April 06, 2026
We compute the \(\bar{\nu}\)-invariant of homogeneous nearly-parallel \(G_2\)-structures on Aloff–Wallach spaces \(N_{k,l} = SU(3)/S^1_{k,l}\). Using Goette’s formulas for the \(\eta\)-invariants of homogeneous spaces, we derive an explicit expression for \(\bar{\nu}\) in terms of representation-theoretic data and show that for the two homogeneous nearly-parallel structures \(\varphi^\pm\) on \(N_{k,l}\) one has \[\bar{\nu}(\varphi^\pm) = \mp 41.\] Additionally, we compare the \(\bar{\nu}\)-invariants of the nearly-parallel \(G_2\)-structures arising from the 3-Sasakian structure.
In Crowley2015? the authors introduced a new \(\mathbb{Z}_{48}\)-valued invariant \(\nu\) of \(G_2\)-structures. Later in Crowley2025? they introduced a \(\mathbb{Z}\)-valued refinement \(\bar{\nu}\) of the \(\nu\)-invariant. Although their original goal was to apply these invariants in the case of parallel \(G_2\)-structures, the \(\bar{\nu}\) invariant is well-suited to study the \(G_2\)-structures inducing metrics of positive scalar curvature, as it is preserved under deformations within this class. One important example of a class of \(G_2\)-structures inducing metrics of positive scalar curvature is the class of nearly-parallel (or weak) \(G_2\)-structures.
An important subclass of examples where these invariants can be computed is provided by Aloff-Wallach spaces \(N_{k,l}\). Each such space admits two non-equivalent nearly parallel \(G_2\)-structures. The problem of computing \(\nu\)-invariants of these structures was suggested by Ball2019?, where the authors showed that nearly parallel \(G_2\)-structure on the same Aloff-Wallach space can be distinguished using \(G_2\)-instantons.
Moreover, some of the Aloff-Wallach spaces are diffeomorphic, providing examples of manifolds admitting two non-equivalent homogeneous structures Kreck1991?. In particular, these examples yield manifolds admitting two pairs of nearly-parallel structures on the same manifolds arising from different homogeneous structures. This naturally leads to the question of whether these structures are homotopic or not.
In this paper, we compute the \(\bar{\nu}\)-invariants of homogeneous nearly-parallel \(G_2\)-structures on Aloff-Wallach spaces. Our approach uses the formulas for the \(\eta\)-invariants of homogeneous spaces proven in Goette2009?, which allow us to compute the \(\bar{\nu}\)-invariants.
Theorem 1. Let \(\varphi_\pm\) denote the two non-equivalent nearly-parallel homogeneous structures on \(N_{k,l}\). Then \[\bar{\nu}(\varphi_\pm) = \mp41.\]
As a consequence, the \(\bar{\nu}\)-invariant (and by extension \(\nu\)-invariant) takes the same value up to sign for all these nearly-parallel homogeneous \(G_2\)-structures on Aloff-Wallach spaces. In particular, it does not distinguish between the two nearly-parallel \(G_2\)-structures induced by different homogeneous structures on \[N_{-4638861,582656} \cong N_{-2594149,5052965}.\] It therefore remains an open question whether these two \(G_2\)-structures are in fact homotopic.
We also compare the \(\bar{\nu}\)-invariants of the nearly-parallel \(G_2\)-structures associated to 3-Sasakian structure \(\varphi_{ts}\) with the squashed nearly parallel \(G_2\)-structure \(\varphi_{sq}\). We note that these structure are in fact homotopic through \(G_2\)-structures inducing the metrics of positive scalar curvature, and hence
\[\bar{\nu}(\varphi_{sq}) = \bar{\nu}(\varphi_{ts}).\]
In particular, this shows that \(\bar{\nu}\)-invariant does not distinguish between these nearly parallel \(G_2\)-structures, even though they can be distinguished using finer gauge-theoretic methods such as deformed \(G_2\)-instantons Lotay2022?.
