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A Bethe Ansatz Type Formula for the Superconformal Index

Abstract

Inspired by recent work by Closset, Kim, and Willett, we derive a new formula for the superconformal (or supersymmetric) index of 4D \({\mathcal {N}}=1\) theories. Such a formula is a finite sum, over the solution set of certain transcendental equations that we dub Bethe Ansatz Equations, of a function evaluated at those solutions.

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Notes

  1. 1.

    There is a small caveat: the IR superconformal R-symmetry must be visible in the UV, i.e., it should not be accidental.

  2. 2.

    The two recent papers [9, 10] also investigate the entropy of BPS black holes in \(\hbox {AdS}_5\).

  3. 3.

    Notice that the superconformal index, up to a change of variables reviewed in Sect. 3.1, is a single-valued function of the fugacities, while the partition function is not [24].

  4. 4.

    An alternative way to obtain the integral formula is to use supersymmetric localization [30]. Indeed, the supersymmetric partition function Z of the theory on a primary Hopf surface \({\mathcal {H}}_{p,q}\simeq S^1\times S^3\) can be computed with localization [31, 32] and it is related to the superconformal index through \(Z=e^{-E_\text {SUSY}}{\mathcal {I}}\), where \(E_\text {SUSY}\) is the supersymmetric Casimir energy [23, 24].

  5. 5.

    In particular, let us stress that the condition \(w\cdot [{{\hat{u}}}] \ne [{{\hat{u}}}]\) in the definition of \({\mathfrak {M}}_\text {BAE}\) could be relaxed with no harm: in that case, we would simply include more poles in the sum, whose residues however combine to zero.

  6. 6.

    In counting the multiplicity one may worry that there could be different choices of rs that give the same \(abm+as+br\) for fixed m. This is equivalent to finding non-trivial solutions to the equation \(as+br=as'+br'\). However, it is easy to see that, as long as \(0\le r, r'\le a-1\) and \(0\le s, s'\le b-1\), such an equation has no non-trivial solution in \(\mathbb {Z}\).

  7. 7.

    The integers pq appearing in this “Appendix” should not be confused with the complex angular fugacities appearing in the rest of the paper.

  8. 8.

    Indeed, \({\hat{\rho _I}}\) is guaranteed to be a weight (and in particular a root) only if the spin \(j_I\) is integer.

  9. 9.

    It is easy to prove that \(\Lambda _{\text {roots},J}\) and \({{\overline{\Lambda }}}_{\text {roots},J}\) are disjoint. Suppose, on the contrary, that there exists some common element \({\hat{\beta }}_J+\ell _J {\hat{\alpha }} = -{\hat{\beta _J}} + k_J {\hat{\alpha }}\) for some \(\ell _J, k_J\). This would imply that \({\hat{\beta }}_J = (k_J-l_J){\hat{\alpha }}/2\), but since \(({\hat{\beta }}_J,{\hat{\alpha }})=0\), then \({\hat{\beta _J}}=0\). Since the only roots proportional to \({\hat{\alpha }}\) are \(-{\hat{\alpha }}\) and \({\hat{\alpha }}\) itself, we have reached a contradiction.

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Acknowledgements

F.B. is supported in part by the MIUR-SIR Grant RBSI1471GJ “Quantum Field Theories at Strong Coupling: Exact Computations and Applications”.

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Correspondence to Francesco Benini.

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Communicated by X. Yin

Appendices

Numerical Semigroups and the Fröbenius Problem

Given a set of non-negative integer numbers \(\{a_1,\dots ,a_r\}\), the Fröbenius problem consists in classifying which integers can (or cannot) be written as non-negative integer linear combinations of those. This problem has deep roots in the theory of numerical semigroups.

A semigroup is an algebraic structure \({\mathcal {R}}\) endowed with an associative binary operation. Analogously to groups, we denote it as \(({\mathcal {R}},*)\). On the other hand, differently from the case of a group, no requirement on the presence of identity and inverse elements is made. A numerical semigroup is an additive semigroup \(({\mathcal {R}},+)\), where \({\mathcal {R}}\) consists of all non-negative integers \(\mathbb {Z}_{\ge 0}\) except for a finite number of positive elements (thus \(0\in {\mathcal {R}}\)). The set \(\{n_1,\dots ,n_t\}\) is called a generating set for \(({\mathcal {R}},+)\) if all elements of \({\mathcal {R}}\) can be written as non-negative integers linear combinations of \(n_1,\dots ,n_t\). We then denote the semigroup with the presentation

(A.1)

Among all possible presentations of \({\mathcal {R}}\), there exists a unique minimal presentation, which contains the minimal number of generators. Such a number is called the embedding dimension\(e({\mathcal {R}})\) of the semigroup. We now define other important quantities associated with numerical semigroups:

  • The multiplicity\(m({\mathcal {R}})\) is the smallest non-zero element of \({\mathcal {R}}\).

  • The set of gaps\({\overline{{\mathcal {R}}}} = \mathbb {N}\setminus {\mathcal {R}}\) is the set of positive integers which are not contained in \({\mathcal {R}}\). Equivalently, the gaps are defined as all natural numbers which cannot be written as non-negative integer linear combination of the generators \(n_1,\dots ,n_t\) of \({\mathcal {R}}\).

  • The set of gaps \({\overline{{\mathcal {R}}}}\) is always a finite set. Its largest element is the Fröbenius number\(F({\mathcal {R}})\). Alternatively, given a presentation , the Fröbenius number is defined as the largest integer which cannot be written as a non-negative integer linear combination of the generators.

  • The genus\(\chi ({\mathcal {R}})\) is the number of gaps, i.e., it is the order of the set of gaps: \(\chi ({\mathcal {R}})=\left| {\overline{{\mathcal {R}}}}\right| \).

  • The weight\(w({\mathcal {R}})\) is the sum of all gaps: \(w({\mathcal {R}})=\sum _{k\in {\overline{{\mathcal {R}}}}}k\).

  • The following inequalities hold:

    $$\begin{aligned} e({\mathcal {R}})\le m({\mathcal {R}})\qquad \qquad F({\mathcal {R}})\le 2\chi ({\mathcal {R}})-1. \end{aligned}$$
    (A.2)

    In particular, if \(x\in {\mathcal {R}}\), then \(F({\mathcal {R}})-x\notin {\mathcal {R}}\).

