N=2 compactifications of M-theory to AdS3 – study throughgeometric algebra

Ioana-Alexandra Coman

Department of Theoretical PhysicsNational Institute of Physics and Nuclear Engineering

Bucharest-Magurele

based on“On N = 2 compactifications of M-theory to AdS3 using geometric algebra techniques”,

submitted to AIP Conference Proceedings

“SigmaXi Showcase Presentation” 08 March 2013

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Outline

1 IntroductionEleven-dimensional supergravity

2 Geometric algebra formulationPreparationsThe Fierz IsomorphismLift to nine dimensionsAnalysis of CGK pinor equations

3 AnalysisLie bracket of V ]1 and V ]3Reducing the result to eight-dimensions

4 Conclusions

5 Acknowledgements

6 References

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Introduction

Original goal

Investigate the most general warped compactification of eleven-dimensional supergravityon eight-manifolds to AdS3 spaces in the presence of non-vanishing four-form flux, whichpreserves N = 2 supersymmetry in three dimensions.

Motivation and context

Study of dualities between compactified supergravity theories with fluxes.

From Heterotic – Type IIA duality to the interest in general compactifications of F-theory.

General compactifications of F-theory on manifolds Y × S have duals in M-theorycompactifications on Y .

F-theory is a geometric twelve-dimensional theory, whose background is a T 2-fiberedmanifold and which represents a non-perturbative limit of M-theory compactifications.

We use geometric algebra techniques to carry out the analysis, as these offer bothconceptual and computational advantages.

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IntroductionEleven-dimensional supergravity

Supergravity action

We start with the low-energy action of M-theory, i.e. the eleven-dimensional bosonicsupergravity action, with metric g of mostly plus Lorenzian signature on M, three-formpotential C with non-trivial four-form field strength G and gravitino ΨM :

S11 =12

∫ [R ? 1−

12

G ∧ ?G −16

G ∧ G ∧ C]

We view the pin bundle S of M as a bundle of simple modules over the Clifford bundle ofT∗M and η ∈ Γ(M, S) is the supersymmetry generator (a Majorana spinor).

The supersymmetry variation of the gravitino must vanish for a supersymmetricbackground δηΨM := DM η = 0 , which defines the ‘supercovariant derivative’ DM :

DMdef.= ∇spin

M −1

288

(GNPQR ΓNPQR

M − 8GMNPQ ΓNPQ). (1)

In the local orthonormal frame (eM )M=0...10 of T M, ∇spinM = ∂M + 1

4 ωMNP ΓNP is theconnection induced on S by the Levi-Civita connection of M, ωMNP are the spin connectioncoefficients and ΓM are the gamma ‘matrices’ of Cl(10, 1), where Γ11 = +idS .

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IntroductionEleven-dimensional supergravity

Compactification ansatz

Compactifications of 11-dimensional supergravity have generated much interest [5]-[8] inliterature; in particular [7] studied compactifications to AdS3 with N = 2 supersymmetry,but not in the most general case, since the internal manifold spinors were Majorana-Weyl.

Make the usual warped product compactification ansatz M = M3 ×M:

ds211 = e2∆(ds2

3 + gmndxmdxn) and G = e3∆(vol3f + F ) ,

Γµ = e∆(γµ ⊗R γ9) and Γm = e∆(1⊗R γm) for µ = 0, 1, 2 , m = 3 . . . 11 .

The supersymmetry condition reduces to the constrained generalized Killing (CGK) spinorequations [3] for the internal part ξ ∈ Γ(M,S) of η, where S is the pin bundle of M:

Dmξ = 0 with Dmdef.= ∇spin

m + Am , Am = −14

fnγnmγ9 +

124

Fmpqrγpqr + κγmγ9 , (2)

Qξ = 0 with Q =12γm∂m∆−

1288

Fmnpqγmnpq −

16

fmγmγ9 − κγ9 . (3)

The differential and algebraic constraints (2)-(3) have (ξ1, ξ2) independent global solutionsfor N = 2 supersymmetric backgrounds. γm transform in the R-representations of Cl(8, 0)and κ ∈ R+ is ∼ the square root of the cosmological constant of the external AdS3 space.

