Dimensional Topological Quantum Field Theory Research Articles

Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory

Symmetry acting on a (2+1)DD topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1)DD on-site symmetry. In this paper, we develop a (3+1)DD topological quantum field theory to calculate the anomaly indicators of a (2+1)DD topological order with a general symmetry group GG, which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1)DD topological quantum field theory on a specific manifold equipped with some GG-bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including \mathbb{Z}_2\times\mathbb{Z}_2ℤ2×ℤ2, \mathbb{Z}_2^T\times\mathbb{Z}_2^Tℤ2T×ℤ2T, SO(N)SO(N), O(N)^TO(N)T, SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T, etc, where \mathbb{Z}_2ℤ2 and \mathbb{Z}_2^Tℤ2T denote a unitary and anti-unitary order-2 group, respectively, and O(N)^TO(N)T denotes a symmetry group O(N)O(N) such that elements in O(N)O(N) with determinant -1−1 are anti-unitary. In particular, we demonstrate that some anomaly of O(N)^TO(N)T and SO(N)\times \mathbb{Z}_2^TSO(N)×ℤ2T exhibit symmetry-enforced gaplessness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for SO(N)SO(N) symmetric topological orders, we derive their SO(N)SO(N) Hall conductance.

Intrinsic sign problems in topological quantum field theories

The sign problem is a widespread numerical hurdle preventing us from simulating the equilibrium behavior of various problems at the forefront of physics. Focusing on an important sub-class of such problems, bosonic $(2+1)$-dimensional topological quantum field theories, here we provide a simple criterion to diagnose intrinsic sign problems---that is, sign problems that are inherent to that phase of matter and cannot be removed by any local unitary transformation. Explicitly, \textit{if the exchange statistics of the anyonic excitations do not form complete sets of roots of unity, then the model has an intrinsic sign problem}. This establishes a concrete connection between the statistics of anyons, contained in the modular $S$ and $T$ matrices, and the presence of a sign problem in a microscopic Hamiltonian. Furthermore, it places constraints on the phases that can be realised by stoquastic Hamiltonians. We prove this and a more restrictive criterion for the large set of gapped bosonic models described by an abelian topological quantum field theory at low-energy, and offer evidence that it applies more generally with analogous results for non-abelian and chiral theories.

Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory.

We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of (2+1)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

Four dimensional topological quantum field theories from $G$-crossed braided categories

We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy $2$-types. If the $G$-BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in $H^4(G,U(1))$ can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of $(3+1)$-TQFTs is a spherical fusion $2$-category. We show that a $G$-BSFC corresponds to a monoidal $2$-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion $2$-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion $2$-category is open.

Orbifolds of n–dimensional defect TQFTs

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincar\'e duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.

State sum invariants of three manifolds from spherical multi-fusion categories

We define a family of quantum invariants of closed oriented [Formula: see text]-manifolds using spherical multi-fusion categories (SMFCs). The state sum nature of this invariant leads directly to [Formula: see text]-dimensional topological quantum field theories ([Formula: see text]s), which generalize the Turaev–Viro–Barrett–Westbury ([Formula: see text]) [Formula: see text]s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the [Formula: see text] approach in that here the labels live not only on [Formula: see text]-simplices but also on [Formula: see text]-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus, we introduce the concept of a SMFC by imposing a weakened version of sphericity. Besides containing the [Formula: see text] theory, our construction also includes the recent higher gauge theory [Formula: see text]-[Formula: see text]s given by Kapustin and Thorngren, which was not known to have a categorical origin before.

Derivation of the Time-Reversal Anomaly for (2+1)-Dimensional Topological Phases.

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2+1)-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T^{2} eigenvalues, generalizing the recent analysis by Barkeshli etal. in the bosonic case.

Congruence subgroups from representations of the three-strand braid group

Ng and Schauenburg proved that the kernel of a (2+1)-dimensional topological quantum field theory representation of SL(2,Z) is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group B3 with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator σ1 is between 2 and 5 the kernel projects onto a congruence subgroup of PSL(2,Z) and compute its level. However, we prove that for three dimensional representations, the projective order is not enough to decide the congruence property. For each integer of the form 2ℓ≥6 with ℓ odd, we construct a pair of non-congruence subgroups associated with three-dimensional representations having finite image and σ1 mapping to a matrix with projective order 2ℓ. Our technique uses classification results of low dimensional braid group representations, and the Fricke–Wohlfahrt theorem in number theory.

From 3D topological quantum field theories to 4D models with defects

(2 + 1) dimensional topological quantum field theories (TQFTs) with defect excitations are by now quite well understood, while many questions are still open for (3 + 1) dimensional TQFTs. Here we propose a strategy to lift states and operators of a (2 + 1) dimensional TQFT to states and operators of a (3 + 1) dimensional theory with defects. The main technical tool is Heegaard splittings, which allow us to encode the topology of a three-dimensional manifold with line defects into a two-dimensional Heegaard surface. We apply this idea to the example of BF theory which describes locally flat connections. This shows in particular how the curvature excitation generating surface operators of the (3 + 1) dimensional theory can be obtained from closed ribbon operators of the (2 + 1) dimensional BF theory. We hope that this technique allows the construction and study of more general models based on unitary fusion categories.

