1. N=2 compactifications of M-theory to AdS3 – study through
geometric algebra
Ioana-Alexandra Coman
Department of Theoretical Physics
National 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
1 / 23

2. Outline
1 Introduction
Eleven-dimensional supergravity
2 Geometric algebra formulation
Preparations
The Fierz Isomorphism
Lift to nine dimensions
Analysis of CGK pinor equations
3 Analysis
Lie bracket of V1 and V3
Reducing the result to eight-dimensions
4 Conclusions
5 Acknowledgements
6 References
2 / 23

3. Introduction
Original goal
Investigate the most general warped compactification of eleven-dimensional supergravity
on eight-manifolds to AdS3 spaces in the presence of non-vanishing four-form flux, which
preserves 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-theory
compactifications on Y .
F-theory is a geometric twelve-dimensional theory, whose background is a T 2 -fibered
manifold and which represents a non-perturbative limit of M-theory compactifications.
We use geometric algebra techniques to carry out the analysis, as these offer both
conceptual and computational advantages.
3 / 23

4. Introduction
Eleven-dimensional supergravity
Supergravity action
We start with the low-energy action of M-theory, i.e. the eleven-dimensional bosonic
˜ ˜
supergravity action, with metric g of mostly plus Lorenzian signature on M, three-form
˜ ˜ ˜
potential C with non-trivial four-form field strength G and gravitino ΨM :
1 ˜ 1 ˜ 1˜ ˜ ˜
S11 = R 1− G∧ G− G∧G∧C
2 2 6
˜ ˜
We view the pin bundle S of M as a bundle of simple modules over the Clifford bundle of
˜ ˜ ˜
T ∗ M and η ∈ Γ(M, S) is the supersymmetry generator (a Majorana spinor).
˜
The supersymmetry variation of the gravitino must vanish for a supersymmetric
˜ ˜ ˜ ˜
background δη ΨM := DM η = 0 , which defines the ‘supercovariant derivative’ DM :
˜
˜ def. 1
DM = ˜ spin −
M
˜ ˜ ˜
GNPQR ΓNPQR M − 8GMNPQ ΓNPQ
˜ . (1)
288
In the local orthonormal frame (eM )M=0...10 of T M, ˜ spin = ∂M + 1 ωMNP ΓNP is the
˜ ˜
M 4
˜ ˜
˜ ˜ ˜
connection induced on S by the Levi-Civita connection of M, ωMNP are the spin connection
˜ ˜
coefficients and ΓM are the gamma ‘matrices’ of Cl(10, 1), where Γ11 = +idS .˜
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5. Introduction
Eleven-dimensional supergravity
Compactification ansatz
Compactifications of 11-dimensional supergravity have generated much interest [5]-[8] in
literature; 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:
s2 2 ˜
d ˜11 = e2∆ (ds3 + gmn dx m dx n ) and G = e3∆ (vol3 f + 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) spinor
equations [3] for the internal part ξ ∈ Γ(M, S) of η , where S is the pin bundle of M:
˜
def. spin 1 1
Dm ξ = 0 with Dm = + Am , Am = − fn γ n m γ9 +
m Fmpqr γ pqr + κγm γ9 , (2)
4 24
1 1 1
Qξ = 0 with Q = γ m ∂m ∆ − Fmnpq γ mnpq − fm γ m γ9 − κγ9 . (3)
2 288 6
The differential and algebraic constraints (2)-(3) have (ξ1 , ξ2 ) independent global solutions
for N = 2 supersymmetric backgrounds. γ m 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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6. Geometric algebra formulation
Preparations
Using current methods from literature to analyse the differential and algebraic constraints
proves cumbersome and inefficient. However, the geometric algebra formalism provides a
clearer, more computationally effective approach.
Given a (pseudo)-Riemannian manifold (M, g) of dimension d, we identify the Clifford
bundle 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:
d d−1
2 2
k
ω η= (−1) ω 2k η+ (−1)k +1 π(ω) 2k +1 η where ω, η ∈ Ω(M) (4)
k=0 k=0
and k are constructed recursively from wedge products and contractions [3]:
1
ω 0 η =ω∧η , ω k +1 η= gab (ιea ω) ∧k (ιeb η) . (5)
k +1
Upshot The recursive description allows for implementation in symbolic computation in [9].
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7. Geometric algebra formulation
Preparations
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 sections
of the latter is the involutive C ∞ (M, R)-linear automorphism of the Kahler-Atiyah algebra
