Three-Form Matter and Supergravity in String Compactifications
Investigating the role of 3-form gauge fields in N=1, D=4 supergravity and matter supermultiplets, exploring their connection to cosmological constant, neutralization mechanisms, and susy breaking. The duality between 3-form fields and cosmological constant is discussed, along with a novel supersymmetric description based on non-linear duality between chiral and 3-form supermultiplets. String compactifications with fluxes play a key role in providing a phenomenologically relevant realization of these phenomena in effective 4D theories.
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N=1, D=4 Three-Form Matter and Supergravity, and String Compactifications with Fluxes Dmitri Sorokin INFN, Padova Section based on arXiv:1706.09422 with F. Farakos, S. Lanza and L.. Martucci SQS, Dubna, July 31, 2017
Introduction and motivation 3-form gauge fields ?3in D=4 do not carry local dynamical degrees of freedoms. On-shell values of their 4-form field strengths ?4=d?3are constant. When coupled to gravity, ?4induce dynamically a non-zero (positive or negative) contribution to the cosmological constant (Duff & van Nieuwenhuizen; Aurilia, Nicolai & Townsend 80) When coupled to membranes, ?3provide a mechanism for neutralizing the cosmological constant via membrane nucleation (Brown & Teitelboim 87) Some models of inflation involve ?3coupled to an inflaton field (e.g. Bousso & Polchinski 00, Kaloper & Sorbo 08) ?3gauge symmetry improves quantum behavior of scalar field potentials. ?3may play the role of auxiliary fields of off-shell supergravity and matter supermultiplets (Stelle & West 78, Ogievetsky & Sokatchev 78, Gates & Siegel 81) and induce spontaneous susy breaking. 2
Introduction and motivation Plenty of three-form gauge fields appear in D=4 upon ?4 ?6 compactifications of 10D type II string theories which contain p-form gauge fields ??(p=-1,0, ,9) with ??+1= ?9 ? Corresponding effective 4D theories with A may provide a phenomenologically relevant realization of phenomena associated with ?3(cosmological constant and its neutralization, corresponding cosmological models, susy breaking, etc.) In particular, Bielleman, Ibanez & Valenzuela 15 assumed that a number of A appearing in D=4 from string compactifications with fluxes may be associated with auxiliary fields of N=1, D=4 chiral matter and supergravity multiplets which are part of the 4D effective field theory The main goal of this talk is to prove this assumption and provide a new manifestly supersymmetric description of the effective 4D theory. It is based on a novel non- linear duality between chiral and 3-form supermultiplets (F. Farakos, S. Lanza, L.. Martucci & D.S. arXiv:1706.09422) 3
Duality between ?3and the (cosmological) constant = = 4 , ( ) F A A x [ ] [ ] mnpq m npq mnp m np It is convenient to work with Hodge-dual quantities: ( ) 1 1 = = = mnpq m * F F F F g A 4 mnpq m ! 4 g g 1 = = m mnpq * A A A A 1 3 npq ! 3 g ( boundary term = F ) = 2 m L g F g A F 1 Lagrangian: m 2 = = 0 , | 0 F c F Field equations: m bound g g = = 2 2 2 L c g c c On-shell Lagrangian: on shell 2 2 4
Duality between chiral and 3-form supermultiplets = + + = 2 ( , ) ( ) ( ) ( ), 0 x x x f x D 1 Chiral N=1 superfield: L L L L 2 = + + 2 2 2 c . . L d d c d c Polonyi 77 Lagrangian: = + + + m m L i f f cf f c = + m i f comp m m m = = = c c m m , f c f c L i On shell: comp m m We would like to generate the parameter c dynamically by dualizing finto F ( ) ( ) = + m m m m L i F F A F A F dual m m m m A = = ( , , * = ) A Double 3-form supermultiplet (Gates 81) : 1 3 = m S D D f F A 1 Special chiral superfield: in which s m 4 = = 2 0 D D is a complex linear superfield where 5
Systematic manifestly-susy dualization procedure = + + 2 2 2 . . c c L d d c d Replace the Lagrangian = + + = 2 2 2 2 2 . ., c 0 L d d d X d d X D c D X with = = : 0 D X X c = = Equations of motion: 2 2 : ( ) X D D D S 1 1 4 4 = 2 : X D S 1 4 = + 2 2 L d d S S L Dual Lagrangian: dual bound ( ) ( = + + 2 2 2 2 ) . . c c L d D d D S 1 1 bound 4 4 Note that the dual Lagrangian does not contain the superpotential and the constant parameter 6
