8. Electric Eigenbranes
9.3. The Affine Grassmannian
The Dimension Of The Space Of Hecke Modifications
For U(N) (or SU(N), or P U(N)), let us determine the dimension of the space of Hecke modifications of an arbitrary type. For weight Lw = (a1, a2, . . . , aN), with a1 ≥ a2 ≥. . .≥aN, the dimension is
∆Lw =X
i<j
(ai−aj). (9.25)
To justify this statement, we considerN sectionssi that generate the trivial rankN bundle E− = O ⊕ O ⊕. . .⊕ O near z = 0. We recall that this means that the si(0) are linearly independent. Then we consider the bundle E+ whose general section is
q = XN i=1
z−aigisi, (9.26)
where the functions gi(z) are holomorphic atz = 0. The bundle E+ is invariant under si →si+X
j≤i
hjsj+X
j>i
zai−ajhjsj, (9.27)
for generic functions hj that are holomorphic at z = 0. Such a transformation of the si
can be absorbed in redefining the functions gi that appear in (9.26). Expanding si in a Taylor series near z = 0, the number of coefficients that cannot be eliminated by the transformation (9.27) is PN
j=i+1(ai−aj). After summing overi, we arrive at the formula (9.25) for the dimension of the space of Hecke modifications.
problem is local, we can fix the curve C to be CP1 and take E− to be trivial. A Hecke modification E of E− at a point p0 ∈ C is a bundle that is endowed with a chosen isomorphism σ :E ∼=E− outside of p0; σ may not extend over p0.
It is natural to regard the space of Hecke modifications for fixed m1, . . . , mN as a subspace of the space of pairs (E, σ), where E is a holomorphic vector bundle on C and σ is its trivialization outside p0. Let us recall an explicit description of the space of such pairs (E, σ), following chapter 8 of [115].
It is convenient to think of C = CP1 as a one-point compactification of the complex z-plane C and to set p0 to be the point z = 0. We also let
U∞ =CP1− {0}, U0 =CP1− {∞}. (9.28) We are given a trivialization σ of E over U∞. Although σ may not extend over U0, we can pick a trivialization σ′ of E over U0. Over U0T
U∞ ≃C∗, σ and σ′ are related by a GL(N,C) gauge transformation g(z). This is a GL(N,C)-valued function whose entries are holomorphic on C∗ but may have poles at 0 and ∞. (σ′ can always be chosen so that the singularities of g(z) are poles.) Let us denote by X the ring of functions holomorphic on C∗ and having a meromorphic extension to CP1; then the group of holomorphic gauge transformations on C∗ can be identified with GL(N,X), and g(z) is an element of this group.
If we changeσ′ by a gauge transformationh0(z) which is holomorphic throughout U0, theng(z) is replaced byg(z)h−10 (z). h0(z) takes values inGL(N,O), the group of invertible matrices whose entries are holomorphic functions onU0 ≃C. The set of holomorphic vector bundles of rank N over CP1 equipped with a fixed holomorphic trivializationσ outside p0 is isomorphic to the quotientGL(N,X)/GL(N,O). The latter space is known as the affine Grassmannian GrN for the groupGL(N). Another name for it is the loop Grassmannian, because it is isomorphic [115] to the space of based loops in U(N).
The definition of the affine Grassmannian admits slight variations. For example, since any function in X can be expanded in a Laurent series, one can embed X into the ring C((z)) of formal Laurent series. Similarly, one can embed the ring of holomorphic functions O into the ring C[[z]] of formal power series. One can show that the quotient GL(N,C((z)))/GL(N,C[[z]]) is isomorphic to the affine Grassmannian GrN. Intuitively, this happens because given any two elements of GL(N,X), one can tell whether they are in the same GL(N,O) orbit by studying a finite number of terms in their Laurent
expansions, and therefore it is immaterial whether the Laurent series have a nonzero region of convergence.
