Lecture II-19: N = 2 Super-symmetric Yang-Mills theories

in dimension four: Part III, Topological Applications

Edward Witten^{∗}

I A survey of N = 2 super-symmetric gauge theories in

dimension four

Let us begin with a brief overview of the range of N = 2 super-symmetric gauge theories. We must specify a gauge group G which will be a compact group. We shall be doing gauge theory, which means our fields include a vector multiplet consisting of the gauge fields and their super partner fields with values in the adjoint bundle of the principal G-bundle. We can also have hypermultiplets. N = 2 super-symmetry forces these hypermultiplets to lie in vector bundles associated to the principal G-bundle by a representation ρ: G → Sp(n). This means that the associated bundles are quaternion bundles. The most important invariant for the qualitative nature of the theory is the β function, and in particular, the one-loop β-function. The formula for the one-loop β-function is

β_{1}(g) = \frac{−g^{3}}{8π^{2}} (4h − c_{2}(ρ)),

where h is the dual coxeter number of the group G and c_{2}(ρ) is the trace of the quadratic Casimir for ρ, normalized so that the defining representation of SU(2) has c_{2} equal to one. Of course, 2h is simply c_{2}(adjoint), the quadratic Casimir of the adjoint representation of G on its Lie algebra. There are three possibilites:

• The one-loop β-function vanishes.

• The one-loop β-function is negative.

∗Notes by John Morgan

• The one-loop β-function is positive.

Since

τ = \frac{θ}{2π} + \frac{4π^{2}i}{g^{2}}

we have the expansion

μ\frac{d}{dμ}(τ) = constant + ∑_{ℓ≥1} C · g^{2ℓ} + ∑ c_{n,ℓ}g^{2ℓ}exp(2πinτ),

where the constant term is the computed from the one-loop β-function, the higher powers of g arise from higher loop contributions and the exponential terms come from instanton corrections (which are of course not seen in perturbation theory). By super-symmetry μ\frac{d}{dμ}(τ) is a holomorphic function, and hence the positive powers of g in this expansion vanish identically. Thus, if the one-loop β function vanishes then β vanishes in perturbation theory. In this case it is believed that β ≡ 0. This result is not completely clear – it is not always the case in N = 2 super-symmetric theories that the exponentially small terms vanish. (Recall that in the pure SU(2)-theory that we studied last time we found instanton corrections to τ which gave exponentially small, but non-zero, corrections to the one-loop β-function.) On the other hand, one case when the one-loop β-function does vanish is when the hypermultiplet representation ρ is the tensor product of the adjoint representation of G with the quaternions. This theory in fact has N = 4 super-symmetry. Using this enhanced super-symmetry one can show that the β-function vanishes identically. The general belief is that when the one-loop β-function vanishes then the β-function is identically zero. When β ≡ 0, the theory has a well-defined τ just like the classical theory. It is believed that in this case there is a subgroup Γ of finite index in SL_{2}(Z) with the property that the theory is invariant under a faithful action of Γ. Of course, when β ≠ 0 the theory has a mass scale and τ is no longer constant. For these theories one would expect, as we saw for the U(1)-theory in Lecture II-17, that there is not an exact duality of the theory, but rather that there is an action of the duality group on the representations of the low energy behaviour of the theory by Lagrangians.

When the one-loop β-function is negative, then the β-function is negative, at least for sufficiently high energy scales. (The pure SU(2)-theory we studied last time is of this type.) In this case, we have an asymptotically free theory which is a good fundamental theory. That is to say we have a well-defined problem (to describe the low energy effective version of the theory).

We solved this problem in the last lecture for G = SU(2) or SO(3) with ρ being the trivial representation.

When the one-loop β-function is positive, we do not have a well-defined problem, since the theory is not asymptotically free. Rather these theories are candidate solutions to the problems posed by the theories with one-loop β-function negative. A theory T_{mac} with positive β-functioin theory is a candidate solution in the sense that its low energy effective theory is well-defined and infrared free. It is a solution to the problem posed by a theory, T_{mic}, with negative β, if T_{mic} flows in the infrared to T_{mac} in the sense that at long distances (or equivalently low energies) the correlation functions of the two theories converge to each other.

The gauge theories we will study today are Lagrangian models that come by quantizing classical physics. But there are more exotic N = 2 theories in four-dimensions, some of which come from string theory. Not all of these have Lagrangian formulations. Whether or not they have Lagrangians, these N = 2 supersymmetric theories should also lead to four-manifold invariants. Those without Lagrangians should lead to four-manifold invariants which do not have a classical description in terms of the moduli space of solutions to some differential equation.

II From Minkowski space to a compact riemannian four-

manifold

Let us fix one of the N = 2 super-symmetric asymptotically free gauge theories (one with β < 0). For example, if we wish to study Donaldson theory we will take pure N = 2 super-symmetric SU(2)-gauge theory. But for a while we can be more general. The theory, as we have discussed it so far, has been on Minkowski four-space. That is to say, we have written down an action which involves the integration over Minkowski space of a Lagrangian function of various fields on this space. We wish to pass from this to a theory defined on a compact riemannian four-manifold. This passage is carried in two steps – i) rotating from Minkowski four-space to Euclidean four space, and ii) globalizing to a compact riemannian four-manifold.

Let us describe how to Wick rotate the theory to Euclidean four-space. Our theory consists of a family of local operators with an operator product expansion (O.P.E). A realization of the theory gives values to the correlation functions of these operators at various points of Minkowski four-space. But there is an open subset of the complexification of Minkowski four-space where the O.P.E. and the correlation functions are defined by analytic continuation. This open subset contains Euclidean four-space. In this way we Wick rotate to define the theory on Euclidean four-space (see, for example, Kazhdan’s lectures from last fall). In our case we are dealing with (supersymmetric) actions

S = ∫_{M^{4}} L

which are integrals of Lagrangians L which are analytic expressions involving fields ϕ on the underlying Minkowski space. These expressions extend to holomorphic expressions on complexified Minkowski space and then can be restricted to the Euclidean subspace to produce a Euclidean Lagrangain L_{E}. The Euclidean action is defined by

S_{E} = ∫_{R^{4}} L_{E}.

The path integral for a correlation function in Minkowski space is

∫ DϕO_{1}(x_{1}) ⋯ O_{t}(x_{t})e^{iS},

and under this process it Wick rotates to the path integral

∫ DϕO_{1}(x_{1}) ⋯ O_{t}(x_{t})e^{−S_{E}}

computing the correlation function in Euclidean space. (Of course, Feynmann diagram computations of path integrals are easier to do in the Eulcidean framework and this process is reversed to give answers in Minkowski space.) For more details on this see the article ‘Actions and Reality’ by D. Freed.

