Twisted supersymmetric 5D Yang-Mills theory and contact geometry (1202.1956v3)

Abstract: We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90's, and in a way it is covariantization of their ideas for a contact manifold.

• The paper introduces a twisted version of 5D Yang-Mills theory on contact manifolds using tools from contact geometry, generalizing 3D Chern-Simons theory.
• It defines 5D contact-instanton equations and constructs a twisted supersymmetric 5D Yang-Mills theory allowing for localization techniques.
• The authors calculate the perturbative partition function on S^5, derive a matrix model, and highlight the significant role of contact geometry in higher-dimensional field theories.

Analyzing Twisted Supersymmetric 5D Yang-Mills Theory and Contact Geometry

The research presented in the paper, "Twisted Supersymmetric 5D Yang-Mills Theory and Contact Geometry," by Johan Kalen and Maxim Zabzine, explores the generalization of 3D Chern-Simons theory to a five-dimensional setting within the framework of twisted supersymmetry and contact geometry. Through an intricate approach that leverages topological and geometrical tools, the authors extend the localization calculations of the Chern-Simons partition function on Seifert manifolds to a more complex five-dimensional context. At the core of this theoretical construction lies the fundamental role of contact geometry, which facilitates the exploration of new physical insights in higher dimensions.

The objective is to define a twisted version of the 5D Yang-Mills theory that admits BRST-exact Lagrangians over contact manifolds, ensuring the mathematical and physical consistency of the model. The construction enjoys broad applicability, with potential extensions to any odd-dimensional manifold admitting a contact structure, offering a fertile ground for exploring supersymmetric gauge theories in higher-dimensional spaces.

Main Contributions and Results

The paper introduces a novel perspective on contact geometry's role in field theories by providing a generalization of instanton equations to five-dimensional contact manifolds. The introduction of a 5D theory is inspired by previous works of Baulieu, Losev, Nekrasov, and others, yet it significantly extends their framework to accommodate contact manifolds. The primary achievements of the research can be summarized as follows:

• Instanton Equations and Localization: The authors define a set of 5D contact-instanton equations similar to the well-known 4D instanton equations, leveraging the vertical and horizontal decompositions of differential forms. Utilizing these, they construct a twisted supersymmetric version of 5D Yang-Mills theory, allowing for localization techniques to compute perturbative partition functions.
• Twisted Supersymmetric Yang-Mills on S^5: The paper performs an explicit calculation of the perturbative partition function on $S^5$ for the twisted supersymmetric Yang-Mills theory, coupled with two different Chern-Simons terms. A significant matrix model emerges from this calculation, representing the theoretical structure of these supersymmetric models.
• Matrix Model and Perturbative Calculations: The localization technique leads to a matrix model describing the full perturbative expansion of the theory's partition function. This result addresses the theoretical tension between nonrenormalizability expectations of 5D supersymmetric Yang-Mills theory and the successful calculation of the perturbative answer using localization methods.

Implications and Future Directions

The findings of the paper have far-reaching implications in both theoretical physics and mathematics. The introduction of contact geometry as an integral element in constructing higher-dimensional supersymmetric field theories opens new avenues for research and might contribute to an improved understanding of gauge theories, potentially revealing deep connections with geometric structures. The extension of the basic framework to include higher-dimensional contact manifolds holds promise for further advancing our grasp of non-trivial S^1-bundles over symplectic manifolds.

Notably, the presented work suggests that the twisted version of supersymmetric Yang-Mills theory could relate to gauge theories with eight supercharges, invoking a topological twist akin to 4D theories. This paves the way for future exploration into the ramifications of these theories in the context of higher-dimensional field theories, as well as the paper of their non-perturbative features.

The conundrum arising from the coexistence of perturbative calculability and theoretical expectations of nonrenormalizability highlights an area of ongoing debate in theoretical physics that presents intriguing questions for further investigation. Potential connections to UV-completion and M-theory invariants, as referenced in related works, offer additional topics ripe for exploration. The abstract promises of contact geometry and its potential impact on the broader landscape of gauge theories and quantum field theories represent a fertile area for future inquiry, which this paper effectively sets the stage for.