1. Dmitri Sorokin, INFN Padova Section based on arXiv:1511.03024 with I. Bandos, L. Martucci and M. Tonin VII Round Table Italy-Russia, Dubna, 25-27 November 2015

2.  Recent(2014-15) burst of interest In applications to cosmology and inflation (S. Ferrara talk).  Recent (2014-15) burst of interest in the study of spontaneous breaking of local supersymmetry and the emergence of positive cosmological constant due to Volkov- Akulov goldstini in D=4 sugra. In applications to cosmology and inflation (S. Ferrara talk).  One of the aims:  One of the aims: provide a solid 4D effective field theory ground for cosmological models constructed in the framework of 10D string theory (e.g. KKLT-like models involving anti- D3-branes to generate dS vacua)  Most of the constructions of D=4 sugra with spontaneously broken susy use constrained (nilpotent) superfield description of the Volkov-Akulov goldstini (scalar partners are composed of goldstino bilinears) Antoniadis, Dudas, Ferrara, Kehagias, Farakos, Kallosh, Linde, Porrati, Sagnotti, dall’Agata, Zwirner; Bergshoeff, Freedman, Kallosh, Van Proeyen; Hasegawa,Yamada; Kuzenko; Antoniadis, Markou;…  Earlier related studies  Earlier related studies of local susy breaking & super-Braut-Englert-Higgs effect in sugra: Volkov & Soroka ‘73, Deser & Zumino ‘77,…, Lindstrom & Rocek ‘79, Samuel & Wess ‘83, Ivanov & Kapustnikov ’84 -’90,… Our goal: Our goal: to generate susy breaking by coupling supergravity to the original VA model which is the most straightforwardly related to the 3-brane constructions 2

3. ORIGINAL VOLKOV-AKULOV MODEL ’72 AS A 3-BRANE  Goldstino as the manifestation of susy breaking 3 Volkov-Akulov superspace SUSY transform. SUSY invariant VA 1-form: spinor field appears on the brane worldvolume non-linear susy transform. Consider a 4d worldvolume (of a 3-brane) placed in superspace susy breaking parameter 3-brane tesnion (Hughes & Polchinski ‘86, Kallosh ’98)

4.  volume of 4d surface in superspace 4 From general theory of non-linear realizations it follows (Ivanov & Kapustnikov ’77) that the VA model is universal: all models of spontaneous susy breaking involving goldstino should be related to the VA model by a non-linear transformation of Kuzenko & Tyler 10’-11’ whose general form was obtained only quite ricently Kuzenko & Tyler 10’-11’ Constrained superfield realization of the VA goldstino (Ivanov, Kapustnikov ’77; Rocek 78’,…,Samuel, Wess ‘83; Casalbuoni et al. 89’,…, Komargodski, Seiberg 09’,..) chiral superfield constrained by: Example:

5. Volkov -Soroka model ‘73 Deser, Zumino ‘77,…, Lindstrom, Rocek ’79; Ferrara et al; Samuel, Wess ’83; Ivanov, Kapustnikov ’84 -’90,… 5 2014-15 2014-15 Antoniadis, Dudas, Ferrara, Kehagias, Farakos, Kallosh, Linde, Porrati, Sagnotti, dall’Agata, Zwirner: Bergshoeff, Freedman, Kallosh, Van Proeyen; Hasegawa,Yamada; Kuzenko; Antoniadis, Markou;…, Schillo, van der Woerd, Wrase Recent constructions are mainly based on the superconformal approach to supergravity () (see Freedman and Van Proeyen Book for a review) e.g. Samuel, Wess’83 Alternatively, one can use «Poincarè» superspace and superfield forlmalism ( e.g. Samuel, Wess’83 ) Kahler potentialsuperpotential chiral measure SYM kin.term Lagrange multiplier - sugra constraints

6.  component pure supergravity + VA goldstino to the second order in χ (upon integrating out the auxiliary fields) (Bergshoeff at. all; Hasegawa,Yamada ‘15) 6 AdS sugra action ( Townsend ‘77 ) In the unitary gauge the remaining first line in the action describes a massive gravitino field coupled to gravity with the cosmological constant

7.  For finding a more direct relation of 4D effective theories with spontaneously broken local susy to 10D stringy constructions which use anti-D3-branes to induce a de Sitter vacuum, it may be useful to couple to supergravity the original Volkov-Akulov 3-brane model, without resorting to the constrained superfields  The action has a suggestive geometric form of the sum of three different volumes in N=1, D=4 curved superspace 7 pullback on 3-brane worldvolume in the static gauge for worldvolume diffeomorphisms: All the couplings of VA goldstino to sugra fields is encoded in. In the WZ gauge we obtain:

8. 8 differ from constrained superfield action superfield action, but should be related by non-linear field redefinitions a la Kuzenko & Tyler

9. 9 1973 Volkov & Soroka o in 1973 Volkov & Soroka gauged non-linearly realized super-Poincaré group Introduced gauge fields associated with local super-Poincarè generators Constructed translation and susy invariant objects with the use of Stueckelberg-like fields: generalization of VA one-form Volkov-Soroka action: ω - is independent field → constant c can be absorbed by Ψ → cΨ= Ψ’ = “de Sitter sugra“ action - does not transform under susy

10.  Freedman, van Nieuwenhuizen & Ferrara ‘76  Deser & Zumino ‘76 10 SYSY transformations: The constructed models of local susy breaking demonstrate that coupling of N=1 D=4 supergravity to Volkov-Akulov goldstino in one way or another results (in the unitary gauge) in the Volkov-Soroka action

11.  We have reviewed models of matter coupled N=1, D=4 supergravity with spontaneously broken supersymmetry triggered by the presence of VA goldstini.  Spontaneous SUSY breaking generates a positive contribution to the vacuum energy thus allowing for the appearance of de Sitter vacua in the theory.  This makes these supergravity models useful and relevant for cosmological and inflationary model building  The realization of the local susy breaking with the use o the space-filling Volkov-Akulov 3-brane coupled to gravity and matter supermultiplets may by useful for finding a more direct connection of these 4d effective field theories to string theory constructions involving anti-D3-branes, like the KKLT model and its refinings 11