M-Theory on Calabi-Yau 5-folds K.S. Stelle Workshop on Holonomy - PowerPoint PPT Presentation

M-Theory on Calabi-Yau 5-folds K.S. Stelle Workshop on Holonomy Groups and applications Hamburg 16 July 2008 Work with Alexander Haupt and André Lukas; previous work with Hong Lü, Chris Pope and Paul T ownsend 1

Motivation CY 3 manifolds provide one of the most important approaches to phenomenological contact between realistic physics and string/M-theory. The standard embedding of an SU(3) spin connection into the heterotic string’s E 8 xE 8 gauge group breaks the YM gauge group down to E 8 xE 6 and E 6 is physically appealing. At the same time, from an M-theory perspective, the 4+7 split is unnatural. A more “democratic” formulation of the spatial dimensions would seem more natural. Cosmology could naturally involve a 1+10 split. All space dimensions would initially be treated as compact, in anticipation of 3 of them expanding. 2

Overview Review of bosonic sector of D=11 supergravity including normalizations Bilal T opological considerations and flux quantization in M- theory topological constraint on compact 10-manifolds CY moduli sigma model 2-component local supersymmetry in D=1 Effect of corrections on CY 5 geometrical structure α ′ Supersymmetry preservation and generalized holonomy 3

D=11 supergravity I 11 = I CJS,B + I CJS,F + I GS + . . . . � � 1 � R ∗ 1 − 1 2 G ∧ ∗ G − 1 G [ 4 ] = dC [ 3 ] I CJS,B = 6 G ∧ G ∧ C 2 κ 2 M 11 4-form field strength for 4 1 � d 11 x √− g � ψ M Γ MNP D N ( ω ) ψ P ¯ I CJS,F = − 3-form gauge field 2 κ 2 11 M � ¯ + 1 G NP QR + (fermi) 4 � ψ M Γ MNP QRS ψ S + 12 ¯ ψ N Γ P Q ψ R � , 96 The above terms combine to form an invariant under the classical supersymmetry transformations δ ǫ g MN = 2¯ ǫ Γ ( M ψ N ) , δ ǫ C MNP = − 3¯ ǫ Γ [ MN ψ P ] , 1 NP QR − 8 δ N M Γ P QR ) ǫ G NP QR + (fermi) 3 . δ ǫ ψ M = 2 D M ( ω ) ǫ + 144( Γ M 4

Variation of the Cremmer-Julia-Scherk action leads to the classical supergravity field equations: R MN = 1 1 M 2 ...M 4 − 144 g MN G M 1 ...M 4 G M 1 ...M 4 12 G MM 2 ...M 4 G N d ∗ G + 1 2 G ∧ G = 0 . Γ MNP D N ( ω ) ψ P + 1 dynamics is encoded in the fermionic action G NP QR + (fermi) 3 = 0 , Γ MNP QRS ψ S + 12 δ MN Γ P Q ψ R � � 96 supersymmetry transformations for all fields are Quantum corrections change these equations in a way that is important for CY 5 compactifications. Among the �������������������������� quantum corrections is a Green-Schwarz β = ( 2 π ) 2 α ′ 3 type term needed for M 5 -brane worldvolume anomaly cancellations. Vafa & Witten Duff, Liu & Minasian This GS term is a superpartner of the effective R 4 µ νρσ action corrections. 5

The classical CJS equation for C [ 3 ] d ∗ G + 1 2 G ∧ G = 0 is accordingly modified by the Green-Schwarz correction I GS = − ( 2 π ) 4 β Z C ∧ X 8 2 κ 2 11 � � 1 − 1 768 ( trR 2 ) 2 + 1 where 192 trR 4 X 8 = ( 2 π ) 4 This gives rise to the quantum-corrected equation d ∗ G + 1 2 G ∧ G + (2 π ) 4 β X 8 = 0 , 6