¨
(Ω(M), ):
d d
π(ω) = (−1)k ω (k) , ∀ω = ω (k ) ∈ Ω(M) , where ω (k) ∈ Ωk (M) .
k =0 k=0
(ea )a=1...d is a local frame of TM and (ea )a=1...d its dual local coframe, satisfying
a
ea (eb ) = δb and g(ea , eb ) = g ab , where (g ab ) is the inverse of the matrix (gab ). The
corresponding contragradient frame (ea )# and coframe (ea )# satisfy (ea )# = g ab eb and
(ea )# = gab eb , 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 involutive
antiautomorphism defined through:
k(k−1)
τ (ω) = (−1) 2 ω , ∀ω ∈ Ωk (M) . (6)
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8. Geometric algebra formulation
The Fierz Isomorphism
We defined pinors over M as sections of the bundle S, with fiberwise module structure
described 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 signature
with d ≡8 0, 1, that allows translating the CGK pinor conditions into conditions on
differential 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 a
sub-bundle ∧γ T ∗ 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 can
be used to define a B-dependent bundle isomorphism E : S ⊗ S → End(S) [3, 13]:
def.
Eξ,ξ (ξ ) = B(ξ , ξ )ξ such that Eξ1 ,ξ2 ◦ Eξ3 ,ξ4 = B(ξ3 , ξ2 )Eξ1 ,ξ4 .
Fierz Isomorphism
ˇ
The Fierz isomorphism is then E = γ −1 ◦ E : S ⊗ S → ∧γ T ∗ M .
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9. Geometric algebra formulation
Lift 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 and
solutions 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 the
ˆ
9-dimensional metric cone M, which allows for a global description of N = 2 supergravity
ˆ
compactifications on eight-manifolds through the reduction of the structure group of M.
ˆ
We consider the metric cone M = R+ × M with squared line element
dscone = dr 2 + r 2 ds2 and note that the canonical normalized one-form θ = dr is a special
2
ˆ
Killing-Yano form with respect to the metric gcone of M.
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10. Geometric algebra formulation
Lift to nine dimensions
Set-up
ˆ
We pull back the Levi-Civita connection of (M, g) through the projection Π : M → M, lift the
ˆ of M and define Π∗ (h) = h ◦ Π for
connection (2) from S to the pin bundle S ˆ
h ∈ C ∞ (M, R) [4].
ˆ ˆ
The morphism γcone : ∧T ∗ M → End(S) determined by γ leads to 2 inequivalent choices
ˆ
γcone (νcone ) = ±idS , where νcone is the volume form of M.
ˆ
ˆ ¨ ˆ
Pick the representation with γcone (νcone ) = +idS ⇒ Kahler-Atiyah algebra (Ω(M), cone )
ˆ ˆ ˆ
can be realized through the truncated model (Ω< (M), ♦cone ) [4]; Ω< (M) = ⊕4 Ωk (M)
k=0
and ♦cone is the reduced geometric product of M.ˆ
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11. Geometric algebra formulation
Lift to nine dimensions
Set-up
ˆ ˆ
Rescale g → 2κ2 g ⇒ the connection induced on S by the Levi-Civita connection of M is:
ˆ
S S ˆ
S
em = em + κγ9m , ∂r = ∂r , where m = 1 . . . 8 .
ˆ def. S,cone + Acone is the pullback of (2) to M; for (econe )
D =
ˆ ˆ
m cone the one-form dual to the
ˆ
base 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: ˆ
ˇ 1 1 cone ˇ 1 1 1
Am = ιem F + (em ) ∧ f ∧ θ and Q = r (d∆) − f ∧ θ −
cone F − κθ .
4 4 2 6 12
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12. Geometric algebra formulation
Analysis of CGK pinor equations
Pinor bilinears and inhomogeneous generators
We are interested in form-valued pinor bilinears written locally as:
ˇ (k) ˇ (k ) 1 ˆ cone ˆ a ...a
E ˆ ˆ ≡ Eij = B(ξi , γa1 ...ak ξj )e+1 k , with a1 . . . ak ∈ {1 . . . 9} , i, j ∈ {1, 2} ,
ξi ,ξj k!
where the subscript ‘+’ denotes the twisted self-dual part [3, 4], and in the weighted sums:
d
ˇ N ˇ (k)
Eij = d Eij .
2
k =0
d
ˇ ˆ
The inhomogeneous differential forms Eij generate the algebra (Ω+ (M), ); N = 2 2 is
ˆ
the real rank of the pin bundle S and d = 9 is the dimension of M. With the properties of
ˆ
the admissible bilinear paring on S, we find:
k (k−1)
ˆ a1 ...a ˆ
B(ξi , γcone k ξj ) = (−1) 2 ˆ a1 ...a ˆ
B(ξj , γcone k ξi ) , ∀i, j = 1, 2 . (7)
We use (7) to construct the non-trivial form-valued bilinears of rank ≤ 4.
12 / 23

