M5-branes from gauge theories on the 5-sphere

Abstract: We use the 5-sphere partition functions of supersymmetric Yang-Mills theories to explore the (2,0) superconformal theory on S^5 x S^1. The 5d theories can be regarded as Scherk-Schwarz reductions of the 6d theory along the circle. In a special limit, the perturbative partition function takes the form of the Chern-Simons partition function on S^3. With a simple non-perturbative completion, it becomes a 6d index which captures the degeneracy of a sector of BPS states as well as the index version of the vacuum Casimir energy. The Casimir energy exhibits the N^3 scaling at large N. The large N index for U(N) gauge group also completely agrees with the supergravity index on AdS_7 x S^4.

• The paper demonstrates that 5-sphere partition functions in 5D SYM capture essential aspects of 6D (2,0) theory, illuminating M5-brane dynamics.
• The research reveals an N^3 scaling of the Casimir energy, aligning with supergravity predictions in large-N limits.
• Localization techniques enable precise computation of indices, firmly establishing a duality between 5D gauge theory and its 6D superconformal counterpart.

Analysis of "M5-branes from gauge theories on the 5-sphere"

The paper investigates the intricate relationship between five-dimensional (5D) supersymmetric Yang-Mills (SYM) theories and six-dimensional (6D) (2,0) superconformal theories, with an emphasis on understanding the nature of M5-branes in M-theory through gauge theories formulated on a five-sphere. This research stands within the sphere of theoretical high-energy physics and aims to bridge the gap between lower-dimensional gauge theories and higher-dimensional M-theory.

Central to the paper's thesis is the utilization of the partition functions computed from 5-sphere supersymmetric Yang-Mills theories to probe the mysterious 6D (2,0) superconformal theories on the product space $S^5 \times S^1$, where $S^5$ is a five-sphere and $S^1$ is a circle. The authors propose that the 5D theories can be viewed as a Scherk-Schwarz reduction of the 6D theories, providing a compactification framework whereby the 5D SYM effectively captures the behavior of the 6D (2,0) theory in a certain sector of BPS states.

Key Mathematical and Physical Insights

1. Partition Functions: In the research, a pivotal role is played by the 5-sphere partition functions. In a specific limit, the perturbative partition function simplifies to mirror the form of a Chern-Simons partition function on $S^3$. This simplification indicates a bridge between different topological and quantum field-theoretic constructs. The partition functions, with non-perturbative corrections, encapsulate 6D physics, such as the degeneracies of a BPS sector and an interpretation of the vacuum Casimir energy.
2. $N^3$ Scaling: Notably, the research identifies that the Casimir energy scales in a manner proportional to $N^3$ at large $N$. This scaling is significant because it aligns with known predictions from the supergravity perspective on $AdS_7 \times S^4$, establishing a theoretical nexus between quantum field theory on the boundary and gravitational theories in higher dimensions. This scaling behavior indicates the extensive degrees of freedom associated with coincident M5-branes.
3. Supergravity Connections: An elegant part of the findings is that the large $N$ index for the $U(N)$ gauge group matches perfectly with what one would calculate using supergravity duals. This concurrence provides a robust check on the theoretical frameworks involved, suggesting that the 5D-6D theory correspondence is not merely speculative but grounded in the consistency of dual formulations.
4. Applications of Localization: By employing localization techniques in computing the path integral over these 5D gauge theories, the paper sidesteps some notorious challenges associated with quantum fluctuations, allowing the researchers to draw precise results about sectors of the 6D theories.

Implications and Future Directions

The implications of these calculations and theoretical insights are profound within the context of non-perturbative string theory and M-theory. The paper opens pathways to understanding intricate properties of M-theoretic constructions such as M5-branes through calculable lower-dimensional gauge theories. Moreover, it challenges researchers to further explore how 5D SYM theories encapsulate rich dual physics.

Future work could expand on the current analysis to cover different gauge groups, or to incorporate additional supergravity theories that could shed light on other aspects of the large $N$ dynamics. Furthermore, these insights may offer pathways to a better understanding of the UV completions of SYM theories, which is an open question in theoretical physics.

To conclude, this paper serves as an intersectional point between gauge theory, superconformal field theory, and M-theory, offering a scaffold upon which deeper theoretical investigations into the nature of higher-dimensional quantum field theories and their string-theoretic counterparts can be developed.