Twisted Topological Field Theories

5. Topological Field Theory In Two Dimensions

5.1. Twisted Topological Field Theories

We now want to study the sigma-model of targetM_H from the viewpoint of topological field theory, so we replaceR^1,1 by a general two-manifold Σ that we eventually will take to have Euclidean signature. We want to discuss what twisted topological field theories on Σ can be constructed from the sigma-model of target M_H, and which of these actually arise by compactifying on the Riemann surface C the four-dimensional topological field theories that we constructed in section 3.

To construct a topological field theory from a sigma-model with a target X, one picks a pair of complex structures (J₊, J₋) on X that obey certain conditions of (2,2) supersymmetry [66].²¹ These conditions have recently been reinterpreted [67] in terms of generalized complex geometry [69]. Once one has (2,2) supersymmetry, there is a standard recipe [70], via the twisting procedure that we reviewed in section 3.1, for constructing a topological field theory.

If X is hyper-Kahler, things simplify. The sigma-model with target X has (4,4) supersymmetry, and a structure of (2,2) supersymmetry can be obtained by picking a (2,2) subalgebra of the (4,4) supersymmetry algebra.²² In terms of choosing complex structures, this amounts to the following. X has a family of complex structures parametrized by a copy of CP¹ that we call CP¹

h, and a (2,2) structure can be defined by picking a pair of points (J+, J₋) ∈ CP¹

h. The conditions in [66] are automatically obeyed for such a pair.

In the case of M_H, this gives the family of topological field theories that is important for the geometric Langlands program.

We can conveniently characterize as follows the topological field theory associated with the pair (J₊, J₋). Let Φ : Σ → M_H be the bosonic field of the sigma-model. Then the equations of unbroken supersymmetry read

(1−iJ+)∂Φ = 0, (5.5)

and

(1−iJ₋)∂Φ = 0. (5.6)

Here∂and∂are the usual (1,0) and (0,1) parts of the exterior derivative on Σ; and∂Φ and

∂Φ are understood as one-forms on Σ with values in the pullback of the tangent bundleT X (on whichJ₊ andJ₋act). The first condition says that the map Φ : Σ→X is holomorphic in complex structure J₊; the second says that it is antiholomorphic in complex structure J₋. Analogously to how we found the conditions of unbroken supersymmetry in section 3.1, these equations arise because there are fermi fields χ₊, χ₋ with {Q, χ₊}= (1−iJ₊)∂Φ, {Q, χ₋}= (1−iJ₋)∂Φ.

21 Generalized complex geometry leads to an extension of this construction, which we will not need here, in which only one of the two generalized complex structures that can be formed [67]

from the pair (J+, J₋) is integrable. This one integrable generalized complex structure is then used to define a topological field theory. See [68] for a systematic explanation of how to do this.

22 Two (2,2) subalgebras that differ by conjugation by Spin(4)_R are considered equivalent.

There is much more information in the equations (5.5) and (5.6) than in their solutions.

To discuss the solutions of the equations, one usually specializes to Σ of Euclidean signature (the most natural case for topological field theory). Then for generic J₊, J₋, they imply that the map Φ : Σ→X must be constant, a result that does not depend on J₊ and J₋. The equations themselves, understood algebraically, and without fixing a real structure on the two-manifold Σ, depend on J+ and J₋.

An important example is the case J₊ = J₋ = Jb(where, in our applications, Jbwill be one of the complex structures Iw of M_H, as described in eqn. (4.6)). The equations combine to (1−iJb)dΦ = 0, which imply thatdΦ = 0. The model is called the B-model in complex structureJb. If insteadJ+ =−J₋ =Jb, we get theA-model in complex structureJb. (This model really depends on the symplectic formω^Jbassociated with complex structureJb rather than the complex structureper se. But calling it the A-model in complex structure Jbis sometimes a convenient shorthand.) For Σ of Euclidean signature, the equations of the A-model are redundant, as eqn. (5.6) is the complex conjugate of (5.5). Either one of them asserts that the map Φ : Σ → M_H is holomorphic in complex structure Jb. If J₋ 6=±J₊, we get a model that is neither anA-model nor a B-model, but can be reduced to an A-model using generalized complex geometry, as we discuss in section 5.2.

We recall from eqn. (4.6) the explicit form of the family of complex structures on M_H parametrized by CP¹

h. In terms of an affine parameter w for CP¹

h, we defined a complex structureI_w in which the holomorphic coordinates areA_z−wφ_z andA_z+w⁻¹φ_z. Explicitly, we found

Iw = 1−ww

1 +wwI+ i(w−w)

1 +ww J+ w+w

1 +wwK. (5.7)

The pair (J₊, J₋) corresponds in this parametrization to a pair (w₊, w₋).

