Differential cohomology in a cohesive infinity-topos (1310.7930v1)

Abstract: We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections, hence higher gauge fields. We discuss various models of the axioms and applications to fundamental notions and constructions in quantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization -- in the sense of extended/multi-tiered quantum field theory -- of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories. We close with an outlook on the cohomological quantization of such higher boundary prequantum field theories by a kind of cohesive motives.

• The paper presents a cohesive ∞-topos formulation that refines differential cohomology and Chern-Weil theory by integrating higher gauge fields and synthetic differential geometry.
• It applies the framework to classify principal ∞-bundles with ∞-connections, providing a robust method for geometric prequantization in quantum and string field theories.
• The approach bridges classical cohomology and modern geometric constructs, paving the way for novel quantization techniques in higher Chern-Simons-type and related field theories.

An Expert Overview of "Differential Cohomology in a Cohesive $\infty$-Topos"

The paper by Urs Schreiber, titled "Differential Cohomology in a Cohesive $\infty$-Topos," embarks on formalizing differential cohomology and Chern-Weil theory within the framework of cohesive $\infty$-toposes. This innovative approach bridges the gap between classical cohomological theories and modern geometric frameworks that align with the complexities of quantum field theory and string theory.

Differential Cohomology and Chern-Weil Theory

Differential cohomology serves as an advancement over conventional cohomology, allowing for a nuanced description of geometric objects like connections on fiber bundles and gauge fields. Specifically, differential cohomology in this context classifies higher principal bundles endowed with cohesive structures. These structures extend beyond traditional topological and smooth frameworks, encompassing synthetic differential and supergeometric intricacies.

The use of cohesive $\infty$-toposes amplifies the general abstract formulation of these theories, enabling the classification of principal $\infty$-bundles and their $\infty$-connections—essentially higher gauge fields. This framework is constructed around a cohesive structure, which integrates aspects of topology, smoothness, synthetic differentials, and more.

Implications in Quantum and String Theory

One of the key insights from the paper is the cohesive and differential refinement of universal characteristic cocycles. These cocycles foster a higher Chern-Weil homomorphism—this homomorphism extends from secondary characteristic classes to the morphisms of higher moduli stacks of gauge fields. Additionally, it presents an extended geometric prequantization relevant to multi-tiered quantum field theories, particularly higher Chern-Simons-type field theories and their Wess-Zumino-Witten-type boundary theories.

Such extended geometric prequantization invokes hierarchies of higher dimensional field theories, incorporating higher codimension defect theories. A paramount implication is the potential for cohomological quantization of these higher boundary prequantum field theories—facilitated by cohesive motives.

Models and Extensions

The paper explores various models aligning with the axioms and discusses applications to foundational constructs in both quantum field theory and string theory. It particularly accentuates a cohesive and differential refinement of these theories, advocating for a cohesive approach to differential cohomology and supporting elements from higher categories.

By leveraging cohesive $\infty$-toposes, it posits that these structures provide synthetic motives that underlie local quantum gauge field theories derived from geometric Lagrangian data. The interaction between the cobordism theorem and these geometric constructs highlights the pivotal role of higher category theory in developing synthetic formulations of quantum theories.

Conclusion

The implications and methodologies discussed in this paper lay foundational work for integrating geometric, homological, and categorical perspectives into the paper of quantum field theories and string interactions. With cohesive $\infty$-toposes offering a synthetic axiomatization of such theories, Schreiber's work invites further exploration into geometric prequantization and motivates developments across multiple dimensions of modern physics.

This paper not only redefines differential geometry's role within theoretical physics but also opens avenues for future exploration along the intersections of geometry, cohomology, and quantum field dynamics, with cohesive $\infty$-toposes as central to these advancements.