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Geometric algebra formulationPreparations

Using current methods from literature to analyse the differential and algebraic constraintsproves cumbersome and inefficient. However, the geometric algebra formalism provides aclearer, more computationally effective approach.

Given a (pseudo)-Riemannian manifold (M, g) of dimension d , we identify the Cliffordbundle of the pseudo-Euclidean vector bundle T∗M with the exterior bundle ∧T∗M.

∧T∗M becomes a bundle of algebras (∧T∗M, ), the Kahler-Atiyah bundle of (M, g)endowed with the geometric product , which can be viewed as a bundle morphism [3, 13]:

: ∧T∗M ×M ∧T∗M → ∧T∗M .

The action of on sections expands into generalized products:

ω η =

[d2

]∑k=0

(−1)kω42k η +

[d−1

2

]∑k=0

(−1)k+1π(ω)42k+1 η where ω, η ∈ Ω(M) (4)

and4k are constructed recursively from wedge products and contractions [3]:

ω40 η = ω ∧ η , ω4k+1 η =1

k + 1gab(ιeaω) ∧k (ιebη) . (5)

Upshot The recursive description allows for implementation in symbolic computation in [9].

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Geometric algebra formulationPreparations

ι : Ω(M)×Ω(M)→ Ω(M) denotes the left generalized contraction (a.k.a. interior product).

π is the main automorphism of the Kahler-Atiyah bundle, whose action on global sectionsof the latter is the involutive C∞(M,R)-linear automorphism of the Kahler-Atiyah algebra(Ω(M), ):

π(ω) =d∑

k=0

(−1)kω(k) , ∀ω =d∑

k=0

ω(k) ∈ Ω(M) , where ω(k) ∈ Ωk (M) .

(ea)a=1...d is a local frame of TM and (ea)a=1...d its dual local coframe, satisfyingea(eb) = δa

b and g(ea, eb) = gab , where (gab) is the inverse of the matrix (gab). Thecorresponding contragradient frame (ea)# and coframe (ea)# satisfy (ea)# = gabeb and(ea)# = gabeb , where the # subscript and superscript denote the (mutually-inverse)musical isomorphisms between TM and T∗M, given by lowering / raising indices with g.

Finally, the main antiautomorphism of the Kahler-Atiyah bundle is the involutiveantiautomorphism defined through:

τ(ω) = (−1)k(k−1)

2 ω , ∀ω ∈ Ωk (M) . (6)

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Geometric algebra formulationThe Fierz Isomorphism

We defined pinors over M as sections of the bundle S, with fiberwise module structuredescribed by a morphism of bundles of algebras γ : (∧T∗M, )→ (End(S), ).

This morphism is surjective and has a partial inverse when (M, g) has Euclidean signaturewith d ≡8 0, 1, that allows translating the CGK pinor conditions into conditions ondifferential forms on M which are constructed as bilinear combinations of sections of S [3].

The Fierz isomorphism encodes this, identifying the bundle of bispinors S ⊗ S with asub-bundle ∧γT∗M of the exterior bundle (the ‘effective domain of definition’ of γ) [3].

The admissible bilinear pairings B on the pin bundle satisfy certain properties [10] and canbe used to define a B-dependent bundle isomorphism E : S ⊗ S → End(S) [3, 13]:

Eξ,ξ′ (ξ′′)

def.= B(ξ′′, ξ′)ξ such that Eξ1,ξ2 Eξ3,ξ4 = B(ξ3, ξ2)Eξ1,ξ4 .

Fierz Isomorphism

The Fierz isomorphism is then E = γ−1 E : S ⊗ S → ∧γT∗M .

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Geometric algebra formulationLift to nine dimensions

Motivation

General N = 2 compactifications on M have two global solutions ξ1, ξ2 ∈ Γ(M,S)satisfying (2) and (3), which have linearly independent values at any point x of M.

(ξ1, ξ2) determine a point on the second Stiefel manifold V2(Sx ) of each fiber Sx of S andsolutions can be classified according to the orbit of the representation of Spin(8) on Sx .

