Branes and generalized complex manifolds Marco Gualtieri MIT - PowerPoint PPT Presentation

Branes and generalized complex manifolds Marco Gualtieri MIT September 4, 2006

“The more outre and grotesque an incident is the more carefully it deserves to be examined, and the very point which appears to complicate a case is, when duly considered and scientifically handled, the one which is most likely to elucidate it.” 1

“How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?” 2

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Generalized complex structures A generalized complex structure is a complex structure J on the bundle T ⊕ T ∗ which is integrable with respect to the Courant bracket. • The Courant bracket [ X + ξ, Y + η ] = [ X, Y ] + L X η − i Y dξ is compatible with the natural split-signature metric on T ⊕ T ∗ in the sense A T � B, C � = � [ A, B ] , C � + � B, [ A, C ] � • Generalized complex structure: O (2 n, 2 n ) → U ( n, n ) 4

� � � − ω − 1 � − J Examples: J J = , J ω = J ∗ ω Local invariant: � � A π • J = , and π is a Poisson structure. − A ∗ σ • k := n − 1 2 rk π , called type of GCS. complex → n, n − 1 , · · · , 1 , 0 ← symplectic 5

Symmetries of the Courant bracket Sym( T ⊕ T ∗ , [ , ]) = Diff( M ) ⊲ < Ω 2 cl ( M ) , where ϕ ∈ Diff( M ) acts via � � ϕ ∗ , ( ϕ ∗ ) − 1 and B ∈ C ∞ ( ∧ 2 T ∗ ) acts via the inclusion ∧ 2 T ∗ ⊂ so ( T ⊕ T ∗ ). � � 1 e B = : X + ξ �→ X + ξ + i X B. B 1 When [ B ] ∈ H 2 ( M, Z ), it is viewed as a B -field gauge transfor- mation, i.e. a gauge transformation of an S 1 -gerbe. 6

We now have exotic examples of generalized complex structures: Theorem: [G. Cavalcanti and M.G.] The multiplicity zero C ∞ log transform of a symplectic 4-manifold along an embedded symplectic 2-torus with trivial normal bundle admits a generalized complex structure with type change along a smooth 2-torus. Applying this to a K 3 surface, we obtain a generalized complex structure on 3 C P 2 #19 C P 2 , which admits neither complex nor symplectic structure, due to vanishing Seiberg-Witten invariants (Gompf-Mrowka). However, what does generalized geometry teach us about usual complex manifolds? Not nothing! 7

1. Altered view of automorphisms − J ϕ � � � � � � � � � 1 1 0 − J � Aut( J J ) = ( ϕ, B ) : = J ∗ ϕ J ∗ B 1 − B 1 = Aut( J ) × Ω 1 , 1 cl ( M ) • This places J , ω on more similar footing, both have infinite dimensional symmetry group. • Restricting to gauge transformations, we have Aut( J J ) = Aut( J ) × Pic( M ) , which is precisely the group of invertible O M −O M bimodules. • Important for the construction of homogeneous coordinate rings A • = Hom ( O , O ⊕ ( B ⊗ O ) ⊕ ( B ⊗ B ⊗ O ) ⊕ · · · ) and the twisting construction of Van den Bergh. 8

Modules (branes) for generalized complex manifolds Natural guess: given a submanifold S ⊂ M , require that TS ⊕ N ∗ S ⊂ TM ⊕ T ∗ M is preserved by J . Problem: This definition is not covariant under B -field symme- Hence we include F ∈ Ω 2 tries. cl ( S ) as part of the brane data. Then we replace the above with τ S,F = { X + ξ ∈ TS ⊕ T ∗ M : ξ | S = i X F } , and require it to be preserved by J . We may view this as a condition on the curvature F of a her- mitian connection on a complex line bundle. 9

• If S = M , i.e. the brane is space-filling, then F is a brane iff � � � � � � − J X Y = , J ∗ FX FY i.e. J ∗ F + FJ = 0, F is of type (1 , 1). • if S ⊂ M then TS must be preserved by J and F must be of type (1 , 1) in the induced complex structure. Hence S is a complex submanifold supporting a holomorphic line bundle. 10