Reduction From Four Dimensions

Now let us compare this to what we get by reduction from four dimensions. We learned in section 3 that gauge theory leads to a family of topological quantum field theories (TQFT’s) defined on any four-manifold M and parametrized by a copy of CP¹ that we will call CP¹

g. This family certainly reduces forM = Σ×C to a subfamily of the family CP¹

h ×CP¹

h that is natural from the two-dimensional point of view. Both families come from a twisting procedure applied to N = 4 super Yang-Mills theory on a four- manifoldM. In this procedure, a topological field theory is obtained from the cohomology of a suitable linear combination Q of the supersymmetries. To get the family CP¹

g, we

requireQ to beSpin^′(4)-invariant so that the construction works for allM. But to get the sigma-model of targetM_H, we specialize to the manifoldM = Σ×C of reduced holonomy, and then, as we discuss more explicitly later, a weaker condition on Q suffices, leading to the larger family CP¹

h×CP¹

h.

We aim to identify the four-dimensional topological field theory family CP¹

g as a subfamily of the sigma-model familyCP¹

h×CP¹

h. The embedding ofCP¹

g inCP¹

h×CP¹

h can be described by functions w₊(t), w₋(t). We can compute these functions by specializing to a convenient configuration or physical state.

A very convenient way to proceed is to simply abelianize the problem, working in a vacuum in which G is broken to its maximal torus by the expectation values of some of the untwisted scalar fields. In the abelian case, from (3.29), the equations of unbroken supersymmetry in four dimensions are

(F +t dφ)⁺ = 0

(F −t⁻¹dφ)⁻ = 0. (5.8)

Here, the one-formφ is a section of T^∗M =T^∗Σ⊕T^∗C, and the gauge field A is locally a one-form or section of T^∗M. To compare to a sigma-model on Σ with target M_H(G, C), we must take Σ to have radius much greater than that of C. In that limit, the parts of A and φ that take values in T^∗Σ are “massive” and can be dropped; so A and φ can be interpreted as sections of T^∗C, slowly varying on Σ.

We write y for a local complex coordinate on Σ and z for a local complex coordinate onC. The two-form dy∧dz is selfdual, whiledy∧dz is anti-selfdual. So the first equation in (5.8) leads to

∂_y(A_z+tφ_z) = 0, (5.9)

and the second leads to

∂_y A_z−t⁻¹φ_z

= 0. (5.10)

By comparing these results to eqns. (5.5) and (5.6), we can read off the functions w+(t) and w₋(t). Eqn. (5.9) asserts that the map Σ → M_H(G, C) is holomorphic if we take on M_H(G, C) the complex structure in which A_z+tφ_z is holomorphic. This agrees with eqn. (5.5), which asserts such holomorphy in complex structure I_w+, if and only if w₊ = −t. Similarly, eqn. (5.10) asserts that the map Σ →M_H(G, C) is antiholomorphic if we take on M_H(G, C) the complex structure in which A_z −t⁻¹φ_z is holomorphic. This

agrees with eqn. (5.6), which asserts antiholomorphy in complex structureIw−, if and only ifw₋ =t⁻¹.

So the embedding of CP¹

g in CP¹

h×CP¹

h is defined by w₊ =−t

w₋ =t⁻¹. (5.11)

Some Simple Considerations

Let us now make a few simple simple observations about this result.

First of all, when do we get a B-model? For a B-model, we want w+ = w₋. So the condition is t⁻¹ =−t, which occurs precisely for t =±i. Since the complex structures I_w for w=∓i coincide with J and −J, we get this way theB-model in complex structure J or −J.

When do we get an A-model? For an A-model, the complex structures Iw+ and Iw−

should be opposite. The mapw→ −1/w mapsI_w to its opposite (this is clear from (5.7)), so we get an A-model in two dimensions if w₋ = −1/w₊, which works out to t = t. In other words, precisely for real t, we get an A-model. For example, for t = 1 or −1, we get the A-model in complex structure K or −K. For t = 0 or ∞, we get the A-model in complex structure I or −I.

Now, let us compare this family to the more complete familyCP¹

h×CP¹

h of the sigma- model. At the end of section 4.2, we described the group U ∼= C^∗ of diffeomorphisms of M_H(G, C); it acts on the family of complex structures I_w by w → λ⁻¹w, λ ∈ C^∗. The topological field theory determined by a pair²³ (w+, w₋) is the same as that determined by (λ⁻¹w₊, λ⁻¹w₋). How many really inequivalent TQFT’s can we define from the two- dimensional point of view? The only invariant we can form from the pair (w₊, w₋) is the ratio q =w+/w₋. We see that

q =−t², (5.12)

so all values of q can be achieved, but not quite uniquely. The points t and −t on CP¹

g

lead to equivalent theories in two dimensions. This equivalence reflects the action of the center of SU(4)_R (recall eqn. (3.14)).

23 This topological field theory depends also on the hyper-Kahler metric andB-field ofM_{H. The} full dependence on all variables is determined in section 5.2 using generalized complex geometry.

Although all values of the invariant q do come from four-dimensional TQFT’s, it is not quite true that all C^∗ orbits on CP¹

h ×CP¹

h have representatives with such an origin.

The missing orbits are the C^∗-invariant points (0,0) and (∞,∞), and also the orbits in which w₊ or w₋, but not both, is 0 or ∞. Particularly notable is the fact that the points (0,0) and (∞,∞) are not equivalent to theories that originate in four-dimensional TQFT’s.

