implies that all $M_p - M_s$ factors vanish from the denominator in this case as well, leaving only denominators of the form $M_s + \sum_{j=1}^{N-1} M_{pj}$.

We have just shown that to order $O(1)$, only the term with $n = B = 0$ is relevant. We can thus rewrite (H.2) as

$$Z_{SU(N)}(M) = \frac{(2\pi)^{N-1}}{N!} \sum_{p_1, \dots, p_{N-1}=1}^{N_F} \left\{ \prod_{i<j<N} (M_{p_i} - M_{p_j})^2 \prod_{j=1}^{N-1} \prod_{s \neq p_j}^{N_F} \frac{\Gamma(iM_{p_j} - iM_s)}{\Gamma(1 - iM_{p_j} + iM_s)} \cdot \prod_{i<N} \left( M_{p_i} + \sum_{j=1}^{N-1} M_{pj} \right)^2 \prod_{s=1}^{N_F} \frac{\Gamma(-iM_s - i \sum_{j=1}^{N-1} M_{pj})}{\Gamma(1 + iM_s + i \sum_{j=1}^{N-1} M_{pj})} \right\} + O(1). \quad (H.7)$$

Terms where $p_i = p_j$ for $1 \le i \ne j \le N$ vanish because of the first product. We then use the relation $\Gamma(x) = \frac{1}{x}\Gamma(1+x)$ to separate the singularities from some Gamma functions. This cancels some factors coming from the vector multiplet determinant, yielding

$$Z_{SU(N)}(M) = \frac{(2\pi)^{N-1}}{N!} \sum_{1 \le p_1 \ne \dots \ne p_{N-1} \le N_F} \left\{ \frac{\prod_{i<N} [iM_{p_i} + \sum_{j=1}^{N-1} iM_{pj}]}{\prod_{s \notin \{p_j\}}^{N_F} [(-iM_s - i \sum_{j=1}^{N-1} M_{pj}) \prod_{j=1}^{N-1} (iM_{pj} - iM_s)]} \cdot \prod_{j=1}^{N-1} \prod_{s \ne p_j}^{N_F} g(iM_{pj} - iM_s) \prod_{s=1}^{N_F} g(-iM_s - i \sum_{j=1}^{N-1} M_{pj}) \right\} + O(1), \quad (H.8)$$

where

$$g(iM) = \frac{\Gamma(1 + iM)}{\Gamma(1 - iM)} = \exp \left( -2\gamma(iM) - 2 \sum_{j=1}^{\infty} \frac{\zeta(2j+1)}{(2j+1)} (iM)^{2j+1} \right) = 1 + O(M). \quad (H.9)$$

Noting that the sums in (H.8) only involve $(N-1)$ flavours $p_i$, we use the relation

$$\prod_{s \notin \{p_i\}}^{N_F} \frac{1}{-iM_s - \sum_{j=1}^{N-1} iM_{pj}} = \sum_{t \notin \{p_i\}}^{N_F} \left[ \frac{1}{-iM_t - \sum_{j=1}^{N-1} iM_{pj}} \prod_{s \notin \{p_i, t\}}^{N_F} \frac{1}{iM_t - iM_s} \right] \quad (H.10)$$

to obtain a sum over all $N$-element subsets $E = \{p_1, \dots, p_{N-1}, p_N = t\} \subseteq \{1, \dots, N_F\}$:

$$Z_{SU(N)}(M) = (2\pi)^{N-1} \sum_{\substack{E \subseteq \{1, \dots, N_F\} \\ \#E=N}} \left[ \frac{\prod_{p \in E} \prod_{s \notin E}^{N_F} \gamma(iM_p - iM_s)}{-i \sum_{p \in E} M_p} \cdot \frac{1}{N} \sum_{t \in E} \left\{ \prod_{p \in E \setminus \{t\}} \frac{iM_p - iM_t + \sum_{s \in E} iM_s}{iM_p - iM_t} \prod_{s=1}^{N_F} \frac{g(iM_s - iM_t)}{g(iM_s - iM_t + i \sum_{p \in E} M_p)} \right\} \right] + O(1), \quad (H.11)$$

