In HW language, the heterotic anomaly cancellation corresponds classically to the condition $Q_{\mathrm{G}4}^{+}+Q_{\mathrm{G}4}^{-}=0$.

Strictly speaking, there is no Kähler potential in 5D theories of supergravity, but on the slice of the moduli space that we are interested in, we can define a function from which the moduli space metric is derived; see Ref. [9] for details.

The EOM for codimension one objects are the same irrespective of the total dimension of spacetime. Here, we use 4D $\mathcal{N}=1$ language as in [22, 23, 24] even though this language is not the most natural in five dimensions.

See Ref. [11] for a similar approach to obtain cosmic string solutions in 4D.

Since the Kähler metric on the moduli space is inherited from the hyperbolic metric on the upper half space, all corrections to $G_{\mathrm{eff}}$ have to be of the form of a Kähler transformation, thereby excluding the addition of modular weight ($-2$) to the classical $(T+\overline{T})$ term appearing in the Kähler potential.

The above is a minimal choice for the dependence of $G_{\mathrm{eff}}$ on the $j$ function. In principle, one could add to $G_{\mathrm{eff}}$ the quotient of two polynomials in $j$ of the same degree without affecting the asymptotics. This could, however, induce additional, unphysical zeros or poles for $G_{\mathrm{eff}}$ within the fundamental domain. It would be interesting to confirm the above form of $G_{\mathrm{eff}}$ by explicitly computing M5-brane instanton corrections to superpotential in 5D gauged supergravity.

The exponent accounts for the fact that $T$ is classically the overall volume of $K3\times T^2$.

Even though the geometric realization of the string is not obvious in this regime, there is still a Bogomol’nyi-Prasad-Sommerfeld string with tension given by the expression on the rhs of (29), which now has to be evaluated at $y\to \infty$.