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Thursday, January 29, 2015

What is generalized geometry?

The following are my notes for a short introductory talk. References below are not intended to be comprehensive!

Math references:

 
Physics references:

 What is geometry?

Before trying to define generalized geometry, we should first decide what we mean by ordinary geometry. Of course, this question doesn't have a unique answer, so there are many ways to generalize the classical notions of manifolds and varieties. The viewpoint taken in generalized geometry is the following: the distinguishing feature of smooth manifolds is the existence of a tangent bundle
TMM
which satisfies some nice axioms. The basic idea of generalized geometry is to replace the tangent bundle with some other vector bundle LM, again satisfying some nice axioms. Different generalized geometries on M will correspond to different choices of bundle LM, as well as auxiliary data compatible with L in some appropriate sense.

Definition. A Lie algebroid over M is a smooth vector bundle LM together with a vector bundle map a:LTM called the anchor map and a bracket [,]:H0(M,L)H0(M,L)H0(M,L) satisfying the following axioms:
  • [,] is  a Lie bracket on H0(M,L)
  • [X,fY]=f[X,Y]+a(X)fY for X,YH0(M,L) and fH0(M,OM)
Note that we can take L to be either a real or complex vector bundle. In the latter case the anchor map should map to the complexified tangent bundle.

Example 1. We can take L to be TM with anchor map the identity.

Example 2. Let σ be a Poisson tensor on M. Then we define a bracket by [X,Y]=σ(X,Y) and an anchor by Xσ(X,). This makes TM into a Lie algebroid.

Example 3. Let M be a complex manifold of and let LTMC be the sub-bundle of vectors spanned by {/z1,,/zn} in local holomorphic coordinates. Then LM is a (complex) Lie algebroid.


Courant Bracket

We'd like to try to fit the preceding examples into a common framework. Let TM=TMTM. This bundle has a natural symmetric bilinear pairing given by
Xα,Yβ=12α(Y)+12β(X)
Note that this bilinear form is of split signature (n,n). We define a bracket on sections of TM by
[Xα,Yβ]=[X,Y](LXβ+12(dα(Y))LYα12d(β(X)))
Note that this bracket is not a Lie bracket. We also have an anchor map a:TMTM which is just the projection.

 Let B be a 2-form on M. Define an action of B on sections of TM by
X+αX+α+iXB

Proposition. This action preserves the Courant bracket if and only if B is closed.

This shows that the diffeomorphisms of M as a generalized manifold are large than the ordinary diffeomorphisms of M. In fact is is the semidirect product of the diffeomorphism group of M with the vector space of closed 2-forms.

Dirac Structures

Definition. A Dirac structure on M is an Lagrangian sub-bundle LTM which is closed under the Courant bracket.

Theorem (Courant). A Lagrangian sub-bundle LTM is a Dirac structure if and only if LM is a Lie algebroid over M, with bracket induced by the Courant bracket and anchor given by projection.

Example 1. TMTM.

Example 2. Take L to be the graph of a Poisson tensor.

Example 3. Take L to be the graph of a closed 2-form.

Admissible Functions

We now let LM be a Dirac structure on M.

Definition. A smooth function f on M is called admissible if there exists a vector field Xf such that (Xf,df) is a section of L.

The Poisson bracket is defined as follows. If f,g are admissible, then define
{f,g}=Xfg.
It is easy to check from the definitions that the bracket on admissible functions is well-defined (independent of choice of Xf) and skew-symmetric. With a little bit of calculation, we find the following.

Proposition. The vector space of admissible functions is naturally a Poisson algebra, and moreover the natural bracket satisfies the Leibniz rule.


Generalized Complex Structures

Definition. A generalized complex structure is a skew endomorphism J of TM such that J2=1 and such that the +i-eigenbundle is involutive under the Courant bracket.

Equivalently: A generalized complex structure is a (complex) Dirac structure LTM satisfying the condition L¯L=0.

Example 1. Let J be an ordinary complex structure on M. Then the endomorphism
[J00J]
defines a generalized complex structure on M.

Example 2. Let ω be a symplectic form on M. Then the endomorphism
[0ω1ω0]
defines a generalized complex structure on M.

Thus, generalized geometry gives a common framework for both complex geometry and symplectic geometry. Such a connection is exactly what is conjectured by mirror symmetry.

Example 3. Let J be a complex structure on M and let σ be a holomorphic Poisson tensor. Consider the subbundle LTM defined as the span of
ˉz1,,ˉzn,dz1σ(dz1),,dznσ(dzn)
Then L defines a generalized complex structure on M.

The last example shows that deformations of M as a generalized  complex manifold contain non-commutative deformations of the structure sheaf.  We also have the following theorem, which shows that there is an intimate relation between generalized complex geometry and holomorphic Poisson geometry.

Theorem (Bailey). Near any point of a generalized complex manifold, M is locally isomorphic to the product of a holomorphic Poisson manifold with a symplectic manifold.


Generalized Kähler Manifolds

Let (g,J,ω) be a Kähler triple. The Kähler property requires that
ω=gJ.
Let I1 denote the generalized complex structure induced by J, and let I1 denote the generalized complex structure induced by the symplectic form ω. We have
I1I2=[J00J][0ω1ω0]=[0g1g0]=I2I1

 Definition. A generalized Kähler manifold is a manifold with two commuting generalized complex structure I1,I2 such that the bilinear pairing (I1I2u,v) is positive definite.

Theorem (Gualtieri). A generalized Kähler structure on M induces a Riemannian metric g, two integrable almost complex structures J± Hermitian with respect to g, and two affine connections ± with skew-torsion ±H which preserve the metric and complex structure J±. Conversely, these data determine a generalized Kähler structure which is unique up to a B-field transformation.

Thus the notion of generalized Kähler manifold recovers the bihermitian geometry investigated by physicists in the context of susy non-linear σ-models.



Generalized Calabi-Yau Manifolds

Definition. A generalized Calabi-Yau manifold is a manifold M together with a complex-valued differential form ϕ, which is either purely even or purely odd, which is a pure spinor for the action of Cl(TM) and satisfies the non-degeneracy condition (ϕ,ˉϕ)0.

Note that (by definition) ϕ is pure if its annihilator is a maximal isotropic subspace. Let LTM be its annihilator. Then it is not hard to see that L defines a generalized complex structure on M, so indeed a generalized Calabi-Yau manifold is in particular a generalized complex manifold.

Example. If M is a complex manifold with a nowhere vanishing holomorphic (n,0) form, then it is generalized Calabi-Yau.

Example. If M is symplectic with symplectic form ω, then ϕ=exp(iω) gives M the structure of a generalized Calabi-Yau manifold.

If (M,ϕ) is generalized Calabi-Yau, then so is (M,exp(B)ϕ) for any closed real 2-form B. In the symplectic case, we obtain
ϕ=exp(B+iω)
This explains the appearance of the B-field (or "complexified Kähler form") in discussions of mirror symmetry.

1 comment:

Unknown said...

In the statement of the theorem by Gualtieri the conclusion mentions integrable almost-complex structures. Isn't that the same as a complex structure?