Friedrichs flux formulas

Sriramkumar's equations (2.12)–(2.16) Sriramkumar2001 use $(+,-,-,-)$, radius $\rho$, and fractional flux $\gamma$. Set $\rho =r,\gamma =\beta$. Its lower-index current is our Theorem thm:current. For its minimally coupled complex tensor, the coefficients are

in

With the covariant stress normalization held fixed, changing the metric sign reverses mixed components. This reproduces (eq:T-complex) at $\xi =0$. Its periodic-gauge two-point expression, equation (2.4), is twice our equal-time kernel after the gauge phase is included.

Spinelly–Bezerra de Mello SBdM2003, equations (30) and (36), agree with $F$ after the flux-even identification $\gamma =\beta$. Their equation (37), as printed in the inspected version, is

That display carries an extra ${\alpha }^{-2}$ relative to (30) and (36). It does not agree with $F$. The earlier sentence in this ledger, which copied (37) with denominator $48{\pi }^2{\alpha }^2{r}^2$ and called it an agreement, was our transcription error. I-B-VP-4 records the same printed discrepancy as SB2. Their equations (54)–(56), in their displayed stress normalization, have half our complex multiplicity; their coupling term agrees after that translation.

discrepancy retained. Their equation (55) prints $\gamma (1-{\gamma }^2)$, rather than ${\gamma }^2(1-\gamma {)}^2$, in the flux-dependent conformal term. The former is not symmetric under $\gamma \mapsto 1-\gamma$: it is $0.273$ at $0.3$ and $0.357$ at $0.7$. The latter is the polynomial obtained here and in Sriramkumar's equation (2.15). We identify an apparent typographical discrepancy in that displayed formula, not a demonstrated error in every result of the paper. No published erratum is asserted.

Bezerra de Mello–Bezerra–Saharian–Harutyunyan BdMBSH2015, equation (4.6), gives the zero-thickness massless current as an image sum plus an integral. For $D=3$, $q=1/\alpha$, their physical component $-j_2/r$, with $j_2=j_{\phi }/\alpha$, equals

in their signature convention. Their equation (4.7) is the $q=1$ limit. The reported numerical comparisons are retained in Appendix app:ABreports; they are not needed to prove the polynomial.

Guimarães Guimaraes1995 gives another explicit zero-temperature comparison. With $B=\alpha ,\gamma =\beta ,\rho =r$,

Its equations (19) and (30) give $F={\omega }_2/(2r^2)$ and, in order $(t,z,r,\phi )$,

Reordering gives (eq:T-complex). This comparison uses the stated overlap of its integral presentation, $B>1/2$; our all-angle result is proved independently.