Supergravitational Heterotic Galileons

Transcription

1 Supergravitational Heterotic Galileons Rehan Deen University of Pennsylvania String Pheno 2017, Virginia Tech July 6, / 30

2 Introduction Collaboration on bouncing cosmology with R.D., Burt Ovrut, Anna Ijjas, Paul Steinhardt, see also R.D., Burt Ovrut] Bouncing cosmologies review: Brandenberger, Peter 16] are an alternative to inflationary scenario : early universe is contracting (ȧ < 0) then bounces and begins expanding (ȧ > 0). Initial singularity problem avoided Bounce can be classical and avoid super-planckian scales Needs Ḣ > 0 - not satisfied by matter, radiation or CC dominated universe. This requires matter which violates Null Energy Condition (NEC): T µνn µ n ν 0, for n µn µ = 0 ρ + p 0 2 / 30

3 Introduction Collaboration on bouncing cosmology with R.D., Burt Ovrut, Anna Ijjas, Paul Steinhardt, see also R.D., Burt Ovrut] Bouncing cosmologies review: Brandenberger, Peter 16] are an alternative to inflationary scenario : early universe is contracting (ȧ < 0) then bounces and begins expanding (ȧ > 0). Initial singularity problem avoided Bounce can be classical and avoid super-planckian scales Needs Ḣ > 0 - not satisfied by matter, radiation or CC dominated universe. This requires matter which violates Null Energy Condition (NEC): T µνn µ n ν 0, for n µn µ = 0 ρ + p 0 2 / 30

4 NEC violation - ghost condensate, P(X = ( φ) 2 ) theories, and Galileons Galileons: higher derivative scalar theories with 2nd order e.o.m L 1 = π L 2 = 1 2 ( π)2 L 3 = 1 2 ( π)2 π L 4 = 1 2 ( π)2 ( ( π) 2 π,µνπ,µν ) L 5 = 1 2 ( π)2 ( ( π) 3 + 2π,µν π,νρ π,ρ µ 3 ππ,µνπ,µν ) 3 / 30

5 NEC violation - ghost condensate, P(X = ( φ) 2 ) theories, and Galileons Galileons: higher derivative scalar theories with 2nd order e.o.m Galileon theories Dvali, Gabadadze, Poratti; Nicolis, Rattazi, Tricherini; De Rham, Tolley; Deffayet et al.; Trodden et al,;... ] can violate the NEC Khoury et al.; Koehn, Lehners, Ovrut], and give rise to a stable classical bounce Vikman et al.; Ijjas, Steinhardt; Koehn, Lehners, Ovrut; ] Galileons arise as a description of the world-volume action of a probe brane in higher dimensionsde Rham, Tolley; Goon, Hinterbichler, Trodden... ] For instance Regular Galileons describe a probe 3-brane in 5d Minkowski space Conformal Galileons describe a probe 3-brane in AdS 5 They inherit a non-linearly realized symmetry from the higher dimensional space, e.g Symmetry: L conformal 3 = 1 2 ( π)2 π 1 4 ( π)4 δˆπ = 1 x µ µˆπ, δ µˆπ = 2x µ + x 2 µˆπ 2x µx ν ν ˆπ. 4 / 30

6 NEC violation - ghost condensate, P(X = ( φ) 2 ) theories, and Galileons Galileons: higher derivative scalar theories with 2nd order e.o.m Galileon theories Dvali, Gabadadze, Poratti; Nicolis, Rattazi, Tricherini; De Rham, Tolley; Deffayet et al.; Trodden et al,;... ] can violate the NEC Khoury et al.; Koehn, Lehners, Ovrut], and give rise to a stable classical bounce Vikman et al.; Ijjas, Steinhardt; Koehn, Lehners, Ovrut; ] Galileons arise as a description of the world-volume action of a probe brane in higher dimensionsde Rham, Tolley; Goon, Hinterbichler, Trodden... ] For instance Regular Galileons describe a probe 3-brane in 5d Minkowski space Conformal Galileons describe a probe 3-brane in AdS 5 They inherit a non-linearly realized symmetry from the higher dimensional space, e.g Symmetry: L conformal 3 = 1 2 ( π)2 π 1 4 ( π)4 δˆπ = 1 x µ µˆπ, δ µˆπ = 2x µ + x 2 µˆπ 2x µx ν ν ˆπ. 4 / 30