The corresponding action on V2(Sx ) fails to be transitive⇒ a generic solution (ξ1, ξ2) to(2)-(3) does not determine a global reduction of the structure group of M.

We simplify the analysis using the construction of [4] to lift the problem to the9-dimensional metric cone M, which allows for a global description of N = 2 supergravitycompactifications on eight-manifolds through the reduction of the structure group of M.

We consider the metric cone M = R+ ×M with squared line elementds2

cone = dr2 + r2ds2 and note that the canonical normalized one-form θ = dr is a specialKilling-Yano form with respect to the metric gcone of M.

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Geometric algebra formulationLift to nine dimensions

Set-up

We pull back the Levi-Civita connection of (M, g) through the projection Π : M → M, lift theconnection (2) from S to the pin bundle S of M and define Π∗(h) = h Π forh ∈ C∞(M,R) [4].

The morphism γcone : ∧T∗M → End(S) determined by γ leads to 2 inequivalent choicesγcone(νcone) = ±idS , where νcone is the volume form of M.

Pick the representation with γcone(νcone) = +idS ⇒ Kahler-Atiyah algebra (Ω(M), cone)

can be realized through the truncated model (Ω<(M),♦cone) [4]; Ω<(M) = ⊕4k=0Ωk (M)

and ♦cone is the reduced geometric product of M.

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Geometric algebra formulationLift to nine dimensions

Set-up

Rescale g → 2κ2g ⇒ the connection induced on S by the Levi-Civita connection of M is:

∇Sem = ∇S

em + κγ9m , ∇S∂r

= ∂r , where m = 1 . . . 8 .

D def.= ∇S,cone + Acone is the pullback of (2) to M; for (econe

m )]cone the one-form dual to thebase vector with respect to the metric on M, lifting and ‘dequantizing’ the connection Am

and endomorphism Q [3, 4] induce the inhomogeneous differential forms on M:

Am =14ιecone

mF +

14

(econem )] ∧ f ∧ θ and Q =

12

r(d∆)−16

f ∧ θ −1

12F − κθ .

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Geometric algebra formulationAnalysis of CGK pinor equations

Pinor bilinears and inhomogeneous generators

We are interested in form-valued pinor bilinears written locally as:

E (k)

ξi ,ξj≡ E (k)

ij =1k!B(ξi , γ

conea1...ak

ξj )ea1...ak+ , with a1 . . . ak ∈ 1 . . . 9 , i, j ∈ 1, 2 ,

where the subscript ‘+’ denotes the twisted self-dual part [3, 4], and in the weighted sums:

Eij =N2d

d∑k=0

E (k)ij .

The inhomogeneous differential forms Eij generate the algebra (Ω+(M), ); N = 2[

d2

]is

the real rank of the pin bundle S and d = 9 is the dimension of M. With the properties ofthe admissible bilinear paring on S, we find:

B(ξi , γa1...akcone ξj ) = (−1)

k(k−1)2 B(ξj , γ

a1...akcone ξi ) , ∀i, j = 1, 2 . (7)

We use (7) to construct the non-trivial form-valued bilinears of rank ≤ 4.

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Geometric algebra formulationAnalysis of CGK pinor equations

Pinor bilinears and inhomogeneous generators

Raising the indices of the local coeffitients with the metric, the pinor bilinears are:

V a1 = B(ξ1, γ

aconeξ1) , V a

2 = B(ξ2, γaconeξ2) , V a

3 = B(ξ1, γaconeξ2), (8)

K ab = B(ξ1, γabconeξ2) , ψabc = B(ξ1, γ

abcconeξ2) ,

φabce1 = B(ξ1, γ

abceconeξ1) , φabce

2 = B(ξ2, γabceconeξ2) , φabce

3 = B(ξ1, γabceconeξ2).

These forms are the definite rank components of the truncated inhomogeneous forms:

E<11 =1

32(1 + V1 + φ1) , E<12 =

132

(V3 + K + ψ + φ3) ,

E<21 =1

32(V3 − K − ψ + φ3) , E<22 =

132

(1 + V2 + φ2) .

