# Vacuum States of the Heterotic String

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

String theory is believed to be a theory capable of describing all the known forces of nature, and provides a solution to the venerable problem of finding a theory of gravity consistent with quantum mechanics. To a first approximation, the world we observe corresponds to a vacuum of this theory. String theory admits many of these vacuum states and the class that is most likely to describe the observed world are the so-called `heterotic vacua'. Analysing these vacua requires the application of sophisticated tools drawn from mathematics, particularly from algebraic geometry. If history is any guide, the synthesis of these mathematical tools with observations drawn from physics will lead not only to significant progress in physics, but also important advances in mathematics. An example of such a major insight in mathematics, that arose from string theory, is mirror symmetry. This is the observation that within in a restricted class of string vacua, these arise in `mirror pairs'. This has the consequence that certain mathematical quantities, which are both important and otherwise mysterious, can be calculated in a straightforward manner. The class of heterotic vacua, of interest here, are a wider class of vacua, and an important question is to what extent mirror symmetry generalises and how it acts on this wider class.

In a more precise description, the space of heterotic vacua is the parameter space of pairs (X,V) where X is a Calabi-Yau manifold and V is a stable holomorphic vector bundle on X. This space is a major object of study in algebra and geometry. String theory tells us that it is subject to quantum corrections. To understand the nature of these corrections is the key research problem in this proposal and any advance in our understanding will have a important impact in both mathematics and physics. By now it is widely understood that string theory and geometry are intimately related with much to be learned from each other, yet this relationship is relatively unexplored in the heterotic string. This fact, together with recent developments that indicate that longstanding problems have recently become tractable, means that the time is right to revisit the geometry of heterotic vacua.

In a more precise description, the space of heterotic vacua is the parameter space of pairs (X,V) where X is a Calabi-Yau manifold and V is a stable holomorphic vector bundle on X. This space is a major object of study in algebra and geometry. String theory tells us that it is subject to quantum corrections. To understand the nature of these corrections is the key research problem in this proposal and any advance in our understanding will have a important impact in both mathematics and physics. By now it is widely understood that string theory and geometry are intimately related with much to be learned from each other, yet this relationship is relatively unexplored in the heterotic string. This fact, together with recent developments that indicate that longstanding problems have recently become tractable, means that the time is right to revisit the geometry of heterotic vacua.

### Planned Impact

The immediate beneficiaries of the proposed research will be mathematicians and physicists working in string theory. There will be a wider benefit to algebraic geometers and to phenomenological physicists who seek to embed models of particle physics in fundamental theory. The beneficiaries will be made aware of the research by the posting of preprints to the arXiv and their publication in journals. Seminars will also be given by the PI, CI, PDRAs and graduate students at national and international conferences and through personal interaction and discussion at these meetings.

Communications and engagement:

For this project, the usual academic channels of advertised seminars, posting of preprints, journal publication and the giving of conference talks, are the most appropriate. We request funds for the PI, CI and PDRAs to attend three relevant international conferences, and for the graduate students to each attend two national and one international conferences. One important reason for going to conferences is to explain the research to users and beneficiaries, through conference talks, discussion and personal interaction.

Activities to be undertaken to ensure good communication and engagement:

Websites such as the arXiv and the Oxford String Theory Group homepage will be used to communicate information and disseminate results. Specialized seminars will be given to communicate results to a target audience. Review seminars and review articles will communicate the results also to a wider community.

Collaboration:

The PI and CI will be responsible for overall management of the project, including advertising the results of the research. There are mathematicians and mathematical physicists who work on topics closely related to those posed in this project with whom we anticipate close collaboration. There are also researchers whose interests are not directly related to these topics, but would be interested in the outcome of this work. It is likely that opportunities for collaboration will arise during the course of this work. These opportunities will be maximised by advertising the results of research as described above.

Exploitation and application:

We do not anticipate any commercial applications of the research. We hope that the new tools we develop will be exploited by other mathematicians and physicists. To ensure this happens, the PI and CI will write short introductory papers about the new technology, as well as long technical papers, and to advertise the new tools in seminars and at conferences. We wish to make the research freely available to anyone who is interested, and will do this by posting it on the internet, and by talking about it in advertised seminars open to everyone. We see no utility to protecting our research outputs.

Capability:

The PI, CI and PDRAs, and to a lesser extent the graduate students, will be undertaking the impact activities, which will consist, primarily, of writing papers, both research and expository, and giving seminar and conference talks. The PI and CI have many years experience in these activities and will mentor the PDRAs and graduate students in academic writing, through commenting on drafts, and in giving talks, and will where possible arrange invitations for them to give seminars, for instance in the Oxford String Seminar series, co-organized by the PI and CI, and to attend relevant conferences. The graduate students will be encouraged to report their results both in the internal String Theory Group seminar and, as the work becomes more complete, to also give seminars in the Oxford String Seminar series and elsewhere.

Communications and engagement:

For this project, the usual academic channels of advertised seminars, posting of preprints, journal publication and the giving of conference talks, are the most appropriate. We request funds for the PI, CI and PDRAs to attend three relevant international conferences, and for the graduate students to each attend two national and one international conferences. One important reason for going to conferences is to explain the research to users and beneficiaries, through conference talks, discussion and personal interaction.