7 Can we incorporate this in a realistic string model? Candidate: Heterotic M-theory Natural 5 dimensional setting Topological M5 branes wrapped on holomorphic curve in Calabi-Yau Choice of vector bundle on observable sector MSSM + 3 R.H. ν 5 / 30

8 Can we incorporate this in a realistic string model? Candidate: Heterotic M-theory Natural 5 dimensional setting Topological M5 branes wrapped on holomorphic curve in Calabi-Yau Choice of vector bundle on observable sector MSSM + 3 R.H. ν 5 / 30

9 Outline for the rest of this talk Construct worldvolume action for probe 3-brane in heterotic M-theory à la Galileons Extend this result to N = 1 SUSY Extend to N = 1 SUGRA Conclusions 6 / 30

10 Bouncing cosmologies Brane world-volume action SUSY SUGRA Conclusions 7 / 30

11 Bouncing cosmologies Brane world-volume action SUSY SUGRA Conclusions 8 / 30

12 Bouncing cosmologies Brane world-volume action SUSY SUGRA Conclusions 9 / 30

13 Bulk space is foliated by time-like hypersurfaces which are Gaussian normal with respect to the bulk metric G AB (X ), de Rham, Tolley; Trodden et al.] G AB (X )dx A dx B = f (X 5 ) 2 g µν(x )dx µ dx ν + (dx 5 ) 2 K µν = X A x µ X B x ν An B ḡ µν = X A x µ X B x ν G AB Brane embedding coordinates X A (x µ ) are five arbitrary functions of the worldvolume coordinates x µ Brane action must be invariant under arbitrary worldvolume diffeomorphisms: δx A = ξ µ µx A, Implies that worldvolume action is composed entirely of the geometrical tensors S = d 4 x ( ḡ L ḡ µν, K µν, µ, R ) µβν α 10 / 30

14 Bulk space is foliated by time-like hypersurfaces which are Gaussian normal with respect to the bulk metric G AB (X ), de Rham, Tolley; Trodden et al.] G AB (X )dx A dx B = f (X 5 ) 2 g µν(x )dx µ dx ν + (dx 5 ) 2 K µν = X A x µ X B x ν An B ḡ µν = X A x µ X B x ν G AB Brane embedding coordinates X A (x µ ) are five arbitrary functions of the worldvolume coordinates x µ Brane action must be invariant under arbitrary worldvolume diffeomorphisms: δx A = ξ µ µx A, Implies that worldvolume action is composed entirely of the geometrical tensors S = d 4 x ( ḡ L ḡ µν, K µν, µ, R ) µβν α 10 / 30

15 Demand: Action yields two-derivative equations of motion We have a finite number of terms we can write down: L 1 = g π dπ f (π ) 4, L 2 = ḡ, L 3 = ḡ K, L 4 = ḡ R, L 5 = 3 2 ḡ KGB with K = ḡ µν K µν, R = ḡ µν R µαν α and K GB is a Gauss-Bonnet boundary term given by K GB = 1 3 K 3 + KµνK 2 2 ( 3 K µν 3 2 Rµν 1 2 Rḡ ) µν K µν Naively, there are five scalar degrees of freedom - but one can use the gauge freedom to set X µ (x) = x µ, X 5 (x) = π(x). Hence, there is really only a single scalar degree of freedom; the function π(x). 11 / 30

16 Demand: Action yields two-derivative equations of motion We have a finite number of terms we can write down: L 1 = g π dπ f (π ) 4, L 2 = ḡ, L 3 = ḡ K, L 4 = ḡ R, L 5 = 3 2 ḡ KGB with K = ḡ µν K µν, R = ḡ µν R µαν α and K GB is a Gauss-Bonnet boundary term given by K GB = 1 3 K 3 + KµνK 2 2 ( 3 K µν 3 2 Rµν 1 2 Rḡ ) µν K µν Naively, there are five scalar degrees of freedom - but one can use the gauge freedom to set X µ (x) = x µ, X 5 (x) = π(x). Hence, there is really only a single scalar degree of freedom; the function π(x). 11 / 30

17 Demand: Action yields two-derivative equations of motion We have a finite number of terms we can write down: L 1 = g π dπ f (π ) 4, L 2 = ḡ, L 3 = ḡ K, L 4 = ḡ R, L 5 = 3 2 ḡ KGB with K = ḡ µν K µν, R = ḡ µν R µαν α and K GB is a Gauss-Bonnet boundary term given by K GB = 1 3 K 3 + KµνK 2 2 ( 3 K µν 3 2 Rµν 1 2 Rḡ ) µν K µν Naively, there are five scalar degrees of freedom - but one can use the gauge freedom to set X µ (x) = x µ, X 5 (x) = π(x). Hence, there is really only a single scalar degree of freedom; the function π(x). 11 / 30

