Infinite Product Decomposition of Orbifold Mapping Spaces

Let G be a finite group. For a G-manifold M , we can consider an orbifold version of the elliptic genus. However, the free loop space L(M/G) on the orbit space is not well behaved. Following [7], we define the orbifold loop space L orb (M/G) by (1.1) L orb (M/G)

where G * is the set of conjugacy classes in G, C G (g) is the centralizer of g in G, and L g M is the space of g-twisted loops in M given by (1.2) L g M = {γ : R → M | γ(t + 1) = g -1 γ(t) for all t ∈ R}.

The centralizer C(g) acts on L g M . Also note that if the order of g is finite and is equal to s, then each twisted loop γ in L g M is in fact a closed loop of length s. Thus, L g M also admits an action of a circle S 1 = R/sZ of length s.

One could use more sophisticated languages on orbifolds (see for example, [11]), but for our purpose, the above definition suffices. Now the orbifold elliptic genus of (M, G), denoted by ell orb (M/G), is defined as the S 1 -equivariant χ y -characteristic of L orb (M/G):

where χ y (L g M ) is thought of as R C(g) -valued S 1 -equivariant χ y -characteristic computed and made sense through a use of localization formulae. Counting the dimension of coefficient vector spaces, we have

where the powers of q are characters of S 1 . Dijkgraaf, Moore, Verlinde and Verlinde [3] essentially proved a remarkable formula for the generating function of orbifold elliptic genera of symmetric products. This was subsequently extended to symmetric orbifold case by Borisov-Libgober [1]. Here, for an integer n ≥ 0, the n-th symmetric product of a space X is defined as SP n (X) = X n /S n , where the n-th symmetric group S n acts on X n by permuting n factors. The DMVV and BL formula for the generating function of orbifold elliptic genera of symmetric orbifolds is given by (1.5) n≥0 p n ell orb SP n (M/G) = n≥1 m≥0 k∈Z (1 -p n q m y k ) -c (mn,k) , where ell orb (M/G) = m≥0 k∈Z c(m, k)q m y k ∈ Z[y, y -1 ][ [q]].

The amazing thing about this formula is that the right hand side of (1.5) is a genus 2 Siegel modular form, up to a simple multiplicative factor. The main motivation of this paper is to understand a geometric origin of this infinite product formula. In fact, we will prove such an infinite product formula on a geometric level, not merely on an algebraic level, as in (1.5).

We can describe this geometric formula in a general context. Let (M, G) be as before, and let Σ be an arbitrary connected manifold with Γ = π 1 (Σ). Instead of a loop space, we consider a mapping space Map(Σ, M/G). As before, this space is not well behaved and the correct space to consider is the orbifold mapping space defined by (1.6)

Here Σ is the universal cover of Σ, and Map θ ( Σ, M ) is the space of θ-equivariant maps α : Σ → M such that α(p • γ) = θ(γ) -1 • α(p) for all p ∈ Σ and γ ∈ Γ. Note here that we regard the universal cover Σ as a Γ-principal bundle over Σ.

For a variable t and a space X, let S t (X) = k≥0 t k SP k (X) be the total symmetric product of X. For convenience, we often write this using the summation symbol as S t (X) = k≥0 t k SP k (X). In this paper, summation symbol applied to topological spaces means topological disjoint union.

Theorem A (Infinite Product Decomposition of Orbifold Mapping Spaces of Symmetric Products). Let M be a G-manifold and let Σ be a connected manifold. Then,

Here the infinite product is taken over all the isomorphism classes of finite sheeted connected covering spaces Σ ′ of Σ, and D(Σ ′ /Σ) is the group of all deck transformations of the covering space Σ ′ → Σ (which is not necessarily Galois). The number of sheets of this covering is denoted by |Σ ′ /Σ|.

We will explain the details of the action of D(Σ ′ /Σ) on Map orb (Σ ′ , M/G) in §2. When Σ = S 1 , the above formula reduces to

where L (r) orb (M/G) is the space of orbifold loops of length r. This is the geometric version of the formula (1.5). This formula itself is relatively easy to prove. See [16].

The above formula (1.8) for orbifold loop space is an “abelian” case since π 1 (S 1 ) ∼ = Z. The formula in Theorem A is, in a sense, a non-abelian generalization of this orbifold loop space case. The most interesting case seems to be the one in which Σ is a 2-dimensional surface (regarding it as a world-sheet of a moving string). Here, the genus of the surface can be arbitrary. In physics literature, elliptic genus itself is computed as a path integral over mapping spaces from torus [3].

Restricting the global decomposition formula (1.7) to the subspace of constant orbifold maps and considering their numerical invariants, we recover our previous result

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