations of the Goldberg-Sachs theorem to higher dimensions.
4.3 V. P. and the PhD student T. Tintěra are studying properties of the Kaluza-Klein reduction from the point of view of the Weyl classification.
[P07] M. Pettini, Geometry and Topology in Hamiltonian Dynamics, Springer (2007).
Algebraic topology and geometry
1. Together with A.A. Voronov (University of Minnesota, Minneapolis), M. Markl will finish their interpretation of Batalin-Vilkovisky algebras in terms of the convolution product. This approach dramatically simplifies previous work of Cieleiebak, Fukaya and others.
2. With M. Batanin, M. Markl plans to develop their theory of Koszul duality for general operadic categories. The expected outcome should be a systematic interpretation of various dualities appearing in algebra and higher category theory.
String field theory, quantum groups and vertex operator algebras
Topics in this field will be mostly studied by M. Markl and A. Zuevsky
1. Markl together with M. Doubek, B. Jurčo (Charles University, Prague) and I. Sachs (LudwigMaximilian University, Munich) will continue his work on the monograph "Algebraic Structures of String Field Theory" that will appear in Springer's Lecture notes in physics.
2. We will apply the theory of vertex operator algebras (VOAs) to problems in algebraic geometry, number theory, and integrable models. We compute and study properties of the partition and correlation functions for various VOAs. The main goal is to establish a higher genus version of the principal algebraic tool – the Zhu reduction procedure (see also [ZT15]).
3. VOAs is a natural source of automorphic forms. We plan an activity to study the structure of vector-valued Jacobi, Siegel, and almost holomorphic modular forms, and generation of various (in particular, Jacobi-Fay) identities at higher genus(see also [441820]).
4. VOAs computations result in algebraic analysis of classical generating kernels (in particular, the Szegő kernel) on higher genus Riemann surfaces. We will study Hardy spaces and enumerate resulting higher genus kernels (see also [Z15a]).
5. We will use the construction of VOA creative intertwining operators in algebraic applications in integrable models [Z15b].
[ZT15]M. P. Tuite, A. Zuevsky, arXiv: 1308.2441, to appear in J. Math. Phys.
[Z15a] A. Zuevsky, Int. J. of Theor. Phys. 2015 (online first)
[Z15b] A. Zuevsky, Homogeneous grading affine Toda models, submitted to J. Geom. and Physics
Geometric structures on manifolds and their deformations, Information geometry and complexity, Symplectic spaces
These topics will be mostly studied by H. V. Lê,
1. We plan to finalize the book Information geometry and continue to work with N. Ay, J. Jost (Max-Planck-Institute for Mathematics in Leipzig) and L. Schwachhöfer (TU Dortmund) on the related problems.
2. We will continue to work with L. Schwachhöfer (TU Dortmund), Y-G. Oh (Institute of Basic Sciences, Center for Geoemtry and Physics in Pohang), L.Vitagliano, A. Tortorella
(Salerno Univesity) on geometric deformation problems related to low-dimensional geometry, special holonomy and Jacobi manifolds. Parts of our results are contained in the following preprints:
H. V. Lê and Y. G. Oh, arXiv:1208.3590, accepted in Asian J. of Math.
H. V. Lê, Y. G. Oh, A. G. Tortorella and L. Vigtaliano, arXiv:1410.8446.
H. V. Lê and L. Schwachhöfer, arXiv:1408.6433.
3. We will work with Kaoru Ono (RIMS, Kyoto University) on periodic solutions of Hamiltonian equations by finding new homotopy invariants for the Floer chain complexes and the Novikov-Floer chain complexes on symplectic manifolds.
4. We plan to continue our work on fields between geometry, topology and computational complexity.In fact, we already published in 2015 some results[441464], [437490] in the proposed research plan.