18 Heterotic M-theory The five dimensional metric takes the form Lukas, Ovrut, Waldram]: ds 2 5 = e 2A(y) η µνdx µ dx ν + e 2B(y) dy 2 We can chose a gauge where the metric is Ovrut, Stokes]: ds 2 5 = (1 2αz) 1/3 η αβ dx α dx β + dz 2 f (z) = (1 2αz) 1/6 The coordinates are chosen so that: z = 0 corresponds to the observable wall z = πρ corresponds to the hidden wall (ρ a fixed reference length, (πρ) GeV) The parameter α has mass dimension one GeV, α = π ( ) 2/3 κ 1 2 4π v β 2/3 with κ the 11-d Newton constant and v is the CY reference volume β = 1 (c 2(V (observable) ) 12 ) v c2(tx ) ω, 1/3 X 12 / 30

19 Heterotic M-theory The five dimensional metric takes the form Lukas, Ovrut, Waldram]: ds 2 5 = e 2A(y) η µνdx µ dx ν + e 2B(y) dy 2 We can chose a gauge where the metric is Ovrut, Stokes]: ds 2 5 = (1 2αz) 1/3 η αβ dx α dx β + dz 2 f (z) = (1 2αz) 1/6 The coordinates are chosen so that: z = 0 corresponds to the observable wall z = πρ corresponds to the hidden wall (ρ a fixed reference length, (πρ) GeV) The parameter α has mass dimension one GeV, α = π ( ) 2/3 κ 1 2 4π v β 2/3 with κ the 11-d Newton constant and v is the CY reference volume β = 1 (c 2(V (observable) ) 12 ) v c2(tx ) ω, 1/3 X 12 / 30

20 Heterotic M-theory The five dimensional metric takes the form Lukas, Ovrut, Waldram]: ds 2 5 = e 2A(y) η µνdx µ dx ν + e 2B(y) dy 2 We can chose a gauge where the metric is Ovrut, Stokes]: ds 2 5 = (1 2αz) 1/3 η αβ dx α dx β + dz 2 f (z) = (1 2αz) 1/6 The coordinates are chosen so that: z = 0 corresponds to the observable wall z = πρ corresponds to the hidden wall (ρ a fixed reference length, (πρ) GeV) The parameter α has mass dimension one GeV, α = π ( ) 2/3 κ 1 2 4π v β 2/3 with κ the 11-d Newton constant and v is the CY reference volume β = 1 (c 2(V (observable) ) 12 ) v c2(tx ) ω, 1/3 X 12 / 30

21 DBI Heterotic Galileons The tadpole, g, gk, gr curvature terms give L = L 1 = 3 (1 2απ)5/3 10α L 2 = (1 2απ) 2/3 1 + (1 2απ) 1/3 ( π) 2 4 c i L i, i=1 L 3 = α 3 (1 2απ) 1/3 5 γ 2] (1 2απ) 1/3 π + γ 2 π 3 ] L 4 = γ (Π] 2 Π 2 ] + 2γ 2 (1 2απ) 1/3 Π]π 3 ] + π 4 ] ]) α 2 γ (1 2απ) 4/3 ( 1 + γ 2 ) αγ(1 2απ) 2/3 ( 4 π + γ 2 π + 4(1 2απ) 1/3 π 3 ] ]) + 2 α 3 γ (1 2απ) 4/3( 1 2γ 2 + γ 4). where γ 1/ 1 + (1 2απ) 1/3 ( π) 2, arbitrary coefficients c 1, c 2, c 3, c 4 have mass dimensions 5, 4, 3 and / 30

22 Derivative expansion Conformal Galileons: total worldvolume Lagrangian isexpanded in powers of ( /M) 2, where R = 1/M is AdS 5 radius Terms of the same order are then grouped together, such that each collection of terms becomes the n-th order conformal Galileon. Due to the symmetry properties, one only needs to consider the expansion up to order ( /M) 8, higher order terms form total divergences. Heterotic case: mass scale associated with the curvature of the five dimensional space is α -hence the appropriate expansion parameter will be ( /α) 2. Truncate derivative expansion at a finite order in the expansion parameter. Define the dimensionless field ˆπ = απ Scale the individual Lagrangians L i and coefficients c i as follows: L i α 2 n L i, c i α n 2 c i 14 / 30