We now study the case where the embedding dimension is \(e({\mathcal {R}})=2\), i.e., the minimal presentation is defined by two positive integers ab with \(\text {gcd}(a,b)=1\). The associated numerical semigroup is denoted by and the set of gaps is \({\overline{{\mathcal {R}}}}(a,b)=\mathbb {N}\setminus {\mathcal {R}}(a,b)\). The multiplicity is simply \(m(a,b)=\min \{a,b\}\), whereas the Fröbenius number is given by

$$\begin{aligned} F(a,b)=ab-a-b. \end{aligned}$$
(A.3)

The genus and the weight are

$$\begin{aligned} \chi (a,b)=\frac{(a-1)(b-1)}{2}\qquad \qquad w(a,b)=\frac{(a-1)(b-1)(2ab-a-b-1)}{12}.\nonumber \\ \end{aligned}$$
(A.4)

Thanks to the properties of \({\mathcal {R}}(a,b)\), one can prove the following identities:

$$\begin{aligned} \begin{aligned}&\prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(zh^{as+br};h^{ab})_\infty = \frac{(z;h)_\infty }{\prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}(1-zh^k)}\\&\prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(z^{-1}h^{ab-as-br};h^{ab})_\infty = (z^{-1}h;h)_\infty \prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}(1-z^{-1}h^{-k}) \;. \end{aligned} \end{aligned}$$
(A.5)

Proof

We begin with the first identity. Using the definition of the q-Pochhammer symbol we can write:

$$\begin{aligned} \prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(zh^{as+br};h^{ab})_\infty = \prod _{n=0}^\infty \prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(1-zh^{abn+as+br})\;. \end{aligned}$$
(A.6)

Using that ab are coprime, the set of integers \(\bigl \{ as + br \,\big |\, r = 0, \dots , a-1,\; s = 0, \dots , b-1 \bigr \}\) covers once and only once every class modulo ab. It follows that the set of exponents \(\{abn + as + br\}\) is precisely \({\mathcal {R}}(a,b)\). Then

$$\begin{aligned} \prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(zh^{as+br};h^{ab})_\infty = \!\! \prod _{k\in {\mathcal {R}}(a,b)} \! (1-zh^{k}) = \frac{\prod _{k=0}^\infty (1-zh^{k})}{\prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}(1-zh^{k})} =\frac{(z;h)_\infty }{\prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}(1-zh^k)},\nonumber \\ \end{aligned}$$
(A.7)

which proves the first equality in (3.36).

The proof of the second identity is a bit trickier. The key point is to notice that the set \(\{as + br\}\) does not contain any element of \(\overline{{\mathcal {R}}}(a,b)\) and thus

$$\begin{aligned} \{as + br\} = \bigl \{ k + \Delta _k ab \,\big |\, k = 0, \dots , ab-1 \bigr \} \qquad \text {with}\qquad \Delta _k = {\left\{ \begin{array}{ll} 0 &{}\text {if } k \in {\mathcal {R}}(a,b) \\ 1 &{}\text {if } k \in \overline{{\mathcal {R}}}(a,b). \end{array}\right. }\nonumber \\ \end{aligned}$$
(A.8)

This implies that \(\{ ab - as - br \} = \bigl \{ -k + (1-\Delta _k) ab \,\big |\, k = 0, \dots , ab-1 \bigr \}\). Finally, including the freedom of choosing \(n \ge 0\), we find that the set of exponents is

$$\begin{aligned} \{ abn + ab - as - ar \} = (-\overline{{\mathcal {R}}}) \cup {\mathbb {Z}}_{>0}. \end{aligned}$$
(A.9)

Then

$$\begin{aligned}&\prod _{r=0}^{a-1}\prod _{s=0}^{b-1}(z^{-1}h^{ab-as-br};h^{ab})_\infty = \prod _{n=0}^\infty \prod _{r=0}^{a-1}\prod _{s=0}^{b-1} \bigl ( 1-z^{-1}h^{ab(n+1)-as-br} \bigr ) \nonumber \\&\quad {} = \prod _{k \in \overline{{\mathcal {R}}}(a,b)} ( 1 - z^{-1} h^{-k}) \times \prod _{k=1}^\infty (1- z^{-1} h^{k}) = (z^{-1}h;h)_\infty \prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}(1-z^{-1}h^{-k}) \;. \end{aligned}$$
(A.10)

This completes the proof of (3.36). \(\square \)

Thanks to the definition of \(\theta _0(u;\omega )\), we can apply (3.36) and we obtain that

$$\begin{aligned} \prod _{r=0}^{a-1}\prod _{s=0}^{b-1} \theta _0 \bigl ( u+(as+br)\omega ;\omega \bigr ) = \prod _{k\in {\overline{{\mathcal {R}}}}(a,b)}\frac{(1-z^{-1}h^{-k})}{(1-zh^k)} \; \theta _0(u;\omega ) = \frac{1}{(-z)^{\chi (a,b)} \; h^{w(a,b)}} \; \theta _0(u;\omega ). \end{aligned}$$
(A.11)

A Dense Set

Here we show that the set of points (pq) such that

$$\begin{aligned} q^a = p^b \qquad \text {for coprime } a,b\in {\mathbb {N}}\end{aligned}$$
(B.1)

is dense in \(\bigl \{ |p|<1,\, |q|<1\bigr \}\). We write the fugacities in terms of chemical potentials, \(p = e^{2\pi i \sigma }\) and \(q = e^{2\pi i \tau }\) with \({\text {Im}}\sigma , {\text {Im}}\tau >0\), and for the sake of this argument we choose the determination on the “strip” \(0 \le {\text {Re}}\sigma , {\text {Re}}\tau <1\). Then the condition (B.1) is equivalent to

$$\begin{aligned} a(\tau + n) = b(\sigma + m) \end{aligned}$$
(B.2)

for some \(m,n\in {\mathbb {Z}}\) and \(a,b\in {\mathbb {N}}\) coprime.

We choose an arbitrary point \((\tau _0, \sigma _0)\) in the strip and ask if we can find another point \((\tau ,\sigma )\), arbitrarily close, that satisfies (B.2). Consider a straight line in the complex plane that starts from 0 and goes through \(\tau _0 + n\) for some integer n. When winding once around the strip, this line has an imaginary excursion

$$\begin{aligned} \Delta y = \frac{{\text {Im}}\tau _0}{{\text {Re}}\tau _0 + n}. \end{aligned}$$
(B.3)

We can make this quantity arbitrarily small by choosing n sufficiently large. We define \(\sigma '\) as the closest point to \(\sigma _0\) that lies on the image of the line on the strip modulo 1, and has \({\text {Re}}\sigma '={\text {Re}}\sigma _0\). It is clear that

$$\begin{aligned} |\sigma ' - \sigma _0| = \bigl | {\text {Im}}\sigma ' - {\text {Im}}\sigma _0 \bigr | \le \Delta y/2, \end{aligned}$$
(B.4)

and, by construction, \((\sigma ' + m) = t (\tau _0 + n)\) for some \(m,n\in {\mathbb {Z}}\) and \(t\in {\mathbb {R}}_+\). We see that \(|\sigma '-\sigma _0|\) can be made arbitrarily small by increasing n. Next, we approximate t by a fraction \(a/b \in {\mathbb {Q}}_+\). This, for a/b sufficiently close to t, defines a point \(\sigma \) in the strip by

$$\begin{aligned} (\sigma + m) = \frac{a}{b} \, (\tau _0 + n). \end{aligned}$$
(B.5)

It is clear that \(\sigma \) can be made arbitrarily close to \(\sigma '\) by approximating t sufficiently well with a/b. We have thus found a pair \((\sigma , \tau =\tau _0)\), arbitrarily close to \((\sigma _0,\tau _0)\), that satisfies the constraint (B.2).