E<ij are the basis elements of the truncated Fierz algebra [3, 4] and satisfy the truncatedFierz identities; dropping the ’cone’ superscript on ♦cone to simplify notation,

E<ij ♦E<kl =12δjk E<il , ∀i, j, k , l = 1, 2 . (9)

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Geometric algebra formulationAnalysis of CGK pinor equations

Constraints

For the analysis of general N = 2 compactifications to AdS3 spaces we need theconstraints implied by the CGK pinor equations on the forms constructed with (8).

The differential constraints (2) are equivalent cf. [3, 4] with:

∇mωa1...ak (ξ, ξ′) = 〈ξ, [Aconem , γconea1...ak

]ξ′〉 ,

which in the language of geometric algebra is:

dE<ij = ea ∧∇aE<ij where ∇aE<ij = −[Aa, E<ij ]−,♦ , ∀i, j ∈ 1, 2 . (10)

The algebraic constraints (3) reduce to 〈ξ, (Qtγconea1,...,ak± γconea1,...,ak

Q)ξ′〉 = 0 , which ingeometric algebra becomes:

Q♦E<ij ∓ E<ij ♦τ(Q) = 0 , ∀i, j ∈ 1, 2 . (11)

We present the set of resulting relations and Fierz identities that we need for the analysisin the next section – the complete set and the full Fierz solution can be found in [2].

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Geometric algebra formulationAnalysis of CGK pinor equations

The relations implied by the truncated Fierz identities and that we need here are:

ιV1 V3 = 0 , ιV1φ3 − ψ + V1 ∧ K = 0 , ιV3φ1 + ψ − V1 ∧ K = 0 . (12)

The equation in (11) with E<12 and the minus sign leads to the following constraints:

ιf∧θK = 0 , (13)

r ιd∆K +13ιf∧θψ −

16ιψF − 2κιθK = 0 , (14)

13ιf∧θφ3 −

16

F 43 φ3 + r(d∆) ∧ V3 + 2κV3 ∧ θ = 0 , (15)

while using the truncated inhomogeneous form E<11 amounts to:

−13

f ∧ θ +13ιf∧θφ1 −

16

F 43 φ1 + r(d∆) ∧ V1 + 2κV1 ∧ θ = 0 . (16)

Expanding (10) for E<11, E<12 gives the covariant derivatives of V1 and V3:

∇mV1n =12

fsθpφ1sp

mn−1

12Fspqmφ1

spqn and ∇mV3n =

12

fsθpφ3sp

mn−1

12Fspqmφ3

spqn .

(17)

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AnalysisLie bracket of V ]

1 and V ]3

We can investigate the geometric implications of the N = 2 supersymmetry condition usingthe differential and algebraic constraints for the form-valued pinor bilinears.

In particular, we can compute the Lie bracket of the vector fields V ]1 , V ]3 dual to V1, V3.

Pick a local orthonormal frame (ea) of (M, gcone) defined above an open subset U ⊂ M ⇒gcone(ea, eb) = δab and expanding V ] = V aea with V a ∈ C∞(U,R), we find:

[V ]1 ,V]3 ] = [V ]1 ,V

]3 ]beb , [V ]1 ,V

]3 ]b = V a

1 ea(V b3 )− V a

3 ea(V b1 ) ∈ C∞(U,R) .

We interpret vector fields defined on U as derivations of the commutative algebraC∞(U,R) and consider the one-form dual to the vector field [V ]1 ,V

]3 ]:

[V ]1 ,V]3 ]] = [V ]1 ,V

]3 ]a(ea)] = [V ]1 ,V

]3 ]beb , where

[V ]1 ,V]3 ]b = gba[V ]1 ,V

]3 ]a = V a

1 ea(V3b)− V a3 ea(V1b) .

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AnalysisLie bracket of V ]

1 and V ]3

Lie bracket of V ]1 and V ]3

ea(Vb) = ∇aVb + ωac

bVc , with ωac

b the coefficients of the Levi-Civita connection

ωabcdef.= gbdωa

dc computed with respect to the local frame (ea) and ωabc = −ωacb .