II.1 Globalizing by twisting – the case of the super-symmetric algebra

Now let us examine the question of putting the quantum field theory on an oriented riemannian four-manifold X. Our supersymmetric theory is defined over four-space by an action S which is an integral of a Lagrangian function of local fields which are spinors and vector representations of certain types under Spin(4). Of course, there are bundles on Minkowski or Euclidean four-space associated to these representations. In order to write the ‘same’ action on a four-manifold X we need to be able to write down the supersymmetric Lagrangian on that manifold. This means that all fields that appear in the Lagrangian must be globally defined (as sections of bundles) on X. This will not in general be the case for the theory as it has been presented so far because the relevant spin bundles may not exist globally on X. In fact one sees that the ‘physically untwisted’ theory on a riemannian four-manifold requires that X be spin and generically breaks all supersymmetry. We have seen before (see Lecture II-??) how to deal with this problem – we twist the theory by an appropriate homomorphism of the spin group Spin(4) to the R-symmetry group. As we shall see an appropriate twist will both allow us to write down the Lagrangian on any oriented riemannian four-manifold and also allow us to preserve a crucial piece of the supersymmetry.

The decomposition of Λ^{2}T^{∗}R^{4} into self-dual and anti-self-dual components is covered by a decomposition of the group Spin(4) as SU(2)_{+} ×SU(2)_{−}. Fortunately, the N = 2 the R-symmetry group is U(2)_{R} = SU(2)_{R} × U(1)_{R}. (The U(1)_{R} is often absent, e.g., if the hypermultiplet has nonzero bare mass. It is also anomalous if β ≠ 0. But even so, we still have SU(2)_{R} as an R-symmetry group.) With this R-symmetry group twists are possible. (For N = 1 the Rsymmetry is U(1)_{R} and there is no possibility for twisting.) Recall from the super-homework that the super-symmetric generators are right-invariant vector fields on R^{4|8} \{Q_{α}^{i}\} and \{\overline{Q}_{jα̇}\} where 1 ≤ i, j, α ≤ 2. Both of these sets are acted on by SU(2)_{R} in the standard fashion on the i, j indices. (The α and α̇ are the spinor indices for SU(2)_{+} × SU(2)_{−}.) The actions of the spin group and the SU(2)_{R} are summarized in the following table.

Transformation under

Charges SU(2)_{+} SU(2)_{−} SU(2)_{R}
\{Q_{α}^{i}\}  \frac{1}{2}  0  \frac{1}{2}
\{\overline{Q}_{jα̇}\}  0  \frac{1}{2}  \frac{1}{2}

We define SU(2)′ as the diagonal subgroup of SU(2)_{+} ×SU(2)_{R} under the obvious identification of these two groups. We shall be interested in the group SU(2)′×SU(2)_{−}. If follows from the information in the table that under SU(2)′ × SU(2)_{−} the {Q_{α}^{i}} transform in the representation

(\frac{1}{2} ⊗ \frac{1}{2}, 0) = (0, 0) ⊕ (1, 0).

This means that the twisted form of the vector fields {Q_{α}^{i}} decompose as a complex-valued vector field and a vector field with values in complex self-dual two-forms. Similarly, the \{\overline{Q}_{jα̇}\} transform in representation

(\frac{1}{2}, \frac{1}{2})

of this same group and thus are tangent vectors, so that the \{\overline{Q}_{jα̇}\} become vector fields with values in the complexified tangent bundle. Thus, both these representations are vector representations of Spin(4) and hence exist globally on any oriented riemannian 4-manifold. Also, notice that these representations are naturally real representations (even though the original ones were only complex). This means that on the riemannian manifold we can take real fields.

In the superspace language, we are working on the split supermanifold whose even part is X and whose odd part is the parity reversed vector bundle ∧^{0}T X ⊕ T X ⊕ ∧_{+}^{2}(T X). On this supermanifold we have a globally defined real vector field Q defined as follows. Suppose that we have local coordinates (x^{1}, …, x^{4}) on an open subset of X. We let θ^{R} be the natural coordinate in the odd ∧^{0}T X-direction and θ^{i} be the coordinates in the odd T X-direction determined by the x^{i}. We denote by ∂_{R^{odd}} and ∂_{i,odd} be the corresponding vector fields in these odd directions. Lastly, ∂_{i,even} is the usual partial derivative in the x^{i}-direction. We write

Q = ∂_{R^{odd}} + θ^{i}∂_{i,even}.

Clearly, as we change local coordinates, this expression is invariant. This gives us a globally defined real vector field Q on the super manifold which is the globalization of the (0, 0) component of the twisted form of the local supersymmetry generators {Q_{α}^{i}}. It is because of the presence of this one global, everywhere nonzero, super-symmetric generator satisfying Q^{2} = 0 that we can do global topological quantum field theory on X.

We also have a globally defined one-form K on X with values in vector fields on the supermanifold derived from the twisted form of \{\overline{Q}_{j,α̇}\}. A formula for it in the same local coordinates is:

K = (∂_{i,odd} + θ^{R}∂_{i,even}) dx^{i}.

One sees directly from the formula that K is invariant under change of coordinates, and hence is a well-defined global object on X.

It is easy to deduce from the super-Poincaré algebra structure, or from the above explicit formulas, that Q^{2} = 0 = {K, K}. The direct computation, or the general super-symmetric algebra structure, also gives

{Q, K} = d,  (1)

where d is the exterior derivative on X.

In fact, by Noether’s theorem there is a local (or unintegrated) form of these equations. Working in Minkowski space and choosing a time-slice Γ, the charge K̃ associated to the operator K can be written as

K̃ = ∫_{Γ} S

for a local operator S, which is the tensor product of three-form on Minkowski space with a one-form on Minkowski space. Letting T be the stress-energy tensor (which will exist once we have managed to formula the theory on a general riemannian manifold) , the unintegrated form of Equation 1 is

{Q, S} = T.

This equation can either be viewed as an equation for Poisson bracket of charges or as an equation for the bracket of the associated vector fields (the symplectic gradients of the charges). Physicists often do not distinguish between the charge and the vector field. In Euclidean space there is the usual relation between the stress-energy tensor and the translations

P_{μ} = ∫_{Γ} d^{3}xT_{μ0}
\overline{Q}α̇ = ∫_{Γ} d^{3}xS_{α̇0}
\{Q_{α}^{i}, \overline{Q}_{jα̇}\} = P_{αα̇}δ^{i}_{j}
{Q_{α}, S_{μα̇}} = T_{μαα̇}.

