5. Topological Field Theory In Two Dimensions
5.2. The Role Of Generalized Complex Geometry
The same reasoning shows that t′ and t′′ transform in precisely the same way. From this and (5.20), it follows that the action ofS-duality onCP1
h×CP1
h is w+ →w+ cτ+d
|cτ+d| w− →w−|cτ +d| cτ +d .
(5.22)
An important special case is that the B-model in complex structure I, which corre- sponds to (w+, w−) = (0,0), is completely invariant under duality transformations. The A-model in complex structure I, which corresponds to (w+, w−) = (0,∞), is likewise in- variant. This and other statements about two-dimensional TQFT’s that are not on the distinguished family CP1
g will be useful auxiliary tools, but as will become clear, the ge- ometric Langlands program is really a statement or collection of statements about the distinguished family.
Dependence On The Metric
In general, a two-dimensional TQFT with targetMH(G, C) will depend on the com- plex structure ofC, because this influences the hyper-Kahler structure ofMH(G, C). How- ever, if a two-dimensional TQFT, such as those parametrized by CP1
g, descends from a four-dimensional one, it must be independent of the complex structure on C. After all, the four-dimensional TQFT on M = Σ×C is independent of the metric on M and in particular on C.
In some cases, we can see directly that these models are independent of the complex structure of C. At t = ±i, we get the B-model in complex structure ±J; this complex structure is independent of the metric of C, as we observed in section 4.1. For real t, we get the A-model in a complex structure that is a linear combination of complex structures I andK. The A-model is determined by the corresponding symplectic structure, which is a linear combination of symplectic structures ωI and ωK; we found these in section 4.1 to be independent of the metric of C.
To go farther, we will use generalized complex geometry.
to clarify a few questions from a two-dimensional point of view. What is the family CP1
g
and why is it independent of the complex structure of C? How, from a two-dimensional point of view, can we understand the canonical parameter Ψ introduced in section 3.5?
And what geometry of MH is really needed in these constructions?
Let T X and T∗X be the tangent and cotangent bundles of a manifold X. We let Tb=T X⊕T∗X and we write a section of Tbas
v ξ
, where v is a section of T X andξ is a section of T∗X. Tb has a natural indefinite metric, in which T X and T∗X are both null and the inner product of T X and T∗X is the natural one, in which the inner product of v
ξ
and v′
ξ′
is viξ′i+v′iξi.
A generalized almost complex structure on a manifold X is a linear transformation I of Tb=T X⊕T∗X that preserves the metric and obeys I2 =−1. If a certain integrability condition is obeyed, it is called a generalized complex structure [69]. (For more detail, see Gualtieri’s thesis [67], as well as [71,68] for applications to sigma-models and [73-75] for applications to supergravity.) One basic example of a generalized complex structure is
IJb=
Jb 0 0 −Jbt
, (5.23)
where Jb : T X → T X is an ordinary complex structure, and Jbt : T∗X → T∗X is its transpose. If ω is a symplectic structure, then a second basic example of generalized complex structure is given by
Iω =
0 −ω−1
ω 0
. (5.24)
Here we regard ω as a linear map from T X toT∗X; ω−1 is the inverse map from T∗X to T X. The topological field theory associated with IJbis the B-model in complex structure Jb, and the one associated with Iω is the A-model with symplectic structure ω.24
In each of these cases, the B-field vanishes. A B-field can be turned on as follows.
For any closed two-form B0, let
M(B0) =
1 0 B0 1
. (5.25)
24 TheA-model is most commonly considered on a Kahler manifold, and then ω is the Kahler form. Because the A-model makes sense on symplectic manifolds more generally, we simply refer toω as the symplectic form.
Then if I is an integrable generalized complex structure, so is
I(B0) =M(B0)IM(B0)−1. (5.26) The transformationI → I(B0) has the effect of shifting theB-field by B0. So in particular the topological field theory derived from IJb(B0) is the B-model with complex structure Jb and B-field B0, and Iω(B0) is similarly related to the A-model with symplectic structure ω and B-field B0. Conjugation by M(B0) is called the B-field transform.
It is shown in chapter 6 of [67] that the conditions [66] of (2,2) supersymmetry are equivalent to the existence of a pair of generalized complex structures obeying a certain compatibility condition. In the case relevant to us that the (2,2) model comes from a hyper-Kahler metric g with a pair of points J+, J− ∈CP1
h, there is a slight simplification because the B-field can be taken to vanish (and restored later by a B-field transform). We let ω± be the two symplectic structures ω± = gJ±. One of the two generalized complex structures determined by the pairJ+,J−with the hyper-Kahler metricgis then, according to eqn. 6.3 of [67],
J = 1 2
J++J− −(ω+−1−ω−−1) ω+−ω− −(J+t +J−t )
. (5.27)
The second one, not relevant for us, is obtained by reversing the sign of J− andω−. Using (5.7) and (w+, w−) = (−t, t−1), we have
J+ = 1−tt
1 +ttI− i(t−t)
1 +tt J − t+t 1 +ttK J− =−1−tt
1 +ttI− i(t−t)
1 +tt J + t+t 1 +ttK.
(5.28)
The associated symplectic structures are ω+ = Imτ
1−tt
1 +ttωI − i(t−t)
1 +tt ωJ − t+t 1 +ttωK
ω− = Imτ
−1−tt
1 +ttωI − i(t−t)
1 +tt ωJ + t+t 1 +ttωK
.
