Noninvertible symmetries in the B model TFT (2504.02023v1)

Abstract: In this paper we explore noninvertible symmetries in general (not necessarily rational) SCFTs and their topological B-twists for Calabi-Yau manifolds. We begin with a detailed overview of defects in the topological B model. For trivial reasons, all defects in the topological B model are topological operators, and define (often noninvertible) symmetries of that topological field theory, but only a subset remain topological in the physical (i.e., untwisted) theory. For a general target space Calabi-Yau X, we discuss geometric realizations of those defects, as simultaneously A- and B-twistable complex Lagrangian and complex coisotropic branes on X \times X, and discuss their fusion products. To be clear, the possible noninvertible symmetries in the B model are more general than can be described with fusion categories. That said, we do describe realizations of some Tambara-Yamagami categories in the B model for an elliptic curve target, and also argue that elliptic curves can not admit Fibonacci or Haagerup structures. We also discuss how decomposition is realized in this language.

Overview of "Noninvertible Symmetries in the B Model TFT"

The paper "Noninvertible Symmetries in the B Model TFT" by Andrei C\u{a}ld\u{a}raru et al. provides a comprehensive exploration of noninvertible symmetries within the framework of topological field theories (TFTs), specifically focusing on the B model involving Calabi-Yau manifolds. The approach taken in this paper intricately connects string theory, topological defects, derived categories, and symmetry operations, offering insights into the rich mathematical structures underpinning these concepts.

Fundamental Concepts and Results

Central to the discussion is the notion of noninvertible symmetries, which differ fundamentally from traditional invertible symmetries described by group actions. In the B model TFT context, these symmetries are represented by topological defects or line operators, which are realized as objects within the derived category of coherent sheaves on the product space $X \times X$. Here, $X$ denotes a Calabi-Yau manifold, and these objects capture both geometric and physical symmetry properties encoded by these defects.

Significantly, the paper extends the conventional category-theoretic approach to include these noninvertible symmetries not easily encapsulated by fusion categories. While traditional fusion categories describe a broad class of symmetries, derived categories offer a richer and more generalized framework. The authors elucidate this distinction by examining how line operators and their fusion products in the B model can be expressed as compositions of integral transforms, making the complex relationship between these operators transparent.

Key Mathematics and Physics

The mathematical framework employed involves the manipulation of objects in derived categories using integral transforms, which are interpreted physically as fusion products of line operators. In this context, derived categories are shown to accommodate a wider class of symmetries than those described by finite group transformations, allowing for the realization of more intricate symmetry operations through the lens of homological algebra.

Noteworthy is the discussion on specific examples of symmetry structures on elliptic curves and K3 surfaces. The paper provides explicit computations showcasing how noninvertible symmetry operations can be characterized at the level of numerical K-theory. The authors' use of tools such as the Hirzebruch-Riemann-Roch theorem further strengthens the algebraic backbone of these constructions, bridging complex algebraic geometry with the topological nuances of field theories.

Applications and Implications

The insights garnered offer potential applications in various domains, notably in enhancing our understanding of dualities and symmetry transformations in string theory and related fields. The exploration of topological interfaces and defects also provides a new vantage point on the paper of boundary conditions and interfaces in two-dimensional quantum field theories (QFTs), potentially influencing future research on modular transformations and symmetries in QFTs.

Moreover, the paper's approach aligns with ongoing efforts to unravel the deeper connections between category theory and quantum symmetries, pointing towards richer categorical frameworks capable of describing a variety of physical phenomena in quantum field theory and quantum gravity.

Speculation on Future Directions

Looking forward, the extension of these ideas to other topological field theories, such as A-model twists and their corresponding mirror formulations, could prove insightful. Additionally, the theoretical implications of these findings suggest potential routes to identifying new classes of noninvertible symmetries in higher-dimensional or non-Calybi-Yau settings. Future research might also explore the interplay between these noninvertible symmetries and the geometric phases of gauge theories, thereby enriching the tapestry of mathematical physics.

In summary, Andrei C\u{a}ld\u{a}raru et al. effectively advance our comprehension of noninvertible symmetry structures within the B model topological field theory, providing a comprehensive treatment that resonates with broader themes in modern theoretical physics and algebraic geometry.