When we pass from Minkowski space to a riemannian manifold, we can no longer view K and Q as operators on Hilbert space. Nevertheless, in this more general context the charges associated to K and Q act by Poisson bracket on the local fields to produce new local fields (or equivalently the associated vector fields act by differentiation to produce new local fields).

II.2 Topological Quantum Field Theory on X

We shall be considering expectation values of products of Q-closed operators. Since the expectation value of any Q-exact operator is trivial (see Lecture II-??), it follows that the expectation value of any product of Q-closed operators is zero provided that at least one of them is Q-exact.

The fact that {Q, T } = 0 means that as we vary the metric, the correlation functions change by the insertion of an operator which is in the image of Q. It then follows that the correlation functions we are considering are independent of the metric. The operators that we have in mind are the analogues in this more general gauge theory setting of the Donaldson classes. The independence of the metric of the correlation functions of Q-closed operators is the physics analogue of the independence of the Donaldson invariants under change of metric.

Let us recall the general scheme of generating operators with values in higher dimensional differential forms out of local operators with values in functions. Here we are generalizing from two-dimensions to four-dimensions the discussion and results of Lecture II-10.

Begin with a polynomial function P of degree d on Lie algebra 𝔤 of the gauge group G, invariant under the adjoint representation. Classically, this produces a Chern form P (F_{A}) of degree 2d when applied to any connection A on a principal G-bundle. Applying this to the universal connection on the universal bundle over the product of the manifold X with the space B of gauge equivalence classes of configurations yields a closed form of degree 2d on X ×B. Recall from the last two lectures that part of the N = 2 supersymmetric gauge multiplet is an N = 1 chiral multiplet which includes a scalar field ϕ with values in the adjoint bundle. Quantum mechanically, the polynomial function P applied to ϕ produces a Q-closed local operator O_{P}^{(0)} in the theory. Applying K repeatedly yields operators O_{P}^{(n)} = K^{n}O_{P}^{(0)}. These are local operators with values in n-forms on X. For an n-cycle Σ^{n} ⊂ X, the operator

∫_{Σ^{n}} O_{P}^{(n)}

is the analogue of taking the slant product of the Chern form P (F_{A}) of the universal connection on X × B with the cycle Σ^{n} to produce a closed form μ_{P} (Σ) of degree 2d − n on B. As we shall see, the expectation value

⟨∏_{i=1}^{s} ∫_{Σ_{i}} O_{P}^{(n_{i})}⟩

computes

∑_{E} ∫_{M_{E}} μ_{P} (Σ_{1}) ∪ ⋯ ∪ μ_{P} (Σ_{s})

where E ranges over the topological types of principal G-bundles over X and where M_{E} ⊂ B_{E} is the moduli space of classical solutions. This integral of course is only taken when the virtual dimension of M_{E} is equal to ∑_{i=1}^{s}(2d − n_{i}).

From the fact that {Q, K} = d and the fact that {Q, O^{(0)}} = 0, we have

{Q, O^{(n)}} = dO^{(n−1)}.

Thus, we see that the expectation value of products these operators are topological in nature in the sense that these expectation values depend only on the homology classes of the Σ_{i} in the four-manifold.

The N = 2 supersymmetric theory that we have been discussing on R^{4} or Minkowski fourspace is induced by dimensionally reducing from a theory on R^{6|8}, i.e., an N = 1 supersymmetric theory in six dimensions. For more details on this, see the superhomework.

II.3 Twisting and globalizing the N = 2 vector multiplet

We still have work to do before we have installed the supersymmetric theories on a riemannian manifold. So far we have studied the effect of twisting on the super Euclidean group, and seen how to implement that on an oriented riemannian manifold. We must still examine the fields in the Lagrangian. In this section we study the vector multiplet.

Let us recall the basic configurations of the (untwisted) fields from an N = 2 vector multiplet in the N = 1 language. First the N = 2 vector multiplet decomposes as an N = 1 vector multiplet

A = (A, λ, D)

and an N = 1 chiral multiplet

Φ = (ϕ, ζ, F).

In the first (vector) multiplet A is a gauge field, λ is a spinor with values in the adjoint bundle and D is a real auxiliary field with values in the adjoint bundle. In the chiral multiplet ϕ is section of the complexification of the adjoint bundle, ζ is a spinor with values in the adjoint bundle, and F is a complex-valued auxiliary field with values in the adjoint bundle. Under the SU(2)_{R}-symmetry the pair (λ, ζ) transform in a two-dimensional representation denoted ψ_{α}^{i} and D^{→}= (D, Re F, Im F) transforms under the adjoint representation of SU(2)_{R}.

The following table summarizes the U(1)_{R}-charge of the various fields

The N = 2 Vector Multiplet

U(1)_{R} charge −2 −1 0 1 2
Field \overline{ϕ} \overline{ψ}_{α̇j} A ψ_{α}^{i} ϕ

D^{→}

Now we twist as described above using SU(2)_{R}. The SU(2)_{R} acts trivially except on D^{→} and on the fermions, thuys these are the only fields which change character when we twist. After twisting the ψi ; α̇ transform in the (1/2, 1/2) representation. This means that after twisting the ψi ; α̇ become a one-form with values in the adjoint representation, denoted ψ^{(1)} on X. Similarly, after twisting the \overline{ψ}_{αj} transform in (\frac{1}{2} ⊗ \frac{1}{2}, 0) and hence decompose as a direct sum of a function ψ^{(0)} and a self-dual two-form ψ^{(2)}. The auxiliary field D^{→} also transforms in (1, 0) and hence becomes a self-dual two-form on X.

We see that, after twisting, all these fields exist globally on X since they have become differential forms. This means that as far as pure gauge theories are concerned, we can write the Lagrangian on any closed oriented 4-manifold perserving the supersymmetric charge Q.

Let us examine the action of Q on these fields. The fields (A, ψ^{(1)}, ϕ) with Q, acting as a differential, form a model of the equivariant cohomology of the gauge group action on the space of connections. To make the notation more suggestive of a differential we let δ denote the action of Q. Then we have

δA = ψ^{(1)}
δψ^{(1)} = −/ ∂_{A}ϕ
δϕ = 0.

