10. The Bogomolny Equations And The Space Of Hecke Modifications
10.1. Boundary Conditions
p1
p2
p3
fixed
E− ιnF= 0
(a)
p1
p2 p3
E+ fixed fixed
E−
φ0 = 0
(b)
Fig. 15: (a) On the three-manifold W =I×C with ’t Hooft operator insertions, a bundle E− is specified on the left, and one wishes to determine possible Hecke modifications E+ that may appear on the right. On the left, the connection A− obeys Dirichlet boundary conditions and φ0 is undetermined; on the right, their roles are reversed. (b) To describe a fiber of the Hecke correspondence, one specifies bothE− andE+ and leaves φ0 undetermined at each end.
by either imposing Dirichlet boundary conditions on A – that is, specifying its boundary values – and leaving the boundary values of φ0 unspecified, or vice-versa. If the boundary values of A are specified, then the Bogomolny equations determine the normal derivative of φ0 at the boundary, and vice-versa. At E−, since we want to specify A, we leave φ0 unspecified. If A− is flat, then the Bogomolny equations require the normal derivative of φ0 to vanish at C−.
AtC+, on the other hand, we want to allow all possible Hecke modificationsE+ that may arise fromE− by solving the Bogomolny equations. So the connectionAis unspecified on C+, and instead we set φ0 = 0 on C+. The Bogomolny equations then require that A obeys covariant Neumann boundary conditions atE+; that is, its curvatureF(A) vanishes when contracted with the normal vector to the boundary. We are only interested in the output bundle E+ up to gauge transformations, so we allow the gauge transformations to be non-trivial on C+. These boundary conditions are summarized in fig. 15(a). In section 10.3, we will understand more deeply the naturalness of these boundary conditions.
The Hecke Correspondence
What we have just described is the boundary condition that we will generally use in employing the Bogomolny equations to study Hecke operators. But we pause to discuss the differential geometric analog of another formulation that is standard in algebraic geometry.
(The details are not needed for the rest of the paper.) Instead of thinking of E− as input and E+ as output, as we have done above, one can treat the two symmetrically by describing the “Hecke correspondence.” In the usual formulation in algebraic geometry, we let BunG be the “stack” of all G-bundles over C, not necessarily stable. The Hecke correspondence of typeLw, z(for a weightLwand a pointz ∈C) is then a varietyQ(Lw, z) that maps to BunG × BunG. The fiber of Q(Lw, z) over E− × E+ ∈ BunG × BunG parametrizes ways of obtainingE+ from E− by a Hecke transformation of type Lw at z.
The relationship between E− and E+ can be stated more symmetrically, as C− and C+ are exchanged by an orientation-preserving automorphism PT that acts trivially on C, exchanges the two ends ofI, and also reverses the time coordinate. Such an automorphism mapsφtoPT∗(φ), and maps an ’t Hooft operator of weightLw to one of weightLw′, where
Lw′ is a dominant weight that is Weyl conjugate to −Lw. SoE+ is produced fromE− by a Hecke transformation of typeLw atz, or equivalentlyE− is produced from E+ by a Hecke transformation of type Lw′ at z.
The fiber of the map Q(Lw, z) to BunG×BunG is generically empty if the weight Lw is small and the genus ofC is large. On the other hand, for anyC, this map is generically a fibration if Lw is large enough. One can extend the definition of Q to allow several Hecke operators with specified positions and weights, in which case the nature of the map Q →BunG×BunG depends on the number and weights of the ’t Hooft operators.
In differential geometry, BunG corresponds to the space A of all connections. A is of course infinite-dimensional, and BunG is an algebro-geometric analog of an infinite- dimensional manifold (it includes bundles that are arbitrarily unstable with no bound on the dimensions of spaces of deformations and automorphism groups). The Hecke cor- respondence Q likewise is infinite-dimensional and will not come from elliptic boundary conditions. A differential-geometric version of the Hecke correspondence is to simply define Q as the space of all solutions of the Bogomolny equations on W =I×C, with specified singularities, modulo gauge transformations that are trivial at each end. The map ofQto A×A is obtained by restricting a solution to the two ends of W.
Q is infinite-dimensional, as in defining it we have specified neither A nor φ0 at either end; these boundary conditions are not elliptic. In differential geometry, it seems
more natural to define the fiber of the Hecke correspondence over a pair of connections A−×A+ ⊂ A×A. This is the gauge theory analog of studying in algebraic geometry the fiber of the map Q → BunG×BunG. For this, we fix A− and A+ at each end, and divide by gauge transformations that are trivial at each end (fig. 15(b)). The space of all solutions of the Bogomolny equations with these boundary conditions is the fiber of the Hecke correspondence over the pairA−×A+. The boundary conditions defining this fiber are elliptic and correspond to simply using at both ends of W the boundary conditions that in fig. 15(a) are imposed only on C−.