Now we have to identify those points in GrN which can be obtained from the trivial vector bundle by a Hecke modification of weightLw= (m1, . . . , mN). A Hecke modification of the trivial vector bundle with this weight is obtained by choosing a trivialization of the trivial vector bundle overU0byN linearly independent holomorphic sectionsf1, f2, . . . , fN, and declaring that E+ is generated overU0 by sections s1, . . . , sN which upon restriction to U∞T
U0 are given by
z−mjfj, j = 1, . . . , N. (9.29) Here we made use of the fact that onU∞ we are given an isomorphism betweenE+ and the trivial vector bundle. Thus if fj = (fj1, fj2, . . . , fjN), then the matrix g(z) corresponding to E+ is given by
gji(z) =z−mjfji(z). (9.30) The simplest choice isfji(z) =δji; all other choices can be obtained from this one by acting on g(z) from the left by an element of GL(N,O). We conclude that the space of Hecke modifications Y(Lw;p0) is the orbit of the point
gji(z) =z−mjδji (9.31)
in the affine Grassmannian under the left action of the groupGL(N,O). One can replace X with C((z)) and O withC[[z]] throughout and get an equivalent result. Note that g(z) given by (9.31) is a homomorphism of C∗ to GL(N,C). In fact, it is the complexification of the homomorphism
ρ:U(1)→U(N), ρ:eiα →
eim1α . . . eim2α . . . ... . . .
eimNα
(9.32)
which enters into the definition of the ’t Hooft operator T(m1, . . . , mN).
We have associated to each set of integers m1, . . . , mN an orbit of GL(N,O) in GrN. It turns out that all GL(N,O) orbits in GrN are obtained in this way [115]. Thus GrN is stratified by spaces Y(m1, . . . , mN), which are called Schubert cells. Equivalently, by applying Hecke modifications to the trivial vector bundle, one can obtain an arbitrary holomorphic vector bundle on CP1 with an arbitrary trivialization on U∞.
The examples of spaces of Hecke modifications discussed in section 9.2 suggest that these spaces are always finite-dimensional. This can be shown in general as follows. Let GrN(i) be the subset of GrN defined by the condition that g(z) has at z = 0 a pole of order not higher thani. The variety GrN(i) is finite-dimensional,GL(N,O)-invariant, and compact, and any point in GrN belongs to GrN(i) for some i. This implies that any orbit of GL(N,O) belongs to some GrN(i) and therefore is finite-dimensional.
It may be helpful to mention at this point that the infinite-dimensional space GrN has another stratification with strata labelled by sets of integersk1, . . . , kN. To define this stratification, we recall that by a theorem of Grothendieck any holomorphic vector bundle of rank N on CP1 is isomorphic to
⊕Ni=1O(p0)ki (9.33)
for some integers k1, . . . , kN. We define the stratum in GrN corresponding to k1, . . . , kN as the set of those pairs (E, σ) whereE is isomorphic to (9.33). In other words, a stratum is obtained by fixingE and varying the trivialization σ. Obviously, the strata are infinite- dimensional in this case, in contrast with the stratification given by Y(m1, . . . , mN). The relation between the two stratifications of GrN is studied in [115].
As we have seen, spaces of Hecke modifications or Schubert cells are noncompact, in general. A natural way to compactify them is to consider the closure of the corresponding orbits in GrN(i) for sufficiently large i. For a given ρ : U(1) → U(N), the closure of the orbit Cρ is in general a singular variety which is a union of Cρ and a finite number of orbits of lower dimension. It is called a Schubert cycle. As discussed in section 9.2, the structure of the closure of Cρ reflects the “mixing” between ’t Hooft operators with different ρ but the same topological type. We also will describe this process in terms of monopole bubbling in section 10.2.
The construction of Gr and the description of the spaces of Hecke modifications as subvarieties in Gr can be generalized to other gauge groups. For any simple compact Lie group G, with complexification GC, one defines GrG as the quotient
GrG =GC(X)/GC(O). (9.34)
An ’t Hooft operator is parametrized by the conjugacy class of a homomorphismρ:U(1)→ G, which can be analytically continued to a homomorphism ρC : C∗ → GC. Obviously, ρC defines a point on GrG, and the orbit of this point under the left action of GC(O) depends only on the conjugacy class ofρ. This orbit is the Schubert cell or space of Hecke modifications Y(ρ).