23 Derivative expansion Conformal Galileons: total worldvolume Lagrangian isexpanded in powers of ( /M) 2, where R = 1/M is AdS 5 radius Terms of the same order are then grouped together, such that each collection of terms becomes the n-th order conformal Galileon. Due to the symmetry properties, one only needs to consider the expansion up to order ( /M) 8, higher order terms form total divergences. Heterotic case: mass scale associated with the curvature of the five dimensional space is α -hence the appropriate expansion parameter will be ( /α) 2. Truncate derivative expansion at a finite order in the expansion parameter. Define the dimensionless field ˆπ = απ Scale the individual Lagrangians L i and coefficients c i as follows: L i α 2 n L i, c i α n 2 c i 14 / 30

24 Analog of Galileons Collecting terms of order ( ˆπ/α) 0, ( ˆπ/α) 2, ( ˆπ/α) 4, ( ˆπ/α) 6 gives us: L = 4 i=1 L T,1 where L T,1 = 3 10 c1(1 2ˆπ)5/3 c 2(1 2ˆπ) 2/3 4 c3(1 2ˆπ) 1/3 3 L T,2 = 1 2 c2(1 2ˆπ)1/3 c 3(1 2ˆπ) 2/3 2 ]( ) ˆπ 2 3 c4(1 2ˆπ) 5/3 α L T,3 = + 1 ]( ) ˆπ 2 2 c3 c4(1 ˆπ 2ˆπ) 1 α α c c3(1 2ˆπ) 1 1 ]( ) ˆπ 4 c4(1 2ˆπ) 2 3 α L T,4 = 1 4 c4(1 2ˆπ) 1/3 µ α + + ( ) ˆπ 2 µ ( ˆπ α α ) 4 ˆπ 19 ( ˆπ 6 c4(1 2ˆπ) 4/3 α α 2 c 3(1 2ˆπ) 1/3 11 ]( ˆπ 3 c4(1 2ˆπ) 4/3 α α ) 2 + c 4(1 2ˆπ) 1/3 ˆπ ˆπ,µ ˆπ,µν ˆπ,ν α 2 α α 2 α )2 ˆπ,µ ˆπ,µν ˆπ,ν α 2 α α 1 16 c2(1 2ˆπ) 1/3 1 3 (1 2ˆπ) 4/3 9 c4(1 2ˆπ) 7/3 4 ]( ) ˆπ 6. α 15 / 30

25 Expanding to first order in ˆπ (necessary for heterotic) yields: L 1 = 3 10 c 1 c c 3 + ( c c c 3 )ˆπ L 2 = L 3 = c 2 c c 4 + ( 1 3 c c 3 20 ]( ) ˆπ 2 9 c 4 )ˆπ α 1 2 c 3 c 4 2c 4ˆπ ]( ˆπ ) 2 ˆπ α 2 α 1 8 c c c 4 + ( 2 3 c c 4)ˆπ 1 L 4 = 4 c c 4ˆπ 19 6 c c 4ˆπ + + ] ( ) ν ˆπ 2 ν α α α ]( ) ˆπ 4 ˆπ α α 2 c c 4 + ( 2 3 c c 4)ˆπ ]( ) ˆπ 4 α ( ) ˆπ 2 + c ] ˆπ α 3 c ˆπ,µ ˆπ,µν ˆπ,ν 4ˆπ α 2 α α 2 α ]( )2 ˆπ ˆπ,µ ˆπ,µν ˆπ,ν α α α 2 α ]( ) ˆπ 6 α 1 16 c c c 4 + ( 1 24 c c c 4)ˆπ (c i s are dimension 4, mass scale set by α) 16 / 30

26 Analog of Galileons Unlike 5d Minkowski or AdS 5 cases, there is no non-linearly realized symmetry here - the higher dimensional space is not maximally symmetric and has no extra Killing vectors. This means that there is nothing telling us how to organize the terms in the derivative expansion - the coefficients from the brane Lagrangian (the c i s) must be used. As we will see, linearization and supersymmetrization will yield constraints between the various c i s. 17 / 30

27 Extension to N = 1 SUSY Since heterotic M-theory is a supersymmetric theory, the heterotic Galileons must be shown to have a supersymmetric completion Similar work has been done in the case of the conformal Galileons has already been completed, see Koehn, Lehners, Ovrut; Farakos et al.] We use the superspace formalism to construct SUSY-invariant Lagrangians which contain the heterotic Galileons Define a chiral multiplet P(x, θ, θ), whose components are the complex scalar A = 1 2 (ˆπ + iχ), a Weyl fermion ψ and the complex scalar F. We will only display the bosonic components to save space - already interesting effects occur. 18 / 30