Weyl Group Fixed Points

In this “Appendix” we prove that \({\mathcal {Z}}_\text {tot}(u; \xi , \nu _R, a\omega , b\omega )\) vanishes when evaluated at a point \({{\hat{u}}}\) which is fixed, on a torus of modular parameter \(\omega \), by a non-trivial element w of the Weyl group \({\mathcal {W}}_G\):

$$\begin{aligned} w \cdot [{{\hat{u}}}] = [{{\hat{u}}}]. \end{aligned}$$
(C.1)

This implies that the solutions to the BAEs (3.9) which are fixed points on the torus of an element of the Weyl group, can be excluded from the set \({\mathfrak {M}}_\text {BAE}\)—as is done in (3.9)—because they do not contribute to the BA formula (3.10) for the superconformal index.

The rank-one case

Let us first consider the case that the gauge group G has rank one, i.e., that \({\mathfrak {g}}= \mathfrak {su}(2)\). Then there are only two roots, \(\alpha \) and \(-\alpha \), and the Weyl group is \({\mathcal {W}}_G = \{1, s_\alpha \} \cong {\mathbb {Z}}_2\) where \(s_\alpha \) is the unique non-trivial Weyl reflection along the root \(\alpha \):

$$\begin{aligned} s_\alpha (u) = -u \qquad \qquad \forall \; u \in {\mathfrak {h}}. \end{aligned}$$
(C.2)

We choose a basis element \(\{H\}\) for the Cartan subalgebra \({\mathfrak {h}}\) such that \(\rho (H) \equiv \rho \in {\mathbb {Z}}\) for any weight \(\rho \in \Lambda _\text {weight}\). In this canonical basis \(\alpha =2\) (while the fundamental weight is \(\lambda = 1\)). The solutions to \(s_\alpha \cdot [{{\hat{u}}}] = [{{\hat{u}}}]\) are given byFootnote 7

$$\begin{aligned} {{\hat{u}}} = \frac{p+q\omega }{2} \qquad \qquad \text {with}\quad p,q \in {\mathbb {Z}}. \end{aligned}$$
(C.3)

Choosing a representative for \([{{\hat{u}}}]\) in the fundamental domain of the torus, the inequivalent solutions are with \(p=0,1\) and \(q=0,1\).

The representations of \({\mathfrak {s}}{\mathfrak {u}}(2)\) are labelled by a half-integer spin \(j\in \mathbb {N}/2\) and their weights are \(\rho \in \big \{ \ell \alpha \,\big |\, \ell = -j, -j +1 , \dots , j - 1, j\big \}\). Therefore, exploiting the expression in (3.58), the function \({\mathcal {Z}}\) reduces to

$$\begin{aligned}&{\mathcal {Z}}(u;\xi ,\nu _R,a\omega ,b\omega ) \nonumber \\&\quad {} = \theta _0\big (\alpha (u);a\omega \big ) \, \theta _0\big ( {-\alpha (u)};b\omega \big ) \prod _{a} \prod _{\ell _a=-j_a}^{j_a} {\widetilde{\Gamma }} \big (\ell _a\alpha (u)+\omega _a(\xi )+r_a\nu _R;a\omega ,b\omega \big ) \;. \end{aligned}$$
(C.4)

Moreover, the function \({\mathcal {Z}}_\text {tot}\) defined in (3.11) is a single sum over \(m=1, \dots , ab\).

We want to prove that \({\mathcal {Z}}_\text {tot}({{\hat{u}}} ;\xi ,\nu _R,a\omega ,b\omega ) = 0\). To do that, we construct an involutive map \(\gamma : m \mapsto m'\) acting on the set of integers \(\{1, \dots , ab\}\) according to

$$\begin{aligned} m' = m \mod b,\qquad \qquad m' = q-m \mod a, \end{aligned}$$
(C.5)

which define \(m'\) uniquely. It will be convenient to introduce the numbers \(r,s \in {\mathbb {Z}}\) such that \(m' = m + sb = q-m + ra\). The map \(\gamma \) has the property that

$$\begin{aligned} m' - q/2 = {\left\{ \begin{array}{ll} m - q/2 &{}\mod b \,, \\ -(m-q/2) &{} \mod a \,, \end{array}\right. } \quad = m- q/2 + sb = -(m - q/2) + ra.\nonumber \\ \end{aligned}$$
(C.6)

We will prove that

$$\begin{aligned} {\mathcal {Z}}\big ( {{\hat{u}}} - m' \omega ; \xi , \nu _R, a\omega , b\omega \big ) = - {\mathcal {Z}}\big ( {{\hat{u}}} - m\omega ; \xi , \nu _R, a\omega , b\omega \big ). \end{aligned}$$
(C.7)

In particular, the sum over m inside \({\mathcal {Z}}_\text {tot}\) splits into a sum over the fixed points of \(\gamma \) and a sum over the pairs of values related by \(\gamma \). The property (C.7) guarantees that each term in those sums vanishes, implying that \({\mathcal {Z}}_\text {tot}\) vanishes.

