String-net models for pivotal bicategories and rational conformal field theories with defects

Two-dimensional conformal field theories are renowned for their richness, in the sense that through studying them, one often reveals connections between different disciplines in both physics and mathematics, at the same time deepens the understanding thereof. In this thesis, with the aim of constructing consistent systems of correlators for worldsheets with topological defects and physical boundaries in rational conformal field theories, we generalize and apply the string-net models for spherical fusion categories, which were first introduced in [LW05] as lattice models of topological phases and later formalized in [KJ11] as 2-dimensional skein theories. The primary theoretical foundation of this project is three-fold: First, it is widely believed [MS89, BK01] (and proven for the cases of genus 0 and genus 1) that through a Riemann-Hilbert type correspondence, the notion of an algebraic modular functor provided by the Reshetikhin-Turaev 3-dimensional topological field theory for a modular fusion category C, is equivalent to that of a complex-analytic modular functor for a rational chiral conformal field theory whose monodromy is governed by the very same modular fusion category C (e.g. as a suitable representation category of a rational chiral vertex operator algebra V). Second, it is demonstrated in a series of works [FRS02, FRS04a, FRS04b, FRS05, FFRS06a] from two decades ago that upon specifying the chiral theory as well as the representation theoretic data in the category C that decorate the (topological) worldsheets, a consistent system of correlators can be fully constructed; by the latter we mean an assignment of an invariant of the appropriate mapping class group in the relevant space of conformal blocks (provided by the 3d TFT) to every worldsheet that satisfies sewing constraints. Third, it has been established that we have the following chain of equivalences of once-extended topological field theories of top-dimension three: SN${}_C$ ≃ TV${}_C$ ≃ RT${}_{Z\left(C\right)}$ (1.0.1) for any spherical fusion category C, where TV${}_C$ stands for the Turaev-Viro state-sum TFT for C and RT${}_{Z\left(C\right)}$ stands for the Reshetikhin-Turaev surgery TFT for the Drinfeld center Z(C), the equivalence between which was proven independently and via different approaches in [Bal11] and [TV13], and SN${}_C$ is the TFT extended from the string-net model for C, whose existence and equivalence to the other two TFTs were established in [KJ11] and [Goo18]. In view of the equivalence of braided tensor categories C${}^{rev}$ ⊠ C ≃ Z(C) which is equivalent to the modularity of C [Shi19], and the fact that one can achieve the necessary procedure of "combining the left and right movers" (which is called holomorphic factorization [Wit92] in the literature) for the construction of correlators by taking the braided double C${}^{rev}$⊠C as the category of chiral data, (1.0.1) suggests that the construction of a consistent system of correlators can be carried out via the string-net model SN${}_C$. The relevant backgrounds as well as a precise formulation of the tasks of establishing the chiral theory as an open-closed modular functor and constructing a consistent system of correlators for all worldsheets with topological defects are provided in Chapter 2, and the concrete construction of correlators in terms of string-nets is given in Chapter 5. As it turned out, not only does the string-net approach to RCFT correlators reduce the technicality of the construction greatly compared to the TFT approach adopted in [FRS02, FRS04a, FRS04b, FRS05, FFRS06a] (for instance, the proof of fulfillment of the sewing constraints in the latter involves rather complicated 3-dimensinal topology), it also verifies (in the semisimple setting) the conjecture proposed in [FS21a] that internal homs, internal natural transformations and their compositions [FS21b] provide the algebraic structures that describe the field contents and their operator products in conformal field theories. These results are presented in Chapter 6. As an additional demonstration of the usefulness of the string-net construction for relating algebraic structures in the braided category Z(C) to the geometric ones provided by the decorated worldsheets, in Chapter 7 we present the proof (and a concrete formulation) of the statement that the vertical and horizontal compositions of the internal natural transformations obey a braided version of the Eckmann-Hilton relation satisfied by the compositions of ordinary natural transformations, which is done by concretizing the observation that the objects of internal natural transformations are braided algebras over the braided colored operad WS${}_C$ of genus-0 worldsheets. Moreover, by including line and point defects into the theory, one realizes that it is necessary to generalize the string-net models to the bicategorical setting: The collection of representation theoretic defect conditions for the worldsheets in an RCFT with fixed chiral data C comprises a pivotal bicategory Fr(C) of simple special symmetric Frobenius algebras, bimodules and bimodule morphisms internal to C, whose composition rules describe the fusion of (point and line) defects. In Chapter 8, we show that worldsheets related by the local composition rules for defects provided by the pivotal bicategory Fr(C) share the same correlator. Put differently: the prescription of string-net correlators gives rise to a family of MCG-intertwiners between string-net spaces for the pivotal bicategory Fr(C) which classify the worldsheets up to local relations, and the string-net spaces for the modular fusion category C which model the spaces of conformal blocks of the theory. We call these intertwiners universal correlators, alluding to the non-linear analog provided by universal bundles and classifying spaces. In this sense, it is the equivalence classes of worldsheets, which form a vector space of Fr(C)-colored string-nets upon fixing a boundary datum for the defect patterns, that are detectable by "observing" the correlators - we therefore refer to these equivalence classes as quantum worldsheets, and define the mapping class groups for them that take the local relations into account. The short proof of Theorem 8.1.1 relies on the careful formulation of the unframed graphical calculi and the string-net models for strictly pivotal bicategories, as well as the discussion of their functoriality under rigid separable Frobenius functors, which are given in Chapter 3 and Chapter 4. It should be noted that our formulations do not require any finiteness of the bicategories. Chapter 9 concludes this thesis with an observation of theoretical interest: our sporadic constructions of string-net modular functors, field functors and universal correlators fit neatly into the framework of double categories: the symmetric monoidal functors of the type Bord${}_{2,o/c}^{or}$ → Prof${}_k$ (which is what the term "open-closed modular functors" means to us in this thesis) provided by the string-net models for pivotal bicategories canonically extend to symmetric monoidal double functors, and the universal correlators along with the field functors comprise a monoidal vertical transformation, whereas various desired properties such as factorization of modular functors, MCG-invariance and the fulfillment of sewing constraints of the correlators correspond directly to the axioms of double functors and vertical transformations, respectively.