Constructing a Family of Vacuum Solutions for the Heterotic G2 System

Constructing a family of vacuum solutions for the heterotic G2 system

Mateo Galdeano

University of Oxford, Mathematical Institute August 20, 2020

All the results described here are joint work with Xenia de la Ossa. Main motivation We want to improve our understanding of heterotic string theory. In particular we want to ﬁnd explicit solutions of the theory reduced to 3 dimensions, which are scarce in the literature. We would also like to see the deformation theory of these solutions working explicitly.

A glimpse at physics [1] Heterotic string theory is a 10-dimensional theory that includes supergravity and a supersymmetric gauge theory at classical level. In order to study the theory in lower dimensions, we perform a compactiﬁcation and put some of the coordinates in a compact manifold

M10 = M10−D × YD (1) Imposing the ansatz (1) reduces the equations the ﬁelds must satisfy to a system on the compact manifold. The Strominger-Hull system When heterotic string theory is reduced to 4 dimensions, we obtain the Strominger-Hull system.

We are interested in the reduction to 3 dimensions. The conditions to have a solution can be encoded as a heterotic G2 system.

Heterotic G2 system [2] A heterotic G2 system is a quadruple [(Y , ϕ), (V , A), (TY , Θ), H] satisfying the following properties:

Y is a 7-dimensional manifold with an integrable G2 structure given by ϕ. We call ψ = ∗ϕ. V is a vector bundle on Y with a connection A that is a G2-instanton, this means that its curvature FA satisfies FA ∧ ψ = 0. TY is the tangent bundle of Y and Θ is a connection which is a G2-instanton, so that its curvature RΘ satisfies RΘ ∧ ψ = 0. H is the torsion 3-form of the G2-structure T(ϕ) on Y , H = T(ϕ). In addition, the following formula must be satisfied: α0 H = dB + (CS(A) − CS(Θ)) , 4 where CS denotes the Chern-Simons form of the corresponding connection and B is the B-field. . Explicit family of solutions 7 7 7 A family of solutions [(S , ϕs), (TS , Θβ), (TS , Θα), H] is given as follows Take Y = S7 the 7-sphere. We see from the Hopf fibration that it is a 3-Sasakian manifold [3] 3 7 4 S −→ S −→ S . This gives an integrable G2 structure to the 7-sphere. Moreover, we can squash the metric along the fibers and this results in a one-parameter family of G2 structures on the sphere ϕs for s > 0. These are the squashed 7-spheres. The 3-form H is determined by the torsion of the G2 structures. In general there is a two-parameter family of metric connections compatible with the G2 structure on the tangent space of any G2 manifold [4]. In this case it reduces to a one-parameter family Θα and all of them are G2 instantons. 7 We can take as vector bundle V the tangent space TS itself and as an instanton another element in the family of connections Θβ. Finally, we must impose the heterotic Bianchi identity α0 dH = (tr FA ∧ FA − tr RΘ ∧ RΘ). 4 This constraints all the parameters and essentially gives a one-parameter family of solutions for each value of the string parameter α0.

References Ralph Blumenhagen, Dieter Lüst,and Stefan Theisen. Basic concepts of string theory. Theoretical and Mathematical Physics. Heidelberg, Germany: Springer, 2013. isbn: 9783642294969. doi: 10.1007/978-3-642-29497-6. Xenia de la Ossa, Magdalena Larfors, and Eirik E. Svanes. “The Infinitesimal Moduli Space of Heterotic G2 Systems”. In: Commun. Math. Phys. 360.2 (2018), pp. 727–775. doi: 10.1007/s00220-017-3013-8. arXiv: 1704.08717 [hep-th]. Charles P. Boyer and Krzysztof Galicki. “3 - Sasakian manifolds”. In: Surveys Diff. Geom. 7 (1999), pp. 123–184. arXiv: hep-th/9810250 [hep-th]. Robert L. Bryant. “Some remarks on G 2-structures”. In: arXiv Mathematics e-prints, math/0305124 (2003), math/0305124. arXiv: math/0305124 [math.DG].