whereτ is the chiral superfield withiC+V /λsin its bottom component,Cbeing the background value of the 3–form gauge field in IIA andV the size of theS3 as before. Ctakes the role of theθ angle and the gauge coupling is promoted to a superfield. The second term in (C.38) stems from instanton corrections. It might seem surprising thatV enters into this superpotential (although the A model is independent of the complex structure modulus), the reason lies within quantum corrections. But the linear term in S is the only coupling that V has to this theory.
We now use that τ is actually a dynamical superfield and can be integrated out from its equation of motion. Requiring ∂τW = 0 gives
S = iλsN αe−τ /N. (C.39)
Solving this forτ and plugging the result back into (C.38) gives an effective superpotential for S
Weff = −N λs
S log
S iN αλs
−S
. (C.40)
We do indeed recover the Veneziano–Yankielowicz superpotential [59]. The scale of the gauge theory Λ can be identified with (N αλs)1/3. The vacuum of the theory exhibits all the known phenomena of gaugino condensation, chiral symmetry breaking and domain walls. This is a remarkable result and the first example where string theory produces the correct superpotential of a gauge theory.
C.4. GENERALIZED COMPLEX GEOMETRY 111 The spaceT⊕T∗ has a number of symmetries, for example the inner product is invariant under SO(n, n), the one which is of most interest to physicists is the B–field transform.
Let B : T → T∗ with B∗ = −B, we can therefore view B as a two–form in V2T∗ via B(X) =ιXB, with the interior product ιX :Vr
(M)→Vr−1
(M) defined as
ιXω(X1, . . . , Xr−1) = ω(X, X1, . . . , Xr−1) (C.42) with vector fieldsXi. This means, for example,
ifXi = ∂
∂xi ιXidxj∧dxk = δji dxk−δki dxj. (C.43) The B–field transform is then defined under the natural splitting T⊕T∗ as a 2n×2nmatrix
exp(B) =
1 0 B 1
(C.44) and acts as X +ξ → X +ξ+ιXB. This means, it acts as a projection onto T and by a
“shearing” transformation onT∗. Consider for exampleX+ξ =∂/∂x+dx andB =b dx∧dy.
The B–field acts on an argument like this as
X+ξ → X+ξ+b dy . (C.45)
This is very reminiscent of the action of “T–duality with B–field along x” we encountered numerous times throughout this thesis. Our hope is therefore that the non–K¨ahler manifolds we constructed in type IIA (and also heterotic and type I) have a natural interpretations in terms of generalized complex structures. See also [91] for similar interpretations.
After this motivation, let us define the basic quantities needed for generalized complex ge-ometry. We will not be very thorough and not aim for completeness, see [89] for a complete introduction to the subject. The above mentioned Courant bracket is defined as
[X+ξ, Y +η] = [X, Y] +LXη− LY ξ−1
2d(ιXη−ιY ξ) (C.46) with X, Y ∈T and ξ, η ∈ T∗. This is a skew–symmetric object but does not satisfy a Jacobi identity. L indicates the usual Lie–derivative
LX = d ιX +ιXd . (C.47)
Note that the Courant bracket reduces to the ordinary Lie bracket on vector fields. The Courant bracket, like the inner product, is not only invariant under diffeomorphisms, but also under the B–field transform. The map eB is an automorphism of the Courant bracket if and only if B is closed, i.e. dB= 0.
In physics, we are not only interested in closed B–fields (vanishing field strength). One can, however, defined “twisted” quantities that differ from the usual ones by terms involvingH =db (with two–form b), so the formalism of generalized complex geometries is still applicable. The twisted Courant bracket, for example, is defined in terms of the usual Courant bracket [,] as
[X+ξ, Y +η]H = [X+ξ, Y +η] +ιYιXH (C.48) whereH is a real5 closed three–form. One can also define a twisted exterior derivativedH that acts on any formη∈V•
T∗ as
dHη = dη+H∧η . (C.49)
5H does not necessarily have to be real, but since the Courant bracket is a real quantity, it makes sense to restrictH to be real here.