The Q-closed operators have representatives modulo Q-exact operators using only these fields. We begin with any operator O_{P}^{(0)} = P (ϕ) given by a gauge-invariant polynomial in ϕ. For example, in the SU(2)-case we take P (ϕ) = Tr ϕ^{2}. This polynomial has U(1)_{R}-charge four and corresponds to the Donaldson class μ(pt) in the fourth cohomology. From these local operators we build operators with values in forms O_{P}^{(j)} = K^{j}O_{P}^{(0)}.

The action of δ on the other fields is:

δ\overline{ϕ} = ψ^{(0)}
δψ^{(0)} = [ϕ, \overline{ϕ}]
δψ^{(2)} = F_{A}^{+}− D^{→}
δ(F_{A}^{+}− D^{→}) = [ϕ, ψ^{(2)}].

Here we are following the notation that the curvature F_{A} is decomposed into its self-dual and anti-self-dual components F_{A}^{+} + F_{A}^{−}.

II.4 Twisting and Globalizing the N = 2 hypermultiplet

Let us turn now to the N = 2 hypermultiplet. We have the following table describing the nature of the fields in this N = 2 multiplet before twisting.

The N = 2 Hypermultiplet

Helicity  −1/2  0  1/2
U(1)_{R} − charge  −1  0  1
SU(2)_{R} − action  trivial  1/2  trivial
Field  \overline{λ}α̇  M  λ_{α}
Type  −chirality Boson +chirality
spinor spinor

The boson and the spinors of both chiralities in the above table all take values in the quaternionic bundle over X associated to the representation ρ.

After twisting the boson M becomes a spinor of type (1/2, 0), that is to say M becomes a plus chirality spinor (a section of the plus spin bundle). The fermions λ and \overline{λ} are unchanged under twisting since the SU(2)_{R} acts trivially on them. Thus, λ remains a spinor of plus chirality and \overline{λ} remains a spinor of minus chirality. Thus, in order to write the part of the Lagrangian involving the hypermultiplet we must be able to make sense of the tensor product of the spin bundles S^{±}(X) of each chirality with the quaternionic representation ρ. (Of course, it is not necessary that the spin bundles actually exist globally, only that these tensor products exist.)

Assuming that we are in this situation, let us describe the action of the differential Q on these fields. Again denoting it by δ we get

δM = λ
δλ = [a, M]
δ\overline{λ} = / ∂(M) + fermions.

III The general form of the high energy computations

At this point we have succeeded in defining a twisted version of our N = 2 super-symmetric gauge theory on any oriented riemannian four-manifold X for which the bundles S^{±}(X) ⊗ ρ are defined. In particular, we have implemented all pure gauge theories on any oriented riemannian four-manifold. Since the ultraviolet behavior of the theory is independent of the metric on the manifold and independent of any global topology, we see that if we began with a fundamental theory (one which has a negative β-function and hence is asymptotically free), then the implementation of the theory on a complete riemannian four-manifold will also be asymptotically free. Thus, it is a well-defined problem to compute the correlation functions in this theory.

We are interested in the correlation functions of the Q-closed local operators in the theory. There is a localization result to the effect that to compute these in the UV limit one does not need to integrate over the entire infinite dimensional space of fields, but rather only over the fixed points of the action of Q on the space of fields (see Lecture II-??). Another way to think about this is that the correlation functions we are computing are independent of the metric and we can compute in the limit when as the metric shrinks to zero, or equivalently by asymptotic freedom, as the coupling constant goes to zero. In the limit we are doing a classical computation over the minima of the Lagrangian – this space of minima is exactly the fixed points of Q.

Let us examine the bosonic part of the fixed point set of Q. From the equations above, ignoring the fermion components and working modulo the action of the gauge group, we see that the fixed points of Q are given by:

d_{A}ϕ = 0
0 = F_{A}^{+}− D^{→}
0 = / ∂_{A}(M)

Now using the equations of motion allows us to integrate out the auxiliary field D^{→} giving the equation

D^{→}= μ(M)

where μ is the hyperkähler moment map. Thus, the above system becomes

d_{A}ϕ = 0  (2)
F_{A}^{+} = μ(M)  (3)
0 = / ∂_{A}(M)  (4)

Away from solutions where A is reducible, the first equation implies that ϕ = 0, and we can ignore ϕ and this equation. At the reducible solutions however this equation is important. Notice that Equations 3 and 4 are the Seiberg-Witten equations in the more general setting where the gauge group is not required to be U(1). Also, notice that if we are considering a pure N = 2 supersymmetric gauge theory (with no hypermultiplets), then M = 0 and these equations become the usual Yang-Mills equations.

What we see is that our correlation functions will localize to be integrals over supermanifolds whose underlying geometric manifold is a disjoint unoin of the moduli spaces

M_{E} = \{ \begin{matrix}d_{A}ϕ = 0 \\ F_{A}^{+} = μ(M) \\ 0 = / ∂(M) \end{matrix}\} /(Modulo gauge),

where the union is taken over all the topological types E for the principal G-bundle. Recall that, assuming that X is compact, there is an index theorem which computes the virtual dimension of M_{E}. It is

virt. dim(M_{E}) = k (4h − c_{2}(ρ)) − \frac{1}{2}dim(G) (χ(X) + σ(X)) − \frac{σ(X)}{4}quaternion dim(ρ),

where k is the instanton number of the bundle E, χ(X) is the Euler characteristic of X and σ(X) is its signature. If G is simply connected and simple then

k = ∫_{X} \frac{Tr(F ∧ F)}{8π^{2}}.

Notice that if β = 0 then the virtual dimension of M_{E} is independent of E. But when β < 0, the virtual dimension of the M_{E} go to infinity as the instanton number of E goes to infinity, so that our localization process becomes an integral over an infinite number of finite dimensional moduli spaces.

As we have said before, we are interested in computing the expectation values of products of operators of the form

∫_{Σ_{i}} O_{P}^{(k_{i})}

for an invariant polynomial P on the Lie algebra 𝔤 and for surfaces Σ_{i} in X. In the case of pure SU(2)-gauge theory these operators correspond to the usual operators in Donaldson theory in the following way. Since the gauge group is SU(2) the space of irreducible polynomials is generated by Tr(ϕ^{2}) = u. If we use this polynomial invariant to define O^{(0)}, then this operator corresponds to what is usually denoted μ(pt) in Donaldson theory. That is to say, the operator corresponds to the cohomology class on the moduli space which is minus one quarter the first Pontrjagin class of the SO(3)-bundle over the moduli space given by the based moduli space. Furthermore, the derived operators

∫_{Σ} O^{(k)}

correspond to the cohomology classes μ([Σ]) ∈ H^{4−k}(M_{E}).

