Orbifold Group Research Articles

String orbifolds and quotient stacks

In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a nonsingular CFT even when the associated quotient space is singular. We also show how to understand twist fields in this language, and outline the derivation of the orbifold Euler characteristic purely in terms of stacks. We also outline why there is a sense in which one naturally finds B≠0 on exceptional divisors of resolutions. These insights are not limited to merely understanding existing string orbifolds: we also point out how this technology enables us to understand orbifolds in M-theory, as well as how this means that string orbifolds provide the first example of an entirely new class of string compactifications. As quotient stacks are not a staple of the physics literature, we include a lengthy tutorial on quotient stacks, describing how one can perform differential geometry on stacks.

Intersecting brane world models from D8-branes on (T2×T4/Bbb Z3)/ΩScript R1type IIA orientifolds

We present orientifold models of type IIA string theory with D8-branes compactified on a two torus times a four dimensional orbifold. The orientifold group is chosen such that one coordinate of the two torus is reversed when applying worldsheet parity. RR tadpole cancellation requires D8-branes which wrap 1-cycles on the two torus and transform non-trivially under the orbifold group. These models are T-dual to orientifolds with D4-branes only which admit large volume compactifications. The intersections of the D8-branes are chosen in such a way that supersymmetry is broken in the open string sector and chiral fermions arise. Stability of the models is discussed in the context of NSNS tadpoles. Two examples with the SM gauge group and two left-right symmetric models are given.

On the fundamental groups at infinity of the moduli spaces of compact Riemann surfaces

In his seminal manuscript Esquisse d’un programme (1984; now available in [GGA]) A.Grothendieck explains that the structure of the tower of all moduli spaces of curves is somehow governed by its “first two levels” (“deux premiers etages”), i.e. the moduli spaces of dimensions 1 and 2. We refer to the Esquisse and to [L] for more context and details about this statement. Let us only mention that speaking in terms of topology, Grothendieck was concerned more precisely with the orbifold fundamental groups of the moduli spaces of curves and he explains that the above “principle” is essentially equivalent to the fact that the orbifold fundamental group of any moduli space of dimension > 2 is equal to its fundamental group at infinity. We do not recall here the notion of orbifold fundamental group, which is due to Thurston in a topological context, because we will be concerned only with the ordinary topological fundamental group, i.e. the fundamental groups of the moduli spaces of curves viewed as manifolds, forgetting about their orbifold structure. In terms of analytic or algebraic geometry, this amounts to viewing them as coarse and not as fine moduli spaces for curves. Before we consider moduli spaces of curves in detail, let us make precise the notion of fundamental group at infinity in a topological context. Note that it is less easy (although feasible) to do it in terms of algebraic geometry because a quasiprojective variety cannot usually be exhausted by an increasing sequence of projective subvarieties, nor is it easy to define tubular neighborhoods of closed subvarieties. So let M be a paracompact differentiable manifold and partially order the compact submanifolds (possibly with boudary) of M by inclusion. Their complements define an obvious inverse

On string models with Scherk-Schwarz supersymmetry breaking

We construct a general class of chiral four-dimensional string models with Scherk--Schwarz supersymmetry breaking, involving freely acting orbifolds. The basic ingredient is to combine an ordinary supersymmetry-preserving Z_N projection with a supersymmetry-breaking projection Z_M' acting freely on a subspace of the internal manifold. A crucial condition is that any generator of the full orbifold group Z_N x Z_M' must either preserve some supersymmetry or act freely in order to become irrelevant in some large volume limit. Tachyons are found to be absent or limited to a given region of the tree-level moduli space. We find several new models with orthogonal supersymmetries preserved at distinct fixed-points. Particular attention is devoted to an interesting Z_3 x Z_3' heterotic example.

Discrete torsion and branes in M-theory from mathematical viewpoint

We study orbifold group actions on locally defined fields upon M-theory branes in a three-form C-fields background. We derive some constraints from the consistency of the orbifold group actions. We show the possibility of the existence of M-theory discrete torsion for the fields on the worldvolume and discuss its features.

