It is possible to transport the definition of an almost complex structure to exact Courant algebroids.
Definition A.4.1. An almost generalized complex structureJ on an exact Courant algebroid
(E, q,[·,·], π)is an almost complex structureJ onE which is orthogonal with respect toq, i.e.
J2 =−1 and (A.4.1)
q(Je1,Je2) =q(e1, e2). (A.4.2)
If the + i-eigenbundle Lof J, acting on the complexification of E, is involutive with respect to[·,·], we callJ an integrable almost generalized complex structure , or simply a generalized complex structure
Remark A.4.2 The second equation in above definition can also be formulated asJ∗ =
−J. Hence, we observe that an almost generalized complex structure onE is in some sense an almost complex structure and an almost symplectic structure at the same time. An immediate consequence of J being orthogonal with respect to q is that the + i- eigenbundle ofJ is isotropic, i.e.
q(e1, e2) =q(Je1,Je2) = i2q(e1, e2) =−q(e1, e2) = 0 ∀e1, e2 ∈L . (A.4.3)
Examples A.4.3 We give two simple examples on the standard Courant algebroid which show that complex and symplectic structures can be embedded into this for- malism.
1. IfI is an almost complex structure, i.e. an endomorphism onT M which squares to−1, it is evident that JI = I 0 0 −I∗ (A.4.4) is an almost generalized complex structure associated toI. The+ i-eigenbundle is given byLI =T M1,0⊕T∗M0,1. Moreover, it is true thatJI is integrable if and
only ifIis integrable as an almost complex structure.
2. If ω is a non-degenerate 2-form on M, we can associate the almost generalized complex structure Jω = 0 −ω−1 ω 0 . (A.4.5)
The+ i-eigenbundle is given byLω = e−iωT M = {X⊕ −iω(X)|X ∈ T M}. It is
A.4. GENERALIZED COMPLEX STRUCTURES AND INTEGRABILITY The connection between Dirac structures and generalized complex structures is
Proposition A.4.4. A generalized complex structure is equivalent to a complex Dirac struc- tureL⊂E⊗Csuch thatL∩L={0}, whereLis the+ i-eigenbundle ofJ.
As a result, the+ i-eigenbundle(L,[·,·], π)defines the structure of a Lie algebroid and we obtain a differential complex
−−→dL −−−→dL −−−→dL −→dL · · · C∞(∧kL∗) C∞(∧k+1L∗) · · · , (A.4.6) where dLω(l1, . . . , lk+1) : = k+1 X i=1 (−1)i+1π(l i)ω(l1, . . . ,ˆli, . . . , lk+1) +X i<j (−1)i+jω([li, lj], l1, . . . ,ˆli, . . . ,ˆlj, . . . , lk+1). (A.4.7)
Proposition A.4.5. The Lie algebroid complex of a generalized complex structure is elliptic.
This provides us with
Corollary A.4.6. The cohomology of the complex(A.4.6), called the Lie algebroid cohomology
H•(M, L), is a finite dimensional graded ring associated to any compact generalized complex manifold.
In order to prepare for a local classification of generalized complex structures, we need
Definition A.4.7. The type of the generalized complex structureJ is the upper semi-continuous function
type(J) = 1
2dimRT
∗M ∩ JT∗M . (A.4.8)
Moreover, we need
Proposition A.4.8. LetJ be a generalized complex structure with+ i-eigenbundleL⊂E⊗C.
ThenP :=π◦J ◦π∗is a Poisson bivector. The distribution∆ =π◦J ◦π∗(T∗M)integrates to a generalized foliation by smooth symplectic leaves with codimension2k, wherek= type(J).
APPENDIX A. GENERALIZED COMPLEX GEOMETRY
Theorem A.4.9. At any point, a generalized complex structure of typek is equivalent, by a choice of a smooth isotropic splitting ofE, to the direct sum of a complex structure of complex dimensionkand a symplectic structure of real dimension2n−2k.
To get an expression in a neighborhood we need
Definition A.4.10. A pointp∈ M in a generalized complex manifold is called regular when the Poisson structureP is regular atp, i.e.type(J)is locally constant atp. A neighborhood in which every point is regular is called a regular neighborhood. If anyp ∈M is regular, we call
J regular.
By corollary A.4.8, a generalized complex structure defines, in a regular neighborhood
U, a foliation F by symplectic leaves of codimension 2k = 2type(J), integrating the distribution∆. We get a complex structure on the leave space
Proposition A.4.11. The leaf spaceU/F of a regular neighborhood of a generalized complex manifold inherits a canonical complex structure.
This enables us to state
Theorem A.4.12(Generalized Darboux theorem). A regular point of typekin a generalized complex manifold has a neighborhood which is equivalent to the product of an open set inCk
with an open set in the standard symplectic space(R2n−2k, ω0).
At the end of this section we should state the definition of a generalized Calabi-Yau structure, a generalized K¨ahler structure and a generalized Calabi-Yau metric geome- try.
Definition A.4.13. The canonical line bundle of a generalized complex structure onTM is the
complex pure spinor line subbundleK ⊂ ∧•T∗⊗Cannihilated by the+ i-eigenbundleLofJ. Definition A.4.14. A generalized Calabi-Yau structure is a generalized complex structure with holomorphically trivial canonical bundle, i.e. admitting a nowhere-vanishing dH-closed
sectionρ∈ C∞(K).
Definition A.4.15. A generalized K¨ahler structure is a pair(J1,J2)of commuting generalized
complex structures such thatG=−J1J2 is a positive definite metric onTM.
Definition A.4.16. A generalized K¨ahler structure (J1,J2) is called a generalized Calabi-
Yau metric geometry if J1 and J2 define a generalized Calabi-Yau structure on their own,
with nowhere vanishing dH-closed sectionsρi ∈ C∞(Ki), and if both sections are related by a
constant, i.e.
