Coexact spectral gaps of flat bundles on homogeneous spherical space forms

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On the round three-sphere and its quotients by finite groups of rotations, the Hodge Laplacian on twisted coexact $`1`$-forms has a lowest nonzero eigenvalue, the spectral gap $`q_\tau^2/R^2`$, where $`q_\tau \ge 2`$ is the lowest coexact level carrying a constituent of the twisting representation $`\tau`$. A constituent at McKay distance $`d`$ enters at level $`d`$ for $`d \ge 2`$ and at level $`2`$ for $`d = 0`$, for any finite $`\Gamma`$; when $`-I \in \Gamma`$, equivalently when the McKay graph is bipartite, a distance-one constituent enters at level $`3`$. Here $`q_\tau`$ is the minimum of these entry levels over the constituents of $`\tau`$, so in particular the gap is $`d_\tau^2/R^2`$ whenever $`d_\tau \ge 2`$.

The adjoint bundle of an irreducible flat $`\mathrm{SU}(2)`$ connection, the coefficient system gauge theory asks for, always has $`d_\tau \ge 2`$, so its gap is the squared adjoint distance. Across the ADE classification of finite subgroups of $`\mathrm{SU}(2)`$ this gap is $`4/R^2`$ for every irreducible flat connection, with one exception: the Galois-conjugate connection on the Poincaré homology sphere $`S^3/2I`$ gives $`36/R^2`$. The exception reflects two related features of $`2I`$, the unique nontrivial perfect finite subgroup of $`\mathrm{SU}(2)`$: its perfectness prevents $`Q^\prime`$ from being a character twist of $`Q`$, while the distinct adjoint $`\mathrm{Sym}^2 Q^\prime`$ separately occupies the distance-six node of affine $`E_8`$, the short branch, rather than the distance-two node every other adjoint occupies. The ingredients are classical: the form spectra of Ikeda and Taniguchi, the representation-theoretic descent of Lauret, Miatello and Rossetti, and the McKay correspondence. The new content is the uniform expression of the coexact gap through the first coexact occurrence level $`q_\tau`$, its McKay-distance form in the bipartite cases, the adjoint application, and the icosahedral case as the one exception.

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1. Introduction

The Poincaré homology sphere $`\Sigma(2,3,5) = S^3/2I`$ holds a distinguished place in gauge theory. Among spherical space forms with nontrivial fundamental group it is the unique integral homology sphere, singled out by Fintushel and Stern [FintushelStern] in the instanton-Floer setting. Adjoint-coefficient spectral-flow computations for it appear in Nasatyr [Nasatyr1992] and Kirk and Klassen [KirkKlassen], with the binary polyhedral character and flat-connection data recorded by Helle [Helle]. Gauge theory typically compares flat connections through global invariants of such adjoint operators, such as spectral flow; the question here is intrinsic and simpler: for a spherical space form $`S^3/\Gamma`$ with $`\Gamma \subset \mathrm{SU}(2)`$ and a flat bundle $`E_\tau`$ determined by a representation $`\tau`$ of $`\Gamma`$, what is the lowest nonzero eigenvalue of the Hodge Laplacian on coexact $`E_\tau`$-valued $`1`$-forms, and how does it depend on $`\tau`$? In the adjoint case that eigenvalue is also the bottom of the gauge-fixed Laplacian on the Coulomb slice at the connection.

The answer is uniform across the ADE classification of finite subgroups of $`\mathrm{SU}(2)`$, with exactly one exception. For the adjoint twist $`\tau = \mathrm{Ad} \circ \rho`$ of an irreducible flat $`\mathrm{SU}(2)`$ connection $`\rho`$, every such connection on every $`S^3/\Gamma`$ has coexact gap $`4/R^2`$, save the Galois-conjugate connection on the Poincaré sphere, whose gap is $`36/R^2`$. The exception reflects two related features of $`2I`$: its perfectness prevents $`Q^\prime`$ from being a character twist of $`Q`$, while the distinct adjoint $`\mathrm{Sym}^2 Q^\prime`$ occupies the distance-six node of affine $`E_8`$, where every other adjoint occupies distance two.

The spectral ingredients are classical. Ikeda and Taniguchi [IkedaTaniguchi] give the round-sphere coexact spectrum, Ikeda [Ikeda1989] its descent to general spherical space forms, and Lauret, Miatello and Rossetti [LauretMiatelloRossetti] the representation-equivalence form of that descent; Bär gives the Dirac analogue [Bar1996] and, by Poincaré-series methods, the untwisted curl spectrum on spherical space forms itself [Bar2019], the $`\tau = \mathbf{1}`$ case of the descent below.

Nearby work extends in two other directions: Henkel and Lauret [HenkelLauret2026] keep the coefficient system trivial and vary the homogeneous metric on $`S^3`$ and $`\mathrm{SO}(3)`$, while Lauret [LauretLens2018, LauretCyclic] fixes constant curvature but allows at most a one-dimensional twist, for the ordinary $`p`$-form spectrum of a lens space and the character-twisted spectrum of functions over a cyclic fundamental group. Here the metric stays round while the coefficient system is an arbitrary finite-dimensional representation of $`\Gamma`$; the adjoint twist $`\mathrm{Sym}^2 \rho`$ on the nonabelian groups is the coefficient system the gauge-theory question above asks for.

Coexact $`1`$-form spectral gaps are studied on hyperbolic manifolds too, by isoperimetric and asymptotic methods suited to their negative curvature and infinite fundamental group, in place of the finite-group representation theory used here: Boulanger and Courtois [BoulangerCourtois] give a Cheeger-type lower bound for coexact $`1`$-forms on closed Riemannian manifolds, sharpened for hyperbolic $`3`$-manifolds by Rudd [Rudd2023].

