Determine the twisted-boundary-condition three-loop coefficient

Determine the analytic form of the three-loop coefficient \(\beta(\eta)\) for the bosonic membrane on \(\mathbb{R}^2\times S^1\) with twist parameter \(\eta\), including its dependence on single-valued polylogarithms or level-two Euler sums, and thereby complete the three-loop result for twisted boundary conditions.

Background

For twisted boundary conditions, the ordinary $\zeta(3)$ factors in the three-loop six-vertex and chain contributions are replaced by a twist-dependent function $\Phi(\eta)$. The remaining basketball contribution is governed by a coefficient $\beta(\eta)$.

The untwisted value $\beta(0)$ is conjecturally expressed in terms of $\zeta(9)$ and $\zeta(3)^3$, while the antiperiodic value involves constants outside the ordinary weight-nine basis because the twist mixes periods and generates level-two Euler sums. The paper leaves the general analytic form of $\beta(\eta)$ unresolved, suggesting an ansatz in terms of single-valued polylogarithms.