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Xenia de la Ossa: Lectures

January 8, 2018
TITLE: The geometry and moduli of heterotic G2 structures

ABSTRACT: A heterotic G_2 system is a quadrupole [(Y, \varphi), (V,A), (TY, \theta), H], where Y is a seven dimensional manifold with an integrable G_2 structure and \varphi is the corresponding associative three form, V is a bundle on Y with an instanton connection A, and \theta is an instanton connection on the tangent bundle TY. H is a three form given in terms of the B field and the Chern-Simons forms of A and \theta (the anomaly cancelation condition) which is further constrained so that it is equal to a natural three form uniquely determined by the G_2 structure on Y. This constraint mixes up the geometry of Y with that of the bundles.

I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative \cal Y acting on forms with values on the bundle {\cal Q} = T^*Y\oplus {\rm End}(V) \oplus {\rm End}(TY) which satisfies \check{\cal D}^2 = 0, for some appropriately defined projection of the operator \cal D. Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancellation condition. We show that the infinitesimal moduli space is given by the cohomology group H^1_{\check{\cal D}} and that it is finite dimensional. Our analysis leads to results that are of relevance to all orders in \alpha. Time permitting, I will comment on work in progress about the finite deformations of heterotic G_2 systems and the relation to differential graded Lie algebras.

From the physics perspective these structures give rise to very interesting three dimensional gauge supergravity theories (on Minkowski or AdS3) with only N=1 supersymmetry. Very little is known about these theories, as opposed to those with N=2 or N=4, however we seem to have just enough supersymmetry to be able to deduce some interesting features of the effective field theories.