Effective N=1, D=4 theories of type IIA string compactifications with RR fluxes D=10 Type IIA bosonic field content = , , gMN H dB NS-NS sector: 3 2 = 9 , 7 , 5 , 3 , 1 , " 1 = " * A p F F RR sector: + 1 10 9 p p p = + B F A ( ) d F e 2 0 = = * + Romans - F mB mass F m F 0 10 = = * F dA 2 1 2 8 + = + = * F dA dA B mB B F 1 4 3 1 2 2 2 6 2 ........ 7
Effective N=1, D=4 theories of type IIA string compactifications with RR fluxes = + B F A ( ) d F e Compactification on ?4 ??3and ?4fluxes on ?4, ~ *( 6 40 0 4 F m F i i CY 2 0 = = + 0 | ) F m F dots stand for contributions of ?2 CY volume form M ~ ~ = = + i i | *( ) 2 2 4 4 ~ e = = + i i | *( ) F e F i=1, ,n dimension of ?2?? ,?4?? i i 4 2 , basis of closed forms 4 4 4 2 CY i i ~ = + | *( ) F 4 0 6 M 4 ~ F ~ F ~ F = = 0 A i = = 0 A i ( , ), ( , ) e e e m m m ( , ), ( , ) F F F 4 4 4 4A 40 4i 0 A i 2n+2 ?4field strengths on ?4 2n+2 internal quantized constant fluxes = = ~ F = + i i i i i ( ) , ( ) , ( ) ( ) ( ) J v x B b x x v x ib x Relevant 4D scalar fields: 2 2 2 2 i i ~ = = A A i i i i * ( , , , ), * ( , , , ) F e m e m 4D on-shell relations: 4 4A A 8
Effective N=1, D=4 Lagrangian for compactifications with internal CY fluxes (c.f. Grimm & Louis 04) Simplified setup in the rigid supersymmetry case ( ) = + + + 2 2 2 A B A ( , ) ( ) . . c c L d d K d e m G A BA = = 0 A i A A A ( , ) ( , , ), ( ) ( ) f G G AB BA 0 - conformal compensator in the super-Weyl invariant description of N=1 SUGRA We would like to find a dual formulation in which constants ?? and ?? are generated dynamically by the on-shell values of ?4field strengths X M d d = + + 2 2 2 A ( , ) . . c c L d d K d X A 2 2 AB ( , ( ) )( ) X D D A A B B ( ) 1 = AB Im ( ) M G AB 9
Dual Lagrangian with double three-form superfields ( 0 ) ( : = B B A X X M D ( ) ( : S O X A A = ) = ) + AB B ( ) X e m G A A BA = = 2 A AB A : ( , ( ) ) , X D M S D 1 A B B B B 4 A = + 2 2 ( , ) L d d K S S bound L dual ~ L = = + A B under 0 = D ( ) S G L Gauge invariance: A D A AB ~ L ~ L = = = 2 2 2 2 A A , 0 0 D L L D A A Bosonic components of ??: S : | = = A A AB ( = , ) | M B ( ) ~ = + 2 A A AB C | : ( , ) ( ) D S f M F G F 1 s B BC 4 real dual 4-form field strengths of ?1= ?3 of ? 10
On-shell values of the 4-form field strengths Component Lagrangian (bosonic sector) = + + A m B A B ( , ) ( , ) comp L K K f f bound L AB m AB S S ( ) 2 ( , ) ~ K = = + A AB m B mC ( , ) , ( , ) ( ) K f M A G A AB s m BC m A B ~ m BA A , ~ = mC Equations of motion of give ~ F the same as in type IIA flux compactifications = A A ( , , , ), ( , , , ) F ?? and ??are integration constants e m e m A A = A B ( , ) ( , ) V K f f bound L Effective scalar potential AB S S 11
Conclusion We have found the non-linear duality transformation of interacting N=1, D=4 matter and supergravity systems which produced their dual description in terms of interacting three-form chiral matter and supergravity multiplets By applying this duality procedure to effective N=1, D=4 theories describing compactifications of type IIA superstring on CY manifolds with Ramond-Ramond fluxes we have proved that 4D four-form field strengths of the RR fields in these models act as auxiliary fields of certain chiral matter supermultiplets and old minimal supergravity. We have thus obtained a new manifestly supersymmetric Lagrangian description of these effective theories in terms of three-form gauge fields. 12
Outlook To apply this dualization procedure to more general, and phenomenologically more relevant, compactification setups in type IIA and IIB theories which include the NS-NS fluxes D-branes (which are required for introducing a gauge field sector into this construction) In the context of the cosmological constant problem, to study within these supersymmetric effective 4D theories the coupling of three-form supermultiplets to membranes generalizing results of Ovrut & Waldram 97 and Bandos & Meliveo 10 Spontaneous susy breaking, non-linear realizations and constrained 3-form superfields (Farakos et. al. 16, E. Buchbinder and Kuzenko 17 ) 13