One can begin with any polynomial function P of degree d on the Lie algebra invariant under the adjoint representation to produce O_{P}^{(0)} and derived operators of O_{P}^{(n)}. In general, we know from topological considerations (the Künneth formula) how to express the products of integrals of operators derived out of a decomposable invariant polynomial function in terms of the integrals of products of operators constructed from indecomposable invariant functions. Thus, in general it suffices to work exclusively with products of higher dimensional operators derived from indecomposable invariant polynomial functions on the Lie algebra. In the case of su(2) there is only indecomposable invariant polynomial Tr ϕ^{2}. Hence, it suffices to compute operators derived from this one local zero dimensional operator. These are the computations we shall do when we come to Donaldson theory.

In general, to compute a correlation function of the type where O_{P}^{(0)} comes from an invariant polynomial P of degree d

⟨∏_{i=1}^{s} ∫_{Σ_{i}} O_{P}^{(b_{i})}⟩

we must do the integration over all components of the supermanifold of solutions whose underlying geometric manifold, M_{E}, satisfies

virt. dim(M_{E}) = ∑_{i=1}^{s}(2d − b_{i}).

Actually, we can do slightly better in the case that G is not simply connected. We can sum over all principal G-bundles E which have a fixed isomorphism class over the two-skeleton of X. (Of course, if G is simply connected, all principal G-bundles on X have isomorphic restriction to the two-skeleton, so in that case we are summing over all E.)

As we have seen before, integration of functions over the odd directions in the supermanifold simply become integration of differential forms over the underlying geometric manifold. The dimension equation ensures that we are integrating a top dimensional form over the geometric manifold M_{E} underlying the supermanifold of solutions.

III.1 A case when β = 0

Let us continue with the case when β = 0. For simplicity, we take the gauge group G to be a connected (compact) semi-simple group. Then, since the M_{E} all have the same formal dimension, our integration is over all M_{E} at once. We have not written down the Lagrangian explicitly on the curved four-manifold, only on flat four-space. In putting the theory on a curved space we can add terms to the action which involve the metric. Since β = 0, there is a well-defined coupling constant τ for these theories. There are two such terms that preserve the topological invariance, namely:

∫_{X} f(τ)Pf(R) + g(τ)L(R)

where R is the riemann curvature tensor of the metric on the four-manifold, Pf denotes the Pfaffian, and L denotes the L-polynomial, and f, g are holomorphic functions of τ. (Since \overline{∂}/∂\overline{τ} = \{Q, ·\}, any topological invariant must be a holomorphic function of τ.) These give topological terms computing universal multiples of χ(X) and σ(X), respectively. Adding in such terms inserts a multiplicative factor in the correlation function. Even if we begin with a classical Lagrangian without these extra terms, they would be added in the renormalization process, and the coefficients with which they appear will depend on the explicit nature of that process. Thus, in the end what we are computing is of the form

q^{a(τ)χ(X)+b(τ)σ(X)} ∑_{k}⟨∏_{i=1}^{s} ∫_{Σ^{b_{i}}} O^{(b_{i})}⟩_{k}q^{k}

for functions a, b derived from f, g. For example, if f, g are linear functions of τ, then a, b are constants. Here, we have fixed the isomorphism class of E over the two skeleton and indicated by k the instanton number. (This action may not be an integer but its value modulo Z is determined by the isomorphism class of the bundles over the two-skeleton.) Also, q = e^{2πiτ}, and the factor q^{k} · q^{a(τ)χ(X)+b(τ)σ(X)} is the value of the exp(−S) on the component of the moduli space of classical solutions with instanton number k. Because of the proposed invariance of such a theory under a subgroup Γ ⊂ SL_{2}(Z) of finite index, this result should be a modular form under Γ possibly with poles at the cusps of the quotient of the upper half-plane by Γ. We can compute this expression by showing that there is k_{0} such that if M_{E} is non-empty then k ≥ k_{0}. Also, if k >> 0 then the index of / ∂ is negative and hence generically at least we expect no solutions to / ∂(M) = 0 except the trivial solution M = 0. Thus, in this range we expect to have to do a computation over the moduli space M_{E}^{instantons} of solutions to the anti-self-dual equations. In this case we are computing

∫_{M_{E}^{instanton}} ∏_{i} ∫_{Σ_{i}} O^{(b_{i})}χ(V)

where V is the obstruction bundle, which in this case is identified with the cokernel of / ∂: S^{+}⊗ρ → S_{−} ⊗ ρ. One case where we can show that there is a vanishing theorem for / ∂ for all instanton number is the case when the gauge group is SU(2) and ρ = su(2)⊗(quaternions) and the manifold is a hyper-Kähler manifold. In this case we have N = 4 supersymmetry, and V is identified with the tangent bundle of the instanton moduli space. In particular, the virtual dimension of M_{E} is zero. Thus, in this case, for dimension reaons there can be no positive dimensional surfaces in the integrals. That is to say, we are computing

q^{C} ∑_{k} q^{k}χ (M_{E_{k}}) ,

for an appropriate constant C. Our conclusion is that this series in q is a modular form (possibly with poles at infinity), for a subgroup Γ ⊂ SL_{2}(Z) of finite index. (In fact, Γ is the 2-congruent subgroup Γ_{0}^{2}.) This prediction has been compared successfully in some detail to mathematical computations [Vafa-Witten, A strong coupling test of S-duality, Nuclear Physics B, 431 (1994) 3-77].

III.2 Donaldson Theory

Now we are ready to specialize to Donaldson theory which is the case of pure SU(2)-theory, pure in the sense that the representaion ρ is trivial so that there are no hypermultiplets. As we remarked above the basic operator is O^{(0)} = u = Tr ϕ^{2}, coming from the quadratic invariant polynomial on su(2), and the derived operators O^{(i)}, i = 1, …, 4 which are to be integrated over i-cycles in X. We denote the various components of the geometric manifold underlying the supermanifold of classical solutions by M_{k} where k ∈ Z is the instanton number, i.e., the value of the second Chern class of the principal SU(2)-bundle. The short distance arguments above show that

⟨∏_{i=1}^{s} ∫_{Σ_{i}} O^{(b_{i})}⟩_{k} = ∫_{M_{k}} μ(Σ_{1}) ∪ ⋯ ∪ μ(Σ_{s})

where M_{k} is the moduli space of instantons on X with second Chern class k, provided that

virt. dim(M_{k}) = ∑_{i=1}^{s}(4 − b_{i}),

or equivalently that

8k − \frac{3}{2} (χ(M) − σ(M)) = ∑_{i=1}^{s}(4 − b_{i}).

