M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills (1012.2882v3)

Abstract: We revisit the relation of the six-dimensional (2,0) M5-brane Conformal Field Theory compactified on a circle to 5D maximally supersymmetric Yang-Mills Gauge Theory. We show that in the broken phase 5D super-Yang-Mills contains a spectrum of soliton states that can be identified with the complete Kaluza-Klein modes of an M2-brane ending on the M5-branes. This provides evidence that the (2,0) theory on a circle is equivalent to 5D super-Yang-Mills with no additional UV degrees of freedom, suggesting that the latter is in fact a well-defined quantum theory and possibly finite.

• The paper demonstrates that the compactified 6D (2,0) M5-brane CFT is equivalent to 5D super-Yang-Mills theory.
• It employs solitonic Kaluza-Klein mode analysis to challenge previous views on 5D SYM non-renormalizability.
• The findings imply that 5D super-Yang-Mills may be UV complete, reshaping our understanding of higher-dimensional quantum field theories.

M5-Branes, D4-Branes and Quantum 5D Super-Yang-Mills

The paper authored by Neil Lambert, Constantinos Papageorgakis, and Maximilian Schmidt-Sommerfeld diligently explores the intricate relationships between the six-dimensional $(2,0)$ conformal field theory (CFT) associated with M5-branes compactified on $S^1$, and the five-dimensional (5D) maximally supersymmetric Yang-Mills (SYM) gauge theory. This work reevaluates previous insights and provides compelling evidence supporting the notion that the $(2,0)$ CFT compactified on a circle is indeed equivalent to the 5D super-Yang-Mills theory, with no additional ultraviolet (UV) degrees of freedom required. The work posits that these theories are not only equivalent in the low-energy limit but that 5D SYM is potentially a well-defined and finite quantum theory, a conjecture with profound implications in the landscape of theoretical physics.

Summary of Key Contributions

1. Theoretical Framework and Motivation:
   • The $(2,0)$ CFT on M5-branes lacks a Lagrangian description and is strongly coupled. It is hypothesized to emerge as the UV fixed point of multiple D4-branes in type IIA string theory. In the weak coupling regime, the dynamics of these D4-branes are described by 5D SYM with $\U(N)$ gauge group.
2. 5D Super-Yang-Mills as a Quantum Field Theory:
   • The authors tackle the long-standing debate about the renormalizability of 5D SYM, historically viewed as non-renormalizable. By investigating the compactification of the $(2,0)$ theory on $S^1$, they furnish evidence that all necessary Kaluza-Klein (KK) modes are represented as solitonic states within 5D SYM without needing extra UV degrees of freedom.
3. KK Modes and Solitonic Spectrum:
   • The paper provides a meticulous analysis of the relationship between KK modes and BPS solitonic states, hypothesizing that dyonic instantons in 5D correspond to wrapped self-dual strings, and BPS monopoles correspond to unwrapped strings with non-zero momentum along the compactified dimension. Crucially, the spectrum of KK states aligns well with the expected compactification of the $(2,0)$ theory, bolstering the paper's central thesis.
4. Implications for UV Finiteness:
   • Affirming the finiteness of the $(2,0)$ CFT leads to the conclusion that its 5D counterpart should share this attribute. This proposal may transform existing perceptions of non-renormalizability in higher-dimensional Yang-Mills theories.

Theoretical and Practical Implications

The findings have profound theoretical significance, potentially reshaping the foundational understanding of higher-dimensional supersymmetric Yang-Mills theories in the context of string theory and M-theory. The assertion that 5D SYM could inherently embody a UV-complete description of the $(2,0)$ theory presents new avenues for exploring the dynamics of quantum field theories in higher dimensions, effectively bridging the gap between non-perturbative effects and perturbative quantum descriptions.

Speculation on Future Developments

Future research could focus on extending these conclusions to gauge groups beyond $SU(2)$, pertinent to scenarios involving more than two M5-branes. Additionally, further work could investigate the implications of these findings for holography and related topics, potentially contributing to a deeper understanding of the AdS/CFT correspondence, and its generalizations. The exploration of novel non-perturbative techniques, perhaps integrating aspects of geometric Langlands, could provide additional insights into the finely balanced intricacies of such high-dimensional theories.

Overall, the paper offers a critical advancement in comprehending the entangled dynamics of M5-branes and their associated field theories, laying foundational ground for future explorations traversing the boundaries of modern theoretical physics. In doing this, it opens up new paths toward a comprehensive understanding of M-theory's complex web of dualities and dynamics.