$C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories

TL;DR

The paper resolves the ambiguity in the 6d (1,0) Weyl–anomaly relations by fixing the free parameter $\xi$ through a new data point: the Weyl coefficient $c_3$ for the higher-derivative vector multiplet, connected to $C_T$ via Renyi entropy. It develops and applies a heat-kernel/Seeley–DeWitt framework to compute $C_T$ and Renyi entropy for higher-derivative conformal fields (scalars, spinors, vectors) in even dimensions, including independent cross-checks on Seeley–DeWitt coefficients on Einstein backgrounds. The results favor the $\xi=-\frac{8}{9}$ solution and yield explicit anomaly coefficients for the 6d vector multiplet and its generalizations, establishing consistency with supersymmetry constraints. These advances enhance exact control over Weyl anomalies in 6d (1,0) theories and demonstrate the utility of $S^1_q\times \mathbb H^{d-1}$ partition-function methods for $C_T$ and Renyi-entropy calculations in higher-derivative conformal theories.

Abstract

In arXiv:1510.02685 we proposed linear relations between the Weyl anomaly $c_1, c_2, c_3$ coefficients and the 4 coefficients in the chiral anomaly polynomial for (1,0) superconformal 6d theories. These relations were determined up to one free parameter $ξ$ and its value was then conjectured using some additional assumptions. A different value for $ξ$ was recently suggested in arXiv:1702.03518 using a different method. Here we confirm that this latter value is indeed the correct one by providing an additional data point: the Weyl anomaly coefficient $c_3$ for the higher derivative (1,0) superconformal 6d vector multiplet. This multiplet contains the 4-derivative conformal gauge vector, 3-derivative fermion and 2-derivative scalar. We find the corresponding value of $c_3$ which is proportional to the coefficient $C_T$ in the 2-point function of stress tensor using its relation to the first derivative of the Renyi entropy or the second derivative of the free energy on the product of thermal circle and 5d hyperbolic space. We present some general results of computation of the Renyi entropy and $C_T$ from the partition function on $S^1 \times \mathbb H^{d-1}$ for higher derivative conformal scalars, spinors and vectors in even dimensions. We also give an independent derivation of the conformal anomaly coefficients of the higher derivative vector multiplet from the Seeley-DeWitt coefficients on an Einstein background.