Compactifications and consistent truncations in supergravity James Liu University of Michigan 7 July 2015 1. Introduction to supersymmetry and supergravity 2. Compactification on manifolds without fluxes 3. AdS×Sphere compactifications and consistent truncations GRaB100 I The original fields can be expanded in a complete set of harmonics on Xn µ i X (a) µ i '(x ; y ) = φ (x )fa(y ) a I By expanding in a complete set, the theory is unchanged, but given a lower-dimensional interpretation { A finite number of D-dimensional fields expand into an infinite number of( D − n)-dimensional fields { Isometries of Xn give rise to symmetries of the lower-dimensional theory The Kaluza-Klein idea I We may obtain a lower-dimensional theory by compactifying a D-dimensional theory on an n-dimensional manifold Xn X M = fx µ; y i g where M =0 ; 1;:::; D − n − 1; D − n;:::; D − 1 µ i JTL The Kaluza-Klein idea I We may obtain a lower-dimensional theory by compactifying a D-dimensional theory on an n-dimensional manifold Xn X M = fx µ; y i g where M =0 ; 1;:::; D − n − 1; D − n;:::; D − 1 µ i I The original fields can be expanded in a complete set of harmonics on Xn µ i X (a) µ i '(x ; y ) = φ (x )fa(y ) a I By expanding in a complete set, the theory is unchanged, but given a lower-dimensional interpretation { A finite number of D-dimensional fields expand into an infinite number of( D − n)-dimensional fields { Isometries of Xn give rise to symmetries of the lower-dimensional theory JTL 1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR I Now expand in Fourier modes on the circle 1 µ X µ iny=R '(x ; y) = φn(x )e n=−∞ I The scalar equation then reduces to 2 2 X 2 2 iny=R (d + @y − mD )' = 0 ) (d − (n=R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R Example: Reduction of a scalar on S 1 I Consider a complex scalar in D-dimensional Minkowski space ∗ M 2 2 2 LD = −@M ' @ ' − mD j'j ) (D − mD )' = 0 JTL I The scalar equation then reduces to 2 2 X 2 2 iny=R (d + @y − mD )' = 0 ) (d − (n=R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R Example: Reduction of a scalar on S 1 I Consider a complex scalar in D-dimensional Minkowski space ∗ M 2 2 2 LD = −@M ' @ ' − mD j'j ) (D − mD )' = 0 1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR I Now expand in Fourier modes on the circle 1 µ X µ iny=R '(x ; y) = φn(x )e n=−∞ JTL Example: Reduction of a scalar on S 1 I Consider a complex scalar in D-dimensional Minkowski space ∗ M 2 2 2 LD = −@M ' @ ' − mD j'j ) (D − mD )' = 0 1 I We compactify on S by demanding periodicity of the last coordinate y = y + 2πR I Now expand in Fourier modes on the circle 1 µ X µ iny=R '(x ; y) = φn(x )e n=−∞ I The scalar equation then reduces to 2 2 X 2 2 iny=R (d + @y − mD )' = 0 ) (d − (n=R) − mD )φne = 0 n or n 2 ( − m2)φ = 0 where m2 = m2 + d n n n D R JTL 2 I The equations of motion( d − mn)φn = 0 can be obtained from the d-dimensional Lagrangian X ∗ µ 2 2 Ld = [−@µφn@ φn − mnjφnj ] n [Obtained by inspection, or by substituting the Fourier expansion of ' into LD and integrating over the extra dimension] The Kaluza-Klein tower I A single complex scalar decomposes into an infinite Kaluza-Klein tower fφng with corresponding masses fmng I Furthermore, internal momentum on the circle pn = n=R can be identified with U(1) charge D-dimensional momentum conservation , d-dimensional charge conservation JTL The Kaluza-Klein tower I A single complex scalar decomposes into an infinite Kaluza-Klein tower fφng with corresponding masses fmng I Furthermore, internal momentum on the circle pn = n=R can be identified with U(1) charge D-dimensional momentum conservation , d-dimensional charge conservation 2 I The equations of motion( d − mn)φn = 0 can be obtained from the d-dimensional Lagrangian X ∗ µ 2 2 Ld = [−@µφn@ φn − mnjφnj ] n [Obtained by inspection, or by substituting the Fourier expansion of ' into LD and integrating over the extra dimension] JTL I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0 I We thus end up with two possibilities '(xµ; y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ fφn(x )g φ0(x ) The zero-mode sector I In general, for an interacting theory, if we had a field with charge n, we would also have to have fields of charge2 n,3 n, etc by charge conservation Once we include at least one field with n 6= 0, we must keep the entire Kaluza-Klein tower JTL I We thus end up with two possibilities '(xµ; y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ fφn(x )g φ0(x ) The zero-mode sector I In general, for an interacting theory, if we had a field with charge n, we would also have to have fields of charge2 n,3 n, etc by charge conservation Once we include at least