Let us adopt the notation

$$\begin{aligned} {\mathcal {Z}}_m \,\equiv \, {\mathcal {Z}}({{\hat{u}}}-m\omega ;\xi ,\nu _R,a\omega ,b\omega ) = {\mathcal {Z}}\bigl ( p/2 - (m-q/2) \omega ;\xi ,\nu _R,a\omega ,b\omega \bigr ).\nonumber \\ \end{aligned}$$
(C.8)

We define the vector multiplet and the chiral multiplet contribution, respectively, as

$$\begin{aligned} {\mathcal {A}}_m= & {} \theta _0\bigl (\alpha (p/2)-\alpha (m-q/2)\omega ;a\omega \bigr ) \, \theta _0\bigl (-\alpha (p/2)+\alpha (m-q/2)\omega ;b\omega \bigr ) \nonumber \\ {\mathcal {B}}_m= & {} \prod _{a} \prod _{\ell _a=-j_a}^{j_a} {\widetilde{\Gamma }}\Bigl ( \ell _a\alpha (p/2)- \ell _a\alpha (m-q/2)\omega +\omega _a(\xi )+r_a\nu _R;a\omega ,b\omega \Bigr ),\nonumber \\ \end{aligned}$$
(C.9)

such that \({\mathcal {Z}}_m = {\mathcal {A}}_m \, {\mathcal {B}}_m\). Then \({\mathcal {Z}}_\text {tot}\) evaluated on \({{\hat{u}}}\) can be expressed as

$$\begin{aligned} {\mathcal {Z}}_\text {tot} \Bigl ( \frac{p+q\omega }{2}; \xi ,\nu _R,a\omega ,b\omega \Bigr ) = \sum _{m=1 \,:\, m'=m}^{ab} {\mathcal {Z}}_m + \sum _{(m,m') \,:\, m' \ne m} \bigl ( {\mathcal {Z}}_m + {\mathcal {Z}}_{m'} \bigr ).\nonumber \\ \end{aligned}$$
(C.10)

Our goal is to show that \({\mathcal {Z}}_{m'} = - {\mathcal {Z}}_m\).

We begin by considering the contribution of \({\mathcal {A}}_m\). Using (C.6) we can write

$$\begin{aligned} {\mathcal {A}}_{m'}= & {} \theta _0 \bigl ( p + (2m-q)\omega - 2ra\omega ; a\omega \bigr ) \, \theta _0\bigl ( -p+ (2m-q)\omega + 2sb\omega ; b\omega \bigr ) \nonumber \\= & {} \theta _0 \bigl ( -p - (2m-q)\omega + (2r+1)a\omega ; a\omega \bigr ) \, \theta _0\bigl ( -p+ (2m-q)\omega + 2sb\omega ; b\omega \bigr ).\nonumber \\ \end{aligned}$$
(C.11)

In the second equality we used the second relation in (3.26). Using the first relation in (3.26), the identity \(2m-q-ra+sb=0\) and reinstating \(\alpha \), with some algebra we obtain

$$\begin{aligned} {\mathcal {A}}_{m'} = - \, e^{-2\pi i \, \alpha (r) \, \alpha (s) \, \nu _R} \; {\mathcal {A}}_m. \end{aligned}$$
(C.12)

Then we turn to \({\mathcal {B}}_m\) and, using (C.6), write

$$\begin{aligned} \begin{aligned} {\mathcal {B}}_{m'}&= \prod _a \prod _{\ell _a=-j_a}^{j_a} \widetilde{\Gamma }\Bigl ( \ell _a p + \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R - 2\ell _a r a \omega ; a\omega , b\omega \Bigr ) \\&= \prod _a \prod _{\ell _a=-j_a}^{j_a} \widetilde{\Gamma }\Bigl ( \ell _a p - \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R + 2\ell _a r a \omega ; a\omega , b\omega \Bigr ). \end{aligned} \end{aligned}$$
(C.13)

We recall that \(j_a\) can be integer or half-integer. In the second equality we simply redefined \(\ell _a \rightarrow -\ell _a\) and shifted the argument by the integer \(2\ell _a p\). Using the identity (3.30) repeatedly and distinguishing the cases \(\ell _a \lessgtr 0\), we obtain

$$\begin{aligned} {\mathcal {B}}_{m'} = \Theta \times {\mathcal {B}}_m \end{aligned}$$
(C.14)

where the factor \(\Theta \) equals

$$\begin{aligned} \Theta = \prod _a \prod _{\ell _a>0}^{j_a} \prod _{k=0}^{2\ell _a r-1} \frac{ \theta _0\bigl ( \ell _a p - \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R + ka\omega ; b\omega \bigr ) }{ \theta _0\bigl ( - \ell _a p + \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R + (k-2\ell _a r) a\omega ; b\omega \bigr ) }.\nonumber \\ \end{aligned}$$
(C.15)

The second product starts from 1 or \(\frac{1}{2}\) depending on \(j_a\) being integer or half-integer. Using \(2m-q-ra+sb=0\) at denominator and shifting the arguments by integers, we rewrite

$$\begin{aligned} \Theta = \prod _a \prod _{\ell _a>0}^{j_a} \prod _{k=0}^{2\ell _a r-1} \frac{ \theta _0\bigl ( \ell _a p - \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R + ka\omega ; b\omega \bigr ) }{ \theta _0\bigl ( \ell _a p - \ell _a(2m-q)\omega + \omega _a(\xi ) + r_a \nu _R + ka\omega - 2 \ell _a sb \omega ; b\omega \bigr ) }.\nonumber \\ \end{aligned}$$
(C.16)

Finally we use the first relation in (3.26) at denominator, to obtain

$$\begin{aligned} \Theta = \prod _a \prod _{\ell _a>0}^{j_a} (-1)^{4\ell _a^2 r s + 8\ell _a^3 r s p} \, e^{-8\pi i \ell _a^2 r s \, \omega _a(\xi )} \, e^{-8\pi i \ell _a^2 r s (r_a-1) \nu _R}. \end{aligned}$$
(C.17)

Reinstating the root \(\alpha \), this factor can be written as

$$\begin{aligned} \Theta = \prod _{a} \prod _{\ell _a>0}^{j_a} (-1)^{\ell _a^3 \alpha (r) \alpha (s) \alpha (p)} \times \prod _{a,\, \rho _a\in {\mathfrak {R}}_a} e^{\pi i \rho _a(r) \rho _a(s) \left( \frac{1}{2} - \omega _a(\xi ) - (r_a-1) \nu _R \right) }.\nonumber \\ \end{aligned}$$
(C.18)

Combining with (C.12), the factor picked up by \({\mathcal {Z}}\) can be expressed in terms of the anomaly coefficients (3.18) and (3.19):

$$\begin{aligned} {\mathcal {Z}}_{m'} = - \, e^{2\pi i \phi } \; e^{\pi i rs \left( \frac{1}{2} {\mathcal {A}}^{ii} - {\mathcal {A}}^{ii\alpha } \xi _\alpha - {\mathcal {A}}^{iiR} \nu _R \right) } \; {\mathcal {Z}}_m. \end{aligned}$$
(C.19)

Here i is the gauge index taking a single value. We recall the anomaly cancelation conditions \({\mathcal {A}}^{ii\alpha } = {\mathcal {A}}^{iiR} = 0\) and \({\mathcal {A}}^{ii} \in 4{\mathbb {Z}}\), implying that the second exponential equals 1. In the first exponential we defined

$$\begin{aligned} \phi = \frac{1}{2} \, \alpha (r) \, \alpha (s)\, \alpha (p) \sum _a \sum _{\ell _a>0}^{j_a} \ell _a^3 = 4rsp \sum _a \sum _{\ell _a>0}^{j_a} \ell _a^3. \end{aligned}$$
(C.20)

It remains to show that \(\phi \in {\mathbb {Z}}\), so that also the first exponential equals 1.

