Notes on Topological Quantum Field Theories

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 derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.

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.

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.

In this book we shall review the systematic approaches and applications of a theory which investigates situations where Feynman path integrals of physical systems can be evaluated exactly leading to a complete understanding of the quantum physics. These mathematical formalisms are in large part motivated by the symmetries present in integrable systems and topological quantum field theories which make these latter examples exactly solvable problems. Besides providing conceptual understandings of the solvability features of these special classes of problems, this framework yields geometric approaches to evaluating the quantum spectrum of generic quantum mechanical and quantum field theoretical partition functions. The techniques that we shall present here in fact motivate an approach to studying generic physical problems by relating their properties to those of integrable and topological field theories. In doing so, we shall therefore also review some of the more modern quantum field theoretical and mathematical ideas which have been at the forefront of theoretical physics over the past two decades.KeywordsPartition FunctionPath IntegralEquivariant CohomologyCoadjoint OrbitSupersymmetric Quantum MechanicThese 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.

Monopoles and topological field theory

We survey results on Frobenius algebras and illustrate their importance to the structure of some generalizations of the notion of Hopf algebra, as well as their connections to topics like monoidal categories, 2-categories, functors, topological quantum field theories, etc.KeywordsFrobenius algebraHopf algebraFrobenius functorTQFT

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.

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.

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.

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

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

In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure (FdHilb). We also provide a detailed discussion of the category which has sets as objects, relations as morphisms, and the cartesian product as its monoidal structure (Rel), and thirdly, categories with manifolds as objects and cobordisms between these as morphisms (2Cob). While sets, Hilbert spaces and manifolds do not share any non-trivial common structure, these three categories are in fact structurally very similar. Shared features are diagrammatic calculus, compact closed structure and particular kinds of internal comonoids which play an important role in each of them. The categories FdHilb and Rel moreover admit a categorical matrix calculus. Together these features guide us towards topological quantum field theories. We also discuss posetal categories, how group representations are in fact categorical constructs, and what strictification and coherence of monoidal categories is all about. In our attempt to complement the existing literature we omitted some very basic topics. For these we refer the reader to other available sources.

Symmetries and their anomalies give strong constraints on renormalization group (RG) flows of quantum field theories. Recently, the identification of a theory’s global symmetries with its topological sector has provided additional constraints on RG flows to symmetry preserving gapped phases due to mathematical results in category and topological quantum field theory. In this paper, we derive constraints on RG flows from ℤ2-valued pure- and mixed-gravitational anomalies that can only be activated on non-spin manifolds. We show that such anomalies cannot be matched by a unitary, symmetry preserving gapped phase without symmetry fractionalization. In particular, we discuss examples that commonly arise in 4d gauge theories with fermions.