28 Extension to N = 1 SUSY Since heterotic M-theory is a supersymmetric theory, the heterotic Galileons must be shown to have a supersymmetric completion Similar work has been done in the case of the conformal Galileons has already been completed, see Koehn, Lehners, Ovrut; Farakos et al.] We use the superspace formalism to construct SUSY-invariant Lagrangians which contain the heterotic Galileons Define a chiral multiplet P(x, θ, θ), whose components are the complex scalar A = 1 2 (ˆπ + iχ), a Weyl fermion ψ and the complex scalar F. We will only display the bosonic components to save space - already interesting effects occur. 18 / 30

29 L 1 : L 2 : L SUSY 2 = K(P, P ) θθ θ θ L 3 : Extension to N = 1 SUSY L SUSY 1 = W (P) + W (P ) θθ θ θ K(P, P ) = (c 2 + 2c c 4) α 2 PP ( 1 3 c c c 4) α 2 (P 2 P + PP 2 ) L SUSY 3 = L SUSY 3,1st term + LSUSY 3,2nd term L SUSY 3,1st term = c c 4 2c 4 (P + P ) L SUSY 3,2nd term = ] DPDP DP DP θθ θ θ 8 c c c ( 2 3 c c 4 ] DPDPD 2 P + h.c.] ] ) P + P θθ θ θ 19 / 30

30 L 4: L SUSY 4 = L SUSY 4,1st term + LSUSY 4,2nd term + LSUSY 4,3rd term + LSUSY 4,4th term + LSUSY 4,5th term L SUSY 4, 1st term = c4 + 1 ] 6 c4(p + P) {D, D}(DPDP){D, D}( DP DP ) 2 θθ θ θ ] L SUSY 4, 2nd term = L SUSY 4, 3rd term = c c4(p + P ) {D, D}(P + P ){D, D}(DPDP) D 2 P + h.c.] 19 ] 76 c c4(p + P ) θθ θ θ 6 DPDP DP DP {D, D}{D, D}(P + P ) θθ θ θ L SUSY 4, 4th term = c c4 + ( 2 3 c3 88 ] 9 c4)(p + P ) {D, D}DPDP DP DP {D, D}P + h.c.] θθ θ θ L SUSY 4, 5th term = c2 1 3 c3 9 4 c4 + 1 ( c2 8 9 c3 21 ] 2 c4)(p + P ) DPDP DP DP {D, D}P{D, D}P θθ θ θ 20 / 30

31 Auxiliary field effects 21 / 30

32 Auxiliary field effects where, for convenience, we define the mass dimension 2 parameters γ, δ, as γ (c 2 + 2c c 4) α 2, δ 1 2 ( 1 3 c c c 4) α / 30

33 Auxiliary field effects In regular (two-derivative) SUSY, the field F appears quadratically in the Lagrangian with no derivatives and acts as a Lagrange multiplier. This means that we are able to use e.o.m to eliminate F, get a Lagrangian invariant under SUSY on-shell. (Elimination works quantum-mechanically too.) Here, we have two new effects: Quartic F -terms - (FF ) 2 Derivative terms in F, e.g. +F F ( A A ) Both of these have been explored elsewhere Koehn et. al; Louis et al. ] Attempt solution as follows - consider L SUSY 2 = 1 2 c 2 c c 4 + ( 1 3 c c c 4)ˆπ ]( ˆπα )2 + ( χα )2 2 FF ] α 2 We will require that F 2 α / 30

34 Auxiliary field effects In regular (two-derivative) SUSY, the field F appears quadratically in the Lagrangian with no derivatives and acts as a Lagrange multiplier. This means that we are able to use e.o.m to eliminate F, get a Lagrangian invariant under SUSY on-shell. (Elimination works quantum-mechanically too.) Here, we have two new effects: Quartic F -terms - (FF ) 2 Derivative terms in F, e.g. +F F ( A A ) Both of these have been explored elsewhere Koehn et. al; Louis et al. ] Attempt solution as follows - consider L SUSY 2 = 1 2 c 2 c c 4 + ( 1 3 c c c 4)ˆπ ]( ˆπα )2 + ( χα )2 2 FF ] α 2 We will require that F 2 α / 30

