Jorgen Ellegaard Andersen, Kenji Ueno
We prove in this paper that the genus zero data of a modular functor determines the modular functor. We do this by establishing that the S-matrix in genus one with one point labeled arbitrarily can be expressed in terms of the genus zero information and we give an explicit formula. We do not assume the modular functor in question has duality or is unitary, in order to establish this.
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We describe the construction of the genus-zero parts of conformal field theories in the sense of G. Segal from representations of vertex operator algebras satisfying certain conditions. The construction is divided into four steps and each step gives a mathematical structure of independent interest. These mathematical structures are intertwining operator algebras, genus-zero modular functors, genus-zero holomorphic weakly conformal field theories, and genus-zero conformal field theories.
In this paper we follow the constructions of Turaev's book [Tu] closely, but with small modifications, to construct of a modular functor, in the sense of Kevin Walker, from any modular tensor category. We further show that this modular functor has duality and if the modular tensor category category is Hermitian or unitary, then the resulting modular functor is also Hermitian or unitary respectively.
We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the tube category are indexed by pairs of simple objects in the underlying MTC, the simple multiplicities of this representation form a candidate modular invariant matrix. In general, this matrix will not be modular invariant, however it will always commute with the T-matrix. Furthermore, under certain additional conditions on the original functor, it is shown …
The Lyubashenko Modular Functor for Drinfeld Centers via Non-Semisimple String-Nets
本文证明了由枢轴有限张量范畴的投射对象构建的有限余完备弦网模函子与由Drinfeld中心构建的Lyubashenko模函子等价，推广了Levin-Wen弦网的结果到非半单情形。
In this paper, we extend the notion of modular functor and fusion category to what we called $G$ equivariant modular functor and $G$ equivariant fusion category, where $G$ is a finite group, and establish a correspondence between between these notions.
For any finite group G with a finite G-set X and a modular tensor category C we construct a part of the algebraic structure of an associated G-equivariant monoidal category: For any group element g in G we exhibit the module category structure of the g-component over the trivial component. This uses the formalism of permutation equivariant modular functors that was worked out in arXiv:1004.1825. As an application we show that the corresponding modular invariant partition function is given by permutation by g.
We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.
It is shown that the quantized Teichm"uller spaces have factorization properties like those required in the definition of a modular functor.
Modular functors from non-semisimple 3d TFTs
本文利用非半单模张量范畴对应的3d TFT，将模函子显式构造为对称幺半2-函子，并恢复Lyubashenko的结果，通过拉回消除粘合异常，得到Drinfeld中心的模函子。
Modular functors from conformal blocks of rational vertex operator algebras
本文证明强有理顶点算子代数的共形块空间构成模函子，进而导出模融合范畴结构，并延拓为三维拓扑场论。
The article is devoted to the $q_R$-conformal modular functors, which being ``deformations'' of the conformal modular functor (the projective representation of the category $Train(Diff_+(S^1))$, the train of the group $Diff_+(S^1)$ of all orientation preserving diffeomorphisms of a circle) in the class of all projective modular functors (the projective representations of the category $Train(PSL(2,R))$, the train of the projective group $PSL(2,R)$), may be regarded as its ``Berezin quantizations''.
This is the second paper in a series of papers aimed at providing a geometric construction of modular functors and topological quantum field theories from conformal field theory building on the constructions in [TUY] and [KNTY]. We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [TUY] by a certain fractional power of the abelian theory first considered in [KNTY] and further studied in our first paper [AU1].
In this note, we give a description of the modular functor associated to the Chern-Simons theory with a finite group from the complex-analytic point of view, i.e. as a vector bundle with a flat connection on the moduli space of punctured curves. We show that it can be obtained from the trivial local system on the moduli space of "admissible G-covers" as a direct image under the forgetful map from moduli space of G-covers to the usual moduli space.
We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D-module on the moduli space of rank 2 vector bundles with parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2),sl(2)" in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl(2),sl(2)"
Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper arXiv:0812.0986 [math.CT].