A Lie algebroid is a vector bundle L on a smooth manifoldM equipped with a Lie bracket [,] on C∞(L) and a smooth bundle map a : L → T, called the “anchor”. The anchor must induce a Lie algebra homomorphism a:C∞(L)→C∞(T), i.e.
a([X, Y]) = [a(X), a(Y)] ∀X, Y ∈C∞(L), (C.50) and satisfy a Leibniz rule [147]. It is useful to think of a Lie algebroid as a generalization of the (complexified) tangent bundle, sine if we take L=T (the tangent bundle) and a=id, the bracket reduces to the ordinary commutator of vector fields and both conditions are obviously satisfied.
A complex structure onM is an endomorphismJ :T →T satisfyingJ2 =−1. A symplectic structure onM is a non–degenerate skew formω ∈V2
T∗. One may viewω as a mapT →T∗ via the interior product
ω : v → ιvω , v∈T . (C.51)
This implies that a symplectic structure on T can be defined as an isomorphism ω : T → T∗ satisfying ω∗=−ω, whereω∗ : (T∗)∗ =T → T∗.
A generalized complex structure on T is an endomorphismJ of the direct sumT⊕T∗ which satisfies
• J is complex, i.e. J2 =−1
• J is symplectic, i.e. J∗ =−J .
The usual complex and symplectic structure are embedded in the notion of generalized complex structure in the following way: If J is a complex structure on M, then the 2n×2n matrix (written w.r.t. the direct sumT ⊕T∗)
JJ =
−J 0 0 J∗
(C.52) is a generalized complex structure on T. Similarly, ifω is a symplectic structure on M, then
Jω =
0 −ω−1
ω 0
(C.53) is also a generalized complex structure. We therefore observe, that diagonal and anti–diagonal generalized complex structures correspond to complex and symplectic structures, respectively.
The interesting aspect of GCG is that it interpolates between the two.
Specifying J is equivalent to specifying a maximal isotropic subspace of (T ⊕T∗)⊗C of real index 0. A subspace L ⊂(T ⊕T∗)⊗C is isotropic when hX, Yi = 0 for all X, Y ∈ L, it is maximal when its dimension is maximal, i.e. n in our case. Its real index r is given by the complex dimension ofL∩L. Every maximal isotropic in T ⊕T∗ corresponds to a pure spinor line bundle. A spinor ϕ is called pure when its null space Lϕ = {v ∈ T ⊕T8 : v·ϕ = 0} is maximal isotropic. The pure spinor line bundle is generated by
ϕL = exp(B+iω) Ω (C.54)
whereB andω are real two–forms and Ω =θ1∧. . .∧θkfor some linearly independent one–forms {θi}. The integer k is called the “type” of the maximal isotropic. The maximal isotropic is of real index zero if and only if
ωn−k∧Ω∧Ω 6= 0. (C.55)
C.4. GENERALIZED COMPLEX GEOMETRY 113 The type of a maximal isotropic is the codimension k of its projection onto T. Then any generalized complex structure of type k = 0 is a B–field transform of a symplectic structure Jω as in (C.53), which determines a maximal isotropic L={X−iω(X) : X ∈T ⊗C} and a pure spinor line generated by ϕL = exp(iω). The B–field transform gives rise to a generalized complex structure
Jk=0 = e−BJωeB =
−ω−1B −ω−1 ω+Bω−1B Bω−1
(C.56) with maximal isotropic ˜L = e−BL = {X−(B +iω)(X) : X ∈ T ⊗C} and pure spinor line ϕe−BL= exp(B+iω).
The extremal typek=nis related to complex structures. Note thatJJ as defined in (C.52) determines a maximal isotropic L = T0,1⊕T1,0∗ (where T1,0 =T0,1 is the +i–eigenspace of J) and a spinor line generated byϕL = Ωn,0 (where Ωn,0 is any generator of holomorphicn–forms on the n–dimensional space (T, J)). Then any generalized complex structure of type k =n is the B–field transform of a complex structure, i.e.
Jk=n = e−BJJeB =
−J 0 BJ +J∗B J∗
(C.57) with maximal isotropic ˜L = e−BL = {X+ξ−ιXB : X+ξ ∈ T0,1⊕T1,0∗ } and pure spinor ϕe−BL= exp(B) Ωn,0. Note that in this case only the (0,2) component of the real two–form B has any effect. B–field transforms of complex structures are always block–lower–diagonal, an observation used in [91].
Let us also note the following integrability considion for generalized (almost) complex struc-tures: A generalized complex structure of type k = n is integrable if and only if the complex structureJ is integrable, a generalized complex structure of typek= 0 is integrable if and only ifd(B+iω) = 0.