13. Geometric algebra formulation
Analysis of CGK pinor equations
Pinor bilinears and inhomogeneous generators
Raising the indices of the local coeffitients with the metric, the pinor bilinears are:
a ˆ ˆ a ˆ a ˆ ˆ a ˆ a ˆ ˆ a ˆ
V1 = B(ξ1 , γcone ξ1 ) , V2 = B(ξ2 , γcone ξ2 ) , V3 = B(ξ1 , γcone ξ2 ), (8)
K = B(ξ1 , γcone ξ2 ) , ψ
ab ˆ ab ˆ abc
= B(ξ1 , γcone ξ2 ) ,
ˆ abc ˆ
φabce = B(ξ1 , γcone ξ1 ) ,
1
ˆ abce ˆ φabce = B(ξ2 , γcone ξ2 ) , φabce = B(ξ1 , γcone ξ2 ).
2
ˆ abce ˆ
3
ˆ abce ˆ
These forms are the definite rank components of the truncated inhomogeneous forms:
ˇ< 1 ˇ< 1
E11 = (1 + V1 + φ1 ) , E12 = (V3 + K + ψ + φ3 ) ,
32 32
ˇ< 1 ˇ< 1
E21 = (V3 − K − ψ + φ3 ) , E22 = (1 + V2 + φ2 ) .
32 32
ˇ<
Eij are the basis elements of the truncated Fierz algebra [3, 4] and satisfy the truncated
Fierz identities; dropping the ’cone’ superscript on ♦cone to simplify notation,
ˇ< ˇ< 1 ˇ<
Eij ♦Ekl = δjk Eil , ∀i, j, k, l = 1, 2 . (9)
2
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14. Geometric algebra formulation
Analysis of CGK pinor equations
Constraints
For the analysis of general N = 2 compactifications to AdS3 spaces we need the
constraints implied by the CGK pinor equations on the forms constructed with (8).
The differential constraints (2) are equivalent cf. [3, 4] with:
ˆ ˆ ) = ξ, [Acone , γa ...a ]ξ
m ωa1 ...ak (ξ, ξ
ˆ m cone ˆ ,
1 k
which in the language of geometric algebra is:
ˇ<
dEij = ea ∧ ˇ<
a Eij where ˇ<
a Eij
ˇ ˇ<
= −[Aa , Eij ]−,♦ , ∀i, j ∈ {1, 2} . (10)
ˆ ˆ
The algebraic constraints (3) reduce to ξ, (Q t γa1 ,...,ak ± γa1 ,...,ak Q)ξ = 0 , which in
cone cone
geometric algebra becomes:
ˇ ˇ<
Q ♦Eij ˇ< ˆ ˇ
Eij ♦τ (Q) = 0 , ∀i, j ∈ {1, 2} . (11)
We present the set of resulting relations and Fierz identities that we need for the analysis
in the next section – the complete set and the full Fierz solution can be found in [2].
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15. Geometric algebra formulation
Analysis of CGK pinor equations
The relations implied by the truncated Fierz identities and that we need here are:
ιV1 V3 = 0 , ιV1 φ3 − ψ + V1 ∧ K = 0 , ιV3 φ1 + ψ − V1 ∧ K = 0 . (12)
ˇ<
The equation in (11) with E12 and the minus sign leads to the following constraints:
ιf ∧θ K = 0 , (13)
1 1
r ιd∆ K + ιf ∧θ ψ − ιψ F − 2κιθ K = 0 , (14)
3 6
1 1
ιf ∧θ φ3 − F 3 φ3 + r (d∆) ∧ V3 + 2κV3 ∧ θ = 0 , (15)
3 6
ˇ<
while using the truncated inhomogeneous form E11 amounts to:
1 1 1
− f ∧ θ + ιf ∧θ φ1 − F 3 φ1 + r (d∆) ∧ V1 + 2κV1 ∧ θ = 0 . (16)
3 3 6
ˇ< ˇ<
Expanding (10) for E11 , E12 gives the covariant derivatives of V1 and V3 :
1 1 1 1
m V1n = fs θp φ1 sp mn − Fspqm φ1 spq n and m V3n = fs θp φ3 sp mn − Fspqm φ3 spq n .
2 12 2 12
(17)
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16. Analysis
Lie bracket of V1 and V3
We can investigate the geometric implications of the N = 2 supersymmetry condition using
the differential and algebraic constraints for the form-valued pinor bilinears.
In particular, we can compute the Lie bracket of the vector fields V1 , V3 dual to V1 , V3 .
ˆ
Pick a local orthonormal frame (ea ) of (M, gcone ) defined above an open subset U ⊂ M ⇒
gcone (ea , eb ) = δab and expanding V = V a ea with V a ∈ C ∞ (U, R), we find:
[V1 , V3 ] = [V1 , V3 ]b eb , [V1 , V3 ]b = V1 ea (V3 ) − V3 ea (V1 ) ∈ C ∞ (U, R) .
a b a b
We interpret vector fields defined on U as derivations of the commutative algebra
C ∞ (U, R) and consider the one-form dual to the vector field [V1 , V3 ]:
[V1 , V3 ] = [V1 , V3 ]a (ea ) = [V1 , V3 ]b eb , where
[V1 , V3 ]b = gba [V1 , V3 ]a = V1 ea (V3b ) − V3 ea (V1b ) .
a a
16 / 23