For each chiral multiplet in the theory, indicized by a, in order to evaluate the second sum in (C.20) we should distinguish different cases:

$$\begin{aligned} \psi _j \,\equiv \, 4\sum _{\ell >0}^j \ell ^3 = {\left\{ \begin{array}{ll} j^2(j+1)^2 &{} \in 4{\mathbb {Z}}\qquad \;\;\, \text {if } j \in {\mathbb {Z}}\\ 2(k+1)^2(8k^2+16k+7) &{} \in 2{\mathbb {Z}}\qquad \;\;\, \text {if } j = 2k+\tfrac{3}{2} \in 2{\mathbb {Z}}+ \tfrac{3}{2} \\ \frac{1}{2} \, (2k+1)^2(8k^2+8k+1) &{} \in 4{\mathbb {Z}}+ \tfrac{1}{2} \quad \text {if } j = 2k+\tfrac{1}{2} \in 2{\mathbb {Z}}+ \tfrac{1}{2}. \end{array}\right. }\nonumber \\ \end{aligned}$$
(C.21)

Therefore, chiral multiplets whose gauge representation has spin \(j \in {\mathbb {Z}}\) or \(j \in 2{\mathbb {Z}}+\frac{3}{2}\) give integer contribution to \(\phi \). On the other hand, chiral multiplets with \(j \in 2{\mathbb {Z}}+ \frac{1}{2}\) can give half-integer contribution. However, because of the Witten anomaly [34], the total number of such multiplets must be even. This is reproduced by the condition (3.21) on the pseudo-anomaly coefficient \({\mathcal {A}}^{ii}\). Indeed, the contribution of a chiral multiplet to the pseudo-anomaly is

$$\begin{aligned} {\mathcal {A}}^{ii}_{(j)} = \sum _{\ell =-j}^j (2\ell )^2 = \frac{4}{3}\, j(j+1)(2j+1) \,\in \, {\left\{ \begin{array}{ll} 4{\mathbb {Z}}&{} \text { if } j \in {\mathbb {Z}}\text { or } j \in 2{\mathbb {Z}}+ \frac{3}{2} \\ 4{\mathbb {Z}}+ 2 &{} \text {if }j \in 2{\mathbb {Z}}+ \frac{1}{2}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(C.22)

and the condition \({\mathcal {A}}^{ii} \in 4{\mathbb {Z}}\) requires that the total number of chiral multiplets with \(j\in 2{\mathbb {Z}}+\frac{1}{2}\) be even. This implies that \(\phi \in {\mathbb {Z}}\), and thus that \({\mathcal {Z}}_{m'} = - {\mathcal {Z}}_m\). In turn, using (C.10), this implies that

$$\begin{aligned} {\mathcal {Z}}_\text {tot}({{\hat{u}}}; \xi , \nu _R, a\omega , b\omega ) = 0 \end{aligned}$$
(C.23)

whenever \({{\hat{u}}}\) is fixed on the torus by the non-trivial element \(s_\alpha \) of the Weyl group of \(\mathfrak {su}(2)\).

The higher-rank case

Let us now move to the case of a generic semi-simple gauge algebra \({\mathfrak {g}}\) of rank \(\text {rk}(G)\). The Weyl group \({\mathcal {W}}_G\) is a finite group generated by the Weyl reflections

$$\begin{aligned} s_\alpha (u) = u - 2 \frac{\alpha (u)}{(\alpha ,\alpha )} \, \widetilde{\alpha }\qquad \qquad \forall \; u \in {\mathfrak {h}}, \end{aligned}$$
(C.24)

where \(\widetilde{\alpha }\) is the image of the root \(\alpha \) under the isomorphism \({\mathfrak {h}}^* \rightarrow {\mathfrak {h}}\) induced by the non-degenerate scalar product \(( \cdot , \cdot )\) on \({\mathfrak {h}}^*\). Suppose that there exists a non-trivial element w of \({\mathcal {W}}_G\) such that \(w \cdot {{\hat{u}}} = {{\hat{u}}}\). It is a standard theorem that the Weyl group acts freely and transitively on the set of Weyl chambers. Therefore, \({{\hat{u}}}\) cannot belong to a Weyl chamber but must instead lie on a boundary between two or more chambers. Such boundaries are the hyperplanes fixed by the Weyl reflections, \(\{u | s_\alpha (u) = u\}\), and their intersections. We conclude that there must exist at least one root \({{\hat{\alpha }}}\) such that \(s_{{\hat{\alpha }}}({{\hat{u}}}) = {{\hat{u}}}\).

On the other hand, we are interested in points \({{\hat{u}}}\) such that their equivalence class on the torus is fixed by a non-trivial element of the Weyl group, \(w \cdot [{{\hat{u}}}] = [{{\hat{u}}}]\). In this case, for each w we can always identify (at least) one root \({\hat{\alpha }}\) such that \(s_{{\hat{\alpha }}} [{{\hat{u}}}] = [{{\hat{u}}}]\), and moreover we can choose a set of simple roots that contains \({\hat{\alpha }}\). Let us fix a basis of simple roots \(\{\alpha _l\}_{l=1,\dots ,\text {rk}(G)}\) for \({\mathfrak {g}}\) that contains \({\hat{\alpha }}\). The fundamental weights \(\lambda _l\) are defined by

$$\begin{aligned} 2 \, \frac{(\lambda _k,\alpha _l)}{(\alpha _l,\alpha _l)}= \delta _{kl} \;. \end{aligned}$$
(C.25)

We choose a basis \(\{H^i\}\) for the Cartan subalgebra \({\mathfrak {h}}\) such that the fundamental weights have components \({\lambda _l}^i = \lambda _l(H^i) = \delta ^i_l\). In this basis \(\rho (H^i)\equiv \rho ^i\in \mathbb {Z}\) for any weight \(\rho \in \Lambda _\text {weight}\). Moreover, the double periodicity of the gauge variables \(u = u_i H^i\) is \(u_i\sim u_i + 1 \sim u_i + \omega \). From (C.24), the fixed points should satisfy

$$\begin{aligned} 2 \frac{{\hat{\alpha }}({{\hat{u}}})}{({\hat{\alpha }}, {\hat{\alpha }})} \, {{\widetilde{\alpha }}} = p + q \omega \qquad \text {for } \quad p = p_i H^i,\quad q = q_i H^i \quad \text { and }\quad p_i,\,q_i\in \mathbb {Z}.\qquad \quad \end{aligned}$$
(C.26)