35 Auxiliary field effects In regular (two-derivative) SUSY, the field F appears quadratically in the Lagrangian with no derivatives and acts as a Lagrange multiplier. This means that we are able to use e.o.m to eliminate F, get a Lagrangian invariant under SUSY on-shell. (Elimination works quantum-mechanically too.) Here, we have two new effects: Quartic F -terms - (FF ) 2 Derivative terms in F, e.g. +F F ( A A ) Both of these have been explored elsewhere Koehn et. al; Louis et al. ] Attempt solution as follows - consider L SUSY 2 = 1 2 c 2 c c 4 + ( 1 3 c c c 4)ˆπ ]( ˆπα )2 + ( χα )2 2 FF ] α 2 We will require that F 2 α / 30

36 Auxiliary field effects L SUSY(0) F (0) W (0) W = F + F A A + γ + 2 2δˆπ ] F (0) F (0) Choose as our superpotential where β i R V (ˆπ, χ) = L SUSY(0) F = β2 1 γ + 1 2γ γ 2 F (0) 1 = ] W γ + 2 2δˆπ A W (A) = β 1 A + β 2 A 2 + β 3 A 3 ] 4β1 2 δ + 4β 1β 2 γ ˆπ + 1 ] 2β 22 γ 3β 1β 3 χ 2 + 9β2 3 ] 12β 1 β 3 δ 8β2 2 δ + 6β 2β 3 γ ˆπχ β3 2δ 2 γ 2 ˆπχ4 In arriving at this expression, we have had to assume that δ γ 1 4γ χ4 24 / 30

37 Matching with L 1 = 3 10 c1 c2 4 3 c3 + ( c c2 )ˆπ 8 9 c3 tells us β1, β 2 in terms of the c i s. Taking χ = µχ = 0 ˆπ m 2 χ = 2 V χ 2 0, χ=0 4β2 2 6β 1β 3 γ 0, 6β 1 β 3 δ 4β 2 2 δ + 3β 2β 3 γ 0 For F (0) to be constant and small requires Ghost free kinetic energy 2β 1 δ β 2 γ = 0, β 1 γ c 2 c c 4 + ( 1 2 c c c ) β1 2 4 γ 2 + ( 1 4 c c ) β c 4 γ 4 < 0, ( 1 3 c c c 4) + ( 8 3 c c ) β1 2 4 γ 2 + ( 1 6 c c ) β c 4 γ 4 < 0 25 / 30

38 N = 1 SUGRA extension There is a known prescription for extending our SUSY action to N = 1 SUGRA Wess and Bagger]. Higher derivative SUGRA work has been looked at before Koehn, Lehners, Ovrut; Baumann, Green; Farakos et al.; Ciupke] - L SUGRA 1 = L SUGRA 2 = M 2 P d 2 Θ 2E W (P) + h.c. d 2 Θ 2E 3 ] 8 (D2 8R)e K(P,P )/3MP 2 + h.c L SUGRA 3 = L 3,I + L 3,II L 3,I = 1 d 2 Θ 2E (D 2 8R) 1 ] 2 c c 4 2c 4 (P + P ) DPDPD 2 P + h.c. L 3,II = h.c. 1 d 2 Θ 2E (D 2 8R) 8 c c c ( c 3 4 ] 3 c 4)(P + P ) DPDPDP DP 26 / 30

39 27 / 30

40 SUGRA extension Auxiliary fields of supergravity - M, and b µ - can be integrated up to L 3. At L SUGRA 4, we will find higher orders in M, b µ as well as b. Coupling to curvature terms arise L SUGRA 4 c ] c 4(A + A ) 4 RFF (A + A ) 9 ] 8 FF R µν µ A ν (A + A ). 28 / 30

41 Conclusions We have found an N = 1 SUSY action for a probe brane in heterotic M-theory, by constructing the analog of Galileons. No symmetry, but truncated due to natural scale απρ. Linearization and supersymmetrization leads to constraints between coefficients. Higher derivative lagrangian leads to interesting effects with auxiliary fields in both SUSY and SUGRA - so far only perturbative approach has been taken To do list Solve the equations of motion for ˆπ coming from the real scalar Lagrangian - sources a 4d a eff. Examine full properties of propagating auxiliary fields Include non-perturbative superpotential for branes in heterotic M-theory - this gives rise (in 2-derivatives SUGRA) to an exponential potential à la ekpyrosis: W exp( T 2 Y) 29 / 30

42 Bouncing cosmologies Brane world-volume action SUSY SUGRA Conclusions Thank you! 30 / 30