A modular tensor category (also called a modular fusion category) is a type of tensor category that plays a role in the areas of topological quantum field theory, conformal field theory, and quantum algebra. Modular tensor categories were introduced in 1989 by the physicists Greg Moore and Nathan Seiberg in the context of rational conformal field theory. In the context of quantum field theory, modular tensor categories are used to store algebraic data for rational conformal field theories in 2 dimensional spacetime, and topological quantum field theories in 3 dimensional spacetime. In the context of condensed matter physics, modular tensor categories play a role in the algebraic theory of topological quantum information, as they are used to store the algebraic data describing anyons in topological quantum phases of matter.
Mathematically, a modular tensor category is a rigid, semisimple, braided fusion category with a non-degenerate braiding, ensuring a well-defined notion of topological invariance. These categories naturally arise in quantum groups, representation theory, and low-dimensional topology, where they are used to construct knot and three-manifold invariants.
Description.
The term 'modular tensor category' was coined by Igor Frenkel in 1989. The interpretation in terms of category theory was introduced by Vladimir Turaev in 1992. His definition, however, is slightly more general than the modern definition, in the sense that it does not require the category to have every object as a direct sum of finitely many simple objects. The word 'modular' refers to the fact that every modular tensor category has an associated modular group representation. The word 'tensor' refers to the fact that modular tensor categories were originally not defined as abstract categories, but were instead defined in terms of a compatible collection of tensors.
There are several equivalent alternative ways of defining modular tensor categories. One definition is as follows: a modular tensor category is a braided spherical fusion category with non-degenerate braiding. In the presence of a braiding, Deligne's twisting lemma states that a spherical structure is equivalent to a ribbon structure, so modular tensor categories can be equivalently defined as non-degenerate ribbon fusion categories. The Bruguières modularity theorem asserts that a braided spherical fusion category has non-degenerate braiding if and only if its S-matrix is non-degenerate (invertible). Thus, a modular tensor category can be equivalently defined as a braided spherical fusion category with non-degenerate S-matrix. Modular tensor categories can also be defined using skeletonization.
There are several theorems about modular tensor categories, such as the existence of the modular group representation, the Bruguières modularity theorem, the Verlinde formula, the rank-finiteness theorem, the Schauenburg-Ng theorem, and Müger's theorem.
Definition.
A modular tensor category formula_1 consists of the following pieces of data:
To form a modular tensor category, the pieces of data are required to satisfy the following axioms:
formula_23
5. (Non-degeneracy) Let formula_24 denote the braiding on formula_1. For all objects formula_19, if formula_27 for every formula_28, then there exists some natural number formula_29 such that formula_30.
These axioms are motivated physically as follows:
Relationship to other notions.
There are various intermediate notions which can be defined using only a subset of the structures and axioms of a modular tensor category.
Relationship to topological quantum field theory.
The relationship between modular tensor categories and topological quantum field theory is codified in the Reshetikhin–Turaev construction, which was introduced in 1991 by Vladimir Turaev and Nicolai Reshetikhin. This construction was introduced to serve as a mathematical realization of Edward Witten's proposal of defining invariants of links and 3-manifolds using quantum field theory. The Reshetikhin-Turaev construction assigns to every modular tensor category a (2+1)-dimensional topological quantum field theory. In one interpretation of the theory, the Reshetikhin-Turaev construction induces a bijection between once-extended anomalous (2+1)-dimensional topological quantum field theories valued in the 2-category of formula_2-linear categories, and modular multi-tensor categories equipped with a square root of the global dimension in each factor. Here, a modular multi-tensor category refers to a modular tensor category with the possibility that formula_40.
Relationship to rational conformal field theory.
The relationship between modular tensor categories and rational conformal field theory was introduced by Greg Moore and Nathan Seiberg. After a series of papers studying the algebraic relations between the basic chiral pieces of data in rational conformal field theories (primary fields), Moore and Seiberg discovered that the structure into which these pieces of data naturally assemble is a modular tensor category. This data is now referred to as the Moore-Seiberg data of a rational conformal field theory. This data is not entirely enough to specify a conformal field theory; in particular, some non-chiral data is needed to arrive at a full theory with local correlation functions. This additional necessary data was studied by Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, corresponds to the data of a symmetric special Frobenius algebra object in the Moore-Seiberg modular tensor category.
The connection between rational conformal field theory and modular tensor categories can also be understood in the language of vertex operator algebras. There is a well-established theory that associates to every conformal field theory a vertex operator algebra. When this vertex operator algebra is rational and satisfies certain algebraic conditions, its category of representations is naturally equipped with the structure of a modular tensor category.
Constructions of modular tensor categories.
There are various constructions of modular tensor categories from across the mathematical and physical literature.
From finite groups.
One construction comes from finite group theory. This construction assigns to every finite group formula_41 a modular tensor category formula_42 referred to as the quantum double of formula_41. This category is defined as the Drinfeld center of the category of (complex) representations of formula_41. That is, formula_45. Alternatively, formula_42 can be defined as the Drinfeld center of the category of formula_41-graded (complex) vector spaces. That is, formula_48. It is a non-trivial fact that these two definitions are equivalent, which is referred to as a categorical Morita equivalence between formula_49 and formula_50. In this context, two monoidal categories are called Morita equivalent if there is an equivalence of braided monoidal categories between their Drinfeld centers.