One rule carries the computation, extended here to an arbitrary finite-dimensional twist $`\tau`$ rather than only the trivial or one-dimensional twists above: the first surviving level $`q_\tau`$ is determined constituentwise by McKay distance, with the rule needed below given in Proposition 4.2. For adjoint twists of irreducible flat $`\mathrm{SU}(2)`$ connections, the relevant constituents have McKay distance at least two, and in this range the first coexact occurrence level equals the McKay distance, so the coexact gap is the squared distance to the nearest constituent. The three classical ingredients above, together with the McKay correspondence [McKay], combine into that rule, and the adjoint classification with its one exception follows from it.

The objects throughout are flat bundles on spherical space forms carrying their round metrics, and the spectrum studied is that of the Hodge Laplacian on the coexact $`1`$-form summand. Section 2 fixes the metric, the bundles, and the operator. Section 3 records the $`S^3`$ coexact spectrum and the descent to $`S^3/\Gamma`$. Section 4 proves the gap for a general flat bundle and derives the adjoint comparison across the ADE classification, indexed by the affine Dynkin diagrams $`\widetilde{A}`$, $`\widetilde{D}`$, $`\widetilde{E}`$ that the McKay correspondence attaches to each $`\Gamma \subset \mathrm{SU}(2)`$ (Section 2.1).

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2. Setup

2.1 The space and the McKay graph

Identify $`S^3`$ with $`\mathrm{SU}(2)`$ carrying the bi-invariant metric of constant sectional curvature $`1/R^2`$. Let $`\Gamma \subset \mathrm{SU}(2)`$ be a finite subgroup acting on $`S^3`$ by left translation, and write $`X = S^3/\Gamma`$ for the resulting spherical space form, a closed oriented $`3`$-manifold of constant curvature $`1/R^2`$ with $`\pi_1(X) = \Gamma`$; since right translation still acts transitively on $`X`$, this is a homogeneous spherical space form in the sense of Ikeda [Ikeda1995]. The finite subgroups are the cyclic groups $`\mathbb{Z}_n`$, the binary dihedral groups $`D_n^*`$ of order $`4n`$, and the binary tetrahedral, octahedral, and icosahedral groups $`2T, 2O, 2I`$ of orders $`24, 48, 120`$.

Let $`V_j = \mathrm{Sym}^j \mathbb{C}^2`$ be the irreducible $`\mathrm{SU}(2)`$ representation of dimension $`j+1`$, so $`V_1`$ is the defining representation. The McKay graph of $`\Gamma`$ has the irreducible representations of $`\Gamma`$ as vertices, the trivial representation as distinguished vertex, and $`m_{\sigma\sigma^\prime} = \dim \mathrm{Hom}_\Gamma(\sigma^\prime,\, V_1\vert_\Gamma \otimes \sigma)`$ edges between $`\sigma`$ and $`\sigma^\prime`$. The McKay correspondence [McKay] identifies it with the affine Dynkin diagram attached to $`\Gamma`$.

The graph is bipartite exactly when $`-I \in \Gamma`$. If $`-I \in \Gamma`$, then $`-I`$ acts by a scalar on each irreducible $`\Gamma`$-representation, and tensoring with $`V_1`$ changes this scalar; the two central characters give the bipartition, so the scalar on a vertex at distance $`d`$ from the trivial vertex is $`(-1)^d`$. The subgroups omitting $`-I`$ are exactly the odd cyclic groups $`\mathbb{Z}_n`$, since every binary family contains $`-I`$ and $`\mathbb{Z}_n`$ contains $`-I`$ if and only if $`n`$ is even; for these groups the McKay graph is the odd cycle $`\widetilde{A}_{n-1}`$. Graph distances $`d(\sigma)`$ from the trivial vertex are taken on the underlying unweighted graph, and for a representation $`\tau`$ we set

$$d_\tau = \min\lbrace\, d(\sigma) : \sigma \text{ a constituent of } \tau \,\rbrace.$$

2.2 Flat bundles and the adjoint

A finite-dimensional unitary representation $`\tau \colon \Gamma \to \mathrm{U}(V)`$ determines a flat Hermitian bundle $`E_\tau = (S^3 \times V)/\Gamma`$ on $`X`$, with the flat connection $`\nabla`$ descended from the trivial connection upstairs. Gauge-equivalence classes of flat $`\mathrm{SU}(2)`$ connections on $`X`$ are identified with conjugacy classes of homomorphisms $`\rho \colon \Gamma \to \mathrm{SU}(2)`$; the real adjoint bundle $`\mathrm{ad}_\rho`$ has fibre $`\mathfrak{su}(2)`$ and holonomy $`\mathrm{Ad} \circ \rho`$. We work with the complexification, the flat bundle $`E_\tau`$ for

$$\tau = (\mathrm{ad}_\rho)_{\mathbb{C}} \cong \mathrm{Sym}^2 \rho \quad \text{as a complex } \Gamma\text{-representation,}$$

the isomorphism being the identification of $`\mathfrak{su}(2)_{\mathbb{C}} = \mathfrak{sl}(2, \mathbb{C})`$ with the spin-1 representation $`V_2`$ of the $`\mathrm{SU}(2)`$ acting through $`\rho`$.

2.3 The operator and positivity

Let $`d_\nabla`$ be the flat exterior derivative of $`E_\tau`$ and $`\Delta_\tau = d_\nabla^* d_\nabla + d_\nabla d_\nabla^*`$ the twisted Hodge Laplacian. For the flat unitary connection the Hodge decomposition reads

$$\Omega^1(X; E_\tau) = \mathrm{im} d_\nabla\ \oplus\ \mathrm{im} d_\nabla^*\ \oplus\ \mathcal{H}^1(X; E_\tau),$$

with $`\Delta_\tau`$ preserving each summand; the coexact $`1`$-forms are $`\mathrm{im} d_\nabla^*`$.