Here \(\widetilde{\alpha }\) is dual to \({\hat{\alpha }}\). It is clear that pq should be aligned with \(\widetilde{\alpha }\), therefore we set

$$\begin{aligned} p = \frac{2{{\hat{p}}}}{({\hat{\alpha }}, {\hat{\alpha }})} {{\widetilde{\alpha }}},\qquad \qquad q = \frac{2{{\hat{q}}}}{({\hat{\alpha }}, {\hat{\alpha }})} {{\widetilde{\alpha }}}, \qquad \quad \text { with } {{\hat{p}}},\,{{\hat{q}}}\in \mathbb {Z}. \end{aligned}$$
(C.27)

In the basis \(\{H^i\}\) we have choosen, the components of \({{\widetilde{\alpha }}}\) are \((\lambda _i, {\hat{\alpha }}) = \delta _{il} \, ({\hat{\alpha }}, {\hat{\alpha }})/2\), where l is such that \({\hat{\alpha }}=\alpha _l\) and we have used (C.25). Only one component of \({{\widetilde{\alpha }}}\) is non-zero, which implies that the integer components of \(p,\,q\) are \(p_i = {{\hat{p}}}\, \delta _{i l}\) and \(q_i = {{\hat{q}}}\, \delta _{i l}\). This proves integrality of \({{\hat{p}}},\, {{\hat{q}}}\). The general solution to (C.26) can then be written as

$$\begin{aligned} {{\hat{u}}} = {{\hat{u}}}_0 + \frac{p + q \omega }{2}, \end{aligned}$$
(C.28)

where \({{\hat{u}}}_0\) is such that \({\hat{\alpha }}({{\hat{u}}}_0) = 0\).

Now, consider the explicit expression (3.11) for \({\mathcal {Z}}_\text {tot}\), in terms of \({\mathcal {Z}}\) given in (2.12). Given any representation \({\mathfrak {R}}\) of \({\mathfrak {g}}\), we can always decompose it into irreducible representations of the \({\mathfrak {s}}{\mathfrak {u}}(2)_{{\hat{\alpha }}}\) subalgebra associated with \({\hat{\alpha }}\). The set of weights (with multiplicities) \(\Lambda _{\mathfrak {R}}\) corresponding to \({\mathfrak {R}}\) can be organized as a union \(\Lambda _{\mathfrak {R}}= \cup _I \Lambda _{{\mathfrak {R}},I}\) of subsets \(\Lambda _{{\mathfrak {R}},I}\), each corresponding to a representation of \({\mathfrak {s}}{\mathfrak {u}}(2)_{{\hat{\alpha }}}\). Concretely, each \(\Lambda _{{\mathfrak {R}},I}\) is associated to a representation of \({\mathfrak {s}}{\mathfrak {u}}(2)_{{\hat{\alpha }}}\) of spin \(j_I\), so that its elements can be expressed as an \({\hat{\alpha }}\)-chain:

$$\begin{aligned} \Lambda _{{\mathfrak {R}},I}=\bigl \{{\hat{\rho }}_I+ \ell _I{{\hat{\alpha }}} \;\big |\; \ell _I=-j_I,-j_I+1,\dots ,j_I-1, j_I\bigr \}. \end{aligned}$$
(C.29)

Here \({\hat{\rho _I}}\) is the central point, which is orthogonal to \({{\hat{\alpha }}}\), i.e., such that \(({\hat{\rho }}_I,{{\hat{\alpha }}})=0\). Notice that, in general, \({\hat{\rho }}_I\) is not a weight.Footnote 8 The product over all weights \(\rho \) of the representation \({\mathfrak {R}}\) can then be expressed as a product over the representations of \({\mathfrak {s}}{\mathfrak {u}}(2)_{{\hat{\alpha }}}\) contained in \({\mathfrak {R}}\). In particular we can write

$$\begin{aligned}&\prod _{a}\prod _{\rho _a\in {\mathfrak {R}}_a} {\widetilde{\Gamma }}\bigl ( \rho _a(u)+\omega _a(\xi )+r_a\nu _R;a\omega ,b\omega \bigr ) \nonumber \\&\quad {} = \prod _{a, I} \prod _{\ell _{aI} = - j_{aI}}^{j_{aI}} {\widetilde{\Gamma }}\bigl ( {\hat{\rho }}_{aI}(u) + \ell _{aI} {\hat{\alpha }}(u)+\omega _a(\xi )+r_a\nu _R;a\omega ,b\omega \bigr ). \end{aligned}$$
(C.30)

When specifying \({\mathfrak {R}}\) to the adjoint representation, we obtain a similar decomposition for the roots of \({\mathfrak {g}}\). Besides the roots \({\hat{\alpha }}\) and \(-{\hat{\alpha }}\) of \(\mathfrak {su}(2)_{{\hat{\alpha }}}\), the other roots organize into \({\hat{\alpha }}\)-chains that we indicate as

$$\begin{aligned} \Lambda _{\text {roots},J} = \bigl \{ {\hat{\beta _J}} + \ell _J {\hat{\alpha }} \;\big |\; \ell _J = - j_J , -j_J+1,\dots ,j_J-1, j_J \bigr \}, \end{aligned}$$
(C.31)

where \({\hat{\beta }}_J\) is the non-vanishing central point orthogonal to \({\hat{\alpha }}\) (once again, \({\hat{\beta _J}}\) is in general not a weight). Notice that, for each subset \(\Lambda _{\text {roots},J}\) of the set of roots, there is a disjoint conjugate subset \({{\overline{\Lambda }}}_{\text {roots},J}\) with the same spin \(j_J\) but opposite central point \(-{\hat{\beta _J}}\).Footnote 9 For this reason, we have that

$$\begin{aligned}&{\mathcal {Z}}(u;\xi ,\nu _R,a\omega ,b\omega ) = \theta _0 \bigl ( {\hat{\alpha }}(u);a\omega \bigr ) \, \theta _0\bigl ( -{\hat{\alpha }}(u);b\omega \bigr ) \nonumber \\&\quad {} \times \frac{ \prod _{a, I} \prod _{\ell _{aI}=-j_{aI}}^{j_{aI}} {\widetilde{\Gamma }} \bigl ( {\hat{\rho }}_{aI}(u)+\ell _{aI}{\hat{\alpha }}(u)+\omega _a(\xi )+r_a\nu _R;a\omega ,b\omega \bigr ) }{ \prod _{J}\prod _{\ell _J=-j_J}^{j_J} {\widetilde{\Gamma }} \bigl ( {\hat{\beta _J}}(u)+\ell _J{\hat{\alpha }};a\omega ,b\omega \bigr ) \, {\widetilde{\Gamma }} \bigl ( -{\hat{\beta _J}}(u)+\ell _J{\hat{\alpha }};a\omega ,b\omega \bigr )}. \end{aligned}$$
(C.32)