This paper has been withdrawn by the author(s). The material contained in the paper will be published in a subtantially reorganized form, part of it is now included in math.QA/0510174
In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras were solved implicitly earlier in [H3] (q-alg/9505019). In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [H1], [H2], [HL1] (hep-th/9301009), [HL2] and [H6] to the notion of intertwining operator…
Construction of Hodge structures on the $\mathrm{SO}(3)$ modular functors
本文证明亏格0的SO(3)模函子具有几何起源，并基于同调模型构造其上的整值霍奇结构变体，同时用Künneth公式几何地恢复粘合性质。
We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) the weight of any nonzero homogeneous elements of V is nonnegative and the weight zero subspace is one-dimensional, (ii) every N-gradable weak V-module is completely reducible and (iii) V is C_2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is complete…
We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule …
A Classification of Modular Functors via Factorization Homology
本文通过因子化同调给出模函子的分类，证明其等价于满足连通性条件的自对偶平衡辫子代数，简化了验证条件，重现相关构造并给出新例子与应用。
In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox-Jurisich's presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into simple modules over the three-point affine algebras of genus zero. As a corollary, vertex representations are constructed for the three-point affine algebra of genus zero using vertex operators. Moreover, we construct a Fock module for certain quotient of three-point Virasoro algebra of genus zero.
Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories with additional structure (Turaev and Yetter), and modular functors (Walker and Kontsevich). These constructions are very closely related. We take a unifying categorical approach based on a natural 2-dimensional generalization of a topological field theory in the sense of Atiyah and Segal, and show that the axioms defining these complex algebraic structures a…
Modular functors, cohomological field theories and topological recursion
本文从 Walker 拓扑模函子构造向量丛，证明其 Chern 类与 ψ-类的交可由拓扑递归计算，并由此重新得到 Verlinde 公式，涵盖有限群主丛计数与 WZW 共形场论的 Verlinde 丛两个例子。
A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore--Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given an (eventually non-semisimple) k-linear abelian ribbon braided category C with some finiteness conditions we construct a functor from a central extension of N with the set of labels ObC to k-vector spaces. Composing the functors we get a modular functor from a ce…
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 theorie…
Open modular functors from non-finite tensor categories
本文提出了一种基于非有限张量范畴的映射群表示，展示了其与三维全息原理的关联，并证明了其在共形块研究中的延续性。
In [AU2] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [TUY] and the abelian analog in [AU1]. We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT constructed first by Witten-Reshetikhin-Turaev from the quantum group associated to SL(N).
We construct the genus two (or two loop) partition function for meromorphic bosonic conformal field theories. We use a sewing procedure involving two genus one tori by exploiting an explicit relationship between the genus two period matrix and pinching modular parameters. We obtain expressions for the partition function for the chiral bosonic string, even rank lattice theories and self-dual meromorphic conformal field theories including the Moonshine Module. In particular, we find that for self-dual theories with central charge 24, the genus two partition function multiplied by a universal hol…
The algebraic modular functor conjecture in type $A_n$ quantum Teichmüller theory
该研究针对Fock与Goncharov提出的高阶泰希米勒理论量子化的代数模函子猜想，通过拓展Fock-Goncharov框架，证明了$G = \mathrm{PGL}_{n+1}$（即$A_n$型）情形下猜想成立，构造了等变典范切割同构以重构量子代数。
Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness
本文构造超出刚性与有限性条件下由顶点算子代数得到的带奇点开-闭模函子，证明其满足切除性质等，还将其应用于三重态模型的关联函数研究。
The Conway--Norton conjectures unexpectedly related the Monster with certain special modular functions (Hauptmoduls). Their proof by Borcherds et al was remarkable for demonstrating the rich mathematics implicit there. Unfortunately Moonshine remained almost as mysterious after the proof as before. In particular, a computer check - as opposed to a general conceptual argument - was used to verify the Monster functions equal the appropriate modular functions. This, the so-called `conceptual gap', was eventually filled; we review the solution here. We conclude by speculating on the shape of a new…
Hodge structures on conformal blocks
本文证明了模函子上复Hodge结构的存在唯一性，利用非阿贝尔Hodge对应和Ocneanu刚性，并给出SU(2)模函子在特定级下的显式Hodge数公式。
For a certain kind of tensor functor $F: \mathcal{C} \to \mathcal{D}$, we define the relative modular object $χ_F \in \mathcal{D}$ as the "difference" between a left adjoint and a right adjoint of $F$. Our main result claims that, if $\mathcal{C}$ and $\mathcal{D}$ are finite tensor categories, then $χ_F$ can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension $A/B$ of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type …
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