This summand carries the only part of the twisted $`1`$-form spectrum that lies beyond functions: the exact and harmonic summands are already determined. On the exact summand $`\mathrm{im} d_\nabla`$, the identity $`\Delta_\tau(d_\nabla f) = d_\nabla(\Delta_\tau f)`$ sends each positive twisted $`0`$-form eigenvalue to the same $`1`$-form eigenvalue, while $`d_\nabla`$ annihilates the zero modes. Thus the exact summand reproduces the positive part of the twisted function spectrum. The harmonic summand $`\mathcal{H}^1(X; E_\tau)`$ vanishes for every finite-dimensional $`\tau`$, since $`\Gamma`$ is finite and the coefficients have characteristic zero, as in Lemma 2.1. The coexact summand $`\mathrm{im} d_\nabla^*`$ therefore carries the entire twisted $`1`$-form spectrum beyond functions.

On that summand the curl operator $`\ast d_\nabla`$ is formally self-adjoint and satisfies the standard three-dimensional curl identity [Bar2019, CapoferriVassiliev], which extends formally to flat unitary bundles

$$(\ast d_\nabla)^2 = d_\nabla^* d_\nabla = \Delta_\tau \qquad \text{on coexact 1-forms},$$

so the coexact spectrum of $`\Delta_\tau`$ is the set of squares of the $`\ast d_\nabla`$ spectrum, and the bottom coexact eigenvalue is the square of the smallest $`\lvert\ast d_\nabla\rvert`$.

Lemma 2.1 (Vanishing of harmonic $`1`$-forms). For every finite-dimensional $`\tau`$, $`H^1(X; E_\tau) = 0`$. Consequently the coexact $`1`$-forms coincide with the coclosed $`1`$-forms, and $`\Delta_\tau`$ is strictly positive on them.

Proof. By de Rham with local coefficients $`\mathcal{H}^1(X; E_\tau) \cong H^1(X; E_\tau)`$. Since the universal cover $`S^3`$ has $`H^1(S^3) = 0`$, the Cartan-Leray spectral sequence for the covering $`S^3 \to X`$ collapses in degree one, so $`H^1(X; E_\tau) \cong H^1(\Gamma; \tau)`$, computed by crossed homomorphisms. Since $`\Gamma`$ is finite and the coefficients have characteristic zero, averaging over $`\Gamma`$ gives $`H^1(\Gamma; V) = 0`$ for every finite-dimensional $`\Gamma`$-module $`V`$; hence $`H^1(X; E_\tau) = \mathcal{H}^1(X; E_\tau) = 0`$. The kernel of $`\Delta_\tau`$ on $`1`$-forms is exactly $`\mathcal{H}^1`$, so $`\Delta_\tau`$ is strictly positive on $`\Omega^1(X; E_\tau)`$, and in particular on the coexact summand; and with $`\mathcal{H}^1 = 0`$ that summand $`\mathrm{im} d_\nabla^*`$ is the full space $`\ker d_\nabla^*`$ of coclosed $`1`$-forms. $`\square`$

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3. The coexact spectrum and its descent

3.1 The coexact spectrum on $`S^3`$

Write $`L^2(S^3) = \bigoplus_{k \ge 0} V_k \boxtimes V_k^*`$ for the Peter-Weyl decomposition of functions, fixing the convention that $`\mathrm{SU}(2)_L`$ acts by left translation on the first factor and the commuting right $`\mathrm{SU}(2)`$-action on the second. Trivialize $`T^* S^3`$ by the left-invariant coframe, so $`\Omega^1(S^3; \mathbb{C}) \cong C^\infty(S^3) \otimes \mathfrak{su}(2)^*_{\mathbb{C}}`$, the coframe carrying the trivial left action and the right adjoint representation $`V_2`$. Peter-Weyl then gives

$$\Omega^1(S^3; \mathbb{C}) \cong \bigoplus_{k \ge 0} V_k \boxtimes (V_k \otimes V_2),$$

and Clebsch-Gordan splits $`V_k \otimes V_2 \cong V_{k+2} \oplus V_k \oplus V_{k-2}`$ for $`k \ge 2`$, with $`V_1 \otimes V_2 \cong V_3 \oplus V_1`$ and $`V_0 \otimes V_2 \cong V_2`$. The exterior derivative $`d \colon C^\infty(S^3) \to \Omega^1(S^3)`$ is $`\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R`$-equivariant and injective on nonconstants, and for $`k \ge 1`$ the middle summand $`V_k \boxtimes V_k`$ occurs once in $`\Omega^1`$, so $`\mathrm{im} d`$ is exactly the middle summands $`V_k \boxtimes V_k`$, $`k \ge 1`$. The harmonic $`1`$-forms vanish since $`H^1(S^3) = 0`$, so the coexact $`1`$-forms are the outer summands $`V_k \boxtimes V_{k \pm 2}`$. Reindexing, the coexact level $`m \ge 2`$ is

$$E_m = \bigl(V_m \boxtimes V_{m-2}\bigr)\ \oplus\ \bigl(V_{m-2} \boxtimes V_m\bigr),$$

the first factor carrying left translation and the second the commuting right action. Each summand is a distinct $`\mathrm{SU}(2)_L \times \mathrm{SU}(2)_R`$ irreducible, so by Schur the curl $`\ast d`$ is scalar on each; its value is $`\pm m/R`$ by Ikeda and Taniguchi [IkedaTaniguchi], and with $`(\ast d)^2 = \Delta`$ the eigenvalue is $`m^2/R^2`$. The two summands are the $`\pm m/R`$ eigenspaces of $`\ast d`$, up to the orientation convention. The lowest level $`m = 2`$ gives $`E_2 = (V_2 \boxtimes V_0) \oplus (V_0 \boxtimes V_2)`$ and eigenvalue $`4/R^2`$.