The ratios of $g$ have the form

$$\frac{g(x)}{g(x + i \sum_{p \in E} M_p)} = 1 + \left[ \sum_{p \in E} iM_p \right] \quad (\text{series in powers of M}). \quad (H.12)$$
All terms beyond the zeroth order have a factor of $\sum_{p \in E} M_p$, cancelling the corresponding pole in (H.11). Those terms thus only have poles of the form $1/(iM_p - iM_t)$, and we have proven earlier that those poles cannot remain. Thus, replacing the ratios (H.12) by 1 only changes the partition function by terms which are regular at $M \to 0$, in other words, $O(1)$ terms. The sum over $t$ in (H.11) is then

$$ \frac{1}{N} \sum_{t \in E} \prod_{p \in E \setminus \{t\}} \frac{iM_p - iM_t + \sum_{s \in E} iM_s}{iM_p - iM_t} = \frac{1}{N \sum_{s \in E} iM_s} \oint \frac{dz}{2\pi i} \prod_{p \in E} \frac{z + iM_p + \sum_{s \in E} iM_s}{z + iM_p} = 1. \qquad (H.13) $$

Inserting this result back into (H.11) gives (7.3), up to the normalization factor $(2\pi)^{N-1}$.

This concludes the proof of (7.3), which in turn implies that the partition function of the $SU(N)$ and $SU(N_F - N)$ theories, with a particular matching of the mass parameters, are equal at order $O(1)$ in the limit of small masses and R-charge.

For any given value of $N$ and $N_F$, the Gamma functions appearing in (H.2) can be expanded in power series using

$$ \frac{(-1)^n \Gamma(x-n)}{\Gamma(1-x+n+|B|)} = \frac{1}{x} \frac{\exp \left[ -2\gamma x - 2 \sum_{j=1}^{\infty} \frac{\zeta(2j+1)}{(2j+1)} x^{2j+1} \right]}{\prod_{j=1}^{n} [j-x] \prod_{j=1}^{n+|B|} [j-x]} \qquad (H.14) $$

The partition function of the $SU(2)$ gauge theory with $N_F = 3$ fundamental chiral multiplets was computed in this manner up to order $O(M^2)$ and is, as expected, equal to the partition function of the theory of three free chiral multiplets, with masses given by (7.4). The signs coming from the chiral multiplet one-loop determinants (4.16) are crucial: the matching would otherwise fail with a difference of order 1.

The study of the $M \to 0$ limit which was just performed highlights the value of considering limits where the partition function has a pole.

The Gamma functions appearing in the series expression (H.2) of $Z_{SU(N)}(M)$ have poles at

$$ iM_t - iM_u = k \geq 0 \qquad (H.15) $$

$$ \sum_{j=1}^{N} iM_{s_j} = k \geq 0, \qquad (H.16) $$

where $k$ is an integer, and $t, u$ and $s_j$ are flavour indices. We ignore in this paper the poles (H.15): in fact, those poles cancel amongst the various terms in the full partition function, which is thus regular at $iM_t - iM_u = k$. We concentrate on the poles (H.16), labelled by a choice of $N$ chiral multiplets $E \subseteq \{1, \dots, N_F\}$, $\#E = N$ and a total vorticity $k$. For definiteness, we choose $E = \{1, \dots, N\}$, that is, $s_j = j$.

Since the $N = (2, 2)$ $SU(N)$ gauge theories with $N_F > N$ massless fundamental chiral multiplets flow to an infrared fixed point, the R-charge of each chiral multiplet should be
non-negative. However, note that $\sum_{s=1}^{N} iM_s = k \ge 0$ implies that the sum of the R-charges of the N chiral multiplets is $-2k \le 0$. Hence only the poles with $k=0$ are in the physical parameter space. The poles corresponding to a non-zero vorticity $k > 0$ can however be reached by analytically continuing with respect to the complex masses M.

The terms in (H.2) which are singular when $\sum_{s=1}^{N} iM_s = k$ are precisely those for which $1 \le p_1 \ne \dots \ne p_{N-1} \le N$, and for which the integer $n_N$ defined by

$$ \sum_{i=1}^{N} \left( n_i + \frac{|B_i|}{2} \right) = k \qquad (H.17) $$

is non-negative. As we will see, the number of such terms is finite. Defining $p_N$ by $\{p_1, \dots, p_N\} = \{1, \dots, N\}$, the residue of the partition function at this pole takes a symmetric form, where the Gamma functions involving sums of masses are recast in terms of the mass $iM_{p_N}$. Following the same procedure as in section 4.2, we introduce the coordinates $k_i^{\pm} = n_i + \frac{1}{2}|B_i| \pm \frac{1}{2}B_i \ge 0$ which factorize the summand into two identical contributions, leading to the expression