The Green-Schwarz correction term is necessary for cancelation of anomalies on the d=6 worldvolumes of 1 5-branes: T 5 = 5-brane tension β = ( 2 π ) 3 T 5 One also has the Dirac quantization condition T 2 T 5 = 2 π T 2 = 2-brane tension 2 κ 2 and the condition 11 T 5 = 1 2 π T 2 de Alwis 2 which is needed, e.g., for invariance under large 3- form gauge transformations. Lavrinenko, Lü, Pope & K.S.S Kalkkinen & K.S.S Putting these together, have � 2 κ 2 � 1 / 3 � 2 / 3 � 2 π 2 and 2 κ 2 11 = (2 π ) 8 ( α ′ ) 9 / 2 . 11 β = T 2 = κ 2 ( 2 π ) 5 11 7

T opological considerations A. Haupt, A. Lukas & K.S.S. Corrected 3-form field equation: d ∗ G + 1 2 G ∧ G +( 2 π ) 4 β X 8 = 0 where �� p 1 � X 8 = 1 � 2 − p 2 48 2 � 1 � 2 p 1 = − 1 tr R 2 1st & 2nd 2 2 π � 1 Pontriagin classes � 4 � p 2 = 1 ( tr R 2 ) 2 − 2tr R 4 � 8 2 π Now specialize to and M 11 = R × CY 5 simplify above relations: p ( T ( R × CY 5 )) = p ( T ( R )) ∧ p ( T ( CY 5 )) so is given by p ( T ( M 10 )) p ( T ( CY 5 )) p ( T ( R )) = 1 8

Now, for complex manifolds, there are relations between Pontriagin and Chern classes: p 1 = c 2 1 − 2 c 2 T . Hübsch p 2 = 2 c 4 − 2 c 1 c 3 + c 2 2 so for the case of a Calabi-Yau manifold with � 2 � p 1 one has c 1 = 0 − p 2 = − 2 c 4 2 X 8 = − 1 and consequently 24 c 4 1 Define and use the corrected field g = ( 2 π ) 2 β 1 / 2 G equations together with the fact that is exact to d ∗ G � 1 � deduce giving the 2 G ∧ G +( 2 π ) 4 β X 8 = 0 topological constraint c 4 ( CY 5 ) − 12 [ g ] ∧ [ g ] = 0 9

4-form flux quantization 2-branes couple to the background via C [ 3 ] Z Z S 2 br ∂ D 4 = W 3 WZ = T 2 C → T 2 G W 3 D 4 This gives the flux quantization condition [ g ] − p 1 4 ∈ H 4 ( CY 5 , Z ) Witten or, for , c 1 = 0 g = T 2 [ g ]+ c 2 2 ∈ H 4 ( CY 5 , Z ) 2 π G Thus, depending on the value of the 2 nd Chern class , the normalized flux is quantized in integer or g c 2 half-integer units. Happily, this is consistent with the topological constraint c 4 ( CY 5 ) − 12 [ g ] ∧ [ g ] = 0 10

For complete intersection compact , analysis CY 5 shows that requiring so CY c . i . � � c 4 > 0 [ g ] � = 0 5 4-form flux must be turned on at order � β However, one can make orbifold constructions with . c 4 = 0 Non-compact can also have . CY 5 c 4 = 0 In cases with , the flux is turned on at order β c 4 = 0 11

CY 5 moduli Supersymmetric sigma model D = 1 CY 5 Hodge diamond: 1 0 0 h 1 , 1 0 0 h 1 , 2 0 0 h 1 , 2 h 1 , 3 h 2 , 2 h 1 , 3 0 0 1 h 1 , 4 h 2 , 3 h 2 , 3 h 1 , 4 1 Hirzebruch-Riemann-Roch theorem with : c 1 = 0 11 h 1 , 1 − 10 h 1 , 2 − h 2 , 2 + h 2 , 3 + 10 h 1 , 3 − 11 h 1 , 4 = 0 so there are 6-1=5 independent Hodge numbers. The corresponding harmonic forms contribute D = 1 massless Kaluza-Klein modes. 12