The virtual complex dimension of the fiber of the Hecke correspondence can be computed via index theory, by modifying the discussion presented below. It equals
−(g−1)dim(G) plus the dimension ∆ (also calculated below) of the space of Hecke mod- ifications of type Lw. If the virtual dimension is a negative number −d, it means that generically Q is a subvariety of A×A of codimension d. If it is a positive integer d′, it means that generically Q fibers over A×A with fibers of dimension d′.
For the rest of our analysis, however, we concentrate on describing the space of possible Hecke modifications of a given bundle E−, and hence we use the boundary conditions of fig. 15(a). Before investigating more detailed properties, first we will see what can be learned from index theory.
Index Theory And The Virtual Dimension
We letZbe the moduli space of solutions of the Bogomolny equations with the elliptic boundary conditions of fig. 15(a). From our discussion of the relationship between ’t Hooft and Hecke operators in section 9, we anticipate that Z is a complex manifold. We demonstrate this in section 10.3. Linearization of the Bogomolny equations leads to an elliptic complex studied in [103,104]:
0→Ω0(ad(E)) d1
−−→Ω1(ad(E))⊕Ω0(ad(E))d2
−−→Ω2(ad(E))→0. (10.1) (Here Ωi(ad(E)) is the space of ad(E)-valued i-forms. d1 is the map from an element of Ω0(ad(E)), the Lie algebra of gauge transformations, to the linearization of A and φ, which take values in Ω1(ad(E)) and Ω0(ad(E)), respectively. d2 is the linearization of the Bogomolny equations.) The tangent space to Z is the first cohomology group of this complex, namely H1 = ker(d2)/im(d1). The index of this elliptic complex is called the virtual dimension of Z, and coincides with the actual dimension if the other cohomology groups H0 = ker(d1) and H2 = coker(d2) vanish. H0 consists of covariantly constant
sections of Ω0(ad(E)), which generate unbroken gauge symmetries. With our boundary conditions, there are none, since we require the gauge transformations to be trivial on one end ofW. As we will see, with the boundary conditions of fig. 15(a), the virtual dimension is nonnegative. This being so, H2 vanishes away from singularities of Z, and the smooth part of Z is a manifold whose dimension equals the virtual dimension or the index of the complex.
We first analyze the Bogomolny equations in the absence of any ’t Hooft operators.
We assume for simplicity that E− is stable and represented by a flat connection with F = 0. (We can always deform to this situation without changing the index.) In this case, a standard sort of argument shows that a solution of the Bogomolny equations with any of the boundary conditions above is a pullback from C, so in particular E+ and E− are isomorphic. This is actually a special case of the vanishing theorems of section 3.3, but the argument is so simple that we present it separately here.
If the Bogomolny equations F −⋆Dφ0 = 0 are obeyed, then 0 =R
W Tr (F −⋆Dφ0)∧
⋆(F −⋆Dφ0). Expanding this out and integrating by parts, we have
− Z
W
Tr (F ∧⋆F +Dφ0∧⋆Dφ0) =−2 Z
∂W
Trφ0F. (10.2)
The boundary term vanishes on a component of ∂W on which either φ0 or F vanishes.
In fig. 15(a), we have F = 0 at one end and φ0 = 0 at the other, so the boundary term vanishes at each end. But the left hand side of (10.2) is positive semi-definite and can only vanish if F = Dφ0 = 0. So, given that E− is flat, any solution of the Bogomolny equations with these boundary conditions and without ’t Hooft operators is given by a flat connection that is pulled back from C, withφ0 = 0.
In particular, E+ is isomorphic to E−, and Z is a point, of dimension zero. One can further verify that, in expanding around such a trivial solution with the boundary condition of fig. 15(a), H2 vanishes. Thus, in this case the virtual dimension of Z is the same as the actual dimension, namely zero. Now, what happens if ’t Hooft operators are included? Each singularity associated with an ’t Hooft operator T(LR) shifts the virtual dimension of the moduli space by an amount that only depends on the representation LR of the dual group (and not the details of the three-manifold in which the ’t Hooft operator is inserted, or the topology of the bundle E, or the possible presence of other ’t Hooft operators). This follows from general excision properties of index theory [116], as shown by Pauly [117]. Moreover, Pauly computed, in our language, the contribution of an ’t Hooft
operator to the dimension of moduli space for G = P SU(2) =SO(3), LG =SU(2). The answer42 is that an ’t Hooft operator associated with a representation of highest weight
Lw = (a/2,−a/2), with positive integer a, shifts the virtual (complex) dimension of the moduli space Z of solutions of the Bogomolny equations by a.