2. Altered view of deformation theory Deformations of a complex structure as a generalized complex structure have tangent space H 0 ( ∧ 2 T ) ⊕ H 1 ( T ) ⊕ H 2 ( O ) . In particular, a section β ∈ C ∞ ( ∧ 2 T 1 , 0 ) deforms the complex structure J if and only if ∂β = 0 and [ β, β ] = 0, i.e. β is a holomorphic Poisson structure. If β = P + iQ then the deformed structure is � � J Q J J,Q = − J ∗ Note that J 2 = − 1 since JQ = QJ ∗ , i.e. Q is of type (2 , 0) + (0 , 2). This is a slightly different viewpoint from the understanding that β is a class in the Hochschild cohomology of O ( M ) which defines an infinitesimal deformation as a sheaf of associative algebras. 11

What are branes in the holomorphic Poisson manifold ( J, β ) ? The space-filling branes are defined by the condition � � � � � � J Q X Y = , − J ∗ FX FY which holds iff FJ + J ∗ F + FQF = 0 . Defining I = J + QF , this equation is equivalent to the general- ized (1 , 1) condition FI + J ∗ F = 0 . Note that � � � � J Q I Q e F = e − F , − J ∗ − I ∗ proving that I is a complex structure compatible with Q as well. 12

Therefore, when we turn on β = P + iQ , branes cease to be objects which are holomorphic with respect to J , and become objects which connect different complex structures, all of whom agree with the real Poisson structure Q . F ij � J j J i , F ij + F jk + F ki = 0 . � � � � � � � � � � � � F ki � F jk � � � � � J k The category of space-filling branes of rank 1 forms a groupoid but has no tensor structure; this is to be expected for modules over a noncommutative ring. We can of course tensor by a bimodule or symmetry of the geometry: J ϕ Q ϕ � � � � � � � � � 1 1 0 J Q � Aut( J J,Q ) = ( ϕ, B ) : = . − J ∗ ϕ − J ∗ B 1 − B 1 13

Aut( J J,Q ) = { ( ϕ,B ) ∈ Diff( M ) × Ω 2 cl such that Q ϕ = Q • J ϕ − J = QB • J ∗ B + BJ ϕ = 0 } • Proposition: Let f ∈ C ∞ ( M, R ), and let X = Qd f be its Hamil- Then the flow ϕ t of X satisfies Q ϕ = Q . tonian vector field. Also, J ϕ t − J = QB t , J ∗ B t + B t J ϕ t , for � t 0 dd c B t = t f. d dt J ϕ t = L X J ϕ t = − [ J ϕ t , X ] = − [ J ϕ t , Qd f ] = Qdd c Proof: t f . 14

Another source of examples for Poisson branes: Observation: ( F ( I + J )) ∗ = − ( I ∗ + J ∗ ) F = F ( I + J ), so g = F ( I + J ) is symmetric. Furthermore, defining ω I = gI , ω J = gJ , one has Lemma: d c I ω I + d c dd c J ω J = 0, I ω I = 0. If g is positive definite, the brane is said to be positive and the above equations define a generalized K¨ ahler structure : Theorem: The pair of operators − ( ω − 1 ∓ ω − 1 � � J A/B = 1 J ) I ± J I − ( I ∗ ± J ∗ ) ω I ∓ ω J 2 define commuting generalized complex structures with positive definite product G = −J A J B , i.e. a generalized K¨ ahler structure. 15

Remarks about the correspondence between positive Pois- son branes and generalized K¨ ahler structures: • Since g = F ( I + J ) is nondegenerate, so is F = g ( I + J ) − 1 = − ( ω − 1 + ω − 1 J ) − 1 , I implying that J B has symplectic type, with symplectic struc- ture F . Hence [ F ] � = 0 in cohomology. • Can use recent quotient construction for generalized K¨ ahler structures developed by Bursztyn-Cavalcanti-M.G. and Lin- Tolman to produce examples of Poisson branes. • Recently Hitchin observed that the Hamiltonian deformation argument can be used to produce interesting positive branes on Del Pezzo surfaces by using functions f with singularities which are cancelled by the zeros of Q . In this way one avoids the problem that dd c f is exact for smooth f . 16

Next time, I hope to tell you more about morphisms between branes! Thank you and happy birthday Nigel! 17