17. Analysis
Lie bracket of V1 and V3
Lie bracket of V1 and V3
ea (Vb ) = c c
a Vb + ωa b Vc , with ωa b the coefficients of the Levi-Civita connection
def. d computed with respect to the local
ωabc = gbd ωa c frame (ea ) and ωabc = −ωacb .
b
The Lie bracket becomes [V1 , V3 ]a = V1 b
− V3 b c b c
+ (V1 V3 − V3 V1 )ωbca .
b V3a b V1a
b
Compute the V1 b
− V3
b V3a b V1a component substituting the covariant derivatives (17):
1
[V1 , V3 ] = ιf ∧θ (ιV1 φ3 ) + ι(ιV F ) φ3 − ιf ∧θ (ιV3 φ1 ) − ι(ιV F ) φ1 . (18)
2 1 3
We omit in what follows the term with spin connection coefficients and rewrite:
ι(ιV F ) φ1 = −ιV3 (F 3 φ1 ) + ι(ιV φ1 ) F and ι(ιV F ) φ3 = −ιV1 (F 3 φ3 ) + ι(ιV φ3 ) F .
3 3 1 1
Eliminate ιV1 φ3 and ιV3 φ1 with the Fierz relations (12), simplify F 3 φ3 and F 3 φ1
through (15)-(16) recalling that ιV1 V3 = 0 due to (12) and eliminate ιf ∧θ ψ with (14).
17 / 23

18. Analysis
Lie bracket of V1 and V3
Finally, expanding ιf ∧θ (V1 ∧ K ), the Lie bracket (18) becomes:
1
[V1 , V3 ] = 3r ιd∆ K − (6κ + ιf V1 )ιθ K + (ιV1 θ)ιf K + 2 ιψ F − ιV1 ∧K F
−3(r ιd∆ V1 − 2κιV1 θ)V3 + 3(r ιd∆ V3 − 2κιV3 θ)V1 − (ιf V3 )θ + (ιV3 θ)f .(19)
Reducing the result to eight-dimensions
ˆ
Any k-form ω ∈ Ωk (M) decomposes uniquely as ω = θ ∧ ω + ω⊥ for ω ∈ Ωk−1 (M)
and ω⊥ ∈ Ωk (M), where θ ω = θ ω⊥ = 0 [4] ⇒ V1 = rU1 + w1 θ , V3 = rU3 + w3 θ.
ˆ
θ = dr is the canonical one-form of M, U1 , U3 are (pull-backs of) one-forms on M and
w1 , w3 are smooth functions on M. Since θ = ∂r ⇒
[V1 , V3 ] = [rU1 , rU3 ] + [rU1 , w3 ∂r ] + [w1 ∂r , rU3 ] + [w1 ∂r , w3 ∂r ] .
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19. Analysis
Reducing the result to eight-dimensions
Using the fact that wk = dwk ,
[V1 , V3 ] = r 2 [U1 , U3 ] + w1 U3 − w3 U1 + r (U1
b
b w3
b
− U3 b w1 )θ. (20)
Following the same steps used to derive (19), we find:
b b 1
r [U1 (w3 ) − U3 (w1 )] = r (U1 b w3 − U3 b w1 ) =r ιθ ι(ιU F ) φ3 − ι(ιU F ) φ1
2 1 3
= 3w1 r 2 ιd∆ U3 − 3w3 r 2 ιd∆ U1 − r ιU3 f ,
which matches the expression obtained directly from (19) by substituting V1 , V3 :
([V1 , V3 ] ) = 3w1 r 2 ιd∆ U3 − 3w3 r 2 ιd∆ U1 − r ιU3 f − 3r 2 ιd∆ (ιθ K ) + rw1 ιf (ιθ K ) .
19 / 23

20. Conclusions
We have presented some applications of the general theory developed in [3, 4, 13] to
N = 2 compactifications of M-theory on eight-manifolds.
We focused on the computation of a certain quantity of geometric interest — the Lie
bracket of V1 and V3 .
We note that we moreover find similar results working with V2 instead of V1 and that it is
also possible to compute [U1 , U2 ] .
The Lie brackets discussed above have various geometric implications related to the
existence of circle fibrations on the compactification manifold, as opposed to mere
foliations.
Some aspects of these geometric implications could be of interest in connection with
F-theory.
20 / 23

21. Acknowledgements
My contribution to this work was supported by the CNCS project PN-II-RU-TE contract number
77/2010 and I also acknowledge my Open Horizons scholarship from the Dinu Patriciu
Foundation, which financed part of my studies.
21 / 23

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