Discrete torsion orbifolds and D-branes II

The consistency of the orbifold action on open strings between D-branes in orbifold theories with and without discrete torsion is analysed carefully. For the example of the C^3/Z_2 x Z_2 theory, it is found that the consistency of the orbifold action requires that the D-brane spectrum contains branes that give rise to a conventional representation of the orbifold group as well as branes for which the representation is projective. It is also shown how the results generalise to the orbifolds C^3/Z_N x Z_N for which a number of novel features arise. In particular, the N>2 theories with minimal discrete torsion have non-BPS branes charged under twisted R-R potentials that couple to none of the (known) BPS branes.

Large AdS/CFT correspondence and D1/D5 systems in theories with 16 supercharges

We discuss spectra of $AdS_3$ supergravities, arising in the near horizon geometry of D1/D5 systems in orbifolds/orientifolds of type IIB theory with 16 supercharges. These include models studied in a recent paper (hep-th/0012118), where the group action involves also a shift along a transversal circle, as well as IIB/$\Omega I_4$, which is dual to IIB on $K3$. After appropriate assignements of the orbifold group eigenvalues and degrees to the supergravity single particle spectrum, we compute the supergravity elliptic genus and find agreement, in the expected regime of validity, with the elliptic genus obtained using U-duality map from (4,4) CFTs of U-dual backgrounds. Since this U-duality involves the exchange of KK momentum $P$ and D1 charge $N$, it allows us to test the (4,4) CFTs in the $P < N/4$ and $N < P/4$ regimes by two different supergravity duals.

D-branes, discrete torsion and the McKay correspondence

We analyze the D-branes of a type IIB string theory on an orbifold singularity including the possibility of discrete torsion following the work of Douglas et al. First we prove some general results about the moduli space of a point associated to the "regular representation" of the orbifold group. This includes some analysis of the "wrapped branes" which necessarily appear when the orbifold singularity is not isolated. Next we analyze the stringy homology of the orbifold using the McKay correspondence and the relationship between K-theory and homology. We find that discrete torsion and torsion in this stringy homology are closely-related concepts but that they differ in general. Lastly we question to what extent the D-1 brane may be thought of as being dual to a string.

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher cyclic traces on $K$-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes-Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in our case is that the Connes-Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. The set of possible fractions has been determined, and is compared with recently available experimental data.

Recent developments in discrete torsion

In this Letter we briefly review some recent developments in understanding discrete torsion. Specifically, we give a short overview of the highlights of a group of recent papers which give the basic understanding of discrete torsion. Briefly, those papers observe that discrete torsion can be completely understand simply as the choice of action of the orbifold group on the B field. We summarize the main points of that work.

Discrete torsion orbifolds and D-branes

D-branes on orbifolds with and without discrete torsion are analysed in a unified way using the boundary state formalism. For the example of the Z2 x Z2 orbifold it is found that both the theory with and without discrete torsion possess D-branes whose world-volume carries conventional and projective representations of the orbifold group. The resulting D-brane spectrum is shown to be consistent with T-duality.

D=4,N=1,type IIA orientifolds

We study D=4, N=1, type IIA orientifold with orbifold group $Z_N$ and $Z_N \times Z_M$. We calculate one-loop vacuum amplitudes for Klein bottle, cylinder and Mobius strip and extract the tadpole divergences. We find that the tadpole cancellation conditions thus obtained are satisfied by the $Z_4$, $Z_8$, $Z'_8$, $Z'_{12}$ orientifolds while there is no solution for $Z_3$, $Z_7$, $Z_6$, $Z'_6$, $Z_{12}$. The $Z_4 \times Z_4$ type IIA orientifold is also constructed by introducing four different configurations of 6-branes. We argue about perturbative versus non-perturbative orientifold vacua under T- duality between the type IIA and the type IIB $Z_N$ orientifolds in four dimensions.

TWISTED INDEX THEORY ON GOOD ORBIFOLDS, I: NONCOMMUTATIVE BLOCH THEORY

We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in two and four dimensions, related to generalizations of the Bethe–Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.

Symmetry breaking boundaries I. General theory

We study conformally invariant boundary conditions that break part of the bulk symmetries. A general theory is developed for those boundary conditions for which the preserved subalgebra is the fixed algebra under an abelian orbifold group. We explicitly construct the boundary states and reflection coefficients as well as the annulus amplitudes. Integrality of the annulus coefficients is proven in full generality.

