String theory origin of constrained multiplets

Published for SISSA by Springer Received: August 1, 2016 Accepted: September 1, 2016 Published: September 12, 2016 Renata Kallosh,a Bert Vercnockeb and Timm Wrasec a Department of Physics, Stanford University, 382 Via Pueblo, Stanford, CA 94305, U.S.A. b Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands c Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8–10, A-1040 Vienna, Austria E-mail: , , Abstract: We study the non-linearly realized spontaneously broken supersymmetry of the (anti-)D3-brane action in type IIB string theory. The worldvolume fields are one vector Aµ , three complex scalars φi and four 4d fermions λ0 , λi . These transform, in addition to the more familiar N = 4 linear supersymmetry, also under 16 spontaneously broken, nonlinearly realized supersymmetries. We argue that the worldvolume fields can be packaged into the following constrained 4d non-linear N = 1 multiplets: four chiral multiplets S, Y i that satisfy S 2 = SY i = 0 and contain the worldvolume fermions λ0 and λi ; and four chiral multiplets Wα , H i that satisfy SWα = S D̄α̇ H̄ ı̄ = 0 and contain the vector Aµ and the scalars φi . We also discuss how placing an anti-D3-brane on top of intersecting O7-planes can lead to an orthogonal multiplet Φ that satisfies S(Φ − Φ̄) = 0, which is particularly interesting for inflationary cosmology. Keywords: D-branes, Superstring Vacua, Supersymmetry Breaking ArXiv ePrint: 1606.09245 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP09(2016)063 JHEP09(2016)063 String theory origin of constrained multiplets Contents 1 2 Evidence from different N = 1 truncations 2.1 Goldstino plus vector 2.2 Goldstino plus fermions 2.3 Goldstino plus complex scalar 3 3 5 5 3 The D3-brane action and its non-linear supersymmetries 3.1 The generic D3-brane action and its symmetries 3.2 The DBI-VA model 3.3 Identifying the non-linear transformations 3.4 A different κ-symmetry gauge fixing 7 7 9 10 13 4 Discussion 15 A Expanding β 16 B Non-linear realizations and constrained multiplets B.1 Non-linear realization of supersymmetry B.2 Superfield constraints to remove components B.2.1 Goldstino B.2.2 Fermions B.2.3 Scalars B.2.4 Vector 17 17 18 18 19 19 20 1 Introduction In attempts to describe the observable universe one finds that non-linearly realized supersymmetry in string theory and supergravity is a helpful tool. Good examples are de Sitter vacua in 4-dimensional N = 1 supergravity that describe dark energy. Such de Sitter supergravities, without scalar fields, were recently constructed in [1–4] by promoting the Volkov-Akulov (VA) model with global non-linear supersymmetry [5, 6] to supergravity with local supersymmetry. The corresponding chiral nilpotent Goldstino multiplet S [7–13] constrained by the condition S2 = 0 , (1.1) is equivalent to the VA theory via a non-linear local field redefinition [14]. This kind of multiplet is present on a D3-brane as well as on an anti-D3-brane [15] in a gauge of the local fermionic κ-symmetry where the Wess-Zumino term vanishes [16, 17]. This is in agreement with the fact [18] that in the absence of scalars and vectors the D3-brane as well as the anti-D3-brane have a Volkov-Akulov type action. –1– JHEP09(2016)063 1 Introduction 1 The coupling of S and Y i to supergravity was studied in [43]. A general approach to couple constrained superfields to gravity was developed recently in [44] and a universal way of obtaining constrained multiplets was derived in [45]. 2 A related class of κ-symmetry gauges for a D3-brane in a supergravity backgrounds was studied in [46–52]. –2– JHEP09(2016)063 In the context of the KKLT construction of de Sitter vacua in string theory [19] a different choice of κ-symmetry fixing, compatible with an anti-D3-brane placed on the top of an orientifold plane, is useful. In this more appropriate gauge, the analysis of [20–24] allows an interpretation of the KKLT construction of de Sitter vacua within a four-dimensional supersymmetric theory. This builds upon early investigations on supersymmetry breaking in string theory [25–32], and it transpires that the low energy effective action for an antiD3-brane on top of an O3-plane in a supersymmetric GKP background [33] is just the VA action [20, 21]. Such an anti-D3-brane on top of an O3-plane can arise in many warped throats [22–24] (including the Klebanov-Strassler throat [34]). An early argument that D-branes have to be associated with spontaneously broken, rather than explicitly broken supersymmetry, was given by J. Polchinski in his book [35] (brane actions as effective descriptions of partially spontaneously broken supersymmetry even go back to [36]). A more specific prediction with regards to anti-D3-branes was presented in a series of papers by S. Kachru and his collaborators in [37–40]. It was argued there that the system must be viewed as spontaneous breaking of supersymmetry, because it can tunnel to a supersymmetric vacuum. Recent holographic studies point towards spontaneous breaking as well [41]. The nilpotent N = 1 multiplet is the beginning of the explicit realization of the expectation in [35, 37–40] and it is natural to look at anti-D3branes to find other constrained superfields which transform under the non-linearly realized spontaneously broken N = 1 supersymmetry. This is the goal of this paper. In [42] it was already shown, using the same type of gauge-fixing as in [20, 21], that, in absence of vectors and scalars on the brane, in addition to a 4d nilpotent multiplet S 2 = 0, there is also a triplet of ‘scalar-less’ chiral multiplets Y i present, that satisfies SY i = 0, i = 1, 2, 3. It was also conjectured there that the worldvolume vector field Aµ and the transverse complex scalars φi can be packaged into constrained multiplets that satisfy SWα = 0 and SDα̇ H̄ ı̄ = 0, respectively.1 The purpose of this paper is to establish that indeed all world-volume fields on the anti-D3-brane, which under linearly realized supersymmetry represent a 4d N = 4 vector multiplet, a vector Aµ , 6 real scalars φIr and 4 spinors λA , can be packaged into constrained N = 1 superfields with a non-linearly realized supersymmetry. For this purpose we will use the κ-symmetry gauge fixing where the Wess-Zumino term vanishes [16, 17].2 This leads to the gauge-fixed Dirac-Born-Infeld-Volkov-Akulov action which is the same for the D3brane and for the anti-D3-brane. It has 16 linear supersymmetries, the usual linear N = 4 supersymmetry preserved by an (anti-) D3-brane in flat space, and another 16 non-linear supersymmetries of Volkov-Akulov type that are spontaneously broken. We are, in particular, motivated here by the issue of cosmology, where constrained superfields proved to be very useful [15, 53, 54], see also the reviews [55–57]. We would like to find out which particular N = 1 superfields, in addition to S 2 = 0, live on a D- (...truncated)