The integral on the right-hand-side is the value of the usual Donaldson polynomial invariant

D([Σ_{1}], …, [Σ_{s}]) = ∫_{M_{k}} μ(Σ_{1}^{b_{1}}) ∪ ⋯ ∪ μ(Σ_{s}^{b_{s}}).

IV Low Energy Computations for Donaldson thoery

From now on we concentrate exclusively on Donaldson theory. To get something interesting we must compare this high energy computation, which we have just showed limits to the usual Donaldson polynomial invariants, with a computation using the low energy effective limit of the theory. In terms of metrics, we imagine a one-parameter family of metrics t^{2}g_{0} for some fixed metric g_{0}. As we have already remarked, since we are commuting correlation functions of Q-cohomology classes of operators, the answers are independent of the metric. As t ↦ ∞ we are recovering the short distance (or high energy) computations we sketched above and we find that the results are the Donaldson polynomial invariants. As t ↦ 0 we can recover the correlation functions by computing in terms of the space of vacua of the theory.

Thus, we obtain a result of the form

⟨∏_{i} ∫_{Σ_{i}} O^{(b_{i})}⟩ = ∫_{u plane} E(Σ_{1}, …, Σ_{s}).dμ

for an appropriate expression E that depends on the dimensions of the operators and a measure dμ on the u-plane that depends on the four-manifold and the path of metrics going to infinity. (Recall that all our operators O^{(b_{i})} are higher dimensional operators derived from the invariant polynomial Tr ϕ^{2}.) It will turn out that as long as b_{2}^{+}(X) > 1, that this integral over the u-plane becomes a sum of two computations, one at each of u = ±1. When b_{2}^{+}(X) = 1, then the answer has three contributions – discrete contributions at u = ±1 as before and an actual integral of a function over the u-plane. In this case the terms are not topological invariants. As we vary the metric in a generic one-parameter family (which of course is all that is necessary to check independence of the metric) there will be a finite set of points where things degenerate and on the two sides of each of these points the terms will have changed. The changes at u = ±1 are the wall-crossing formulas for the Seiberg-Witten invariants whereas the sum of the changes of the three terms is the wall-crossing formula for the Donaldson polynomial invariants.

When we do the integral over the u-plane, everywhere except at u = ±1, we are computing in a pure U(1)-theory, but at u = ±1 we cannot ignore the hypermultiplets which become massless at these points. At u = ±1 the classical equations are

F_{A}^{+} = (M\overline{M})^{+}
/ ∂(M) = 0.

Here A is a U(1)-connection on a line bundle detS^{+} ⊗ L and M is a section of S^{+} ⊗ L. Of course, we must sum over all complex line bundles. Actually, as we remarked earlier, it is not necessary that S^{+} exist, only that S^{+} ⊗ L exist. Thus, we are really summing over Spin_{c}- structures on X (which form a principal homogeneous set for the group of isomorphism classes of complex line bundles on X).

The appearance of M only at u = ±1 is of course the reason that there are delta-function contributions to the measure at these points.

IV.1 Operators in the effective low energy theory

In order to do the low energy effective computations we need to know the image of our operators (defined in the fundamental or high energy theory) in the low energy effective theory. In defining the low energy effective version of an operator we must integrate out all the high energy modes. This operation does not commute with taking products. Thus, while we shall see that several of the elementary operators will correspond in the obvious way, funny things will happen to products. In the high energy theory we have the operator u = Tr ϕ^{2}. We have seen that this operator flows in the low energy limit to an operator u which we know explicitly. Its expectation values are used to parameterize the vacua of the low energy theories. Of course, local operators at the point x in the high energy theory map to local operators at the same point x in the low energy theory, and hence the derivative of a local operator O in the high energy theory maps to derivative of the image \overline{O} of O in the low energy theory and similarly ∫_{Σ} O in the high energy theory flows to ∫_{Σ} \overline{O}. Since the low energy theory is still N = 2 supersymmetric, it has the operators \overline{Q} and \overline{K}, which are the images of the operators Q and K in the high energy theory. Thus, the image in the low energy theory of K^{n}u is \overline{K}^{n}u. Thus, as long as Σ_{1}, Σ_{2} are disjoint cycles in X, the operation of taking products commutes with integrating over the high energy modes. This means that the high energy operator

∫_{Σ_{1}} K^{n_{1}}u · ∫_{Σ_{2}} K^{n_{2}}u

goes in the low energy theory to the product of the images of the individual operator:

∫_{Σ_{1}} \overline{K}^{n_{1}}u · ∫_{Σ_{2}} \overline{K}^{n_{2}}u

as long as

Σ_{1} ∩ Σ_{2} = ∅.

Where products will not be preserved in passing to the low energy effective theory is when the cycles intersect. In this case the general form of the answer is that the product

∫_{Σ_{1}} K^{n_{1}}u · ∫_{Σ_{2}} K^{n_{2}}u

of high energy operators flows in the low energy limit to an operator of the form

∫_{Σ_{1}} \overline{K}^{n_{1}}u · ∫_{Σ_{2}} \overline{K}^{n_{2}}u + ∫_{Σ_{1}∩Σ_{2}} \overline{K}^{n_{1}+n_{2}}f(u)

for some function f(u). If we take products of more than two operators of this form then there will be higher corrections over triple and higher intersections of the cycles. The reason for all this is that if O_{i}(x) ↦ \overline{O}_{i}(x) then

O_{1}(x)O_{2}(y) ↦ \overline{O}_{1}(x)\overline{O}_{2}(y) + δ(x − y)Õ_{12}(x)

for an appropriate operator Õ(x) in the low energy theory. In our case if O_{i} = K^{n_{i}}u, then it must be the case that Õ_{12} = \overline{K}^{n_{1}+n_{2}}f(u). Integrating this result for local operators gives us the formula above for the image in the low eneryg theory of the product.

