Project Summary

In this project we will apply Killing spinor analysis (Tod, 1983, 1995; Gauntlett et al., 2003) to find and classify the supersymmetric solutions of some three-dimensional supergravity (SUGRA) theories admitting an Anti de Sitter (AdS) vacuum, to compare these with their analogs in higher dimensions and to study their physical properties in detail. We will especially try to find BTZ (Banados et al., 1992) and Lifshitz-type black hole (Ayon-Beato et al., 2009; Sarıoğlu, 2011), black string (Değer and Sarıoğlu, 2004, 2006), warped AdS (Anninos et al., 2009; Değer et al., 2014; Değer and Moutsopoulos, 2016), Kundt-type (Chow et al., 2010; Değer and Sarıoğlu, 2015), pp-wave (Gibbons et al., 2008) and Lifshitz solutions (Balasubramanian and McGreevy, 2008) that have been extensively studied in recent years.

Supersymmetric solutions, which form the main subject of this project, are pivotal in the study of supergravity theories since they possess stability properties that survive quantum deformations. According to the celebrated “AdS/CFT conjecture” (Maldacena, 1998), there is equivalence between string theory on an AdS space and a conformal field theory (CFT) defined on the boundary of the AdS space. Using this conjecture, one can determine physically important quantities on the theory with strong-coupling using perturbative methods on the weakly-coupled side. This conjecture is arguably one of the most remarkable ideas ever put forward by string theory, and has found a wide range of applications including condensed matter theory. Although it has not been rigorously proven, it has been verified several times by various tests. The low energy limit of string theory on an AdS background is generally a SUGRA theory, and among those, the three-dimensional ones (Marcus and Schwarz, 1983; Achucarro and Townsend, 1986) are especially important, since a lot is already known on two-dimensional CFTs (Di Francesco et al., 1997). Moreover, three-dimensional gravity theories provide good toy models for testing various ideas in quantum gravity since they share several of the physically important properties of the four-dimensional gravitational models, including specifically Einstein’s General Relativity, but lack thereof many of the complications inherent in their counterparts (Carlip, 1998). Unfortunately, few explicit D=3 SUGRA models are known to date, and hence the number of known exact supersymmetric solutions is not much (de Wit et al., 2003b). There is little information regarding the connection of these models to higher dimensions and we will contribute to these issues via the new solutions that will be found in this project. For these reasons, it is important to work on the problems described in detail below.

In an earlier paper (Değer et al., 2015a), we found the embedding of three-dimensional SO(4)×R⁶ gauged N=4 supergravity with quaternionic target space SO(4,4)/(SO(4)×SO(4)) into the anomaly-free D=6 N=(1,0) supergravity theory (Nishino and Sezgin, 1986), which has many interesting applications in cosmology, coupled to a single chiral tensor multiplet through a consistent reduction on AdS₃×S³. This work is important since it contains the detailed construction of the aforementioned N=4 gSUGRA model and proves that this model can be obtained by a consistent S³ reduction from 6-dimensions. The meaning of a consistent sphere reduction is that any solution found in 3-dimensions automatically corresponds to a solution of the theory in 6-dimensions. The number of such consistent sphere reductions known to date is very few (Cvetic et al., 2000). Our first goal is to find and classify all supersymmetric solutions of this N=4 model. After this stage, it would also be interesting to study the renormalization group (RG) flows and to make predictions on the relevant CFTs as well. 

Our second goal is to find the supersymmetric solutions of D=3 N=(2,0) U(1)-gauged matter coupled supergravity models, constructed in Abou-Zeid and Samtleben (2002), which has a scalar sector that lives on an arbitrary Kähler manifold with a U(1) isometry. Specifically, we will identify all those that admit warped AdS spaces and study the effects of Fayet-Iliopoulos terms on the solutions. The warped AdS spacetimes are important since they allow for relatively new applications of the AdS/CFT conjecture. 

In a recent paper (Değer et al., 2015b), we found and described in detail the classification of all supersymmetric solutions, which possess a timelike Killing vector, of the ungauged version of the D=3 N=8 (half-maximal) SUGRA model (Nicolai and Samtleben, 2001b). Even though there are many papers about finding supersymmetric solutions in various dimensions and in diverse models, this work is important since it was here that all possible solutions were for the first time given exactly. In an older work (Değer et al., 2010), all supersymmetric solutions of the same model that have a null Killing vector were already constructed. Taken together, the problem of finding all the supersymmetric solutions of the ungauged version of the D=3 N=8 gSUGRA model has been completed in these papers. Our third goal is to work on the gauged version, i.e. the version with non-vanishing vector fields, of this model and to find and classify all supersymmetric solutions that have timelike and null Killing vectors. Since the six-dimensional origins (Romans, 1986) of this theory are known (Değer et al., 1998; de Boer, 1999; Balasubramanian et al., 2001; Nicolai and Samtleben, 2003b), this would be an important contribution to the literature.

Besides gSUGRA theories, we also want to consider off-shell supergravity models admitting an AdS vacuum in this project. Such models contain auxiliary fields subject to algebraic equations and their supersymmetry algebra closes without using the field equations. These allow for more freedom in the sense that one may construct supersymmetric invariant pieces with different number of derivatives and consider additions of them with arbitrary coefficients. Moreover, the algebraic and differential identities coming from the Killing spinor analysis remain valid for any combination of higher derivative invariants. In particular, we are planning to work on N=2 off-shell supergravity theories (Alkaç et al., 2015a, 2015b; Değer et al., 2014; Değer and Moutsopoulos, 2016). 

We expect that the duration of this project will be three years in total for studying the open problems briefly mentioned above. In addition to several high-quality international publications, we also aim to have one PhD and one MSc thesis as our project’s outcome. This project is a natural continuation of our quite successful 3-year old Tübitak-1001 project, titled “Gauged Supergravities in Three Dimensions” with project number 113F034, that will be completed by September 2016. Most of the problems briefly mentioned above are direct consequences of our current project.