3.2 Descent to $`X = S^3/\Gamma`$

Fix the associated-bundle convention $`\gamma \cdot (x, v) = (\gamma x, \tau(\gamma) v)`$ on $`E_\tau = (S^3 \times V_\tau)/\Gamma`$, so sections of $`E_\tau`$ identify with $`\Gamma`$-invariant $`V_\tau`$-valued forms on $`S^3`$ for the diagonal action, $`\Gamma`$ acting by left translation on the form and by $`\tau`$ on $`V_\tau`$. Left translation commutes with the curl and preserves each $`E_m`$. Pulling $`E_\tau`$ back to $`S^3`$ trivializes it with the product flat connection, so the lifted twisted Laplacian is $`\Delta_{S^3} \otimes 1`$ on $`V_\tau`$-valued forms. Thus the level-<i>m</i> twisted eigenspace on $`S^3/\Gamma`$ is

$$\bigl(E_m \otimes V_\tau\bigr)^\Gamma, \qquad \text{eigenvalue }\ m^2/R^2.$$

This is the descent of the $`p`$-spectrum of a constant-curvature space form, due to Ikeda [Ikeda1989] and given its representation-equivalence form by Lauret, Miatello and Rossetti [LauretMiatelloRossetti], with the coefficient $`V_\tau`$ inserted by the diagonal action.

Under left translation only the first factor of each piece of $`E_m`$ is acted on, so the right factors are flat multiplicity spaces:

$$E_m\big\vert_\Gamma = \bigl(V_m\big\vert_\Gamma\bigr)^{\oplus(m-1)}\ \oplus\ \bigl(V_{m-2}\big\vert_\Gamma\bigr)^{\oplus(m+1)},$$

the left types $`V_m\vert_\Gamma`$ and $`V_{m-2}\vert_\Gamma`$ carrying the right-factor multiplicities $`\dim V_{m-2} = m-1`$ and $`\dim V_m = m+1`$. The dimensions close, $`(m-1)(m+1) + (m+1)(m-1) = 2(m^2-1)`$, the Ikeda-Taniguchi multiplicity of the level-<i>m</i> coexact eigenvalue. Computing the invariants by restriction and the tensor-hom adjunction,

$$\bigl(E_m \otimes V_\tau\bigr)^\Gamma \cong \mathrm{Hom}_\Gamma\!\bigl(\tau^*,\, E_m\vert_\Gamma\bigr),$$

nonzero exactly when $`\tau^*`$ shares a constituent with the left content $`W_m := V_m\vert_\Gamma \oplus V_{m-2}\vert_\Gamma`$. Each $`V_j`$ is self-dual, so $`W_m`$ is self-dual as a $`\Gamma`$-module; hence $`\tau^*`$ meets $`W_m`$ iff $`\tau`$ meets $`W_m^* = W_m`$, and the criterion reads

$$\bigl(E_m \otimes V_\tau\bigr)^\Gamma \ne 0 \iff \text{a constituent of } \tau \text{ occurs in } W_m.$$

The reduction uses the self-duality of the left content, not of $`\tau`$, so it holds for arbitrary $`\tau`$; the adjoint case is self-dual only incidentally.

Proposition 3.1 (Twisted coexact spectrum). For $`m \ge 2`$, the multiplicity of the eigenvalue $`m^2/R^2`$ in the $`E_\tau`$-valued coexact $`1`$-form spectrum on $`X`$ is

$$\mu_\tau(m) = (m-1)\dim \mathrm{Hom}_\Gamma\!\bigl(\tau^*, V_m\vert_\Gamma\bigr) + (m+1)\dim \mathrm{Hom}_\Gamma\!\bigl(\tau^*, V_{m-2}\vert_\Gamma\bigr).$$

As a set of eigenvalues the twisted coexact spectrum is

$$\bigl\lbrace\, m^2/R^2 :\ m \ge 2,\ \mathrm{Hom}_\Gamma\!\bigl(\tau^*, W_m\bigr) \ne 0 \,\bigr\rbrace,$$

equivalently those $`m`$ at which a constituent of $`\tau`$ occurs in $`W_m = V_m\vert_\Gamma \oplus V_{m-2}\vert_\Gamma`$. Its bottom is $`q_\tau^2/R^2`$, where

$$q_\tau = \min\bigl\lbrace\, m \ge 2 :\ \text{a constituent of } \tau \text{ occurs in } W_m \,\bigr\rbrace.$$

Proof. The multiplicity formula follows from the decomposition of $`E_m\vert_\Gamma`$ and the tensor-hom adjunction above, and the eigenvalue criterion by forgetting the right-factor multiplicities. It remains to see that the defining set for $`q_\tau`$ is nonempty: $`V_1\vert_\Gamma`$ is faithful, so by Burnside-Brauer every irreducible representation of $`\Gamma`$ occurs in some tensor power $`V_1^{\otimes k}\vert_\Gamma`$, and Clebsch-Gordan decomposes these into symmetric powers $`V_j\vert_\Gamma`$. Hence every irreducible occurs in some $`V_m\vert_\Gamma`$, and $`q_\tau`$ is finite; the sharp first occurrence in the symmetric-power tower, at $`a = d(\sigma)`$, is proved in Lemma 4.1. $`\square`$

For the trivial bundle, where $`\tau = \mathbf{1}`$, the descent recovers the classical untwisted coexact $`1`$-form spectrum, the $`\Gamma`$-invariant part $`(E_m)^\Gamma`$ of each round-sphere level $`E_m`$. The round-sphere levels are computed by Ikeda and Taniguchi [IkedaTaniguchi], the $`p`$-form spectra of spherical space forms by Ikeda [Ikeda1989], and those of lens spaces by Lauret [LauretLens2018]. The trivial representation occurs in $`W_2 = V_2\vert_\Gamma \oplus V_0\vert_\Gamma`$ through its $`V_0`$ summand, which is present for every $`\Gamma`$, so $`q_{\mathbf{1}} = 2`$ and the untwisted coexact gap is $`4/R^2`$ on every $`S^3/\Gamma`$ considered here.

