One-loop Chern-Simons terms in five dimensions

TL;DR

This work computes one-loop corrections to the five-dimensional gauge and gravitational Chern-Simons terms induced by integrating out massive charged fields, including spin-1/2, spin-3/2, and complex self-dual tensors, and connects these corrections to six-dimensional anomalies via circle compactification. The authors perform explicit field-theory loop calculations for $k_{AFF}$ and $k_{ARR}$, analyze non-minimal couplings, and then verify the results through geometric M-/F-theory reductions as well as a heterotic string computation on $K3\times S^1$. The finite, one-loop coefficients are shown to be quantized and independent of the heavy masses, consistent with topological Chern-Simons structure and anomaly inflow. The cross-checks across field theory, geometry, and string theory reinforce the interpretation of these five-dimensional terms as probes of higher-dimensional consistency and demonstrate their potential utility in understanding M5-brane dynamics and quantum corrections to black hole entropy.

Abstract

We compute one-loop corrections to five-dimensional gauge and gravitational Chern-Simons terms induced by integrating out charged massive fields. The considered massive fields are spin-1/2 and spin-3/2 fermions, as well as complex two-forms with first order kinetic terms. Consistency with six-dimensional gravitational anomalies of (1,0) and (2,0) theories is shown by interpreting the massive fields as excited Kaluza-Klein modes in a circle compactification. The results are in accordance with the geometric predictions of the M-theory to F-theory duality as well as the comparison with an explicit one-loop computation in heterotic string theory compactified on K3 x S^1.

Figures (2)

• Figure 1: One-loop Feynman diagram involved in the computation of the Chern-Simons coefficient $k_{AFF}$. The external lines are three vectors $A$ with incoming momenta $p_1$, $p_2$, $p_3$ and polarisation vectors $e_1$, $e_2$, $e_3$. The internal lines can represent a massive spin-1/2 fermion, a massive self-dual tensor, or a massive spin-3/2 fermion. The loop momentum $k$ flows in the direction of the arrow.
• Figure 2: One-loop Feynman diagrams involved in the computation of the Chern-Simons coefficient $k_{ARR}$. The external line on top represents a vector $A$ with incoming momentum $p_0$ and polarisation vector $e_0$. The other external lines are gravitons $h$ with incoming momenta $p_1$, $p_2$ and symmetric polarisation tensors $e_1$, $e_2$. The internal lines can represent a massive spin-1/2 fermion, a massive self-dual tensor, or a massive spin-3/2 fermion.