In principle, the dictionary is not unique (there are $2^3=8$ possible dictionaries) reflected in the symmetries $a_0\leftrightarrow -a_0$, $a_1\leftrightarrow -a_1$, and $(\mu ,\epsilon )\leftrightarrow -(\mu ,\epsilon )$. Here, we specify a choice that will then determine uniquely the ingoing solution at the event horizon and the outgoing one at infinity, as specified in (11).

This is a convergent series [38] whose coefficients can be computed explicitly (see [23, 24] for details).

These quantities ${\mu }_1$ and ${\mu }_2$, together with the parameter $\mu$, are the masses of the hypermultiplets in the fundamental representation from the gauge theory point of view.

Strictly speaking the GY formula computes the regularized determinant without zero modes ${\det }^{\prime }$. For $s=1$, 2 the operator we consider does not display zero modes so that ${\det }^{\prime }=\det$.

From the gauge theory viewpoint, this procedure corresponds to the holomorphic decoupling of the hypermultiplet mass ${\mu }_2$, and it produces the $N_f=2$ theory from the original $N_f=3$ one. The new parameter $\mathrm{\Lambda }\equiv -8M^2\omega (\omega -m{\mathrm{\Omega }}_H^{\mathrm{ext}})$, with ${\mathrm{\Omega }}_H^{\mathrm{ext}}=1/(2M)$, corresponds to the instanton-counting parameter of the $N_f=2$ theory.

In detail, this extra factor can be seen as the combination of an overall factor ${\epsilon }^{\frac{1}{2}-\mu }$ in the normalization of the irregular semiclassical block around infinity and a factor ${\epsilon }^a$ coming from the irregular semiclassical block around zero with shifted intermediate momentum, arising in the confluence procedure between the singularities at $z=0$ and $z=1$. Specifically, the first factor can be seen by taking the semiclassical limit of the $\mathrm{\Lambda }$ factors in the first line of formula (3.2.7) in [40]. In particular, considering the semiclassical limit $b\to 0$, the term $b\mathrm{\Lambda }$ corresponds to our parameter $\epsilon$, and the semiclassical limit of ${\mathrm{\Delta }}_{2,1}-\theta b\mu$ in [40] becomes $-\frac{1}{2}+\mu$ in our notation. This, consistently with our choices of the local solution around $z=\infty$, cancels the factor ${\epsilon }^{\frac{1}{2}-\mu }$. The other factor can be seen by taking the semiclassical limit of the ${\mathrm{\Lambda }}_2$ term in formula (3.4.4) in [40], when considering the shift of the intermediate momentum. Indeed, ${\mathrm{\Delta }}_{\sigma }\to \sigma a$ in the semiclassical limit, and this simplifies with the corresponding factor in the connection formula (12). This, in turn, implies the simplification of the term ${\epsilon }^a$ in (28) after taking the confluent limit.

The Matsubara frequencies associated with the anti-QNMs are obtained by taking the asymptotic behavior ${\psi }_{+}^{(1)}$ in Eq. (1) in Supplemental Material, which, together with the corresponding normalization, amounts to impose $2a_1=k$, $k\ge 0$, so that ${\omega }_k^{(M)}(-s)={\overline{\omega }}_k^{(M)}(s)$.