Similarly to the rank one case, we want to prove that \({\mathcal {Z}}_\text {tot}({{\hat{u}}};\xi ,\nu _R,a\omega ,b\omega ) = 0\) for \({{\hat{u}}}\) in (C.28). Thus, we construct an involutive map \(\gamma : m \mapsto m'\), acting on the set \({\mathcal {M}}\) of vectors \(m = m_i H^i\) with integer components \(1 \le m_i \le ab\). The map is constructed in such a way that it leaves m invariant along the directions orthogonal to \({{\widetilde{\alpha }}}\), whereas it shifts the component parallel to \({{\widetilde{\alpha }}}\) by an integer amount. To be precise, take two vectors \(r,\,s\in {\mathfrak {h}}\) such that

$$\begin{aligned} r = \frac{2{{\hat{r}}}}{({\hat{\alpha }}, {\hat{\alpha }})} \, {{\widetilde{\alpha }}},\qquad \qquad s = \frac{2{{\hat{s}}}}{({\hat{\alpha }}, {\hat{\alpha }})}\, {{\widetilde{\alpha }}},\qquad \qquad \text {with } {{\hat{r}}},\, {{\hat{s}}} \in {\mathbb {Z}}, \end{aligned}$$
(C.33)

meaning that \(r,\,s\) are parallel to \({{\widetilde{\alpha }}}\) and have integer components \(r_i = {{\hat{r}}} \, \delta _{il}\), \(s_i = {{\hat{s}}} \, \delta _{il}\). Then, we construct \(m'\) as

$$\begin{aligned} m' = m + s \,b, \end{aligned}$$
(C.34)

which implies that \(m'\) differs from m only by integer shifts along the direction of \({{\widetilde{\alpha }}}\). For \({{\hat{s}}}\) we take the unique integer such that \(m' \in {\mathcal {M}}\) and

$$\begin{aligned} {\hat{\alpha }}(m') = {\hat{\alpha }}(m) + {\hat{\alpha }}(s) \, b = {\hat{\alpha }}(q - m) + {\hat{\alpha }}(r) \, a. \end{aligned}$$
(C.35)

Indeed, consider the following equation in r and s: \(2{\hat{\alpha }}(m) - {\hat{\alpha }}(q) = {\hat{\alpha }}(r) \, a - {\hat{\alpha }}(s) \, b\). Using (C.27) and(C.33), it reduces to \({\hat{\alpha }}(m) - {{\hat{q}}} = {{\hat{r}}} a - {{\hat{s}}} b\). Since ab are coprime, this equation always admits an infinite number of solutions in the pair \(({{\hat{r}}}, {{\hat{s}}})\), which can be parametrized as \(({{\hat{r}}}_0 + kb, {{\hat{s}}}_0 + ka)\) with \(k\in {\mathbb {Z}}\). There is however one and only one solution such that \(m'\) has components \(1\le m'_i \le ab\). We define \(\gamma (m) = m'\) in such a way. One can easily check that it is an involution.

As in the rank-one case, we adopt the notation

$$\begin{aligned} {\mathcal {Z}}_m \,\equiv \, {\mathcal {Z}}\bigl ( {{\hat{u}}}-m\omega ;\xi ,\nu _R,a\omega ,b\omega \bigr ) = {\mathcal {Z}}\bigl ( {{\hat{u}}}_0 + p/2 - (m-q/2) \omega ;\xi ,\nu _R,a\omega ,b\omega \bigr ), \end{aligned}$$
(C.36)

and, for later convenience, split \({\mathcal {Z}}\) into the vector multiplet and chiral multiplet contributions:

$$\begin{aligned} {\mathcal {A}}_m&= \theta _0 \bigl ( {\hat{\alpha }}(p/2) - {\hat{\alpha }} (m-q/2)\omega ;a\omega \bigr ) \, \theta _0\bigl (-{\hat{\alpha }}(p/2) + {\hat{\alpha }}(m-q/2)\omega ; b\omega \bigr )\nonumber \\ {\mathcal {C}}^{\pm }_m&= \prod _J\prod _{\ell _J=-j_J}^{j_J} {\widetilde{\Gamma }}\bigl ( \pm {\hat{\beta _J}}({{\hat{u}}}_0 - m\omega ) + \ell _J {\hat{\alpha }}(p/2) - \ell _J {\hat{\alpha }}(m-q/2) \omega ; a\omega , b\omega ) \nonumber \\ {\mathcal {B}}_m&= \prod _{a,I} \prod _{\ell _{aI}=-j_{aI}}^{j_{aI}} {\widetilde{\Gamma }}\Bigl ( {\hat{\rho }}_{aI}({{\hat{u}}}_0 - m \omega ) + \ell _{aI} {\hat{\alpha }}(p/2)- \ell _{aI} {\hat{\alpha }}(m-q/2) \omega + \omega _a(\xi ) + r_a\nu _R; a\omega , b\omega \Bigr ) \end{aligned}$$
(C.37)

such that \({\mathcal {Z}}_m = {\mathcal {A}}_m \, {\mathcal {B}}_m / {\mathcal {C}}^+_m \, {\mathcal {C}}^-_m\). We will prove that \({\mathcal {Z}}_{m'} = -{\mathcal {Z}}_m\), which implies that \({\mathcal {Z}}_\text {tot}({{\hat{u}}})\) vanishes because \(\gamma \) is an involution.

