- The paper establishes a new framework linking supersymmetric field theories to generalized cohomology, bridging gaps between geometry and physics.
- It demonstrates that partition functions of 2|1-dimensional theories are modular functions, providing a physical grounding for elliptic cohomology and TMF.
- The work formalizes concordance classes in Euclidean and twisted field theories and posits periodicity conjectures, influencing both topology and string theory.
Supersymmetric Field Theories and Generalized Cohomology
The paper "Supersymmetric Field Theories and Generalized Cohomology" by Stephan Stolz and Peter Teichner presents a comprehensive study of the interplay between physics-inspired mathematical structures and advanced concepts in algebraic topology. Primarily, it explores how supersymmetric field theories can be utilized to model and investigate generalized cohomology theories, with a specific focus on elliptic cohomology and topological modular forms.
Overview
This work extends the pioneering ideas of Graeme Segal, which establish a conceptual framework linking field theories and cohomology theories. Segal's proposal, wherein 2-dimensional conformal field theories correspond to elliptic cohomology, serves as an intellectual backdrop for the investigations into how higher-dimensional and supersymmetric field theories map onto more complex cohomological frameworks, such as topological modular forms (TMF).
Key Concepts and Results
- Euclidean Field Theories and Concordance:
- The authors formalize the notion of Euclidean field theories, emphasizing their role as geometric cocycles in cohomology theories. Concordance classes of these theories correspond directly to cohomological data.
- The paper outlines how 1-dimensional field theories are equivalent to vector bundles with connections over manifolds, establishing an important bridge to traditional geometrical objects.
- Supersymmetric Extensions:
- Incorporating supersymmetry, the authors consider field theories on manifolds augmented with supermanifold structures. These supersymmetric field theories expand the toolkit available for constructing and interpreting elliptic cohomology classes.
- A significant advancement is the demonstration that partition functions of 2|1-dimensional supersymmetric field theories are modular functions, revealing deep connections to number theory.
- Twisted Field Theories and TMF:
- The authors introduce twisted field theories to account for more intricate cohomology theories that allow twists in their formulation, capturing phenomena in twisted K-theory and beyond.
- They conjecture that the 2|1-dimensional Euclidean field theories associated with local (extended) field theories correspond to TMF, suggesting a periodicity of 48, which would refine to 576 in its fully extended form.
- Modular Functions and Periodicity:
- A critical achievement is showing that certain supersymmetric field theories yield holomorphic modular functions. This result parallels classical modular forms, providing tools to probe the structure of TMF through field-theoretic lenses.
- The paper affirms the periodicity phenomena in these theories through explicit constructions and conjectures on the role of modular forms determined by known cohomological invariants like the Chern character.
Implications and Future Directions
The implications of this research are manifold, spanning pure mathematics and theoretical physics. By grounding cohomology theories in physically-motivated constructs such as field theories, the authors pave the way for a reciprocal flow of ideas. This interaction not only deepens our understanding of cohomological operations but also equips theoretical physicists with sophisticated mathematical tools to tackle problems in string theory and quantum field theory.
The exploration of higher-level categories to describe local field theories continues to be an open area of research. The holistic framework outlined in this paper promises to unify discrete aspects of geometry, topology, and physics under a shared language of modularity and supersymmetry.
The work concludes with aspirations to refine the definition of fields in varying geometries and extend these ideas into broader topological settings. The ongoing quest involves harnessing the full potential of category theory to capture the locality and non-trivial twistings underpinning prominent physical theories and mathematical conjectures.
In summary, Stolz and Teichner's paper marks a significant stride towards a unified theory of geometric and topological transformations framed within the context of supersymmetric field theories. It sets the stage for future explorations into the interfaces of mathematics and theoretical physics, providing a formidable structure to engage with enduring challenges in both disciplines.