Coupling a QFT to a TQFT and Duality

Abstract: We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On R^d the new theory differs from the original one by the spectrum of operators. Sometimes the local operators are the same but there are different line operators, surface operators, etc. The effects of the added topological degrees of freedom are more dramatic when we compactify R^d, and they are crucial in the context of electric-magnetic duality. We explore several examples including Dijkgraaf-Witten theories and their generalizations both in the continuum and on the lattice. When we couple them to ordinary quantum field theories the topological degrees of freedom allow us to express certain characteristic classes of gauge fields as integrals of local densities, thus simplifying the analysis of their physical consequences.

• The paper demonstrates that coupling QFT with TQFT alters the spectrum of line and surface operators, thereby influencing electric-magnetic duality.
• It uses concrete examples like Dijkgraaf-Witten theories and 3D Chern-Simons-matter models to explore changes in gauge symmetries and operator classifications.
• The analysis bridges continuum and lattice formulations, offering practical insights on non-perturbative effects and dualities in quantum systems.

Overview of "Coupling a QFT to a TQFT and Duality"

The paper by Anton Kapustin and Nathan Seiberg addresses the implications of coupling an ordinary quantum field theory (QFT) to a topological quantum field theory (TQFT). The main consideration is how the incorporation of a TQFT, which is characterized by its finite number of degrees of freedom, in contrast to the potentially infinite degrees of freedom in an ordinary QFT, can substantially alter the behavior and symmetries of the field theory. This coupling leads to significant modifications in the spectrum of observables such as line operators and surface operators. The analysis here becomes noteworthy in the context of electric-magnetic duality, lending insights into the role of topological degrees of freedom.

The paper approaches the subject through several examples, with a particular focus on Dijkgraaf-Witten theories, and extends its reach to formulations both in continuum and lattice descriptions. Central to the author's investigation is the transformation and categorization of various operator types under this coupled system, offering a pathway for examining gauge symmetries influenced by TQFT.

Key Contributions and Results

1. Change in Operator Spectra: The coupling of a TQFT to a QFT modifies the spectrum of operators. While local operators might remain unaltered, line, surface, and other higher-dimensional operators undergo significant transformations, relevant to electric-magnetic duality considerations.
2. Examples of Coupled Theories: The authors explore 2d orbifolds and 3d Chern-Simons-matter theories as key examples where coupling results in increased interaction complexities within the dynamics of the resulting quantum field theories.
3. Gauge Symmetries and Class Analysis: Gauge symmetries are deeply analyzed, employing advanced classifications involving q-form gauge symmetries and general global symmetries. The authors leverage these to understand and demonstrate how coupling allows the characterization and alteration of higher-form global symmetries in the theory.
4. Topo Obstructions and Field Configurations: When examining compactified theories, distinct topological sectors emerge, often necessitating the definition of sectors across continuous and discrete labels. This analysis is eloquently expanded in their examination of partition functions and characteristic classes, notably using integrals of local densities.
5. Lattice Formulations and Duality: A substantial portion of the analysis explores lattice gauge theories, elucidating on duality transformations in discrete systems, showing parallel insights into dualities of both spin and gauge systems. The analysis reflects precision in attention to boundary terms, particularly in regard to the implications of topological sectors.
6. Comparison with Spin Systems: The paper also revisits duality in Z_n spin systems, exploring parallels with lattice gauge theory and expanding on conventional duality observations in such systems.

The rigorous mathematical investigation aligns well with anticipated developments in both theoretical and practical domains, particularly in understanding confinement, deconfinement phases, and spontaneous symmetry breakings due to topological effects. Additionally, the theoretical implications of such couplings may influence methods in condensed matter physics and quantum computing, where effective field theories often emerge enriched by topological considerations.

Implications and Speculation for Future Work

Looking forward, the exploration of coupling QFTs to TQFTs raises several possibilities. Advances may appear in better defining gauge theories within the constraints set by topological sectors, potentially assisting in the refinement of duality transformations. Furthermore, as interplay between topology and quantum field theories becomes more accessible, one might expect fresh insights into non-perturbative effects, particularly in high-dimensional theories and complex systems in condensed matter physics.

The coupling theory can also inform the design of computational models that leverage topological invariants and dualities to overcome challenges in simulating high-complexity quantum systems. Continuing to investigate the lattice formulations of these dualities could unveil unprecedented algorithms for numerical simulations in quantum computing, solidifying the practical applicability of theoretical high-energy physics.

In conclusion, Kapustin and Seiberg's exploration of QFTs with TQFTs not only deepens the theoretical understanding of field theories but also opens doors for potential breakthroughs in the application fields dependent on such frameworks. Their rigorous approach sets groundwork for future endeavors in bridging the gap between theoretical constructs and practical applications in diverse scientific areas.