Topological Field Theory, Higher Categories, and Their Applications (1004.2307v2)

Abstract: It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geometric nature, while others are related to representation theory. I outline two application of higher-dimensional TFTs. One is related to the problem of classifying monoidal deformations of the derived category of coherent sheaves, and the other one is geometric Langlands duality.

• The paper demonstrates how extended topological field theories are naturally framed via higher categories, especially in dimensions greater than two.
• It shows the application of these theories to monoidal deformations of coherent sheaves, deepening insights into quantum group representations.
• The work bridges extended TFTs with geometric Langlands duality, linking algebraic geometry and representation theory for advanced quantum field models.

Overview of "Topological Field Theory, Higher Categories, and Their Applications" by Anton Kapustin

Anton Kapustin's paper "Topological Field Theory, Higher Categories, and Their Applications" explores the interaction between topological field theories (TFTs) and the mathematical structures known as higher categories. This research addresses important questions in both theoretical physics and advanced mathematics, focusing on how TFTs in dimensions higher than two are naturally framed using higher-dimensional categories, specifically $n$-categories where $n > 1$. This paper not only constructs specific examples of extended topological field theories (Extended TFTs) but also offers significant applications, including monoidal deformations of coherent sheaves and geometric Langlands duality.

Higher Categories and Topological Field Theories

TFTs provide a simplified framework for the analysis of functional integrals in quantum field theory (QFT). While initially formulated in terms of $n$-categories, where $n = 2$, Kapustin extends this notion to higher dimensions, unveiling the significance of $n$-categories as pivotal in formulating these theories when $n > 2$. The framework of higher categories is crucial, especially as one moves to Extended TFTs which consider categories associated with objects of dimension less than the field theory itself.

In Extended TFTs, boundary conditions are interpreted using categories (and higher categories) at yet lower dimensions. For a 3d TFT, for instance, boundary conditions assume the structure of a 2-category.

Applications

Kapustin's research showcases the potential for applying Extended TFT to solve high-complexity problems in mathematics:

1. Monoidal Deformations: The research borrows from the Rozansky-Witten model to tackle monoidal deformations in coherent sheaves. These deformations shed light on quantum groups and involve intricate operations that affect the tensor product of categories. This is particularly relevant for examining quantum deformations akin to quantum group representations.
2. Geometric Langlands Duality: The theory is extended to include the Geometric Langlands Program, linking categories associated with compact Riemann surfaces. Here, TFTs serve as a bridge between the world of algebraic geometry and representation theory, suggesting equivalences of categories corresponding to different dualities. Electric-magnetic duality, a classic in four-dimensional gauge theories, is foundational in understanding this duality within the context of TFTs.

Implications and Future Developments

Kapustin’s exploration of Extended TFTs not only provides a bridge to potential insights in the mathematical classification of manifold invariants and quantum invariants but also opens pathways for their application in broader contexts such as string theory, condensed matter physics, and possibly quantum computation. A particularly intriguing future direction involves exploiting the formalism of $(\infty,n)$-categories, which might provide a robust theoretical underpinning for complex systems within theoretical physics.

By linking higher-dimensional category theory with TFTs, this research positions itself as a critical stepping stone towards understanding the deep connections between quantum field theories and abstract algebraic structures, promising rich areas for further exploration, including higher category theory’s possible influence on emerging quantum technologies and theories of quantum gravity.