As a consequence, we now know all the \(\bar{\nu}\)-invariants of the homogeneous proper nearly-parallel \(G_2\) manifolds. Namely, up to the sign defined by choice of orientation:
By Example 1 \[\bar{\nu}(\varphi_{sq}(S^7)) = 1,\]
By Theorem 1 \[\bar{\nu}(\varphi(N_{k,l})) = 41,\]
By [Crowley2025?, Example 1.8] \[\bar{\nu}(\varphi(SO(5)/SO(3))) = 1 .\]
In Section 2, we review the basics of \(G_2\)-structures and recall the definition of the \(\bar{\nu}\)-invariant. In Section 3, we discuss the homogeneous \(G_2\)-structures and provide the formula for the \(\bar{\nu}\)-invariant in the homogeneous case. In Section 4, we specify the previous discussion to the Aloff-Wallach spaces and explain how to compute the \(\bar{\nu}\)-invariant. In Section 5 and appendix [appx], we carry out explicit computations of the terms constituting the \(\bar{\nu}\)-invariant. In Section 6, we compare the nearly parallel \(G_2\)-structures given by 3-Sasakian structures with the associated squashed nearly-parallel proper \(G_2\)-structures. Finally, in Section 7, we gather some results about first Pontryagin class of nearly-parallel \(G_2\)-manifolds and note that first Pontryagin class of all known examples is a torsion class.
Definition 1. A \(G_2\)-structure on a manifold \(M^7\) is a choice of a 3-form \(\varphi\), which is pointwise equivalent to the form \[dx^{123} + dx^{145} + dx^{167} + dx^{246} - dx^{257} - dx^{347} - dx^{356}.\] Equivalently, the \(G_2\)-structure is determined by the choice of the orientation, metric \(g\) and a unit spinor \(s \in \Gamma(SM)\).
Definition 2. The nearly-parallel \(G_2\)-structure is a \(G_2\)-structure given by a 3-form \(\varphi\), satisfying
\[d\varphi = \lambda *\varphi,\] for some \(\lambda \neq 0\).
The metric induced by the nearly-parallel \(G_2\)-structure is Einstein with constant \(7\cdot 24 \lambda^2\) Friedrich1997?.
Definition 3 (Crowley2025?). Let (g,s) be a \(G_2\)-structure on the closed manifold \(M^7\). Let \(g^{SM}\) be the metric on the spinor bundle \(SM\) and \(\nabla^{SM}\) be the connection on \(SM\) induced by the Levi-Civita connection on \(TM\). The \(\bar{\nu}\)-invariant is defined as: \[\bar{\nu}(\varphi) = 2\int_M s^*\psi(\nabla^{SM},g^{SM}) - 24\eta(D_M) + 3\eta(B_M).\]
Here \(\psi\) is the Mathai-Quillen current on the bundle \(SM\) Bismut1992?, \(D_M\) is the Dirac operator, and \(B_M\) is the odd signature operator.
This expression is invariant under the deformations of the \(G_2\)-structures preserving positive scalar curvature. Moreover, \[\bar{\nu}(\varphi) + 24\dim \ker(D_M) \equiv \nu(\varphi) \mod \;48\] is the homotopy invariant of \(G_2\)-structures Crowley2025?.
We recall some of the properties of the Mathai-Quillen current, for more detailed discussion one can use Bismut1992?.
Lemma 1 (Bismut1992?). The Mathai-Quillen current satisfies the following transgression formula:
\[\label{eq:trans} \psi(\nabla^1,g) - \psi(\nabla^2,g) = \pi^*\widetilde{e}(\nabla^2,\nabla^1, g) \text{ \textrm{modulo exact currents,}}\qquad{(1)}\] where \(\widetilde{e}(\nabla^2,\nabla^1, g)\) is the second characteristic form associated to the Euler class.
We will also need the following lemma
Lemma 2 (Crowley2025?). Let \(s \in \Gamma(SM)\), if \(s\) is parallel with respect to \(\nabla\), then \[s^*\psi(\nabla, g) = 0.\]
Let \(G/H\) be a homogeneous space. Any vector bundle \(E\) over \(G/H\) is of the form \(E = G \times_\kappa V\) for some \(H\)-representation \(\kappa : H \to \mathop{\mathrm{Aut}}(V)\) Goette1999?. Any homogeneous section \(s\) of such bundle can be identified with \(H\)-equivariant map \(\hat{s}: G \to V\) via
\[s([g]) = [g,\hat{s}(g)].\]
The reductive connection is defined as Goette2009?: \[\widehat{\nabla^0_Vs}(g) = \widehat{V}\hat{s}(g) = \frac{d}{dt}\bigg|_{t=0}\hat{s}(ge^{t\widehat{V}}).\]
Lemma 3. Any homogeneous section \(s \in \Gamma(E)\) is parallel with respect to \(\nabla^0\).