$$ \underset{iM_1+\dots+iM_N=k}{\mathrm{res}} Z_{SU(N)}(M) = (2\pi)^{N-1}(-1)^{N_F k} \left[ \prod_{p=1}^{N} \prod_{s=N+1}^{N_F} \gamma(-iM_s + iM_p) \right] [Z_k(\{1, \dots, N\}, M)]^2 \qquad (H.18) $$

with

$$ Z_k(\{1, \dots, N\}, M) = \sum_{k_1+\dots+k_N=k} \left[ \prod_{p=1}^{N} \frac{1}{k_p!} \prod_{p \ne s}^{N} \frac{1}{(iM_p - iM_s - k_p)_{k_s}} \prod_{p=1}^{N} \prod_{s=N+1}^{N_F} \frac{1}{(1+iM_s - iM_p)_{k_p}} \right]. \qquad (H.19) $$

The Seiberg-like duality (7.5) between $SU(N)$ and $SU(N_F - N)$ gauge theories on $S^2$ is satisfied in this limit if

$$ Z_k(\{1, \dots, N\}, M) = \pm Z_k(\{N+1, \dots, N_F\}, M'), \qquad (H.20) $$

with the mass matching (7.4).

The case of a zero total vorticity $k$ is elementary: since $(x)_0 = 1$,

$$ Z_0(\{1, \dots, N\}, M) = 1. \qquad (H.21) $$

The duality is shown in the case of a total vorticity $k=1$ using a one-dimensional contour integral: the sum over partitions $(k_1, \dots, k_N)$ of $k=1$ simply ranges over $N$ terms, interpreted as the residues at $N$ poles of a complex function with $N_F$ poles on the Riemann sphere. The resulting contour integral is thus equal to the sum over residues of the $N_F - N$
remaining poles, and this reproduces the desired dual object.

$$
\begin{align}
Z_1(\{1, \dots, N\}, \mathrm{M}) &= \sum_{p=1}^{N} \frac{1}{\prod_{s \neq p}^{N} (i\mathrm{M}_s - i\mathrm{M}_p) \prod_{s=N+1}^{N_F} (1 + i\mathrm{M}_s - i\mathrm{M}_p)} \nonumber \\
&= \oint \frac{\mathrm{d}z}{2\pi i} \frac{1}{\prod_{s=1}^{N} (i\mathrm{M}_s + z) \prod_{s=N+1}^{N_F} (1 + i\mathrm{M}_s + z)} \tag{H.22} \\
&= \sum_{p=N+1}^{N_F} \frac{(-1)^{N_F}}{\prod_{s=1}^{N} (1 + i\mathrm{M}'_s - i\mathrm{M}'_p) \prod_{N+1 \le s \ne p \le N_F} (i\mathrm{M}'_s - i\mathrm{M}'_p)} \nonumber \\
&= Z_1(\{N+1, \dots, N_F\}, \mathrm{M}'). \nonumber
\end{align}
$$

Since the objects in consideration are rather explicit finite sums, it should be possible to prove (H.20) for arbitrary $k \ge 0$. If one can additionally show that the two partition functions have the same asymptotic behavior at infinity, then the exact Seiberg-like duality between $Z_{SU(N)}(M)$ and $Z_{SU(N_F-N)}(M')$ follows, for arbitrary masses and R-charges, by noting that their difference is a bounded entire function, hence is a constant.

A different approach to studying the $SU(N)$ partition function is to note that integrating over $\xi$ and $\vartheta$ in (4.21) constrains $B$ and $a$ to be traceless, hence reproducing the corresponding $SU(N)$ partition function in the Coulomb branch representation,

$$ Z_{SU(N)}(M) = \int d\xi d\vartheta Z_{U(N)}(M, \tau). \qquad (H.23) $$

Integrating instead the Higgs branch representation of $Z_{U(N)}(M, \tau)$ singles out the Fourier components where the exponents of $z$ and $\bar{z}$ in (4.27) vanish. The $\vartheta$ constraint imposes that the total vorticity at the north pole be the same as that at the south pole. The $\xi$ constraint relates the masses to the vorticity $k$ precisely as $\sum_{s \in E} iM_s = k$ for a set $E$ of $N$ flavours. The residues (H.18) of $Z_{SU(N)}(M)$ at the corresponding poles are reproduced by the coefficient of $(z\bar{z})^{-\sum_{s \in E} iM_s + k}$ in the Higgs branch representation of the $U(N)$ partition function, which explains the appearance of the $k$-vortex partition functions in the factorization of $Z_{SU(N)}(M)$.
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