Let us simplify the dicussion by supposing that b_{1} = 0. Then the only operators that we need to consider are u, K^{2}u, K^{4}u. But there is never any problem with K^{4}u since it is a numerical invariant which is simply the second Chern class, so that inserting it in a correlation function simply multiples the expectation value on the bundle by the second Chern class. From now on we drop this operator from consideration. Among the remaining operators there are no triple intersections assuming, as we always can by topological invariance, that the cycles over which we integrate are in general position. Furthremore, the only double intersections are finite sets of points where a pair of two-cycles meet transversely. In this case we have that the image of product of the high energy operators

∫_{Σ_{1}} K^{n_{1}}u · ∫_{Σ_{2}} K^{n_{2}}u

in the low energy effective theory will be of the form

∫_{Σ_{1}} \overline{K}^{n_{1}}u · ∫_{Σ_{2}} \overline{K}^{n_{2}}u + (Σ_{1} ∩ Σ_{2})T (u),

where T is a holomorphic function of u and where Σ_{1} ∩ Σ_{2} denotes the algebraic intersection of Σ_{1} and Σ_{2}. The reason that T (u) is holomorphic is that since Q-closed operators map to Q-closed operators, we must have that the correction term in formula is Q-closed. On the other hand, up to Q-exact terms any Q-closed function of u is holomorphic.

IV.2 Warm-up with U(1)-theory with a hypermultiplet

We shall now do a practice computation that will help in learning to relate the SU(2) and the U(1)-theories. But we shall do this practice computation in a simpler theory – the U(1)-theory with a hypermultiplet and a minimal Lagrangain, minimal in the sense that we will omit certain couplings. We simply take the Lagrangian for the vector multiplet and hypermultiplet as we wrote them down in flat space with τ constant. We set q = exp(2πiτ). The only invariant polynomials for the Lie algebra u(1) are polynomials in a. We are assuming that H^{1}(X) = 0 so that the operators under discussion are the zero-dimensional operator O^{(0)} = a and the twodimensional operator O^{(2)} = c_{1}(L). Once again insertion of this latter operator simply multiplies the correlation function by the first Chern class of the line bundle, and hence can be ignored. That is to say the only operators whose expectation values we need to compute are holomorphic (or polynomial) functions P (a).

Recall that in this theory the bosonic part of the hypermultiplet field M becomes after twisting a section of S^{+} ⊗ L. This means that L should be though of as a Spin_{c}-structure on X rather than as a complex line bundle. (We shall come back to this point in the next lecture.)

Let us suppose that we wish to compute ⟨P (a)⟩_{λ} for some Spin_{c}-structure λ. The high energy arguments outlined above show that this expectation value concentrates along the geometric manifold M_{λ} underlying the moduli space of solutions to the classical equations for the Spin_{c}- structure λ. Since the value of exp(−S) on M_{λ} is q^{λ^{2}}, we have:

⟨P (a)⟩_{λ} = q^{λ^{2}} ∫_{M_{λ}} P (a).

Since a is a two-form on M_{λ}, this integral is equal to

q^{λ^{2}}c_{n_{λ}} ∫_{M_{λ}} a^{n_{λ}},

where n_{λ} is one-half the virtual dimension of M_{λ} and c_{n_{λ}} is the coefficient of a^{n_{λ}} in the Taylor series for P (a) at a = 0. Of course,

SW_{X}(λ) = ∫_{M_{λ}} a^{n_{λ}}

is by definition the Seiberg-Witten invariant of the manifold X evaluated on the Spin_{c}-structure λ. (By convention, all of these integrals are interpreted to be zero if n_{λ} is not an integer.) Thus, we have evaluated the expectation value ⟨P (a)⟩_{λ} to be the coefficient of a^{n_{λ}} in the Taylor series for P (a) times q^{λ^{2}}SW_{X}(λ). It is convenient for some of the later computations to rewrite this as

⟨P (a)⟩_{λ} = q^{λ^{2}}SW_{X}(λ) · Res_{a=0} (\frac{da}{a^{1+n_{λ}}} P (a)) .

Summing over all Spin_{c}-structures yields

⟨P (a)⟩ = ∑_{λ}⟨P (a)⟩ = ∑_{λ} q^{λ^{2}}SW_{X}(λ) · Res_{a=0} (\frac{da}{a^{1+n_{λ}}} P (a)) .

There is a similar formula when we allow a two-dimensional operator as well.

⟨P (a) · exp (α ∫_{Σ} O^{(2)})⟩ = ∑_{λ} q^{λ^{2}} · α^{⟨c_{1}(λ),Σ⟩}⟨P (a)⟩_{λ}
= ∑_{λ} q^{λ^{2}} · α^{⟨c_{1}(λ),Σ⟩}SW_{X}(λ) · Res_{a=0} (\frac{da}{a^{1+n_{λ}}} P (a)) .

Now let us generalize this U(1)-gauge theory computation by allowing more general terms that preserve topological invariance. The most general form for the Lagrangian on a curved manifold X, preserving topoloogical invariance, is

L(a, A, M) = {Q, ·} + ∫ K^{4}e(a) + ∫ g(a)Pf(R_{X}) + ∫ h(a)L(R_{X})

where R_{X} is the riemannian curvature tensor of X. By supersymmetry, topological invariance implies that the functions e, g, h must be holomorphic functions. Of course ∫ Pf(R_{X}) gives a multiple of the Euler characteristic whereas ∫ L(R_{X}) gives a multiple of the signature of X.

The expectation value is

∑_{λ} SW (λ)Res_{a=0} \{\frac{da}{a^{1+n_{λ}}} (E(a)^{λ^{2}}G(a)^{χ(X)}H(a)^{σ(X)})\} ,

where G = e^{−g}, H = e^{−h}, and E = exp(−∂^{2}e/∂a^{2}). The reason for the last equation is that since K(a) = ψ and K(ψ) = F, we have K(e(a)) = e′(a)ψ and

K^{2}(e(a)) = e′′ψ ∧ ψ + e′(a)K(ψ) = e′′ψ ∧ ψ + e′(a)F.

Continuing the differentiation, we find that

K^{4}(e(a) = e′′(a)F ∧ F + ψ ∧ (⋯).

In our case all the terms involving ψ vanish since b_{1}(X) = 0.

IV.3 The case of SU(2)

Now we are ready to return to the pure SU(2)-theory and understand the low energy effective computation in terms of the U(1)-computations we just did. Suppose that we fix α ∈ C and a class Σ ∈ H^{2}(X; C). We wish to compute

⟨exp (αu + β ∫_{Σ} O^{(2)})⟩_{SU(2)}.

