BRANE INDUCED SUSY BREAKINGAND DE SITTER SUPERGRAVITY

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

MOTIVATION 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: 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 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: to generate susy breaking by coupling supergravity to the original VA model which is the most straightforwardly related to the 3-brane constructions

3

ORIGINAL VOLKOV-AKULOV MODEL ’72 AS A 3-BRANE Goldstino as the manifestation of susy breaking

termsnonlinear )()(')( xxx

• Volkov-Akulov superspace 1,2, 3,2,1,0 ),,( mxz mM

matrices Pauli , , , mmmm iix SUSY transform.

SUSY invariant VA 1-form: mmmm iddidхЕ 0

)( )( ),()( 11 xfxxfx spinor field appears on the brane worldvolume

)()()()( 11 xiffxxfx mmm

mm

non-linear susy transform.

),(

mx

)(, xxm

Consider a 4d worldvolume (of a 3-brane) placed in superspace )4,4(M

susy breaking parameter3-brane tesnion

(Hughes & Polchinski ‘86, Kallosh ’98)

4

VOLKOV-AKULOV ACTION volume of 4d surface in superspace

))(()]([)(

....).())((det

022

0

2 0

420000!4

44

2

xEdxifdxddifdxE

ccifxExdfEEEES

mn

nmnn

mmn

nmmmm

mm

M

mnmnpq

qp

M

nmfVA

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 whose general form was obtained only quite ricently Kuzenko & Tyler 10’-11’

)(x

Constrained superfield realization of the VA goldstino(Ivanov, Kapustnikov ’77; Rocek 78’,…,Samuel, Wess ‘83; Casalbuoni et al. 89’,…, Komargodski, Seiberg 09’,..)

FFFFf

FFixdccSdzdfSSzdS m

mLNS )(.. 448

)()()(),( 221

LLLL xFxxsxS chiral superfield constrained by:F

sS 02

Example:

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COUPLING VA GOLDSTINO TO N=1, D=4 SUGRA, SUPER-BEHIGGS EFFECT AND DE SITTER VACUA

• Volkov -Soroka model ‘73• Deser, Zumino ‘77,…, Lindstrom, Rocek ’79; Ferrara et al; Samuel, Wess ’83; Ivanov, Kapustnikov ’84 -’90,…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)

Alternatively, one can use «Poincarè» superspace and superfield forlmalism (e.g. Samuel, Wess’83)

262

),,,(84

3 ),g( ),( Ber 232

2 SSSWzdeEzdS LmSSeKA

MNS

V

WWtrE

Kahler potential superpotentialchiral measure SYM kin.termLagrangemultiplier

Sugra coupled to scalar Φ(z), vector V(z), (z), and goldstino S(z) superfields

abba

aab

abaaa

AB

BAAAM

MA

TEEzREiEzGEiETEiET

EdETzEdzE

21

)()()( ,

),(

- sugra constraints),,( mM xz

6

COUPLING VA GOLDSTINO TO N=1, D=4 SUGRA, SUPER-BEHIGGS EFFECT AND DE SITTER VACUA component pure supergravity + VA goldstino to the second order in χ

(upon integrating out the auxiliary fields) (Bergshoeff at. all; Hasegawa,Yamada ‘15)

),...,(.

.2

26),(

84234

24

22242

1

2

2

2

O

R

ccexd

ccmifexd

fmmeexdS

fcabc

ba

aai

aa

bab

anmlmn

l

AdS sugra action (Townsend ‘77)

In the unitary gauge the remaining first line in the action describes a massivegravitino field coupled to gravity with the cosmological constant

0

2

22 3

mf

7

VOLKOV-AKULOV BRANE COUPLED TO SUPERGRAVITY 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

))((det Ber 4262

84

322

Ma

iLmA

MVASG zdfzdEzdS E

),,()(),,()(),,()())((E

xExExExz a

ia

iam

mi

Mai

pullback on 3-braneworldvolume

in the static gauge for worldvolume diffeomorphisms: )()( ,)( 1 xfx mi

im

All the couplings of VA goldstino to sugra fields is encoded in . In the WZ gauge we obtain:))((E xzMai

..)()()(

))( ())((2.).)(2()()(

2643

][

ccTTdOO

xRRexGeGeccixixedxzE

cdabcdiab

bi

acab

bac

caaam

mMa

RiGiGieeT babacc

banmn

bm

aab )~(2)~(2 ][][][][

8

Integrating out the auxiliary fields R(x) and (x) from the action we get

VOLKOV-AKULOV BRANE COUPLED TO SUPERGRAVITY

),...,(.

.2

26),(

84234

24

22242

13

2

2

2

O

R

ccexd

ccmifexd

fmmeexdS

fcabc

ba

aai

aa

bab

anmlmn

lbrSG

differ from constrainedsuperfield action, but should be related by non-linear field redefinitionsa la Kuzenko & Tyler

9

VOLKOV-SOROKA MODEL OF THE SUPER-HIGGS EFFECT

o in 1973 Volkov & Soroka gauged non-linearly realized super-Poincaré group • Introduced gauge fields associated with local super-Poincarè generators

)( )()(

)( )()(

)( )()(

xlxdxx

xxdxx

xaxedxxe

ababm

mab

mm

aam

ma

local super-Poincarè transform: ΩG+ dG(x)

• Constructed translation and susy invariant objects with the use of Stueckelberg-like fields:

)()()()(

).2()( )()()(

xxx

ccixXxexE aaaaa

generalization of VA one-form

)()()( ),()( aaaa ixaxXxx

Volkov-Soroka action: ..)()( det 24

21

2 ccmcRExdS bab

anmlmn

lVS

ω - is independent field → constant c can be absorbed by Ψ → cΨ= Ψ’

= “de Sitter sugra“ action

)( )()(

)()()()(

)()(2)( )()(

xlx

xlxx

xleixaxe

abab

ab

baaaa

- does not transform under susy

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GENUINE N=1, D=4 SUPERGRAVITY Freedman, van Nieuwenhuizen & Ferrara ‘76 Deser & Zumino ‘76

..)()(det42

12 ccRexdS nm

lmnlSG

.. )(

)()(

)(2)(

][ ccieix

xx

ixe

cnpbpnc

am

abnppnm

abm

aaa

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

VSVASG SSS 0|)(

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CONCLUSION 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

))((det ))(,,(

)g( )( Ber 42

62

),(84

3matter 2

32

2

Mai

LmeK

VASG

zVdf

WzdeEzdSV

F

E WWtr