Cluster decomposition, T-duality, and gerby cfts. (English) Zbl 1156.81039

Hellerman, Simeon; Henriques, André; Pantev, Tony; Sharpe, Eric; Ando, Matt

Adv. Theor. Math. Phys. 11, No. 5, 751-818 (2007).

The authors describe a T-duality between strings propagating on gerbes and strings propagating on disconnected union of spaces. This duality solves a basic problem with the notion of strings propagating on gerbes, namely that the massless spectrum violates cluster decomposition. Indeed, a sigma model on a disconnected union of spaces also violates cluster decomposition, but, in a certain sense, in the mildest possible way; in particular CFTs associated to disjoint sums of spaces are consistent. Hence, the T-duality described in the paper means that also the CFTs associated to gerbes must be consistent, despite violating cluster decomposition.
More precisely, the authors consider a \(G\)-gerbe over \(M\), presented as \([X/H]\), where \[ 1\to G\to H\to K\to 1, \] and the set \(\widehat{G}\) of irreducible represtentations of \(G\). The group \(K\) naturally acts both on the space \(X\) and on the set \(\widehat{G}\), and one can form the stacky quotient \(Y=[(X\times \widehat{G})/K]\). The connected components of \(Y\) are in bijection with the \(K\)-orbits on \(\widehat{G}\); namely, if \(K_1,\dots,K_n\) are the stabilizers in \(K\) of the various orbits, then \[ Y=[X/K_1]\sqcup\cdots\sqcup [X/K_n]. \] Furthermore, \(Y\) is naturally equipped with a flat \(U(1)\)-gerbe, representing the obstruction to making a certain natural \(G\)-equivariant bundle on \(Y\) be \(H\)-equivariant. More precisely, each component \([X/K_i]\) of \(Y\) is equipped with some discrete torsion coming from a \(U(1)\)-valued \(2\)-cocycle on \(K_i\).
The authors conjecture that the CFT of a string on the gerbe \([X/H]\) is the same as the CFT of a string on \(Y\), and so is the same as a string on copies and covers of \(M=[X/K]\), with variable \(B\)-fields (using the relation between \(B\)-fields and discrete torsion [E. Sharpe, Phys. Rev. D 68 126003 (2003); Phys. Lett., B 498, No. 1–2, 104–110 (2001; Zbl 0972.81157)]). In the special cases of banded gerbes, i.e., when the principal \(\text{Out}(G)\)-principal bundle associated to the gerbe is trivial, this conjecture simplifies somewhat, since the \(K\)-action on \(\widehat{G}\) is trivial in this case and so \(Y=M\times \widehat{G}\). Thus the conjecture predicts in this case that the CFT of a banded \(G\)-gerbe over \(M\) is T-dual to a disjoint union over irreducible representations of \(G\) of CFTs on \(M\) with falt \(B\)-fields.
The authors exhibit several types of evidence for this decomposition conjecture.

1.

For gerbes presented as global quotients by finite (noneffectly acting) groups, explicit computation of partition functions (at arbitrary genus), massless spectra and operator products support the conjecture.

2.

For gerbes presented as global quotients by nonfinite groups, one can see the decomposition directly in the structure of the nonperturbative sectors which distinguish the physical theory on the gerbe from the physical theory on the underlying space.

3.

At the open string level, D-branes on gerbes decompose according to the irreducible representation of the band, and there are only massless open string states between D-branes in the same irreducible representation. A mathematical consequence of this fact is that the \(H\)-equivariant \(K\)-theory of \(X\) is the same as the twisted \(K\)-equivariant \(K\)-theory of \(X\times \widehat{G}\).

4.

Mirror symmetry fo gerbes generates Landau-Ginzburg models with discrete-valued fields, which can be interpreted in terms of a sum over components, consistent with the decomposition conjecture.

5.

Additional evidence is provided noncommutative geometry-based arguments, by general aspect of T-duality and by the structure of quantum cohomology rings (in particular by Batyrev’s conjecture for quantum cohomology of toric varieties).

In the final part of the paper the authors discuss why the above described decomposition should be understood as a T-duality. Also, they discuss applications of the ideas presented in the main body of the paper. One important set of applications is to curve counting: the decomposition conjecture allows predictions for quantum cohomology and Gromov-Witten invariants, and seems to be consistent with the general aspects of the Gromov-Witten-Donaldson-Thomas correspondence. Finally, it is discussed how part of the physical picture of the geometric Langlands correspondence can be described using results of the paper, by giving a direct physical relationship between different descriptions of geometric Langlands in tems of disconnected spaces and in terms of gerbes.

Reviewer: Domenico Fiorenza (Roma)

MSC:

81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
19M05	Miscellaneous applications of \(K\)-theory
55U99	Applied homological algebra and category theory in algebraic topology
14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45	Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Keywords:

Gerbes; string theory; conformal field theory; K-theory; geometric Langlands

Citations:

Zbl 0972.81157