Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe $c$ on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} $c$. We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of $c$-twisted sheaves on $\hat{\Y}$. When $\Y$ is symplectic, we show, by a combination of techniques from noncommutative geometry and symplectic topology, that the Chen-Ruan orbifold cohomology of $\Y$ is isomorphic to the $c$-twisted orbifold cohomology of $\hat{\Y}$ as graded algebras.
Deep Dive into Duality theorems for etale gerbes on orbifolds.
Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe $c$ on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} $c$. We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of $c$-twisted sheaves on $\hat{\Y}$. When $\Y$ is symplectic, we show, by a combination of techniques from noncommutative geometry and symplectic topology, that the Chen-Ruan orbifold cohomology of $\Y$ is isomorphic to the $c$-twisted orbifold cohomology of $\hat{\Y}$ as graded algebras.
was first introduced in [58] under the name "V -manifold," and was introduced in algebraic geometry in [26], and is now called a Deligne-Mumford stack. The term "orbifold" was coined by Thurston [60] during his study of 3-dimensional manifolds. Orbifolds are geometric objects that are locally modeled on quotients of manifolds by actions of finite groups. Introductory accounts about orbifolds can be found in [5], [40], and [47].
Besides being interesting in its own right, the theory of orbifolds can be applied in numerous areas, such as the study of moduli problems and quotient singularities. Moreover, there has been an increase of activities in the study of the stringy geometry of orbifolds. See [5], [56], and [57] for expository accounts.
In this paper, we study a special kind of orbifolds called gerbes. Let G be a finite group and BG = [pt/G] the classifying orbifold of G. Roughly speaking, one can think of a G-gerbe over an orbifold B as a BG-bundle over B. Then in order to define a G-gerbe Y over B, one starts with an open cover {U i } of B and specifies the following data: ϕ ij ∈ Aut(G) for each double overlap U ij := U i ∩ U j , and g ijk ∈ G for each triple overlap U ijk := U i ∩ U j ∩ U k , (1.1) so that the following constraints are satisfied:
Here, Ad g : G → G denotes the map of conjugation by g. The data in (1.1) are then used to glue U i × BG together to form a G-gerbe Y together with an associated map Y → B.
We easily see that BG is the unique G-gerbe over a point. Gerbes arise naturally from ineffective group actions. For example, let M be a manifold and H a compact group that acts on M with finite stabilizers at every point. The quotient space [M/H] is an orbifold. Suppose that G is a finite normal subgroup of H such that the induced action of G on M is trivial. Then there is an induced action of the quotient group
In general, gerbes play an important role in the structure theory of orbifolds. For example, given an orbifold X there is a finite group G and a reduced orbifold X ′ such that X is a G-gerbe over X ′ . See [13,Proposition 4.6]. Introductory accounts about gerbes can be found in [27], [31], and [42].
The purpose of this paper is to study the geometry and topology of G-gerbes. Our study is motivated and inspired by results in the physics paper [34]. Given a G-gerbe Y → B, the authors of [34] construct a disconnected space Y with a map Y → B and a flat U (1)-gerbe c on Y. This construction is reviewed in Sec. 1.2 below. The main point of [34] is the conjecture which asserts that the conformal field theories on the G-gerbe Y are equivalent to the corresponding conformal field theories on Y twisted by the B-field c. 1 Throughout this paper we consider orbifolds which are not necessarily reduced. This conjecture suggests the existence of a certain duality between the G-gerbe Y and the pair ( Y, c). Our viewpoint toward this conjecture is the following claim:
(⋆) The geometry/topology of the G-gerbe Y is equivalent to the geometry/topology of Y twisted by c.
The claim (⋆) reveals a deep and highly nontrivial connection between different geometric spaces. Let us look at the simplest G-gerbe, namely, a G-gerbe over a point, (i.e., B = pt and Y = [pt/G] = BG). The dual orbifold Y is the discrete set G, the space of isomorphism classes of irreducible unitary Grepresentations, with a trivial U (1)-gerbe c on Y. In this case, the claim (⋆) states that the geometry/topology of the classifying space BG is equivalent to the geometry/topology of the discrete set G. Such a relationship is not clear at all at the level of spaces. For example, when G = Z 2 , there does not seem to be any obvious geometric connection between the space BZ 2 (interpreted either as an orbifold [pt/Z 2 ] or as the space RP ∞ ) and the space Z 2 . In general, to our best knowledge there is no known geometric relation at the level of spaces between a G-gerbe Y and the orbifold Y with the U (1)-gerbe c.
One observes that a natural place where both BG and G appear is representation theory, since BG encodes information about principal G-bundles, and G is defined to be the set of isomorphism classes of irreducible G-representations. Noncommutative geometry is a powerful modern approach to representation theory. Thus, it makes sense to consider possible relations between BG and G in noncommutative geometry. In noncommutative geometry, BG is represented by the group algebra CG, and G is represented by the commutative algebra, C( G), of functions on G. By a classical result, the group algebra CG is Morita equivalent to the algebra C( G). We can interpret this as saying that the two spaces BG and G are “equivalent” from the viewpoint of noncommutative geometry. This observation strongly suggests that noncommutative geometry naturally relates the two geometries of Y and ( Y, c), which appear to be very different in the classical geometric/topological viewpoints.
Indeed, the main theme of this paper is using tools from nonc
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