Abstract
Abstract We construct rigid supersymmetric theories for interacting vector and tensor multiplets on six-dimensional Riemannian spin manifolds. Analyzing the Killing spinor equations, we derive the constraints on these theories. To this end, we reformulate the conditions for supersymmetry as a set of necessary and sufficient conditions on the geometry. The formalism is illustrated with a number of examples, including manifolds that are hermitian, strong Kähler with torsion. As an application, we show that the path integral of pure super Yang-Mills theory defined on a Calabi-Yau threefold $ {{\mathcal{M}}_6} $ localizes on stable holomorphic bundles over $ {{\mathcal{M}}_6} $ .
Highlights
Consistency of this limit requires the existence of solutions of the corresponding Killing spinor equations which in turn poses non-trivial constraints on the background fields
We show that the path integral of pure super Yang-Mills theory defined on a Calabi-Yau threefold M6 localizes on stable holomorphic bundles over M6
One way to construct these theories is to start from the Euclidean version of off-shell supergravity coupled to YM and tensor multiplets, and take the rigid limit in which the fields in the supergravity multiplet are frozen in a manner consistent with supersymmetry
Summary
Our starting point is the off-shell formulation of six-dimensional supergravity obtained from superconformal tensor calculus with the dilaton Weyl multiplet coupled to a linear multiplet after particular gauge fixing [19,20,21]. As discussed in the introduction, when deriving rigid supersymmetric theories from supergravity one has to impose the vanishing of the supersymmetry variation of all fermionic fields from the supergravity multiplet. This defines the Killing spinor and poses constraints on the bosonic background fields. In our conventions (cf appendix A) and switching from Minkowsi to Euclidean signature, the Killing spinor equations imposing the vanishing of the gravitino fields’ variation, read. In the following we will construct rigidly supersymmetric field theories on backgrounds that allow for non-trivial solutions of the Killing spinor equations (2.1), (2.3). Anticipating the result of the full analysis that the background field strength Vm◦n vanishes (cf. equation (5.15) below), shows that the scalar combination ξH is constant
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