The general form of the answer will be

⟨exp(αu + β ∫_{Σ} O^{(2)}⟩_{u=1} + ⟨exp(αu + β ∫_{Σ} O^{(2)}⟩_{u=−1} + ∫_{u−plane} · · · .

Let us see what operators we obtain in the low energy effective theory. By our above computations of K^{2}(e(a)) we have

∫_{Σ} O^{(2)} ↦ ∫_{Σ} K^{2}u = ∫_{Σ} \frac{∂u}{∂a}F + ∫_{Σ} \frac{∂^{2}u}{∂a^{2}}ψ ∧ ψ.

Here, this expression is valid in some region of the u-plane and we have picked one of the possibilities of the U(1)-variables a (remeber there is an ISL_{2}(Z)-family of such a) which is a local parameter in this region. Near u = ∞ we use the usual a but near u = 1 we have to use the a different a. If b_{1}(X) = 0 then we can drop the second term so that we have

∫_{Σ} O^{(2)} ↦ ∫_{Σ} \frac{∂u}{∂a}F.

Now using the rules for multiplication of the two-dimensional operators worked out above we see

exp (αu + β ∫_{Σ} O^{(2)}) ↦ exp (αu + β \frac{∂u}{∂a} ∫_{Σ} F + β^{2}Σ^{2}T (u)) .

Now let us compute the u = 1 contribution to the Donaldson invariant. The a we use here is the one produced by the monopole field that is massless at u = 1. We translate this coordinate so that u = 1 corresponds to a = 0. The effective low energy theory is a U(1)-theory and is a special case of the one we discussed in the last section with soem unkonwn universal functions e, g, h (or equivalently unknown functions E, G, H).

So, we have

⟨exp (αu + β \frac{∂u}{∂a}(Σ, λ) + β^{2}Σ^{2}T (u))⟩

= 2 ∑_{λ} SW (λ)Res_{a=0} \{\frac{da}{a^{1+n_{λ}}} exp (αu + β \frac{∂u}{∂a}(Σ, λ) + β^{2}Σ^{2}T (u)) E^{λ^{2}}H^{σ(X)}K^{χ(X)}\} .

The factor of 2 is introduced because of the following reason. The group of gauge transformations is SU(2)-bundle automorphisms, but its action on the space of connections is not faithful. There is a central group of order 2 which acts trivially. In performing the path integrals in gauge theory one has to normalize by dividing out by the volume of the group of gauge transformations. The fact that this group is twice as big as the group that acts faithfully introduces a factor of two in the comparison of the path integral with the topological formulas obtained by integrating over the moduli space.

Of course, by the index theorem, the one-half the dimension of the moduli space, n_{λ}, is a linear function of λ^{2}, χ(X), σ(X).

Let us suppose that the manifold X is of simple type (Seiberg-Witten simple type). By definition this means that all the invariants SW_{X}(λ) vanish except those for which n_{λ} = 0. Our expression then simplifies to

2 ∑_{λ} SW_{X}(λ)Res_{a=0} \frac{da}{a}exp(⋯)

where the unwritten expression is a regular expression in a. We can evaluate at u = 1 to obtain

2 ∑_{λ} SW_{X}(λ)exp (αu|_{u=1} + (Σ, λ)\frac{du}{da}|_{u=1} + Σ^{2}T (1)) G̃^{χ(X)}H̃^{σ(X)}.

Here G̃ and H̃ are complex numbers derived from E, G, H by setting u = 1 and using the index theorem (recall n_{λ} = 0) to express λ^{2} as a linear compbination of χ(X), σ(X). To complete determine the answer we must evaluate

u(1), \frac{∂u}{∂a}|_{u=1} , G̃, H̃, T (1).

Of course, we also have the contribution at u = −1 which differs from this one by the involution u ↦ −u.

In any event, assuming that, as we shall show in the next lecture, the integral over the uplane does not contribute to the answer when b_{2}^{+}(X) > 1, in this case the Donaldson polynomial invariant of X is completely determined by the function SW_{X}(λ) on Spin_{c} structures and the explicit determination involves five universal constants that must be computed. Fortunately,there is a list of four-manifolds whose Donaldson invariants are explicitly known, and whose Seiberg-Witten function is easy to compute, elliptic surfaces. From these known answers it is an easy computation to determine the constants. In fact, there is more than enough information available to determine these constant since there are infinitely many manifolds for which both SW and the Donaldson invariants are known. This gives infinitely redundant determinations of the constants and gives then infinitely many checks on the method. All these computations produce the same values for the five unknown constants in the above discussion.

The final result is that for manifolds X with b_{2}^{+}(X) > 1 which are of Seiberg-Witten simple type we have:

⟨exp (∑_{a} β_{a} ∫_{Σ_{a}} O^{(2)} + αO^{(0)})⟩ = 2^{1+ \frac{1}{4} (7χ+11σ)} (exp (\frac{v^{2}}{2} + 2α) ∑_{λ} SW_{X}(λ)e^{v·c_{1}(λ)}

+i^{Δ}exp (−\frac{v^{2}}{2} − 2α) ∑_{λ} SW_{X}(λ)e^{−iv·c_{1}(λ)}) ,

where v = ∑_{a} β_{a}[Σ_{a}] and where Δ = (χ(X) + σ(X))/4. The first of the two terms on the right-hand-side is the contribution at u = 1 and the second is the contribution at u = −1. It can be derived from the first by using the involution.

The factor of 2 in front of the α seems to appear mysteriously, and is not consistent with the formulas that we developed because in those formulas the coefficient was always one. The reason for this descrepancy is that we made an implicit assumption throughout this discussion which now has to be corrected in order to compare with Donaldson theory. Recall that in the low energy theory there is a mass scale Λ^{2} and that the monopoles become massless at u = ±Λ^{2}. We have implicitly assumed that the monopoles become massless at u = ±1. But there is no reason that this value of Λ^{2} should be the one that is compatible with topology. So we should do the computations with an extra unknown variable ±Λ^{2} and then determine this variable by comparison with topological computations in examples. Since ±Λ^{2} is the expectation value of u at the two special points, the way that ±Λ^{2} enters the formulas derived from the low energy computations is as a multiplicative factor in front of α (the two different signs appearing in the two different terms). It also changes the points at which we evaluate the unknown functions, but these values were not determined directly in terms of ±Λ^{2}, but rather by comparing with known answers. The fact that ±Λ^{2} = ±2 in order for the field theory computations to give the usual Donaldson polynomial answers is a reflection of the fact that the simple type condition implies that u^{2} = 4 in Donaldson theory.