Euclidean supergravity

Abstract

Supergravity with eight supercharges in a four-dimensional Euclidean space is constructed at the full non-linear level by performing an off-shell time-like reduction of five-dimensional supergravity. The resulting four-dimensional theory is realized off-shell with the Weyl, vector and tensor supermultiplets and a corresponding multiplet calculus. Hypermultiplets are included as well, but they are themselves only realized with on-shell supersymmetry. We also briefly discuss the non-linear supermultiplet. The off-shell reduction leads to a full understanding of the Euclidean theory. A complete multiplet calculus is presented along the lines of the Minkowskian theory. Unlike in Minkowski space, chiral and anti-chiral multiplets are real and supersymmetric actions are generally unbounded from below. Precisely as in the Minkowski case, where one has different formulations of Poincaré supergravity upon introducing different compensating supermultiplets, one can also obtain different versions of Euclidean supergravity.

One-loop determinants for black holes in 4d gauged supergravity

“…We choose coordinates 6) and turn on the relevant gauge and auxiliary fields in the Weyl multiplet to support half-BPS solutions, as well as a set of electric and magnetic charges and scalar fields in the vector multiplets fixed on-shell by the attractor mechanism of gauged supergravity. The complete half-BPS near-horizon solution was derived in [27,28] in the superconformal formalism and rewritten in [1] in the Euclidean formulation of supergravity [29]. We gather the essential features of the near-horizon solution in appendix A.…”

“…In this section we perform the steps required for the evaluation of the one-loop determinant for generic abelian vector multiplets, starting from the rewriting of the supersymmetry variations in terms of twisted variables, then performing the gauge-fixing, and proceeding with the evaluation of the D 10 operator and its index using three different methods. As in section 1.1, we refrain from reviewing the full 4d N = 2 Euclidean conformal supergravity formalism used in this paper, instead referring the reader to [29] for a thorough presentation. We merely gather some relevant aspects of the near-horizon background in appendix A (see [1] for more details).…”

“…We consider n V + 1 vector multiplets coupled to the conformal supergravity background. The transformation rules for a Euclidean vector multiplet V I under the localizing supercharge Q parameterized by commuting symplectic-Majorana Killing spinors (ξ i , κ i ) (see (A.6) for their explicit expressions) are given by [29] QX I ± = ±ξ i± Ω i I ± ,…”

“…We will denote this linear combination by ξ I V I . To be explicit, recall that the off-shell transformation rules of the compensating hypermultiplet fields under the localizing supercharge Q are [1,29] QA…”

“…withF I µν field strengths of the vector fieldsW I µ . In the Euclidean superconformal formalism, the coupling of vector multiplets to the gravity background is specified by two homogeneous functions each of degree two in the scalar fields X I + and X I − [29]. At the two derivative level, such prepotentials F ± (X ± ) completely determine the action of the theory under consideration.…”