(5.29)
(The factor of Imτ was obtained in (4.15).) So we compute that the generalized complex structure determined by this data is
It = 1 1 +tt
−i(t−t)J −(Imτ)−1((1−tt)ωI−1−(t+t)ωK−1) Imτ((1−tt)ωI −(t+t)ωK) i(t−t)Jt
. (5.30)
(To get this formula, it helps to know that on a hyper-Kahler manifold, ifa2+b2+c2 = 1, then (aωI +bωJ+cωK)−1 =aωI−1+bω−1J +cωK−1.)
The first important observation is that I, K, and ωJ have disappeared. It depends only onJ, ωK, andωI, or equivalently on J and the holomorphic two-form ΩJ =ωK+iωI (and its complex conjugate). But this data, as we noted in section 4.1, is independent of the metric ofC. This explains, from the two-dimensional point of view, why the family of topological field theories that we get by dimensional reduction from four dimensions does not depend on the metric.
An analogous family of generalized complex structures can be constructed on any hyper-Kahler manifold X and is described in section 4.6 of [67]. As is explained there (Proposition 4.34), the generic element in such a family is a B-field transform of a gener- alized complex structure derived from a symplectic structure. Indeed, a small calculation shows that
It = M(B0)
0 −ω0−1 ω0 0
M(B0)−1, (5.31)
where
ω0 = Imτ 1−t2t2 (1 +t2)(1 +t2)
ωI − t+t 1−ttωK
B0 =−Imτ i(t2−t2) (1 +t2)(1 +t2)
ωI + 1−tt t+t ωK
.
(5.32)
Therefore, for t 6= ±i the TQFT derived from the generalized complex structure It is equivalent to an A-model with symplectic structure ω0 and B-field B0. This fact will enable us to understand from a two-dimensional point of view the canonical parameter Ψ introduced in section 3.5.
In general, theA-model with symplectic formω0 and B-fieldB0 depends only on the cohomology class [B0+iω0]. (On a Kahler manifold, this is called the complexified Kahler class.) In the present problem, since the cohomology class ofωK vanishes (as we explained in section 4.1), we have
[B0+iω0] =−iImτ
t−t−1 t+t−1
[ωI]. (5.33)
Thus, when the four-dimensional θ angle vanishes (we took it to vanish by starting with It rather than a B-field transform thereof), the model depends on Im τ andt only in the combination that appears in (5.33).
The four-dimensional θ angle induces, as we explained at the end of section 4.1, an additional contribution B′ = −(θ/2π)ωI = −(Reτ)ωI to the two-dimensional B-field.
This can be incorporated in the generalized complex structure simply by conjugating with M(B′). The resulting model depends on the cohomology class [B0+B′ +iω0], which is
−[ωI] times
Ψ = Reτ +iImτ
t−t−1 t+t−1
. (5.34)
This gives a two-dimensional interpretation of why the model depends on t and τ only in the combination Ψ.
A Few Loose Ends
Finally, let us wrap up a few loose ends.
First of all, ift is real, the TQFT determined by complex structuresJ+(t), J−(t) is an A-model to begin with, and the above argument showing that Ψ is the only relevant pa- rameter did not really require generalized complex geometry. The advantage of generalized complex geometry is that it enables us to understand the general case.
Second, in the above analysis, we used the fact that theA-model only depends on the cohomology class ofB+iω. This is proved by writing the action as{Q, V}+R
ΣΦ∗(ω−iB), where Φ : Σ→X is the sigma-model map.
In the above derivation, the B-field is not just of type (1,1) (in a complex structure in which the symplectic form is Kahler); it also has components of type (2,0)⊕(0,2). In fact, the A-model is sensitive to all components of the B-field, including the part of type (2,0)⊕(0,2), but this point may require some clarification.
If Φ : Σ→X is a holomorphic map, andB is a form onX of type (2,0)⊕(0,2), then Φ∗(B) = 0. This makes the (2,0)⊕(0,2) part of B appear irrelevant, if one interprets the A-model purely as a mechanism for computing correlation functions by summing over holomorphic maps. But if one considers branes (as we most definitely will do to under- stand the geometric Langlands program), one immediately sees that all parts of the Hodge decomposition of B are relevant. A Lagrangian submanifold N ⊂ X endowed with a Chan-Paton line bundle Lof curvatureF gives anA-brane if F+B|N = 0. This condition is certainly sensitive to the (2,0)⊕(0,2) part of B.
Finally, our analysis showing that Ψ is the only relevant parameter is really not valid at t = ±i, because of poles in the formulas. At these values of t, the model is actually not the B-field transform of an A-model; it is a B-model in complex structure ±J. To complete our analysis for these values of t, we will argue directly using the B-model.
At t=±i, we have Ψ =∞, independent ofτ, so to complete the demonstration that Ψ is the only relevant parameter, we must show that τ is completely irrelevant at t=±i.
Im τ controls the Kahler class of MH, and this is certainly irrelevant in the B-model.
Varying Reτ adds to theB-field a multiple ofωI. To show that this term is irrelevant in the B-model with complex structureJ, we writeωI =−iΩJ+iωK. The contribution fromωK
is irrelevant because the form ωK is exact. The contribution from ΩJ is irrelevant because ΩJ is a form of type (2,0). But in theB-model, the B-field contribution−iR
ΣΦ∗(B) can be written as {Q, . . .} if B is of type (1,1) or (2,0). (By contrast, a (0,2) component of theB-field does affect the category ofB-branes [71].) Of course, in complex structure−J, we make the same argument, starting with ωI =iΩJ −iωK.