These points correspond to the B-models in complex structures I and −I.

The B-model in complex structure I has been the starting point in mathematical efforts – briefly surveyed in the introduction – to interpret the geometric Langlands pro- gram in terms of the geometry of M_H. Because the Hitchin fibration is holomorphic in complex structure I, the T-duality on the fibers of the Hitchin fibration (whose relation to four-dimensional S-duality we review soon) maps the B-model of complex structure I to itself, acting on D-branes via the Fourier-Mukai transform that is the starting point in the mathematical description. Although the point (0,0) corresponding to thisB-model is not in the family CP¹

g that comes from four-dimensional TQFT’s, it is interesting that it can be infinitesimally perturbed to give points on the C^∗ orbits corresponding to almost any point on CP¹

g. We simply perturb (0,0) to (α, β) for infinitesimal α, β; the invariant q =w₊/w₋ is thenq =α/β, and can take any value for arbitrarily smallα, β. Perhaps this fact will lead eventually to an understanding of the geometric Langlands program based on perturbing from the B-model in complex structureI. Our approach, however, will rely on the family CP¹

g that comes directly from four dimensions.

Although certain C^∗ orbits, such as the point (0,0), do not arise by specializing a four-dimensional TQFT toM = Σ×C, this does not meant that they cannot be described in four-dimensional gauge theory. In section 3.1, we obtained the family CP¹

g using the supersymmetry generator

ǫ=uǫℓ+vǫr (5.13)

that was constrained to be Spin^′(4)-invariant. If we specialize to M = Σ×C, there is no need to ask for Spin^′(4) invariance. The holonomy group of Σ×C is SO(2)×SO(2) ⊂ SO(4), and Spin^′(2)×Spin^′(2) invariance is enough. This means that we can generalize (5.13) to

b

ǫ = (u+euΓ^∗)ǫ_ℓ+ (v+evΓ^∗)ǫ_r, (5.14) where

Γ^∗ =

Γ₀₁ in Lorentz signature

iΓ₀₁ in Euclidean signature (5.15)

is the operator that distinguishes the two chiralities of two-dimensional spinors. We will adopt Euclidean signature here, as this is more natural for topological field theory.

The supersymmetry generators in (5.14) are precisely the ones that obey the following conditions:

Γ₂₃₆₇ǫ=ǫ

Γ₀₁₄₅ǫ=ǫ. (5.16)

The first is eqn. (5.3), which says that the supersymmetry generated by ǫ is unbroken by the curvature of C; the second says that it is similarly unbroken by the curvature of Σ.

With this more general starting point, it is possible to get the whole family CP¹

h ×CP¹

h

of two-dimensional TQFT’s from four-dimensional gauge theory, though not from a four- dimensional TQFT.

It actually is convenient to rewrite (5.14) in an eigenbasis of Γ^∗: b

ǫ = 1

2(1−Γ^∗) (u^′ǫ_ℓ+v^′ǫ_r) + 1

2(1 + Γ^∗) (u^′′ǫ_ℓ+v^′′ǫ_r). (5.17) Here (u^′, v^′) and (u^′′, v^′′) are, respectively, homogeneous coordinates for the two factors of CP¹

h×CP¹

h. Let δb_T be the extended topological symmetry generated by bǫ. To determine the two complex structures (J₊, J₋), or equivalently, to determine the pair (w₊, w₋), we just need to computebδTχ⁺_{y z} andbδTχ⁻_yz. Setting these to zero will give the generalization of (5.8). To determine the generalization of the usual formulas from eqn. (3.27), all we need to know is that if χe⁺ = Γ^{y z}χ⁺_{y z} and χe⁻ = Γ^yzχ⁻_yz, then χe⁺Γ^∗ = χe⁺ and χe⁻Γ^∗ = −eχ⁻. Using this, we get

bδ_Tχ⁺_{y z} =u^′(F −φ∧φ)_{y z}+v^′(Dφ)_{y z}

bδ_Tχ⁻_yz =v^′′(F −φ∧φ)_yz−u^′′(Dφ)_yz. (5.18) If, therefore, t^′ =v^′/u^′ and t^′′ =v^′′/u^′′, then the generalizations of eqns. (5.9) and (5.10) are

∂y(Az+t^′φz) = 0

∂y(Az−(t^′′)⁻¹φz) = 0. (5.19) This determines the two complex structures:

(w₊, w₋) = (−t^′,(t^′′)⁻¹). (5.20) Now we can determine how S-duality acts on the full family CP¹

h ×CP¹

h. We learned in eqn. (3.13) that a duality transformation

a b c d

∈SL(2,R) acts on t by t →t cτ+d

|cτ+d|. (5.21)

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 targetM_H(G, C) will depend on the com- plex structure ofC, because this influences the hyper-Kahler structure ofM_H(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 ω_I and ω_K; we found these in section 4.1 to be independent of the metric of C.

To go farther, we will use generalized complex geometry.