The fact that the singularity increases the complex dimension by a agrees with the result we found in section 9.2 by considering the space of Hecke modifications. There we showed that for U(2) weight (a,0), or equivalentlySU(2) weight (a/2,−a/2), the space of Hecke modifications has complex dimension a.
Pauly’s proof actually generalizes immediately to give the result for any compact Lie group G. To explain this, we must review the technique behind Pauly’s proof, which was introduced by Kronheimer [106]. The basic idea is to consider instantons on the four-manifold43 C2 ∼= R4 that are invariant under the action of F = U(1) on C2 by (z1, z2)→(exp(iθ)z1,exp(iθ)z2). The quotient C2/F is isomorphic to R3. F does not act freely – it has a fixed point at the origin,z1 =z2= 0 – but nonetheless the quotient C2/F is a manifold R3. The map from (z1, z2)∈C2 to ~x∈R3 is
~x=z~σz, (10.3)
where ~σ are the 2 × 2 traceless hermitian matrices (normalized to Trσiσj = 2δij and known as the Pauli spin matrices). This suggests that F-invariant instantons onC2 might be related to interesting objects on R3, but we should expect something special to happen at the origin of R3, which corresponds to the fixed point at the origin of C2.
The description ofF-invariant instantons onC2 is somewhat subtle. If the action ofF on C2 is lifted to an action on a G-bundle Eb →C2, then in particularF acts on the fiber Eb0 of Eb at the fixed point. Such an action is characterized by a homomorphism ρ : F ∼= U(1)→G. We recall that such a homomorphism can be interpreted as a weight ofLGand determines an ’t Hooft operatorT(ρ). Kronheimer considers F-invariant instantons onC2 with a given choice ofρ, and shows that they are equivalent to solutions of the Bogomolny
42 This is Theorem I of [117], withkcorresponding to what we calla/2. The theorem is stated for evena, as the gauge group is taken to beG=SU(2) rather thanP SU(2), but the proof also works for odda and in fact for allG, as we will discuss.
43 Kronheimer actually considers a more general situation withC2 replaced by a more general four-dimensional hyper-Kahler manifold with U(1) action, and gets a description of solutions of the Bogomolny solutions onR3 with a more general set of singularities.
equations on R3 with a singularity at the origin which in our language represents the insertion of the operatorT(ρ).
Pauly then shows that the contribution ∆ρ of a singularity of type ρ to the virtual dimension of Z can be computed from the contribution of the fixed point at the origin in C2 to the F-equivariant index of the linear operator that computes the deformations of instantons onC2. Let ad(E) be the adjoint bundle derived fromb E, and let ad(b Eb)0 denote its fiber at the origin. The fixed point contribution to the index involves a trace in ad(E)b 0. (The adjoint representation comes in because deformation theory of instantons involves an elliptic operator acting on the adjoint bundle.) In view of Pauly’s computation in section 4.2, the result can be described as follows. The action of F ∼= U(1) on ad(E)b 0, which is obtained by composing ρ : F → G with the adjoint representation of G, decomposes as a sum of characters. Any character of F takes the form exp(iθ) → exp(imθ) for some integer m. As the adjoint representation of G is real, the nonzero integers appearing in the decomposition of ρ into characters come in pairs ck,−ck, where we can takeck to be positive. Then the contribution of the ’t Hooft operator T(ρ) to the virtual dimension of Z is
∆ρ =X
k
ck. (10.4)
This generalizes the case ofG=P SU(2), in which only a single integer cappears. Pauly’s analysis immediately extends fromSU(2) to anyGbecause the equivariant index theorem works for any G. The computation involves a trace in the Lie algebra, which leads to (10.4).
With no ’t Hooft operators at all, the virtual dimension with our boundary conditions is zero. With a single ’t Hooft operator T(ρ), the virtual dimension is therefore ∆ρ. But with a single ’t Hooft operator, Z is just the space Y(ρ) of Hecke modifications of type ρ. The formula (10.4) for the dimensions of spaces of Hecke modifications agrees with the result (9.25) for U(N) (or SU(N) or P SU(N)) and generalizes it to arbitrary compactG.
Since the contribution of an ’t Hooft operator to the index theory is local, we can immediately write the general result for the dimension of the moduli space Z(ρ1, . . . , ρn) of solutions of the Bogomolny equations with an arbitrarily prescribed set of ’t Hooft operator insertions. If there are no ’t Hooft operators, the dimension ofZ is zero. Each ’t Hooft operator T(ρi) associated with a homomorphism ρi :U(1) →G contributes ∆ρi to the dimension, so in general the dimension ofZ(ρ1, . . . , ρn) is
∆ =X
i
∆ρi. (10.5)