Introduction

This graduate/research level text describes in a unified fashion the statistical mechanics of random walks, random surfaces and random higher dimensional manifolds with an emphasis on the geometrical aspects of the theory and applications to the quantisation of strings, gravity and topological field theory. With chapters on random walks, random surfaces, two- and higher dimensional quantum gravity, topological quantum field theories and Monte Carlo simulations of random geometries, the text provides a self-contained account of quantum geometry from a statistical field theory point of view. The approach uses discrete approximations and develops analytical and numerical tools. Continuum physics is recovered through scaling limits at phase transition points and the relation to conformal quantum field theories coupled to quantum gravity is described. The most important numerical work is covered, but the main aim is to develop mathematically precise results that have wide applications. Many diagrams and references are included.

Monopoles and topological field theory

It is a long-standing question to extend the definition of 3-dimensional Chern-Simons theory to one which associates values to 1-manifolds with boundary and to 0-manifolds. We provide a solution in case the gauge group is a torus. We also develop from different points of view an associated 4-dimensional invertible topological field theory which encodes the anomaly of Chern-Simons. Finite gauge groups are also revisited, and we describe a theory of path integrals as a general construction for a certain class of finite topological field theories. Topological pure gauge theories in lower dimension are presented as a warm-up.

We show that recently proposed generalized Chern-Simons action can be identified with the field theory action of a topological point particle. We find the crucial correspondence which makes it possible to derive the field theory actions from a special version of the generalized Chern-Simons actions. We provide arguments that the general coordinate invariance in the target space and the flat connection condition as a topological field theory can be accommodated in a very natural way. We propose series of new gauge invariant observables.

We show that the fundamental time-reversal invariant (TRI) insulator exists in $4+1$ dimensions, where the effective-field theory is described by the $(4+1)$-dimensional Chern-Simons theory and the topological properties of the electronic structure are classified by the second Chern number. These topological properties are the natural generalizations of the time reversal-breaking quantum Hall insulator in $2+1$ dimensions. The TRI quantum spin Hall insulator in $2+1$ dimensions and the topological insulator in $3+1$ dimensions can be obtained as descendants from the fundamental TRI insulator in $4+1$ dimensions through a dimensional reduction procedure. The effective topological field theory and the ${Z}_{2}$ topological classification for the TRI insulators in $2+1$ and $3+1$ dimensions are naturally obtained from this procedure. All physically measurable topological response functions of the TRI insulators are completely described by the effective topological field theory. Our effective topological field theory predicts a number of measurable phenomena, the most striking of which is the topological magnetoelectric effect, where an electric field generates a topological contribution to the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant $\ensuremath{\alpha}={e}^{2}∕\ensuremath{\hbar}c$. Finally, we present a general classification of all topological insulators in various dimensions and describe them in terms of a unified topological Chern-Simons field theory in phase space.

In a recent article Turaev and Viro [TV] have constructed nontrivial “quantum” invariants of closed compact 3-manifolds M 3 in the form of state sums (called partition functions in statistical physics and vacuum functionals in quantum field theory) associated with the quantized universal enveloping algebra Uq(sl(2, C)) when q is a complex root of unity of a certain degree 2r > 4. The state sum is first defined for a given triangulation X of M and then shown to be independent of the triangulation X of M thus giving rise to a well defined invariant of M, which thus depends on q. This result may be viewed as a rigorous mathematical construction of what is called a topological quantum field theory. In fact, in the language of physicists, a triangulation corresponds to the introduction of a high-energy cut-off. Now topological quantum field theories have trivial dynamics, in other words they only deal with the vacuum sector, are scale invariant and more generally independent of any metrics. Invariance under subdivision is just the statement that the associated renormalization group transformation is trivial. This result suggests that the familiar techniques from algebraic topology should become useful to construct and discuss other topological quantum field theories. Moreover, similar combinatorial techniques could become helpful in discussing particle structures of low dimensional quantum field theories, whose vacuum sector is given in terms of a topological quantum field theory.

