Andrea Borghese The role of photons in Supergravity

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1 Based on: A.B., A.Guarino, D.Roest, [ ] A.B., G.Dibitetto, A.Guarino, D.Roest, O.Varela, [ ] A.B., A.Guarino, D.Roest, [ ]

2 PLAN Describe the spin-1 sector in supergravity theories Describe the genesis of a new class of maximal supergravities Holography WHY? desitter hunting N =8 Finiteness Higher dimensional origin

3 MAXWELL S EQUATIONS E = ρ e 0 Coulomb s law E + B t = 0 Faraday-Neumann-Lenz law B = 0 absence of magnetic charges B + µ 0 0 E t = µ 0 j m Ampère s law (modified by Maxwell)

4 COVARIANT MAXWELL S EQUATIONS E + B t E = ρ e 0 = 0 B = 0 F 0i E i, F ij ijk B k J µ i ρ e, j m inhomogeneous equations ν F µν = J µ B + µ 0 0 E t = µ 0 j m homogeneous equations µνρσ ν F ρσ = 0

5 COVARIANT MAXWELL S EQUATIONS Locally F µν = µ A ν ν A µ F 0i E i, F ij ijk B k J µ i ρ e, j m A µ Φ, A i B = A E = Φ A t inhomogeneous equations ν F µν = J µ homogeneous equations µνρσ ν F ρσ = 0

6 EM DUALITY 1.0 ν F µν = J µ µνρσ ν F ρσ = 0 ν F µν = 0 F µν i 2 µνρσ F ρσ ν F µν = 0

7 EM DUALITY 1.0 ν F µν = J µ µνρσ ν F ρσ = 0 ν F µν = 0 F µν i 2 µνρσ F ρσ ν F µν = 0 this set of equations is invariant if we rotate the field strength into each others F µν i F µν non local field redefinition E B, B E

8 EM DUALITY 2.0 F IJ µν 2 [µ A IJ ν] G µν IJ e µνρσ L F IJ ρσ I,J,K,L 8 of SU(8) [µ F IJ νρ] = 0 [µ G IJ νρ] = 0 BI EoM

9 EM DUALITY 2.0 F IJ µν 2 [µ A IJ ν] G µν IJ e µνρσ L F IJ ρσ I,J,K,L 8 of SU(8) [µ F IJ νρ] = 0 [µ G IJ νρ] = 0 BI EoM invariant under F IJ µν G µν IJ S F IJ µν G µν IJ S Sp(56, R) Sp(56, R) transformations modify the Lagrangian but leave invariant the set of EoM plus BI

10 DUALITY IN SUPERGRAVITY 128 = 128 d.o.f. field spin [ Cremmer, Julia 79 ; de Wit, 82 ] 1( 2) a e µ 2 8( 2) I ψ µ 3/2 28 ( 2) IJ A µ 1 56 ( 2) χ IJK 1/2 70 ( 1) φ IJKL 0

11 DUALITY IN SUPERGRAVITY 128 = 128 d.o.f. field spin [ Cremmer, Julia 79 ; de Wit, 82 ] 1( 2) e µ a 2 8( 2) ψ µ I 3/2 28 ( 2) A µ IJ 1 56 ( 2) χ IJK 1/2 70 ( 1) φ IJKL 0 φ IJKL form a coset space E 7 SU(8) Cremmer & Julia discovered that the scalar fields coming from sphere reduction of eleven dimensional Supergravity transform under E 7

12 DUALITY IN SUPERGRAVITY 128 = 128 d.o.f. field spin 1( 2) a e µ 2 8( 2) I ψ µ 3/2 28 ( 2) IJ A µ 1 A µ M [ Cremmer, Julia 79 ; de Wit, 82 ] electric vectors can be supplemented with their magnetic counterpart E 7 M =1,...,56 56 ( 2) χ IJK 1/2 70 ( 1) φ IJKL 0 φ IJKL form a coset space E 7 SU(8) Cremmer & Julia discovered that the scalar fields coming from sphere reduction of eleven dimensional Supergravity transform under E 7

13 CHARGES IN SUPERGRAVITY Some vector fields in the spectrum of the theory can be used to promote a subgroup of the global duality group to a local gauge group G g E 7 schematic form of covariant derivatives U(1) covariant derivatives D µ µ A µ M Θ M α t α D µ µ A µ q e generators of the duality group t α adj E 7 = 133 embedding tensor (constant set of charges) selects which linear combination of vectors must be used to gauge a transformation associated to a generator of the duality group ΘM α E 7 covariant formulation of the theory [ de Wit, Samtleben, Trigiante ]

14 EM DUALITY vs DUALITY E 7 transformations modify the Lagrangian but leave invariant the set of EoM plus BI Sp(56, R) rotate among themselves electric and magnetic vectors and also modify the set of EoM plus BI thus changing the physics E 7 Sp(56, R)

15 EM DUALITY vs DUALITY E 7 transformations modify the Lagrangian but leave invariant the set of EoM plus BI Sp(56, R) rotate among themselves electric and magnetic vectors and also modify the set of EoM plus BI thus changing the physics E 7 Sp(56, R) U(1) EM rotation is NOT inside the duality group of the theory! F µν i F µν

16 EM DUALITY vs DUALITY U(1) EM rotation is NOT inside the duality group of the theory! These transformations amount to a different choice of physical vectors among the 56 electric + magnetic ones

17 EM DUALITY vs DUALITY U(1) EM rotation is NOT inside the duality group of the theory! These transformations amount to a different choice of physical vectors among the 56 electric + magnetic ones α SO(8) gauged supergravity fix specific values for Θ M There is a one parameter family of gauged maximal supergravity theories SO(8) ω phase [ Dall Agata, Inverso and Trigiante, 12 ]

18 NEW THEORIES WITH DIFFERENT FEATURES different vacuum structure vacuum = {configuration of fields which solves the EoM}

19 NEW THEORIES WITH DIFFERENT FEATURES different vacuum structure vacuum = {configuration of fields which solves the EoM} new vacua (some of them stable and non-supersymmetric) have been found in the SO(8) gauged theories [ A.B., A.Guarino and D. Roest, 12; A.B., G.Dibitetto, A. Guarino, D.Roest and O.Varela, 12; A.B., A.Guarino and D.Roest, 13 ]

20 NEW THEORIES WITH DIFFERENT FEATURES different vacuum structure vacuum = {configuration of fields which solves the EoM} new vacua (some of them stable and non-supersymmetric) have been found in the SO(8) gauged theories [ A.B., A.Guarino and D. Roest, 12; A.B., G.Dibitetto, A. Guarino, D.Roest and O.Varela, 12; A.B., A.Guarino and D.Roest, 13 ] In the SO(4, 4) gauged theories ds vacua have been found which solve the η - problem of supergravity m 2 tach V = η = O(1) [ Dall Agata and Inverso, 12 ]

21 THANK YOU!