Activities to be undertaken to ensure good communication and engagement:

Websites such as the arXiv and the Oxford String Theory Group homepage will be used to communicate information and disseminate results. Specialized seminars will be given to communicate results to a target audience. Review seminars and review articles will communicate the results also to a wider community.

Collaboration:

The PI and CI will be responsible for overall management of the project, including advertising the results of the research. There are mathematicians and mathematical physicists who work on topics closely related to those posed in this project with whom we anticipate close collaboration. There are also researchers whose interests are not directly related to these topics, but would be interested in the outcome of this work. It is likely that opportunities for collaboration will arise during the course of this work. These opportunities will be maximised by advertising the results of research as described above.

Exploitation and application:

We do not anticipate any commercial applications of the research. We hope that the new tools we develop will be exploited by other mathematicians and physicists. To ensure this happens, the PI and CI will write short introductory papers about the new technology, as well as long technical papers, and to advertise the new tools in seminars and at conferences. We wish to make the research freely available to anyone who is interested, and will do this by posting it on the internet, and by talking about it in advertised seminars open to everyone. We see no utility to protecting our research outputs.

Capability:

The PI, CI and PDRAs, and to a lesser extent the graduate students, will be undertaking the impact activities, which will consist, primarily, of writing papers, both research and expository, and giving seminar and conference talks. The PI and CI have many years experience in these activities and will mentor the PDRAs and graduate students in academic writing, through commenting on drafts, and in giving talks, and will where possible arrange invitations for them to give seminars, for instance in the Oxford String Seminar series, co-organized by the PI and CI, and to attend relevant conferences. The graduate students will be encouraged to report their results both in the internal String Theory Group seminar and, as the work becomes more complete, to also give seminars in the Oxford String Seminar series and elsewhere.

### Publications

Blesneag S
(2016)

*Holomorphic Yukawa couplings in heterotic string theory*in Journal of High Energy Physics
Braun A
(2015)

*De Sitter vacua from a D-term generated racetrack potential in hypersurface Calabi-Yau compactifications*in Journal of High Energy Physics
Braun A
(2016)

*Heterotic-type IIA duality and degenerations of K3 surfaces*in Journal of High Energy Physics
Braun A
(2016)

*Box graphs and resolutions II: From Coulomb phases to fiber faces*in Nuclear Physics B
Braun Andreas P.
(2016)

*Tops as building blocks for G$_{2}$ manifolds*in JHEP
Braun V
(2014)

*F-theory on genus-one fibrations*in Journal of High Energy Physics
Braun V
(2015)

*Complete intersection fibers in F-theory*in Journal of High Energy Physics
Braun Volker
(2015)

*Two One-Parameter Special Geometries*
Candelas P
(2016)

*Hodge numbers for CICYs with symmetries of order divisible by 4*in Fortschritte der Physik
Candelas P
(2015)

*Type IIB flux vacua from G-theory I*in Journal of High Energy PhysicsDescription | A key finding is in the 2014 paper by X de la Ossa and E Svanes in which we solved a long standing problem regarding the tangent space of the moduli space of certain string theories with less supersymmetry (therefore their geometries are less constrained and the problem is harder) including perturbative quantum corrections. These are instanton moduli spaces of gauge theories on manifolds with a 3-form field. This work has already had impact in geometry, as the moduli spaces for the case of compactifications on six dimensional manifolds have been reinterpreted in terms of holomorphic currant algebroids. Moreover, this work has lead to generalisations: the collaboration between X de la Ossa, M Larfors and E Svanes, in which a more general structure of the moduli spaces is in terms of differential graded Lie algebras. Another key finding is the paper in 2016 by P Candelas, X de la Ossa and J McOrist. The analysis of this paper was technically difficult and only became practical on developing a calculus of covariant derivatives for the various fields, with respect to the parameters. For the Yang Mills field, for example, it has been known for a long time how to do this. For other fields of a more recondite nature, such as the Kalb-Ramond B field, the construction of the covariant derivative was new and far from trivial. This calculus of covariant derivatives is likely to remain useful in future investigations. Two papers authored by P Candelas, A. Constantin and C. Mishra completed a program of finding Calabi-Yau manifolds with small Hodge numbers. The idea here was to examine all freely acting symmetries that act on so called, complete intersection Calabi-Yau manifolds. The freely acting symmetries themselves had been found by V. Braun. Our contribution was the calculation of the Hodge numbers for each of the quotients. Not all the manifolds are interesting, but among this collection there are some very remarkable manifolds with very small Hodge numbers. These, arguably, are among the simplest Calabi-Yau manifolds and may well be useful for constructing interesting string theory vacua. Finally, A Braun paper in 2016 constructing manifolds with G2 holonomy will remain useful in future investigations, particularly in the context of compactifications of M-theory. |

Exploitation Route | Our work, and one of the goals of the grant, is to study the quantum corrections to the moduli spaces of certain string theories. Our work represents good progress towards this goal, and will certainly remain useful in future investigations. In the last year, together with my student MA Fiset, we are studying the conformal field theories associated to minimally supersymmetric string theories. This was, in part, one of the aims of the research award, and we now have the tools to begin taking steps toward the understanding the fully quantum corrected theory. |

Sectors | Other |