Rigid 4D \( \mathcal{N}=2 \) supersymmetric backgrounds and actions
JHE
= 2 supersymmetric backgrounds and
Daniel Butter 0 1
Gianluca Inverso 0 1
Ivano Lodato 0 1
Nikhef Theory Group 0 1
0 Open Access , c The Authors
1 Science Park 105 , 1098 XG Amsterdam , The Netherlands
We classify all N = 2 rigid supersymmetric backgrounds in four dimensions with both Lorentzian and Euclidean signature that preserve eight real supercharges, up to discrete identifications. Among the backgrounds we find specific warpings of S3
actions; Supersymmetric gauge theory; Extended Supersymmetry; Superspaces
-
AdS3 × R, AdS2 × S2 and H2
× S2 with generic radii, and some more exotic geometries.
We provide the generic two-derivative rigid vector and hypermultiplet actions and analyze
the conditions imposed on the special K¨ahler and hyperka¨hler target spaces.
1 Introduction
2 The general rigid Lorentzian supersymmetry algebra
3 Lorentzian backgrounds
General comments
The supercoset construction
AdS4 spacetime
Round and squashed spatial three-spheres (R × S3)
The round R × S3 and SU(2|2) supersymmetry
The squashed S3
SU(2|1) × SU(2|1) supersymmetry
Warped AdS3 spaces (wAdS3 × R)
Timelike stretched AdS3 × R
Spacelike squashed AdS3 × R
Lightlike warped AdS3 × R
Other geometries
Lightlike S3
Warped Lorentzian S3 × R
‘Overstretched’ AdS3 (Heis3 × R)
4 Rigid supersymmetric actions
Vector multiplets
Conformal supergravity and the origin of rigid actions
4.4 A simple example: the N = 2∗ action
5 The general Euclidean supersymmetry algebra
6 Euclidean backgrounds
6.2 Squashed and stretched S3 × R and S3 × S1 Warped one-sheeted H3 × R The Heis3 × R limit
The two-sheeted H3
6.7 Deformed supersymmetry in flat space – i – 7 Rigid Euclidean supersymmetric actions 7.1
Vector multiplets
7.2 Hypermultiplets
8 Discussion and conclusions
A Conventions
A.1 Lorentzian signature
A.2 Euclidean signature
B General action principle in rigid superspace
C Details of Lorentzian backgrounds
Warped AdS3 spaces (wAdS3 × R)
C.4 Lightlike S3
C.5 ‘Overstretched’ AdS3
C.6 Plane waves
D Details of Euclidean backgrounds
D.1 S4 and H4
D.2 Squashed S3
D.3 The one-sheeted H3 × R
D.5 The two-sheeted H3
Introduction
In the last few years, there has been a great deal of interest in supersymmetric field theories
on rigid curved backgrounds, beginning with the seminal work of Pestun [1]. These efforts
have exploited the principle of supersymmetric localization to evaluate path integrals and
compute certain supersymmetric observables in various rigid backgrounds.
A systematic approach to addressing such curved spaces at the component level was
initiated by Festuccia and Seiberg [2]. Taking the point of view that a rigid supersymmetric
theory could be understood as a supergravity theory with the metric and other bosonic
components frozen to some background values, they investigated the conditions required in both
supersymmetries existed. Other aspects, such as the weaker requirements imposed by fewer
supercharges in both signatures, were addressed in later work [3–9]. In the case of extended
analyzed by Gupta and Murthy [10] and by Klare and Zaffaroni [11]. The analysis of [11]
determined the main geometric criterion in either Euclidean or Lorentzian signature: the
spacetime must admit a conformal Killing vector. In the presence of such a vector, one
supercharge may be kept by turning on background values for the R-symmetry gauge fields.
Our goal in this paper is to perform a complementary analysis to that of [10, 11]. First,
independent Killing spinors — in both Lorentzian and Euclidean signatures and classify
the possible smooth backgrounds up to discrete identifications. Second, we construct the
general vector and hypermultiplet actions on such spaces and find the conditions on the
allowed target spaces.
Our analysis, some of which appeared in a different context in [12], leads to several
AdS4, R × S3, AdS3 × R, and an Hpp-wave arising as a Penrose limit of the last two
and Hpp-wave admitting two alternative N
R-symmetry groups — and several cases requiring extended supersymmetry:
• a timelike stretched, spacelike squashed, or null warped AdS3 × R,
• a warped S3 × R where the S1 fiber of S3 is either timelike or null,
• AdS2 × S2 with generic radii, with two different SUSY realizations for each choice of
to S4, H4, a two-sheeted H3
supersymmetry plays a major role:
• a squashed or stretched S3 × R,
• a Heis3 × R group manifold,
as well as an Heis3 × R space and Hpp-wave variants where the background fields become
null. (Some of these correspond to Penrose limits of other cases.) Each of the resulting
supersymmetry algebras can be identified as a massive deformation of the super-Poincar´e
algebra, and indeed each possesses a supercoset structure permitting the straightforward
construction of each of the Killing spinors, which we compute explicitly.1
The options in Euclidean signature are summarized in table 2 of section 6. In addition
× R, and S3 × R, we find several geometries where extended
1It is shoul (...truncated)