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.”

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

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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 ] + LXη − iY dξ is compatible with the natural split-signature metric on T ⊕T ∗ in the sense ATB, C = [A, B], C + B, [A, C]

• Generalized complex structure: O(2n, 2n) → U(n, n)

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Examples: JJ =

• −J

J∗

• ,

Jω =

• −ω−1

ω

• Local invariant:
• J =
• A

π σ −A∗

• , and π is a Poisson structure.
• k := n − 1

2rk π, called type of GCS.

complex →n, n − 1, · · · , 1, 0← symplectic

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Symmetries of the Courant bracket Sym(T ⊕ T ∗, [, ]) = Diff(M) ⊲ < Ω2

cl(M),

where ϕ ∈ Diff(M) acts via

• ϕ∗

(ϕ∗)−1

• ,

and B ∈ C∞(∧2T ∗) acts via the inclusion ∧2T ∗ ⊂ so(T ⊕ T ∗). eB =

• 1

B 1

• : X + ξ → X + ξ + iXB.

When [B] ∈ H2(M, Z), it is viewed as a B-field gauge transfor- mation, i.e. a gauge transformation of an S1-gerbe.

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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 K3 surface, we obtain a generalized complex structure on 3CP 2#19CP 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!

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• 1. Altered view of automorphisms

Aut(JJ) =

• (ϕ, B) :
• 1

B 1 −Jϕ J∗ϕ 1 −B 1

• =
• −J

J∗ = Aut(J) × Ω1,1

cl (M)

• This places J,ω on more similar footing, both have infinite

dimensional symmetry group.

• Restricting to gauge transformations, we have

Aut(JJ) = Aut(J) × Pic(M), which is precisely the group of invertible OM −OM 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.

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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- tries. Hence we include F ∈ Ω2

cl(S) as part of the brane data.

Then we replace the above with τS,F = {X + ξ ∈ TS ⊕ T ∗M : ξ|S = iXF}, 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.

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• If S = M, i.e. the brane is space-filling, then F is a brane iff
• −J

J∗ X FX

• =
• Y

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.

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• 2. Altered view of deformation theory

Deformations of a complex structure as a generalized complex structure have tangent space H0(∧2T ) ⊕ H1(T ) ⊕ H2(O). In particular, a section β ∈ C∞(∧2T1,0) 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 JJ,Q =

• 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.

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What are branes in the holomorphic Poisson manifold (J, β)? The space-filling branes are defined by the condition

• J

Q −J∗ X FX

• =
• Y

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 e−F

• J

Q −J∗

• eF =
• I

Q −I∗

• ,

proving that I is a complex structure compatible with Q as well.

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Therefore, when we turn on β = P + iQ, branes cease to be

• bjects which are holomorphic with respect to J, and become
• bjects which connect different complex structures, all of whom

agree with the real Poisson structure Q. Ji

Fij

Jj

Fjk

• Jk

Fki

• ,

Fij + Fjk + Fki = 0. The category of space-filling branes of rank 1 forms a groupoid but has no tensor structure; this is to be expected for modules

• ver a noncommutative ring.

We can of course tensor by a bimodule or symmetry of the geometry: Aut(JJ,Q) =

• (ϕ, B) :
• 1

B 1 Jϕ Qϕ −J∗ϕ 1 −B 1

• =
• J

Q −J∗ .

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Aut(JJ,Q) = {(ϕ,B) ∈ Diff(M) × Ω2

cl such that

• Qϕ = Q
• Jϕ − J = QB
• J∗B + BJϕ = 0}

Proposition: Let f ∈ C∞(M, R), and let X = Qd f be its Hamil- tonian vector field. Then the flow ϕt of X satisfies Qϕ = Q. Also, Jϕt − J = QBt, J∗Bt + BtJϕt, for Bt =

t

0 ddc tf.

Proof:

d dtJϕt = LXJϕt = −[Jϕt, X] = −[Jϕt, Qd

f] = Qddc

tf.

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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 ωI = gI, ωJ = gJ, one has Lemma: dc

IωI + dc JωJ = 0,

ddc

IωI = 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 JA/B = 1 2

• I ± J

−(ω−1

I

∓ ω−1

J )

ωI ∓ ωJ −(I∗ ± J∗)

• define commuting generalized complex structures with positive

definite product G = −JAJB, i.e. a generalized K¨ ahler structure.

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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

I

+ ω−1

J )−1,

implying that JB 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

• n 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 ddcf is exact for smooth f.

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

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