Dn quivers from branes

D-branes can end on orbifold planes if the action of the orbifold group includes (-1)^{F_L}. We consider configurations of D-branes ending on such orbifolds and study the low-energy theory on their worldvolume. We apply our results to gauge theories with eight supercharges in three and four dimensions. We explain how mirror symmetry for N=4 d=3 gauge theories with gauge group Sp(k) and matter in the antisymmetric tensor and fundamental representations follows from S-duality of IIB string theory. We argue that some of these theories have hidden Fayet-Iliopoulos deformations, not visible classically. We also study a class of finite N=2 d=4 theories (so-called D_n quiver theories) and find their exact solution. The integrable model corresponding to the exact solution is a Hitchin system on an orbifold Riemann surface. We also give a simple derivation of the S-duality group of these theories based on their relationship to SO(2n) instantons on R^2\times T^2.

Stable non-BPS bound states of BPS D-branes

S-duality symmetry of type IIB string theory predicts the existence of a stable non-BPS state on an orbifold five plane of the type IIB theory if the orbifold group is generated by the simultaneous action of (-1)^{F_L} and the reversal of sign of the four coordinates transverse to the orbifold plane. We calculate the mass of this state by starting from a pair of D-strings carrying the same charge as this state, and then identifying the point in the moduli space where this pair develops a tachyonic mode, signalling the appearance of a bound state of this configuration into the non-BPS state.

Four-dimensional N = 1 ZN × ZM orientifolds

We calculate the tadpole equations and their solutions for a class of four-dimensional orientifolds with orbifold group Z N × Z M , and we present the massless bosonic spectra of these models. Surprisingly, we find no consistent solutions for the models with Z 2 × Z 4 and Z 4 × Z 4 orbifold groups.

Minimal index torsion-free subgroups of Kleinian groups

A Kleinian group Γ is a discrete subgroup of PSL(2, C), the full group of orientation-preserving isometries of 3-dimensional hyperbolic space. In the language of [T1] Q = H3/Γ is a hyperbolic 3-orbifold; that is a metric 3-orbifold in which all sectional curvatures are -1, and for which Γ is the orbifold fundamental group (see [T1] for further details). A Fuchsian group is a discrete subgroup of PSL(2, R) and as such acts discontinuously on the hyperbolic plane. We define Γ to be of finite co-volume (resp. co-area) if the volume (resp. area) of the quotient orbifold Q is finite. By Selberg’s Lemma, if Γ is a Kleinian (resp. Fuchsian) group of finite covolume (resp. co-area) it contains a torsion-free subgroup of finite index. By definition any torsion-free subgroup cannot contain any finite subgroups of Γ , so the index must be a multiple of the lowest common multiple of all orders of finite subgroups of Γ (see [CFJR] for a proof). Γ being of finite co-volume implies there are only finitely many conjugacy classes of finite subgroups. Thus, we make the following

Matrix theory and heterotic strings on tori

We consider compactifications of the matrix model of M-theory on S 1/ Z 2 × T d for d > 0 , and interpret them as orbifolds of the supersymmetric U( N) Yang-Mills theory on R × T d+1. The orbifold group acts both on the gauge group and on the T d+1, reduces the gauge group to O( N) over 1 + 1-dimensional fixed-point submanifolds, and breaks half of the supersymmetry. We clarify some puzzling aspects of the gauge anomaly cancellation in the presence of spacetime Wilson lines; in general, the Yang-Mills theory requires certain Chern-Simons couplings to supergravity background fields. We discuss the possibility that D8-branes are present as certain matrix configurations in the yang-Mills theory, and the fundamental fermions emerge as zero-modes. Finally, we point out that the correspondence between matrix theory and string theory suggests the existence of a multitude of non-trivial RG fixed points and dualities in orbifold Yang-Mills theories with eight supercharges in various dimensions.

Quasi-isometries preserve the geometric decomposition of Haken manifolds

We prove quasi-isometry invariance of the canonical decomposition for fundamental groups of Haken 3-manifolds with zero Euler characteristic. We show that groups quasi-isometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by-product we describe all 2-dimensional quasi-flats in the universal covers of non-geometric Haken manifolds.