A generalized almost complex structure is said to be atwistedgeneralized complex structure when its +i–eigenbundle is involutive w.r.t. theH–twisted Courant bracket,Hbeing the closed real three–form introduced around equation (C.48). Given any integrableH–twisted generalized complex structure J, its conjugate eBJe−B is integrable w.r.t. the H+dB–twisted Courant bracket, for any smooth two–formB. This means that the space of twisted generalized complex structures depends only on the cohomology class [H]∈H3(M,R).
We close this section we a few remarks about generalized K¨ahler and generalized Calabi–Yau manifolds, as they are of particular importance in string theory.
Since the bundle T⊕T∗ has a natural inner producth,i, it has structure groupO(2n,2n).
The specification of a positive definite metric G (G2 = 1) that is compatible with this inner product is equivalent to the reduction of the structure group toO(2n)×O(2n). If the manifold allows for a generalized complex structure J this means a reduction of the structure group U(n, n)⊂O(2n,2n) toU(n)×U(n) if the metricGcommutes withJ. Note that sinceG2 = 1 andJG=GJ, the mapGJ squares to−1, i.e. it defines another generalized complex structure.
Gualtieri [89] is therefore led to the following definition of ageneralized K¨ahlermanifold: A generalized K¨ahler structure is a pair (J1,J2) of commuting generalized complex structures such that G =−J1J2 is a positive definite matric on T ⊕T∗. In accordance with the observations above this means that the existence of a generalized K¨ahler structure is equivalent to a reduction of the structure group to U(n)×U(n). This has been exploited to extend string theory on manifolds ofSU(3) structure (recall chapter 4) to manifolds withSU(3)×SU(3) structure [94].
A special case of generalized K¨ahler structures is of course the usual K¨ahler structure, since a K¨ahler manifold has both a complex and a symplectic structure. So, we can define two generalized complex structures (C.52) and (C.53), which obviously commute, and
G = −JJJω =
0 g−1 g 0
(C.58) whereg is the Riemannian metric on the K¨ahler manifold.
Any B–field transform of a generalized K¨ahler structure (J1B,J2B) = (BJ1B−1, BJ2B−1) is also generalized K¨ahler. In the case with J1 = JJ and J2 = Jω, the metric after B–field transform becomes
GB =
−g−1B g−1 g−Bg−1B Bg−1
, (C.59)
showing that the generalized K¨ahler metric need not be diagonal.
According to [89], any generalized K¨ahler metric is uniquely specified by the existence of a Riemannian metric g and a two–form b, where b does not have to be closed. Therefore, a generalized K¨ahler metric is not simply a B–field transform of a Riemannian metric (for a B–
field transform we require B to be closed). The torsion of the generalized K¨ahler structure is given byh=db.
Moreover, and this has been exploited in the formulation of generalized topological sigma models (see section C.5), any generalized K¨ahler structure is determined by a bi–Hermitian structure on the manifold M. A U(n)×U(n) structure is equivalent to the specification of (g, b, J+, J−) with Riemannian metricg, two–formband two Hermitian complex structuresJ±. One could call this a “bi–Hermitian structure with b–field”. Letω±be the two–forms associated to the complex structures J±, i.e.
ω± = gJ±. (C.60)
Then the maps b±g determine the metric G via G =
−g−1b g−1 g−bg−1B bg−1
= 1 0
b 1
0 g−1 g 0
1 0
−b 1
. (C.61)
The generalized complex structures J1,2 are found to be J1,2 = 1
2 1 0
b 1
J+±J− −(ω−1+ ∓ω−−1) ω+∓ω− −(J+∗ ±J−∗)
1 0
−b 1
. (C.62)
Note that the degenerate case where J+ = ±J− corresponds to (J1,J2) being the B–field transform of a genuine K¨ahler structure.
For a generalized K¨ahler structure the torsionh is of type (2,1)+(1,2) w.r.t. both complex structures J±. Also note the following important corollary: For a generalized K¨ahler structure with data (g, b, J±) the following are equivalent
• h=db= 0
• (J+, g) is K¨ahler
• (J−, g) is K¨ahler.
In other words, in the absence of torsion the bi–Hermitian structure reduces to a bi–K¨ahler structure. This does not necessarily implyJ+ =J−, though.
C.5. GENERALIZED TOPOLOGICAL SIGMA MODELS 115