For the full untwisted $`1`$-form spectrum we compare the coexact gap with the exact summand. The scalar level $`k`$ on $`S^3/\Gamma`$ is present precisely when $`V_k\vert_\Gamma`$ contains the trivial representation. If $`\Gamma \ne \lbrace 1\rbrace`$, then $`V_1\vert_\Gamma`$ has no nonzero invariant vector: such a vector would be fixed by every element of $`\Gamma`$, in particular by a nonidentity $`\gamma`$; but an element of $`\mathrm{SU}(2)`$ fixing a nonzero vector has eigenvalue $`1`$, and since $`\det \gamma = 1`$ both eigenvalues are $`1`$, so $`\gamma = I`$, a contradiction. The level-one scalar eigenvalue $`3/R^2`$ is therefore absent, the exact $`1`$-forms can only begin at level $`k \ge 2`$, and their eigenvalues are at least $`8/R^2`$. The coexact gap $`4/R^2`$ is thus the gap of the full untwisted $`1`$-form Laplacian for every nontrivial $`\Gamma \subset \mathrm{SU}(2)`$. For $`\Gamma = \lbrace 1\rbrace`$, the level-one scalar harmonics on $`S^3`$ give exact $`1`$-forms with eigenvalue $`3/R^2`$.

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4. The gap and the ADE classification

4.1 First occurrence

Let $`A`$ be the McKay adjacency matrix, with entries $`A_{\sigma\sigma^\prime} = \dim \mathrm{Hom}_\Gamma(\sigma^\prime, V_1\vert_\Gamma \otimes \sigma)`$, so that multiplication by $`V_1`$ in the representation ring acts as $`A`$. The symmetric powers satisfy $`V_{a+1} = V_1 V_a - V_{a-1}`$, hence $`V_a = U_a(A)`$ applied to the trivial node, where $`U_a`$ is the degree-<i>a</i> polynomial of leading coefficient one with $`U_a(2\cos\theta) = \sin((a+1)\theta)/\sin\theta`$.

Lemma 4.1 (First occurrence). For an irreducible $`\Gamma`$-representation $`\sigma`$ at McKay distance $`d(\sigma)`$, the least $`a`$ with $`\sigma \subset V_a\vert_\Gamma`$ is $`a = d(\sigma)`$. If moreover $`-I \in \Gamma`$, equivalently the McKay graph is bipartite, then $`\sigma \subset V_a\vert_\Gamma`$ forces $`a \equiv d(\sigma) \pmod 2`$.

Proof. The multiplicity of $`\sigma`$ in $`V_a\vert_\Gamma`$ is the matrix element $`(U_a(A))_{0\sigma}`$. Since $`U_a`$ has degree and parity $`a`$ and leading coefficient one, this is a combination of $`(A^a)_{0\sigma}, (A^{a-2})_{0\sigma}, \dots`$, and $`(A^k)_{0\sigma}`$ counts walks of length $`k`$ from $`0`$ to $`\sigma`$, which vanishes for $`k < d(\sigma)`$. Hence the multiplicity vanishes for $`a < d(\sigma)`$. At $`a = d(\sigma)`$ the lower terms $`(A^{d(\sigma)-2})_{0\sigma}, (A^{d(\sigma)-4})_{0\sigma}, \ldots`$ vanish, since their exponents are below $`d(\sigma)`$. The unit-coefficient leading term remains, so the multiplicity is $`(A^{d(\sigma)})_{0\sigma}`$, the positive count of walks of length $`d(\sigma)`$ from $`0`$ to $`\sigma`$. When $`-I \in \Gamma`$, the preceding bipartition gives that $`-I`$ acts on $`\sigma`$ by $`(-1)^{d(\sigma)}`$, while it acts on $`V_a\vert_\Gamma`$ by $`(-1)^a`$. Hence occurrence of $`\sigma`$ in $`V_a\vert_\Gamma`$ forces $`a \equiv d(\sigma) \pmod 2`$. $`\square`$

4.2 The gap

Proposition 4.2 (Gap from first occurrence). Let $`\tau`$ be a finite-dimensional $`\Gamma`$-representation and $`d_\tau = \min\lbrace d(\sigma) : \sigma \text{ an irreducible constituent of } \tau \rbrace`$. For an irreducible constituent $`\sigma`$, let $`e(\sigma)`$ be the least $`m \ge 2`$ for which $`\sigma`$ occurs in $`W_m = V_m\vert_\Gamma \oplus V_{m-2}\vert_\Gamma`$. Then

$$e(\sigma) = 2 \quad \text{if } d(\sigma) = 0, \qquad e(\sigma) = d(\sigma) \quad \text{if } d(\sigma) \ge 2,$$

with no hypothesis on $`\Gamma`$. If $`-I \in \Gamma`$, then also $`e(\sigma) = 3`$ for $`d(\sigma) = 1`$. Consequently

$$q_\tau = \min_{\sigma \subset \tau} e(\sigma),$$

where the minimum runs over irreducible constituents. In particular, if $`d_\tau \ge 2`$, then $`q_\tau = d_\tau`$ and the bottom of the twisted coexact spectrum is $`d_\tau^2/R^2`$.