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

Distinct quantum vacua of topologically ordered states can be tunneled into\neach other via extended operators. The possible applications include condensed\nmatter and quantum cosmology. We present a straightforward approach to\ncalculate the partition function on various manifolds and ground state\ndegeneracy (GSD), mainly based on continuum/cochain Topological Quantum Field\nTheories (TQFT), in any dimension. This information can be related to the\ncounting of extended operators of bosonic/fermionic TQFT. On the lattice scale,\nanyonic particles/strings live at the ends of line/surface operators. Certain\nsystems in different dimensions are related to each other through dimensional\nreduction schemes, analogous to (de)categorification. Examples include spin\nTQFTs derived from gauging the interacting fermionic symmetry protected\ntopological states (with fermion parity $\\mathbb{Z}_2^f$) of symmetry group\n$\\mathbb{Z}_4\\times \\mathbb{Z}_2$ and $(\\mathbb{Z}_4)^2$ in 3+1D, also\n$\\mathbb{Z}_2$ and $(\\mathbb{Z}_2)^2$ in 2+1D. Gauging the last three cases\nbegets non-Abelian spin TQFT/topological order. We consider situations where a\nTQFT lives on (1) a closed spacetime or (2) a spacetime with boundary, such\nthat the bulk and boundary are fully-gapped and short or long-range entangled\n(SRE/LRE). Anyonic excitations can be deconfined on the boundary. We introduce\nnew exotic topological interfaces on which neither particle nor string\nexcitations alone condensed, but only fuzzy-composite objects of extended\noperators can end (e.g. a string-like composite object formed by a set of\nparticles can end on a special 2+1D boundary of 3+1D bulk). We explore the\nrelations between group extension constructions and partially breaking\nconstructions (e.g. 0-form/higher-form/composite breaking) of topological\nboundaries, after gauging. We comment on the implications of entanglement\nentropy for some of such LRE systems.\n

We discuss connections between certain classes of supersymmetric quiver gauge theories and integrable lattice models from the point of view of topological quantum field theories (TQFTs). The relevant classes include 4d $\mathcal{N} = 1$ theories known as brane box and brane tilling models, 3d $\mathcal{N} = 2$ and 2d $\mathcal{N} = (2,2)$ theories obtained from them by compactification, and 2d $\mathcal{N} = (0,2)$ theories closely related to these theories. We argue that their supersymmetric indices carry structures of TQFTs equipped with line operators, and as a consequence, are equal to the partition functions of lattice models. The integrability of these models follows from the existence of extra dimension in the TQFTs, which emerges after the theories are embedded in M-theory. The Yang-Baxter equation expresses the invariance of supersymmetric indices under Seiberg duality and its lower-dimensional analogs.

In this short paper, we argue that the chiral central charge c− of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry. For example, if a (2+1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T2=(−1)F labeled as ν∈Z16, the TQFT must have c−=1/4 mod 1/2 for odd ν, while c−=0 mod 1/2 for even ν. This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c−=4 mod 8 to fermionic cases. We also study such a constraint for fermionic TQFT with U(1)×CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.Received 5 January 2021Revised 16 March 2021Accepted 12 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023107Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSymmetry protected topological statesTopological field theoriesTopological superconductorsCondensed Matter, Materials & Applied Physics

We examine Atiyah’s Hermitian axiom for two-dimensional complex topological quantum field theories. Building on the correspondence between 2D topological quantum field theories (TQFTs) and Frobenius algebras, we find the algebraic objects corresponding to Hermitian and unitary TQFTs, respectively, and prove structure theorems about them. We then clarify a few older results on unitary TQFTs using our structure theorems.

We show that the four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c$. This result is obtained by taking into account topological non-trivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of a topological quantum field theory and subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear sigma model. The topological quantum field theory part of the four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase of U(1) gauge theory corresponds to the high (resp. low) temperature phase of O(2) nonlinear sigma model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by the vortex condensation. Thus the continuum gauge theory has the direct correspondence to the compact formulation of lattice gauge theory.

Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with old results (first examples of topological quantum field theories were constructed in my papers in late seventies) and I come to some new results, that were not published yet.

We have seen that the equivariant localization formalism is an excellent, conceptual geometric arena for studies of supersymmetric and topological field theories, and more generally of (quantum) integrability. Given that the Hamiltonians in an integrable hierarchy are functionals of action variables alone [106], the equivariant localization formalism might yield a geometric characterization of quantum integrability, and perhaps some deeper connection between quantum-integrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory.KeywordsLoop SpaceEquivariant CohomologySupersymmetric Quantum MechanicLocalization FormulaEquivariant LocalizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The ground states encode the information of the topological phases of a 2-dimensional system, which makes them crucial in determining the associated topological quantum field theory (TQFT). Most numerical methods for detecting the TQFT relied on the use of minimum entanglement states (MESs), extracting the anyon mutual statistics and self statistics via overlaps and/or the entanglement spectra. The MESs are the eigenstates of the Wilson loop operators, and are labeled by the anyons corresponding to their eigenvalues. Here we revisit the definition of the Wilson loop operators and MESs. We derive the modular transformation of the ground states purely from the Wilson loop algebra, and as a result, the modular $S$ and $T$ matrices naturally show up in the overlap of MESs. Importantly, we show that due to the phase degree of freedom of the Wilson loop operators, the MES-anyon assignment is not unique. This ambiguity obstructs our attempt to detect the topological order, that is, there exist different TQFTs that cannot be distinguished solely by the overlap of MESs. In this paper, we provide the upper limit of the information one may obtain from the overlap of MESs without other additional structure. Finally, we show that if the phase is enriched by rotational symmetry, there may be additional TQFT information that can be extracted from overlap of MESs.