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects.This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably.We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

A short survey of quantum computing

Quantum computing is closely related to [Formula: see text]-dimensional topological quantum field theory through its approach in topological quantum computation. The following are notes based on lectures given by the author at the Mathematics Department of George Washington University. The idea is to give a brief introduction and survey of a few basic facts of quantum computing to an audience of non-experts interested in taking a quick look at some of the most important concepts of quantum computing. The author is a topologist with strong interest in the general ideas of quantum computing. The notes touch on how quantum topology enters the field but emphasizes the general ideas of quantum computing in the first place.

Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level $m$, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for $n$-characters on $\infty$-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for $n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.

(3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions

Currently, there is much interest in discovering analytically tractable (3+1)-dimensional models that describe interacting fermions with emerging topological properties. Towards that end we present a three-dimensional tight-binding model of spinless interacting fermions that reproduces, in the low energy limit, a (3+1)-dimensional Abelian topological quantum field theory called BF model. By employing a mechanism equivalent to the Haldane's Chern insulator, we can turn the non-interacting model into a three-dimensional chiral topological insulator. We then isolate energetically one of the two Fermi points of the lattice model. In the presence of suitable fermionic interactions, the system, in the continuum limit, is equivalent to a generalised (3+1)-dimensional Thirring model. The low energy limit of this model is faithfully described by the BF theory. Our approach directly establishes the presence of (2+1)-dimensional BF theory at the boundary of the lattice and it provides a way to detect the topological order of the model through fermionic density measurements.

Self-dual and quasi self-dual algebras

Self-dual algebras are ones with an A bimodule isomorphism A → A ∨op, where A ∨ = Hom k (A, k) and A ∨op is the same underlying k-module as A ∨ but with left and right operations by A interchanged. These are in particular quasi self-dual algebras, i.e., ones with an isomorphism H*(A,A) ≌ H*(A,A ∨ op). For all such algebras H*(A,A) is a contravariant functor of A. Finite-dimensional associative self-dual algebras over a field are identical with symmetric Frobenius algebras; an example of deformation of one is given. (The monoidal category of commutative Frobenius algebras is known to be equivalent to that of 1+1 dimensional topological quantum field theories.) All finite poset algebras are quasi self-dual.

G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.

Chord diagrams, contact-topological quantum field theory and contact categories

We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Mati\'{c} in \cite{HKM08}. The $\Z_2$ $SFH$ of these solid tori forms a "categorification of Pascal's triangle", and contact structures correspond bijectively to chord diagrams, or sets of disjoint properly embedded arcs in the disc. Their contact elements are distinct and form distinguished subsets of $SFH$ of order given by the Narayana numbers. We find natural "creation and annihilation operators" which allow us to define a QFT-type basis of $SFH$, consisting of contact elements. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams. We prove that contact elements are in bijective correspondence with comparable pairs of basis elements with respect to a certain partial order. The algebraic and combinatorial structures in this description have intrinsic contact-topological meaning. In particular, the QFT-basis of $SFH$ and its partial order have a natural interpretation in pure contact topology, related to the contact category of a disc: the partial order enables us to tell when the sutured solid cylinder obtained by "stacking" two chord diagrams has a tight contact structure. This leads us to extend Honda's notion of contact category to a "bounded" contact category, containing chord diagrams and contact structures which occur within a given contact solid cylinder. We compute this bounded contact category in certain cases. Moreover, the decomposition of a contact element into basis elements naturally gives a triple of contact structures on solid cylinders which we regard as a type of "distinguished triangle" in the contact category. We also use the algebraic structures arising among contact elements to extend the notion of contact category to a 2-category.

Unoriented topological quantum field theory and link homology

We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.

Curves in Calabi-Yau threefolds and topological quantum field theory

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds. We define relative invariants for local theory which give rise to a (1+1)-dimensional topological quantum field theory (TQFT) taking values in the ring $\mathbb{Q}[[t]]$. The associated Frobenius algebra over $\mathbb{Q}[[t]]$ is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1.

Formula omitted]-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants

In this paper, we establish the general theory of ( 2 + 1 ) -dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev–Viro–Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev–Viro–Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin–Turaev invariant constructed from a tube system, which is a modular category corresponding to the quantum double construction of a C * -tensor category. In Sato and Wakui (J. Knot Theory Ramif. 12 (2003) 543), we will exhibit computations of Turaev–Viro–Ocneanu invariants for several “basic 3-manifolds”. In the appendix, we discuss the relationship between the system of M ∞ - M ∞ bimodules arising from the asymptotic inclusion M ∨ M op ⊂ M ∞ constructed from N ⊂ M and the tube system obtained from a subfactor N ⊂ M .

2+1-D topological quantum field theory and 2-D conformal field theory

We describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory. Some Applications to quantum knot invariants leading to the equivalence of Chern-Simons-Witten and Kohno's approaches are outlined.