There is a more general construction that comes from twisting the associativity relation by a 3-cocycle in group cohomology formula_51, where formula_52 is the circle group. More precisely, given any 3-cochain formula_53 there is an associated spherical fusion category formula_54 which is defined identically to the category of formula_41-graded vector spaces formula_50 except that its associativity relation is twisted by formula_57. Cochains which differ by a coboundary yield equivalent spherical fusion categories, so the spherical fusion category formula_54 is well-defined up to equivalence on cohomology classes in formula_51. Taking the Drinfeld center formula_60 results in a modular tensor category which is determined by a finite group formula_41 and a cohomology class formula_62.
On the level of topological quantum field theory, the group-theoretical modular tensor category formula_42 correspond to discrete gauge theory with finite gauge group formula_41, also called Dijkgraaf-Witten theory, named after Robbert Dijkgraaf and Edward Witten. The 3-cocycle formula_62 corresponds to a choice of Dijkgraaf-Witten action in the Lagrangian. On the level topological order, formula_42 corresponds to the anyons in Kitaev's quantum double model with input group formula_41.
From quantum groups.
Associated to every compact, simple, simply-connected Lie group formula_41 with associated Lie algebra formula_69 and every positive integer formula_70, there is an associated quantum group formula_71 where formula_72 is a certain root of unity associated to formula_73 via the formula formula_74 where formula_75 is the dual Coxeter number of formula_69 and formula_77 is the biggest absolute value of an off-diagonal entry of the Cartan matrix of formula_69. From this quantum group it is possible to define a category called the formula_79, which is defined by performing a certain semi-simplification procedure on the category of representations of formula_71. For choices of formula_69, formula_73 not lying is certain exceptional families, the category formula_79 is modular and is called the quantum group modular category of formula_69 at level formula_73.
On the level of topological quantum field theory, quantum group modular categories correspond to Chern–Simons theory. Chern-Simons theories are specified by a compact simple Lie group formula_41, which corresponds to the gauge group of the theory, and an integer level formula_70 which specifies a coupling constant in the Chern-Simons action. The modular tensor category corresponding to the formula_88 Chern-Simons theory under the Reshetikhin-Turaev construction is formula_79. It was on the physical grounds of Chern-Simons theory that Edward Witten theorized that every compact, simple Lie group and integer level should be associated to invariants of links and 3-manifolds, and it is using the Reshetikhin-Turaev construction associated to formula_79 that Witten's program was completed.
From weak Hopf algebras.
There is a construction of modular tensor categories coming from the theory of weak Hopf algebras. These constructions play on the general theme of Tannaka–Krein duality. It can be shown that the representation category of every finite-dimensional Weak Hopf algebra is a formula_2-linear monoidal category, which is equivalent as a formula_2-linear category to formula_93. It is a theorem of Takahiro Hayashi that the converse is also true - every formula_2-linear monoidal category, which is equivalent as a formula_2-linear category to formula_93 is equivalent to the representation category of some weak Hopf algebra. Adding more structures onto the weak Hopf algebras corresponds to adding more structures on the representation category. For instance, adding a quasitriangular structure to the weak Hopf algebra corresponds to adding a braiding on the representation category. In their original work, Reshetikhin-Turaev introduced the notion of a modular Hopf algebra, which has sufficiently many structures and axioms so that its representation category will be a modular category. In the context of Hopf algebras, it is common to work with the quantum double construction which is defined by taking in an input weak Hopf algebra formula_97 and outputting the doubled Hopf algebra formula_98 which can naturally be equipped with a quasi-triangular structure, and whose representation category will often be a modular tensor category. These sorts of modular Hopf algebras are called 'doubled'. On the level of topological order, the representation categories doubled Hopf algebras correspond to anyons in the generalized Kitaev quantum double model.
From subfactors.
There are relationships between modular tensor categories and subfactors introduced and developed throughout the late 1990s and early 2000s by Adrian Ocneanu, Michael Müger, and other authors. These constructions typically work by first constructing a spherical fusion category and then taking its Drinfeld center, which is modular by Müger's theorem. There are various relevant constructions, depending on the type of the subfactor and the axioms it is required to satisfy. For example, in the case of a type formula_99subfactor formula_100 with finite index and finite depth, the associated spherical fusion category is defined by taking by considering the sub-category of formula_101-formula_101 bimodules generated by formula_103, viewed as an formula_101-formula_101 bimodule. In the case of separable type formula_106factors formula_107, there is an associated spherical fusion category formula_108 whose objects are formula_109-automorphisms of formula_107 and whose morphisms are intertwining maps. Any finite-index subfactor formula_100 naturally gives rise to the structure of a Frobenius algebra in formula_108, and in fact there is a bijection between finite-index subfactors of formula_107 and Frobenius algebras in formula_108.
Using the Reshetikhin-Turaev construction, all of these constructions of modular tensor categories can assigned topological quantum field theories. In the case of type formula_99subfactors formula_100 with finite index and finite depth, there is an alternative approach due to Ocneanu which directly constructs the relevant field theory.