We begin by considering the contribution of \({\mathcal {A}}_{m'}\). Following the same steps as in (C.11) and using (C.35) and (3.26), we can show

$$\begin{aligned} {\mathcal {A}}_{m'} = - e^{-2\pi i \, {\hat{\alpha }}(r) \, {\hat{\alpha }}(s) \, \nu _R} \; {\mathcal {A}}_m. \end{aligned}$$
(C.38)

We also used that \({\hat{\alpha }}(r),\, {\hat{\alpha }}(s),\, {\hat{\alpha }}(q) \in 2\mathbb {Z}\), which is guaranteed by (C.27) and (C.33). We now turn to \({\mathcal {B}}_{m'}\). Eqn. (C.34) implies that \({\hat{\rho }}_{aI}(m') = {\hat{\rho }}_{aI}(m)\) for any \({\hat{\rho }}_{aI}\) orthogonal to \({\hat{\alpha }}\). Using the identity (3.30) repeatedly and distinguishing the cases \(\ell _{aI} \lessgtr 0\), we obtain

$$\begin{aligned} {\mathcal {B}}_{m'} = \prod _{a,I} \prod _{\ell _{aI}>0}^{j_{aI}} (-1)^{\ell _{aI}^3 {\hat{\alpha }}(r) {\hat{\alpha }}(s) {\hat{\alpha }}(p)} \prod _{a,\, \rho _a} (-1)^{\frac{1}{2}\rho _a(r)\rho _a(s)} \, e^{-\pi i \rho _a(r) \rho _a(s) \bigl (\rho _a({{\hat{u}}}_0 - m\omega ) +\omega _a(\xi ) + (r_a-1) \nu _R \bigr )} \, {\mathcal {B}}_m. \end{aligned}$$
(C.39)

The analysis of \({\mathcal {C}}^\pm _m\) is analogous to the one for \({\mathcal {B}}_m\) and it gives the following:

$$\begin{aligned} {\mathcal {C}}_{m'}^\pm = \prod _J \prod _{\ell _J>0}^{j_J} (-1)^{\ell _J^3 {\hat{\alpha }}(r) {\hat{\alpha }}(s) {\hat{\alpha }}(p)} \prod _{\alpha \ne \pm {\hat{\alpha }}} (-1)^{\frac{1}{2}\alpha (r)\alpha (s)} \; e^{\pi i \alpha (r)\alpha (s)\nu _R} \times {\mathcal {C}}^\pm _m.\qquad \qquad \end{aligned}$$
(C.40)

Combining (C.38) with the latter, we obtain that the vector-multiplet contribution is

$$\begin{aligned} {\mathcal {A}}_{m'} / {\mathcal {C}}_{m'}^+ {\mathcal {C}}_{m'}^- = - e^{-2\pi i \sum _{\alpha \in \Delta }\alpha (r)\alpha (s)\nu _R} \; {\mathcal {A}}_m / {\mathcal {C}}_m^+ {\mathcal {C}}_m^-. \end{aligned}$$
(C.41)

We used \({\hat{\alpha }}(r) {\hat{\alpha }}(s) \in 4{\mathbb {Z}}\), as well as \(\sum _{\alpha \in \Delta } \alpha (r) \alpha (s)\in 4\mathbb {Z}\) for any semi-simple Lie algebra \({\mathfrak {g}}\), and that \(2\ell _J^3 {\hat{\alpha }}(r) {\hat{\alpha }}(s) {\hat{\alpha }} (p)\in 2\mathbb {Z}\) for any integer or half-integer spin. Including now also the contribution from \({\mathcal {B}}_m\), the factor picked up by \({\mathcal {Z}}\) can be expressed in terms of the anomaly coefficients (3.18) and (3.19):

$$\begin{aligned} {\mathcal {Z}}_{m'} = - \, e^{2\pi i \phi } \; e^{\pi i\, r_i s_j \left( \frac{1}{2} {\mathcal {A}}^{ij} - {\mathcal {A}}^{ijk}({{\hat{u}}}_0 - m \omega )_k - {\mathcal {A}}^{ij \alpha } \xi _\alpha - {\mathcal {A}}^{ijR} \nu _R \right) } \; {\mathcal {Z}}_m. \end{aligned}$$
(C.42)

The anomaly cancelation conditions \({\mathcal {A}}^{ijk} = {\mathcal {A}}^{ij\alpha } = {\mathcal {A}}^{ijR} = 0\) and \({\mathcal {A}}^{ij} \in 4\mathbb {Z}\) imply that the second exponential equals 1. In the first exponential we defined

$$\begin{aligned} \phi = \frac{1}{2} \, {\hat{\alpha }}(r) \, {\hat{\alpha }}(s)\, {\hat{\alpha }}(p) \sum _{a,I} \sum _{\ell _{aI}>0}^{j_{aI}} \ell _{aI}^3 = 4{{\hat{r}}} {{\hat{s}}} {{\hat{p}}} \sum _{a,I} \sum _{\ell _{aI}>0}^{j_{aI}} \ell _{aI}^3. \end{aligned}$$
(C.43)

Once again, in an anomaly-free theory \(\phi \in {\mathbb {Z}}\). Indeed, labelling the chiral multiplets by a, their \(\mathfrak {su}(2)_{{\hat{\alpha }}}\) representations by I and dubbing their spin \(j_{aI}\), the only non-integer contributions to \(\phi \) come from representations with \(j_{aI} \in 2{\mathbb {Z}}+ \frac{1}{2}\). On the other hand, the contribution of an \(\mathfrak {su}(2)_{{\hat{\alpha }}}\) representation to the pseudo-anomaly coefficient is

$$\begin{aligned} {\mathcal {A}}^{ij}_{a I} = \sum _{\ell = -j_{aI}}^{j_{aI}} ({\hat{\rho }}_{aI} + \ell {\hat{\alpha }})^i({\hat{\rho }}_{aI} + \ell {\hat{\alpha }})^j \;. \end{aligned}$$
(C.44)

Since generic vectors rs (C.33) have integer components, the condition \({\mathcal {A}}^{ij} \in 4{\mathbb {Z}}\) implies that also \({\mathcal {A}}^{ij} r_i s_j \in 4{\mathbb {Z}}\) for any choice of rs. Contracting with the vectors, we obtain

$$\begin{aligned} {\mathcal {A}}^{ij}_{a I} r_i s_j = \frac{4}{3} {{\hat{r}}} {{\hat{s}}} \, j_{aI}(j_{aI} + 1)(2j_{aI} + 1) \,\in \, {\left\{ \begin{array}{ll} 4{\mathbb {Z}}&{} \text {if } j_{aI} \in {\mathbb {Z}}\text { or }j_{aI} \in 2{\mathbb {Z}}+ \frac{3}{2} \\ 4{\mathbb {Z}}+ 2 &{} \text {if } j_{aI} \in 2{\mathbb {Z}}+ \frac{1}{2}. \end{array}\right. }\nonumber \\ \end{aligned}$$
(C.45)

Therefore, the condition \({\mathcal {A}}^{ij} \in 4{\mathbb {Z}}\) requires that the number of \(\mathfrak {su}(2)_{{\hat{\alpha }}}\) representations with \(j_{aI} \in 2{\mathbb {Z}}+ \frac{1}{2}\) be even, and this guarantees that \(\phi = 0\).

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Benini, F., Milan, P. A Bethe Ansatz Type Formula for the Superconformal Index. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-019-03679-y

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