Brane Tensions and Coupling Constants from within M-Theory

TL;DR

The paper reexamines local anomaly cancellation for M-theory on $\mathbb{R}^{10} \times S^1/\mathbb{Z}_2$ to rederive the boundary Yang-Mills coupling and brane tensions, clarifying the roles of the upstairs and downstairs formulations. It derives the upstairs coupling $\lambda^2$ and the associated $\eta$ parameter via anomaly inflow and shows that natural units are best taken in the downstairs picture, where $\lambda^2 = 4\pi (4\pi \kappa^2)^{2/3}$. By incorporating the bulk Chern-Simons and fivebrane contributions, the membrane tension $T_2$ and fivebrane tension $T_5$ are fixed and shown to satisfy specific relations (e.g., $2\kappa^2 T_2 T_5 = \bar{\kappa}^2 T_2 T_5 = 2\pi$ and $T_5/T_2^2 = 1/(2\pi)$), bringing the results into agreement with prior work like Alwis when evaluated in consistent units. The macroscopic membrane tension is tied to the boundary K4 normalization through anomaly inflow, yielding $T_2 = ((2\pi)^2/(2\kappa^2))^{1/3}$ (in downstairs units), thereby cementing the link between bulk dynamics and boundary gauge data.

Abstract

Reviewing the cancellation of local anomalies of M-theory on R^10 x S^1/Z_2 the Yang-Mills coupling constant on the boundaries is rederived. The result is lambda^2 = 2^(1/3) (2 pi) (4 pi kappa^2)^(2/3) corresponding to eta = lambda^6/kappa^4 = 256 pi^5 in the `upstairs' units used by Horava and Witten and differs from their calculation. It is shown that these values are compatible with the standard membrane and fivebrane tensions derived from the M-theory bulk action. In view of these results it is argued that the natural units for M-theory on R^10 x S^1/Z_2 are the `downstairs' units where the brane tensions take their standard form and the Yang-Mills coupling constant is lambda^2 = 4 pi (4 pi kappa^2)^(2/3).