(ρ1, ρ1) = c(ρ2, ρ1), (A.4.9)
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Index
A-brane, 84 actionnonlinear sigma-model on Rieman- nian mfd., 42, 45
SQM on a Riemannian mfd., 28 supersymmetric sigma-model with
H-flux, 60
admissible vector fields along a mapΦ, 138, 144
in almost complex mfds., 148 in almost symplectic mfds., 148 almost complex structure, 172 almost Dirac structure, 171
integrable, 171
almost generalized complex structure, 172 integrable-, 172
associated toω, 172 associated toI, 172
compatible to another almost com- plex structure, 95
nonsingular, 147
tamed by another almost complex struc- ture, 94
almost symplectic structure, 90 anchor, 168
β-transformation, 167 bi-Hermitian structure, 61 BRST cohomology, 25
B-transformation, 71, 167
-of an exact Courant algebroid, 170 canonical transformation, 62, 72
Carleman similarity principle, 126 co-isotropic A-brane, 53
coisotropic submanifold, 78
connection coefficients (notation), 47 connection preserving map, 46 Courant algebroid, 168
exact, 169 exact-, 169 standard-, 170
covariant exterior derivative, 44 covariant Lie derivative, 55
differential of a map between smooth man- ifolds, 43
viewed as a vector bundle valued1- form, 44 Dirac structure, 171 almost-, 171 Dorfman bracket, 170 twisted, 170 (E,J)-holomorphic curve, 93
invariance under orthogonal automor- phisms, 94
(E,J)-holomorphic pair, 93 critical point of-, 127 critical value of-, 127 elliptic regularity, 123
Index
finite deformation, 141 identity theorem, 120
infinitesimal deformation, 141 injective point of-, 127
multiply covered, 127 noninjective point of-, 135 simple, 127
somewhere injective, 127
enhanced solution space of a mapΦ, 53 exact Courant algebroid, 169
fermion-number, 26, 30 fermion-number operator, 25
formal adjoint of covariant derivative, 49
generalized B-model, 65–75
generalized Calabi-Yau manifold,seegen- eralized Calabi-Yau metric geom- etry
generalized Calabi-Yau metric geometry, 59, 65, 174
generalized Calabi-Yau structure, 174 generalized Darboux theorem, 174 generalized energy, 62, 104
invariance under homotopy, 109 invariance under orthogonal automor-
phisms, 104
generalized energy identity, 105 generalized holomorphic map, 90 generalized K¨ahler structure, 61, 96, 174 generalized Levi-Civita connection, 139 generalized Levi-Civita operator, 139 generalized nonlinear Cauchy-Riemann
equation, 119
generalized torsion operator, 138 generalized vertical differential, 142
reduced, 146
geodesic flow, 141–142, 144
grading operator, 69 Hyperk¨ahler manifold, 73
identity theorem of generalized pseudo- holomorphic pairs, 120
inner product of vector bundle valued
k-forms global, 45 local, 44 instanton, 27, 33, 59, 70 instanton corrections, 27 isotropic embedding, 92, 98, 109, 169 extension, 98 isotropic splitting, 93, 98, 169 isotropic subbundle, 91, 172 J-holomorphic curve, 88 J-holomorphic curve, 59 J-holomorphic curve, 93 Langlands correspondence, 11
Lie algebroid cohomology, 67, 70, 173 Lie algebroid complex, 173
Lie algebroid derivative, 173 linear Dirac structure, 167 local observable, 66
localization principle, 26–27
main theorem of local(E,J)-holomorphic pairs, 129
maximal isotropic subbundle seealmost Dirac structure, 171 maximal isotropic subspace, 167 metric
induced by tamed structures, 95 mirror symmetry, 7–11, 73
mode expansion, 57
Index nonlinear sigma-model on a Riemannian
manifold, 41–58 action, 42, 45
equations of motion global, 46, 50–52 local, 42
number operator,seefermion-number op- erator orthogonal automorphism onTMinduced by a diffeomorphism, 170 orthogonal automorphism of E bracket preserving-, 170 perturbative ground state, 34 Poisson bivector, 173
pullback bundle, 42
pullback connection, 46–47 connection coefficients, 48 pullback metric, 44, 48
pullback of almost Dirac structures, 90 pure spinor, 168
pushforward of almost Dirac structures, 90
R-charge, 62, 63 anomaly, 64 R-symmetry, 28, 62
regular generalized complex structure, 174
regular neighborhood, 174 regular point, 174
spin representation, 168 spin represntation, 69
standard Courant algebroid, 170 supercovariant derivative, 61 superfield, 61
supersymmetric ground state, 23, 69, 70
supersymmetric operator, 24 SQM
-on a Riemannian manifold, 27–40 action, 28 algebra of observables, 30 fermion-number, 30 fermion-number operator, 29 Hamiltonian, 31 Hilbert space, 29 instanton, 36
instanton corrected action ofQ, 39 instanton corrections, 34–40 supercharges, 29
supersymmetric ground states, 31 supersymmetry transformations, 28 Witten index, 31 conserved supercharges, 22 definition, 21 grading, 22, 23 non-negative spectrum, 22 supersymmetric ground state, 23 supersymmetry algebra, 22
zero-energy state,seesupersymmet- ric ground state
supersymmetry transformations, 28, 61, 66 target manifold, 41 tension-field, 45 topological twist, 62 A-twist, 63 B-twist, 63
twisted generalized complex map, 59, 71 type
generalized complex structure, 173 maximal isotropic, 168
of a maximal isotropic, 168 vector bundle valued forms, 44
Index
vector field along a map between smooth manifolds, 43
Witten index, 24, 25 world sheet, 41