Space-time geometry of topological phases

The 2 + 1 dimensional lattice models of Levin and Wen (2005) [1] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the Hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so-called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.     

Duality and orbifolds

We construct several examples where duality transformation commutes with the orbifolding procedure even when the orbifolding group does not act freely, and there are massless states from the twisted sector at a generic point in the moduli space. Often the matching of spectrum in the dual theories is a result of nontrivial identities satisfied by the coefficients on one loop tadpoles in the heterotic, type II and type I string theories.     

Permutation orbifolds

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Λ-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S₃ is described to illustrate the general theory.     

Degenerate orbifolds

We present evidence for the existence of new four-dimensional string theories, obtained from a smooth variation of background fields in the twisted sectors of symmetric and asymmetric orbifolds. Flat directions only in the untwisted sector are shown to reproduce previously constructed models in terms of Wilson lines, exhibiting a Three-Higgs-Rule (THR). The new models provide a mechanism to lower the rank of the gauge group, lead to more flexible Yukawa couplings and give a strict separation of hidden and observable sectors, which are usually mixed in (2, 0)-models. Even though Fayet-Iliopoulos terms are induced in some of the models due to the presence of anomalous U(1)'s supersymmetry remains, in general, unbroken. Particular examples of the new models correspond to “blown up” versions of (2, 0)-orbifolds.     

Sphalerons on orbifolds

In this work, we study the electroweak sphalerons in a 5D background, where the fifth dimension lies on an interval. We consider two specific cases: flat space-time and the anti-de Sitter space-time compactified on S ¹/Z ₂. In our work, we take the SU(2) gauge–Higgs model, where the gauge fields reside in the 5D bulk; but the Higgs doublet is confined in one brane. We find that the results in this model are close to those of the 4D Standard Model (SM). The existence of the warp effect, as well as the heaviness of the gauge Kaluza–Klein modes make the results extremely close to the SM ones.     

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.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.     

Global anomalies on orbifolds

Communicated by A. Jaffe     

Constraints on asymmetric orbifolds

It is claimed that besides the left-right matching condition, one has to check the relative phases dictated by modular invariance for an asymmetric orbifold to give rise to an acceptable model.     

Cosmic strings as orbifolds

We construct string theory versions of cosmic strings by considering orbifold compactifications of spacetime down to two dimensions.     

Nonabelian orbifolds and the boson-fermion correspondence

For a positive integerl divisible by 8 there is a (bosonic) holomorphic vertex operator algebra (VOA) associated to the spin lattice _l . For a broad class of finite groupsG of automorphisms of we prove the existence and uniqueness of irreducibleg-twisted -modules and establish the modular-invariance of the partition functionsZ(g, h, ) for commuting elements inG. In particular, for any finite group there are infinitely many holomorphic VOAs admittingG for which these properties hold. The proof is facilitated by a boson-fermion correspondence which gives a VOA isomorphism between and a certain fermionic construction, and which extends work of Frenkel and others.Supported by NSA grant MDA904-92-H-3099.Supported by NSF grant DMS-9122030.     

Open strings on orbifolds

We consider propagation of type I SO(32) superstrings on orbifolds. It is shown that anomaly cancellation requires the existence of “twisted” open strings and we determine the Chan-Paton factors for these strings in a simple example.     

Models of non-abelian orbifolds

Several models of non-abelian orbifolds have been constructed. There are models with three or four families of quarks and leptons, and gauge symmetry SU(3) × SU(2) × SU(2) × U(1)² × SU(3)′ × SO(10)′ × U(1)′ or SU(3) × SU(2) × U(1)³ × SU(4)′ × SO(8)′ × U(1)′.     

Global anomalies on orbifolds

We consider global anomalies for heterotic string theory formulated on orbifolds. The vanishing of certain characteristic classes in group cohomology provides sufficient conditions for the absence of global anomalies. For abelian orbifolds level matching implies these cohomology conditions, so suffices for the absence of anomalies. For nonabelian orbifolds level matching does not suffice, and there are additional constraints. We give some examples to illustrate these new constraints.The first author is partially supported by an NSF Postdoctoral Research Fellowship. The second author is supported in part by the NSF contract no. PHY 82-15249, and in part by a fellowship from the Harvard Society of Fellows     

Monstrous Moonshine from orbifolds

We consider the Monster Module of Frenkel, Lepowsky, and Meurman as aZ ₂ orbifold of a bosonic string compactified by the Leech lattice. We show that the main Conway and Norton Monstrous Moonshine properties, stating that the Thompson series for each Monster group conjugacy class has a modular invariance group of genus zero, follow from an orbifold construction based on an orbifold group composed of Monster group elements. it is shown that a conjectured vacuum structure for the orbifold twisted sectors is sufficient to specify the modular group and the genus zero property for each Thompson series. It is also shown that the Power Map formula of Conway and Norton follows from the same vacuum structure. Finally, we demonstrate the validity of the vacuum conjectures for sectors twisted by Leech lattice automorphisms in many cases.