Proof. The equality $`q_\tau = \min_{\sigma \subset \tau} e(\sigma)`$ is just the definition of $`q_\tau`$ (Proposition 3.1). If $`d(\sigma) = 0`$, then $`\sigma`$ is the trivial representation and occurs in $`V_0`$, hence in $`W_2`$, and no level below $`2`$ is allowed. If $`d(\sigma) \ge 2`$, Lemma 4.1 places $`\sigma`$ first in $`V_{d(\sigma)}\vert_\Gamma`$, so $`W_{d(\sigma)}`$ contains $`\sigma`$ through its $`V_{d(\sigma)}`$ summand. For $`m < d(\sigma)`$, both $`m`$ and $`m-2`$ are below the first occurrence, so $`W_m`$ contains no copy of $`\sigma`$. Thus $`e(\sigma) = d(\sigma)`$. Finally assume $`-I \in \Gamma`$ and $`d(\sigma) = 1`$. The first occurrence is in $`V_1\vert_\Gamma`$, while the parity restriction excludes $`\sigma`$ from the even levels $`V_0\vert_\Gamma`$ and $`V_2\vert_\Gamma`$ appearing in $`W_2`$. Hence the first coexact level is $`m = 3`$, through the $`V_{m-2} = V_1`$ summand. $`\square`$

The bipartite hypothesis enters only for $`d_\tau \le 1`$, where the non-bipartite odd cyclic groups need separate treatment. For $`\Gamma = \mathbb{Z}_n`$ odd with distance-one character $`\chi`$, $`V_2\vert_{\mathbb{Z}_n} = \chi^2 \oplus \mathbf{1} \oplus \chi^{-2}`$, so $`\chi`$ already occurs at $`m = 2`$ exactly when $`\chi^3 = 1`$, i.e. only for $`n = 3`$; for every other odd $`n`$, $`\chi`$ first occurs at $`m = 3`$. The adjoint uses only the $`d_\tau \ge 2`$ case.

4.3 The adjoint

A flat $`\mathrm{SU}(2)`$ connection is irreducible precisely when $`\rho \colon \Gamma \to \mathrm{SU}(2)`$ is an irreducible $`2`$-dimensional representation. Its complexified adjoint is

$$(\mathrm{ad}_\rho)_{\mathbb{C}} \cong \mathrm{Sym}^2 \rho,$$

a $`3`$-dimensional representation of $`\Gamma`$, irreducible for the polyhedral groups $`2T, 2O, 2I`$ and reducible for the binary dihedral groups, where $`\mathrm{Sym}^2 \tau_{2j-1}`$ contains a fixed distance-two sign character $`\varepsilon_D`$.

Proposition 4.3 (Adjoint gap). For an irreducible flat $`\mathrm{SU}(2)`$ connection $`\rho`$ on $`X = S^3/\Gamma`$, the bottom of the adjoint coexact spectrum is $`d_{\mathrm{Sym}^2 \rho}^2/R^2`$, and $`d_{\mathrm{Sym}^2 \rho} \ge 2`$.

Proof. An irreducible $`2`$-dimensional $`\rho`$ forces $`\Gamma`$ nonabelian, hence binary dihedral or polyhedral, so $`-I \in \Gamma`$. Being central, $`-I`$ acts on the irreducible $`\rho`$ by a scalar, necessarily $`\pm I`$ since $`\det \rho = 1`$, so $`\mathrm{Sym}^2 \rho(-I) = I`$: the adjoint factors through $`\Gamma/\lbrace\pm I\rbrace`$, and as $`-I`$ acts on a distance-<i>d</i> node by $`(-1)^d`$, every constituent lies at even distance. The trivial is not among them: since $`\det \rho = 1`$ the representation $`\rho`$ is self-dual, so $`\mathrm{Hom}_\Gamma(\mathbf{1}, \rho \otimes \rho) \cong \mathrm{Hom}_\Gamma(\rho^*, \rho)`$ is one-dimensional by Schur, and as $`\rho \otimes \rho = \mathrm{Sym}^2 \rho \oplus \Lambda^2 \rho`$ with $`\Lambda^2 \rho = \det \rho = \mathbf{1}`$, that unique trivial summand is the determinant, lying in $`\Lambda^2`$. Even distance and no trivial constituent give $`d_{\mathrm{Sym}^2 \rho} \ge 2`$, and Proposition 4.2 returns the bottom $`d_{\mathrm{Sym}^2 \rho}^2/R^2`$. $`\square`$

4.4 The ADE table

The distance $`d_{\mathrm{Sym}^2 \rho}`$ is read from the McKay graph; for the binary polyhedral groups the graph and the flat $`\mathrm{SU}(2)`$ connections are tabulated by Helle [Helle]. Cyclic groups carry no irreducible flat $`\mathrm{SU}(2)`$ connection, their image being abelian. In every other case the nearest constituent of the adjoint sits at distance two, with a single exception.

The table lists only the $`2`$-dimensional irreducibles of determinant one, the flat connections valued in $`\mathrm{SU}(2)`$. For the binary dihedral groups these are the odd-index $`\tau_{2j-1}`$ with $`1 \le 2j-1 \le n-1`$, equivalently $`j = 1, \dots, \lfloor n/2 \rfloor`$; for $`2T`$ only the defining $`Q`$ qualifies, its twists by the nontrivial cube-root characters having determinant a cube root of unity; for $`2O`$ both $`Q`$ and $`Q \otimes \varepsilon`$ qualify, while the remaining $`2`$-dimensional irreducible has determinant the sign character; and $`2I`$, being perfect, has no nontrivial character to twist by, so both $`Q`$ and $`Q^\prime`$ have determinant one and appear.

