On D=5 super Yang-Mills theory and (2,0) theory (1012.2880v2)

Abstract: We discuss how D=5 maximally supersymmetric Yang-Mills theory (MSYM) might be used to study or even to define the (2,0) theory in six dimensions. It is known that the compactification of (2,0) theory on a circle leads to D=5 MSYM. A variety of arguments suggest that the relation can be reversed, and that all of the degrees of freedom of (2,0) theory are already present in D=5 MSYM. If so, this relation should have consequences for D=5 SYM perturbation theory. We explore whether it might imply all orders finiteness, or else an unusual relation between the cutoff and the gauge coupling. S-duality of the reduction to D=4 may provide nonperturbative constraints or tests of these options.

• The paper establishes that the compactification of (2,0) theory on a circle captures all its degrees of freedom in the D=5 MSYM framework.
• The study reveals that supersymmetry and S-duality potentially render D=5 MSYM perturbatively finite despite traditional nonrenormalizability concerns.
• The analysis explores how instantonic particles and non-simply laced gauge groups emerge from twisted boundary conditions, influencing nonperturbative dynamics.

Overview of D = 5 Super Yang-Mills Theory and (2,0) Theory

This paper provides a detailed examination of the parallels between five-dimensional maximally supersymmetric Yang-Mills theory (D = 5 MSYM) and the six-dimensional (2,0) theory, which holds a significant position in the context of high-energy theoretical physics, specifically within the framework of string theory and M-theory.

The main conjecture explored in the paper is that all the degrees of freedom of the (2,0) theory can be captured by compactifying it on a circle to D = 5 MSYM, effectively arguing the reverse relation. The implications of this equivalence are profound, particularly for understanding perturbative finiteness and the necessity for counterterms in D = 5 MSYM.

Key Concepts and Results

1. Reduction and Degrees of Freedom: The paper explores the compactification process of the (2,0) theory, establishing that its equivalence to D = 5 MSYM is valid in the low-energy regime. This conjectural equivalence suggests new insights into the inherent degrees of freedom already present in D = 5 MSYM, potentially obviating the extension to (2,0) degrees for certain calculations.
2. Perturbative Renormalizability: A central focus of the paper is the investigation of perturbative renormalizability. Given that D = 5 MSYM is traditionally considered nonrenormalizable by power-counting arguments, the paper suggests that supersymmetric properties, S-duality, and other advanced techniques may lead to all-orders finiteness. This aspect draws a parallel to similar discussions in lower-dimensional N = 8 supergravity theories.
3. S-duality and Compactification: The compactification of (2,0) theory and its dimensions allow for exploration into S-duality and nonperturbative constraints. For instance, the reduction of (2,0) theory on a torus leads to N = 4 SYM in four dimensions, opening discussions on S-duality invariance and potential computational limits, which are examined in detail through a one-loop analysis.
4. Instantonic Particles and Non-Simply Laced Gauge Groups: Instantons in D = 5 are identified as particles carrying conserved charges linked to instanton number from four-dimensional perspectives. Moreover, the paper entertains discussions around the emergence of non-simply laced gauge groups in D = 5 through twisted boundary conditions, providing a field-theoretic pathway influenced by arguments in string theory.

Implications and Speculations

The theoretical implications of these findings are manifold. If the proposal that D = 5 MSYM is perturbatively finite holds, this could suggest that (2,0) theory, previously defined in the field of M-theory and lacking a concrete action description, might be more inherently understood through its D = 5 incarnation. Further, this would indicate a potential for using D = 5 MSYM to explore the nonperturbative structure of (2,0) theory, particularly when considering ultraviolet (UV) completions.

Practically, while the exact nonperturbative completion remains to be determined, the hypotheses presented provide a fertile ground for future computational and conceptual exploration. Perhaps the most riveting speculation lies in the role of instantonic particles and whether they provide the necessary completion of the theory as self-contained solutions.

Conclusion

In summary, the paper effectively bridges several complex aspects of theoretical physics, offering conjectures that stimulate further inquiry into the nature of higher-dimensional quantum field theories and their interrelations via compactification. The tantalizing notion that D = 5 MSYM may capture all (2,0) degrees of freedom redefines the scope of perturbative and nonperturbative analysis, encouraging deeper investigation into dualities and the fundamental architecture of supersymmetric theories. Future developments, possibly informed by enhanced computational capabilities or novel theoretical insights, could validate or refine the propositions laid out, potentially impacting our understanding of quantum field theories at a structural level.