The Lie bracket becomes [V ]1 ,V]3 ]a = V b

1∇bV3a − V b3∇bV1a + (V b

1 V c3 − V b

3 V c1 )ωbca .

Compute the V b1∇bV3a − V b

3∇bV1a component substituting the covariant derivatives (17):

[V ]1 ,V]3 ]] =

12

(ιf∧θ(ιV1φ3) + ι(ιV1

F )φ3 − ιf∧θ(ιV3φ1)− ι(ιV3F )φ1

). (18)

We omit in what follows the term with spin connection coefficients and rewrite:

ι(ιV3F )φ1 = −ιV3 (F 43 φ1) + ι(ιV3

φ1)F and ι(ιV1F )φ3 = −ιV1 (F 43 φ3) + ι(ιV1

φ3)F .

Eliminate ιV1φ3 and ιV3φ1 with the Fierz relations (12), simplify F 43 φ3 and F 43 φ1through (15)-(16) recalling that ιV1 V3 = 0 due to (12) and eliminate ιf∧θψ with (14).

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AnalysisLie bracket of V ]

1 and V ]3

Finally, expanding ιf∧θ(V1 ∧ K ), the Lie bracket (18) becomes:

[V ]1 ,V]3 ]] = 3r ιd∆K − (6κ+ ιf V1)ιθK + (ιV1θ)ιf K + 1

2 ιψF − ιV1∧K F

−3(r ιd∆V1 − 2κιV1θ)V3 + 3(r ιd∆V3 − 2κιV3θ)V1 − (ιf V3)θ + (ιV3θ)f .(19)

Reducing the result to eight-dimensions

Any k -form ω ∈ Ωk (M) decomposes uniquely as ω = θ ∧ ω> + ω⊥ for ω> ∈ Ωk−1(M)and ω⊥ ∈ Ωk (M), where θyω> = θyω⊥ = 0 [4]⇒ V1 = rU1 + w1θ , V3 = rU3 + w3θ.

θ = dr is the canonical one-form of M, U1,U3 are (pull-backs of) one-forms on M andw1,w3 are smooth functions on M. Since θ] = ∂r ⇒

[V ]1 ,V]3 ] = [rU]1 , rU

]3 ] + [rU]1 ,w3∂r ] + [w1∂r , rU]3 ] + [w1∂r ,w3∂r ] .

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AnalysisReducing the result to eight-dimensions

Using the fact that ∇wk = dwk ,

[V ]1 ,V]3 ]] = r2[U]1 ,U

]3 ]] + w1U3 − w3U1 + r(Ub

1∇bw3 − Ub3∇bw1)θ. (20)

Following the same steps used to derive (19), we find:

r [U]1(w3)− U]3(w1)] = r(Ub1∇bw3 − Ub

3∇bw1) =12

r ιθ[ι(ιU1

F )φ3 − ι(ιU3F )φ1

]= 3w1r2ιd∆U3 − 3w3r2ιd∆U1 − r ιU3 f ,

which matches the expression obtained directly from (19) by substituting V1,V3:

([V ]1 ,V]3 ]])> = 3w1r2ιd∆U3 − 3w3r2ιd∆U1 − r ιU3 f − 3r2ιd∆(ιθK ) + rw1ιf (ιθK ) .

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Conclusions

We have presented some applications of the general theory developed in [3, 4, 13] toN = 2 compactifications of M-theory on eight-manifolds.

We focused on the computation of a certain quantity of geometric interest — the Liebracket of V ]1 and V ]3 .

We note that we moreover find similar results working with V2 instead of V1 and that it isalso possible to compute [U]1 ,U

]2 ]].

The Lie brackets discussed above have various geometric implications related to theexistence of circle fibrations on the compactification manifold, as opposed to merefoliations.

Some aspects of these geometric implications could be of interest in connection withF-theory.

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Acknowledgements

My contribution to this work was supported by the CNCS project PN-II-RU-TE contract number77/2010 and I also acknowledge my Open Horizons scholarship from the Dinu PatriciuFoundation, which financed part of my studies.

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