The ADE classification and the labelling of flat $`\mathrm{SU}(2)`$ connections are standard. The table records the remaining datum needed here: the nearest irreducible constituent of $`\mathrm{Sym}^2 \rho`$ to the trivial node in the corresponding affine Dynkin diagram. The binary dihedral row is checked by characters below, while the binary polyhedral rows are read from the labelled diagrams and character tables, with the $`2I`$ exception verified explicitly after the table.

The binary dihedral row covers every odd index at once. In $`D_n^* = \langle a, b \mid a^{2n} = 1,\ b^2 = a^n,\ bab^{-1} = a^{-1}\rangle`$ the $`2`$-dimensional irreducible $`\tau_\ell`$ has $`\chi_{\tau_\ell}(a^k) = 2\cos(\pi\ell k/n)`$ and $`\chi_{\tau_\ell}(ba^k) = 0`$. From $`(ba^k)^2 = a^n`$ and $`\chi_{\tau_\ell}(a^n) = 2(-1)^\ell`$,

$$\chi_{\mathrm{Sym}^2 \tau_\ell}(a^k) = 1 + 2\cos(2\pi\ell k/n), \qquad \chi_{\mathrm{Sym}^2 \tau_\ell}(ba^k) = (-1)^\ell.$$

For the connections $`\tau_{2j-1}`$, subtracting the distance-two sign character $`\varepsilon_D`$ ($`a \mapsto 1,\ b \mapsto -1`$) leaves a degree-two character, possibly reducible, while no constituent at distance zero or one occurs. The nearest constituent is $`\varepsilon_D`$ at distance two and the gap is $`4/R^2`$. Since $`Q`$ is the distance-one node adjacent to $`\mathbf{1}`$ in each polyhedral diagram, the decomposition $`Q \otimes Q \cong \mathbf{1} \oplus \mathrm{Sym}^2 Q`$ places the nontrivial summand $`\mathrm{Sym}^2 Q`$ at distance two. The two $`2O`$ connections share an adjoint, since $`\varepsilon^2 = \mathbf{1}`$ gives $`\mathrm{Sym}^2(Q \otimes \varepsilon) = \mathrm{Sym}^2 Q`$, again distance two. The binary icosahedral group is the exception: it is perfect, so it has no $`1`$-dimensional character to twist $`Q`$ by, and its only other irreducible $`\mathrm{SU}(2)`$ connection is the Galois conjugate $`Q^\prime`$, whose adjoint is a distinct node. The Galois adjoint $`\mathrm{Sym}^2 Q^\prime`$ sits at distance six, so the Galois connection has gap $`36/R^2`$. Since $`\Lambda^2 Q = \Lambda^2 Q^\prime = \mathbf{1}`$ (as in the proof of Proposition 4.3), $`\chi_{\mathrm{Sym}^2 Q} = \chi_Q^2 - 1`$ and $`\chi_{\mathrm{Sym}^2 Q^\prime} = \chi_{Q^\prime}^2 - 1`$, a direct cross-check against the branching table below. Explicitly, the character of $`V_a = \mathrm{Sym}^a \mathbb{C}^2`$ is

$$\chi_a(\theta) = \frac{\sin((a+1)\theta)}{\sin\theta},$$

with the limiting values at $`\theta = 0, \pi`$. Evaluating it on the nine conjugacy classes of $`2I`$ and decomposing against the $`2I`$ character table (the nine classes have half-angles $`0, \pi/5, 2\pi/5, \pi/3, \pi/2, 2\pi/3, 3\pi/5, 4\pi/5, \pi`$, and the entries are the inner products of $`\chi_a`$ with the irreducible characters of $`2I`$ [Helle]) gives

where $`\mathbf{4}^\prime`$ denotes the second $`4`$-dimensional irreducible. Thus $`\mathrm{Sym}^2 Q`$ first occurs at level two, and $`\mathrm{Sym}^2 Q^\prime`$, the Galois conjugate of $`\mathrm{Sym}^2 Q`$, is absent from $`V_a\vert_{2I}`$ for $`a < 6`$ and first occurs in $`V_6\vert_{2I}`$ (Figure 1).

<img src="https://github.com/dmobius3/mode-identity-theory/blob/main/files/assets/e8-mckay-graph.png" width="80%" alt="The affine E8 McKay graph of 2I: the trivial node, the defining representation Q, and the Galois pair Sym^2 Q, Sym^2 Q', Q', labelled by name and McKay distance">

Figure 1. The affine $`E_8`$ McKay graph of $`2I`$: the trivial node $`\mathbf{1}`$, the defining representation $`Q`$, and the Galois pair $`\mathrm{Sym}^2 Q`$, $`\mathrm{Sym}^2 Q^\prime`$, $`Q^\prime`$, labelled by name and McKay distance. The three unlabelled nodes are the remaining irreducibles of $`2I`$: dimensions $`4, 5, 6`$ from $`\mathbf{1}`$ to the branch node, and the second $`4`$-dimensional irreducible $`4^\prime`$ between the branch node and $`Q^\prime`$.

Every irreducible flat $`\mathrm{SU}(2)`$ connection on a quotient $`S^3/\Gamma`$ with $`\Gamma \subset \mathrm{SU}(2)`$ thus has adjoint coexact gap $`4/R^2`$, with the single exception of the Galois connection on $`S^3/2I`$, whose gap is $`36/R^2 = 6^2/R^2`$.