Metric: ds 2 = − Nd τ 2 + 2 g rs ( x , ϕ I ( τ )) dx r dx s ϕ I ( τ ) = ( t i ( τ ) , z a ( τ ) , z ¯ a ( τ )) h 1 , 1 h 1 , 4 moduli in complex coordinates x r → x µ , x ¯ µ , ¯ ν ν = 1 ,..., 5 ν = δ t i ω iµ ¯ a b ¯ ν = δ z a b a ¯ δ g µ ν = δ z ¯ δ g µ ¯ δ g ¯ ν aµ ν ν µ ¯ µ ¯ i o¯ ρ ¯ σ ¯ ¯ τ χ ¯ b ¯ || Ω || 2 Ω µ ω i ∈ Harm ( 1 , 1 ) aµ ν = o¯ ρ ¯ σ ¯ a ¯ τν (5,0) volume (4,1) harmonic χ a ∈ Harm ( 1 , 4 ) form form 3-form field: δ C = ξ p ( τ ) ν p + c.c. ν p ∈ Harm ( 1 , 2 ) h 1 , 2 13

Fermionic zero modes Expand using the Killing spinor on CY 5 , Ψ M ( τ , x r ) η ( x r ) e.g. η † η = 1 Ψ 0 ( τ , x r ) = ¯ ψ 0 ( τ ) η ( x r )+ cc For the expansion uses the Ψ µ ( x r ) , Ψ ¯ ν ( x r ) (1,1), (2,1), (3,1) and (4,1) harmonic forms: ⊗ ⊗ (1,1) (2,1) µ γ α 1 η ) + 1 µ = ψ i ( τ ) ⊗ ( ω i α 1 ¯ 4 λ p ( τ ) ⊗ ( ν p α 1 α 2 ¯ µ γ α 1 α 2 η ) ψ ¯ + 1 µ γ α 1 ... α 3 η ) − 1 4! ρ x ( τ ) ⊗ ( ̟ x α 1 ... α 3 ¯ 4! κ a ( τ ) ⊗ ( || Ω || − 1 χ a α 1 ... α 4 ¯ µ γ α 1 ... α 4 η ) , (3,1) (4,1) = ( ) µ ) ∗ , ψ µ = ( ψ ¯ The (3,1) species has no bosonic partners, however. This points out a strange feature of supersymmetric life in : on-shell bosonic and fermionic degrees D = 1 Coles & of freedom do not have to balance. Papadopoulos 14

What happens to the other possible types of harmonic forms, e.g. (3,2), (2,2) and (5,0)? These are reabsorbed into the (1,1) and (2,1) harmonic types. T o see this, one needs to use the property γ ¯ µ η = 0 of CY Killing spinors together with the Dirac algebra and Fierz identities to reduce { γ µ , γ ¯ ν } = 2 g µ ¯ ν these species to other types. E.g. the (5,0) type is converted into a (1,1) species, and is the superpartner of the CY volume modulus. 15

Bosonic sigma model gauge N=1 � 1 � Z − → t j + G ( 2 , 1 ) ξ p ˙ 4 G ( 1 , 1 ) q − 4 V ( t ) G ( 4 , 1 ) ¯ q ( t ) ˙ ¯ I B t i ˙ ξ ¯ z a ˙ b d τ ( t ) ˙ b ( z , ¯ z ) ˙ z ¯ p ¯ a ¯ CJS i j M 11 = R × CY 5 K ( 1 , 1 ) = − 1 = ∂ i ∂ j K ( 1 , 1 ) − 25 K i K j G ( 1 , 1 ) 2ln K ij K 2 Z J = t i ω i K = complex structure J ∧ J ∧ J ∧ J ∧ J = d i 1 ... i 5 t i 1 ... t i 5 d i 1 ... i 5 : intersection numbers Z ω i ∧ J ∧ J ∧ J ∧ J = d i j 1 ... j 4 t j 1 ... j 4 K i = K ( 4 , 1 ) = − ln ( i ( G ¯ a − z a ¯ G ( 4 , 1 ) b K ( 4 , 1 ) a z ¯ = ∂ a ∂ ¯ G a )) a ¯ b Z G ( 2 , 1 ) qij t i t j X ν p ∧ ∗ ¯ = − 2 ν ¯ q = i d p ¯ Canonical inner product p ¯ q 16

Notes The (1,1) metric is not a canonical special Kähler metric but it is determined by intersection numbers (topological data), as is the canonical (2,1) metric. The (4,1) metric is the canonical Weil-Peterson metric (very special Kähler) but it is determined by a prepotential (involving non-topological data). The Kähler and complex structure sectors don’t decouple owing to the V(t) factor. 17