one field with n 6= 0, we must keep the entire Kaluza-Klein tower I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0 JTL The zero-mode sector I In general, for an interacting theory, if we had a field with charge n, we would also have to have fields of charge2 n,3 n, etc by charge conservation Once we include at least one field with n 6= 0, we must keep the entire Kaluza-Klein tower I However, the zero mode φ0 is special, since it is charge-neutral It is consistent to truncate the d-dimensional theory by keeping only φ0 I We thus end up with two possibilities '(xµ; y) ¨¨ HH Dimensional reductionKaluza-Kleintower ¨ H ¨¨ HHj µ µ fφn(x )g φ0(x ) JTL I The truncation is consistent if the d-dimensional equations of motion imply the D-dimensional equations of motion For the complex scalar on S 1, the zero-mode reduction ansatz µ µ '(x ; y) = φ0(x ) is a consistent truncation since 2 2 (d − mD )φ0 = 0 ) (D − mD )' = 0 I The zero-mode truncation is straightforward for torus reductions, but can be less obvious for more general Xn We can always obtain a consistent truncation by restricting to singlets under the isometry group of Xn Consistent truncations I If we expand in a complete set of modes on Xn, then we have simply rewritten the original D-dimensional theory in a d-dimensional manner I However, it is often useful to consider a consistent truncation Truncate to a finite set of d-dimensional fields (often just the zero modes) JTL I The zero-mode truncation is straightforward for torus reductions, but can be less obvious for more general Xn We can always obtain a consistent truncation by restricting to singlets under the isometry group of Xn Consistent truncations I If we expand in a complete set of modes on Xn, then we have simply rewritten the original D-dimensional theory in a d-dimensional manner I However, it is often useful to consider a consistent truncation Truncate to a finite set of d-dimensional fields (often just the zero modes) I The truncation is consistent if the d-dimensional equations of motion imply the D-dimensional equations of motion For the complex scalar on S 1, the zero-mode reduction ansatz µ µ '(x ; y) = φ0(x ) is a consistent truncation since 2 2 (d − mD )φ0 = 0 ) (D − mD )' = 0 JTL Consistent truncations I If we expand in a complete set of modes on Xn, then we have simply rewritten the original D-dimensional theory in a d-dimensional manner I However, it is often useful to consider a consistent truncation Truncate to a finite set of d-dimensional fields (often just the zero modes) I The truncation is consistent if the d-dimensional equations of motion imply the D-dimensional equations of motion For the complex scalar on S 1, the zero-mode reduction ansatz µ µ '(x ; y) = φ0(x ) is a consistent truncation since 2 2 (d − mD )φ0 = 0 ) (D − mD )' = 0 I The zero-mode truncation is straightforward for torus reductions, but can be less obvious for more general Xn We can always obtain a consistent truncation by restricting to singlets under the isometry group of Xn JTL I We can work out the linearized Kaluza-Klein spectrum from the linearized Einstein equation ¯ M ¯ D hMN = 0;@ hMN = 0 Kaluza-Klein gravity 1 I Now consider the reduction of pure gravity on S −1 1 e LD = R ) RMN − 2 gMN R = 0 ) RMN = 0 I There are several complications { Tensor decomposition gMN −! gµν ; gµy ; gyy { Gauge degrees of freedom from general coordinate transformations { The Einstein equation is non-linear JTL Kaluza-Klein gravity 1 I Now consider the reduction of pure gravity on S −1 1 e LD = R ) RMN − 2 gMN R = 0 ) RMN = 0 I There are several complications { Tensor decomposition gMN −! gµν ; gµy ; gyy { Gauge degrees of freedom from general coordinate transformations { The Einstein equation is non-linear I We can work out the linearized Kaluza-Klein spectrum from the linearized Einstein equation ¯ M ¯ D hMN = 0;@ hMN = 0 JTL I The zero mode (n = 0) is massless and needs to be treated separately ¯0 µ¯0 µ¯0 d hMN = 0 @ hµν = 0 @ hµy = 0 Kaluza-Klein gravity I Use the standard Kaluza-Klein decomposition ¯ µ X ¯n µ iny=R hMN (x ; y) = hMN (x )e n I The linearized Einstein equation reduces to n ( − m2)h¯n = 0 m = d n MN n R along with the gauge conditions µ¯n ¯n µ¯n ¯n @ hµν + imnhyν = 0 @ hµy + imnhyy = 0 JTL Kaluza-Klein gravity I Use the standard Kaluza-Klein decomposition ¯ µ X ¯n µ iny=R hMN (x ; y) = hMN (x )e n I The linearized Einstein equation reduces to n ( − m2)h¯n = 0 m = d n MN n R along with the gauge conditions µ¯n ¯n µ¯n ¯n @ hµν + imnhyν = 0 @ hµy + imnhyy = 0 I The zero mode (n = 0) is massless and needs to be treated separately ¯0 µ¯0 µ¯0 d hMN = 0 @ hµν = 0 @ hµy = 0 JTL We can transform