These coexact gaps are also the gaps for the full adjoint $`1`$-form spectrum. As in Section 2.3, the Hodge decomposition splits the adjoint $`1`$-forms into exact, coexact, and harmonic parts; the harmonic part vanishes (Lemma 2.1), and the exact part carries the positive twisted function spectrum. On $`S^3`$ the scalar Laplacian has levels $`k(k+2)/R^2`$ for $`k \ge 0`$, with level-$`k`$ left content $`V_k\vert_\Gamma`$ [IkedaTaniguchi], so by Lemma 4.1 the adjoint first occurs in the function spectrum at level $`k = d_{\mathrm{Sym}^2\rho}`$, where the exact bottom is $`d_{\mathrm{Sym}^2\rho}(d_{\mathrm{Sym}^2\rho}+2)/R^2`$. Since $`d_{\mathrm{Sym}^2\rho} \ge 2`$ (Proposition 4.3), this exceeds the coexact bottom $`d_{\mathrm{Sym}^2\rho}^2/R^2`$, so the coexact gap is the gap of the full adjoint $`1`$-form Laplacian:

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5. Conclusion

The coexact spectral gap of a flat bundle on a spherical space form is determined by representation-theoretic data alone: the first level at which the twisting representation meets the round-sphere spectrum, read off as a McKay-graph distance. For the trivial twist this level is universal, and for the adjoint twist of an irreducible flat $`\mathrm{SU}(2)`$ connection it is uniform across the entire ADE classification, giving coexact gap $`4/R^2`$.

The single exception is the Galois-conjugate connection on the Poincaré homology sphere $`S^3/2I`$. Because $`2I`$ is perfect, its Galois conjugate pair $`Q, Q^\prime`$ is not related by any character twist, and the distinct adjoint $`\mathrm{Sym}^2 Q^\prime`$ occupies the distance-six node of affine $`E_8`$, giving gap $`36/R^2`$ where every other adjoint gives $`4/R^2`$. The same gap classification, with the same single exception, holds for the full adjoint $`1`$-form spectrum, not only its coexact summand.

The spectral gap computed here is a static quantity, the bottom of a Laplacian at a fixed connection, rather than a path invariant such as spectral flow. Whether finer spectral data at the connection, such as multiplicities or the sign of the curl eigenvalue, distinguishes flat connections beyond what the bottom eigenvalue already shows is a natural question left open by this note; the eta-invariant methods of Capoferri and Vassiliev [CapoferriVassiliev2025] are one avenue toward the latter.

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References

• [Bar1996] C. Bär, The Dirac operator on space forms of positive curvature. J. Math. Soc. Japan 48 (1996), 69–83.
• [Bar2019] C. Bär, The curl operator on odd-dimensional manifolds. J. Math. Phys. 60 (2019), no. 3, 031501.
• [BoulangerCourtois] A. Boulanger and G. Courtois, A Cheeger-like inequality for coexact 1-forms. Duke Math. J. 171 (2022), 3593–3641.
• [CapoferriVassiliev] M. Capoferri and D. Vassiliev, Beyond the Hodge theorem: curl and asymmetric pseudodifferential projections. J. Lond. Math. Soc. 113 (2026), no. 1, e70431.
• [CapoferriVassiliev2025] M. Capoferri and D. Vassiliev, A microlocal pathway to spectral asymmetry: curl and the eta invariant. arXiv:2502.18307 (2025).
• [FintushelStern] R. Fintushel and R. Stern, Instanton homology of Seifert fibred homology three spheres. Proc. Lond. Math. Soc. (3) 61 (1990), 109–137.
• [Helle] G. O. Helle, Equivariant instanton Floer homology and calculations for the binary polyhedral spaces. arXiv:2203.09471 (2022).
• [HenkelLauret2026] J. Henkel and E. A. Lauret, Hodge Laplacian on $`1`$-forms of homogeneous $`3`$-spheres. arXiv:2605.05406 (2026).
• [Ikeda1989] A. Ikeda, Riemannian manifolds $`p`$-isospectral but not $`(p+1)`$-isospectral. In Geometry of Manifolds (Matsumoto, 1988), Perspect. Math. 8, Academic Press (1989), 383–417.
• [Ikeda1995] A. Ikeda, On the spectrum of homogeneous spherical space forms. Kodai Math. J. 18 (1995), 57–67.
• [IkedaTaniguchi] A. Ikeda and Y. Taniguchi, Spectra and eigenforms of the Laplacian on $`S^n`$ and $`P^n(\mathbb{C})`$. Osaka J. Math. 15 (1978), 515–546.
• [KirkKlassen] P. Kirk and E. Klassen, Computing spectral flow via cup products. J. Differential Geom. 40 (1994), no. 3, 505–562.
• [LauretCyclic] E. A. Lauret, Spectra of orbifolds with cyclic fundamental groups. Ann. Global Anal. Geom. 50 (2016), no. 1, 1–28.
• [LauretLens2018] E. A. Lauret, The spectrum on $`p`$-forms of a lens space. Geom. Dedicata 197 (2018), 107–122.
• [LauretMiatelloRossetti] E. A. Lauret, R. J. Miatello and J. P. Rossetti, Representation equivalence and $`p`$-spectrum of constant curvature space forms. J. Geom. Anal. 25 (2015), no. 1, 564–591.
• [McKay] J. McKay, Graphs, singularities, and finite groups. Proc. Sympos. Pure Math. 37 (1980), 183–186.
• [Nasatyr1992] E. B. Nasatyr, Seifert-fibred homology 3-spheres, V-surfaces and the Floer index. Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 461–485.
• [Rudd2023] C. G. Rudd, Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds. J. Topol. 16 (2023), 588–633.