Proof of Lemma 3. Section \(s\) is homogeneous if and only if it is \(G\)-invariant, that is,
\[l_gs = s \;\forall g \in G.\]
Equivalently, \[\hat{s}(g_0) = \widehat{l_gs}(g_0) = \hat{s}(g^{-1}g_0) \;\forall g \in G.\] Hence, \(\hat{s}\) is constant on \(G\), therefore \[\nabla^0s = 0.\] ◻
Remark 1. The space of sections parallel with respect to \(\nabla^0\) can be identified with the subspace \(V^H \subset V\) consisting of vectors fixed by \(H\) action.
Lemma 4. All of the homogeneous \(G_2\)-structures on \(G/H\) inducing the same orientation are homotopic.
Proof of Lemma 4. Let \((S, \widetilde{\pi})\) and \((T,\kappa)\) be the linear representations of \(H\), such that associated vector bundles give the spinor bundle \(SM\) and tangent bundle \(TM\), respectively.
A homogeneous \(G_2\)-structure with chosen orientation is determined by a homogeneous Riemannian metric \(g \in \Gamma(\odot^2 TM)\) together with a homogeneous unit spinor \(s \in \Gamma(SM)\). By remark 1, the choice of homogeneous sections is in one-to-one correspondence with an \(H\)-invariant pair of unit spinor \(s \in \mathbb{S}(S^H) \subset S\) and a metric \(g \in \odot^2 T^H \subset \odot^2 T\). Since the space of metrics inside \(\odot^2 T^H\) is a convex cone, there always exists a path between two homogeneous metrics \(g_1\) and \(g_2\).
Unless the dimension of the sphere \(\mathbb{S}(S^H)\) is \(0\), we can find a path connecting any two homogeneous spinors \(s_1\) and \(s_2\), such that each element in this path paired with a metric \(g\) determines a homogeneous \(G_2\)-structure and, consequently, we obtain a homotopy of \(G_2\)-structures.
If \(\dim \mathbb{S}(S^H) = 0\), we use the following argument from Crowley2025?. The spinor \(s\) induces an isomorphism \(SM \cong \underline{\mathbb{R}} \oplus TM\) and the Euler class of an oriented 7-manifold vanishes, so \(SM\) contains a trivial 2-plane field \(K \subset SM\) with \(s \in K\), within which \(s\) can be rotated into \(-s\). Note that in this case the path between \(G_2\)-structures may leave the space of homogeneous \(G_2\)-structures. ◻
To compute the \(\eta\)-invariants, we use the following results:
Theorem 2 (Goette2009?). Let \(G/H\) be a homogeneous space with the normal metric \(g\). Then, the following formulas for the Dirac operator \(D\) and the odd signature operator \(B\) hold:
\[\label{eq:etaD} \eta(D) = I_D + 2\int_M \widetilde{\widehat{\mathrm{A}}}(\nabla^0,\nabla^{TM}) + J_D.\qquad{(2)}\]
\[\label{eq:etaB} \eta(B) = I_B + \int_M \widetilde{L}(\nabla^0,\nabla^{TM}) + J_B.\qquad{(3)}\]
Here, \(\widetilde{\widehat{\mathrm{A}}}\) and \(\widetilde{L}\) are the secondary characteristic forms of the \(\widehat{\mathrm{A}}\)-genus and \(L\)-genus.
The terms \(I\) and \(J\) depend purely on the representation-theoretic data of \(G/H\) and are explained in section 5.
It turns out that these formulas are well-suited for the computation of \(\bar{\nu}\)-invariants, which we discuss in the next section.
We derive the formula for the \(\bar{\nu}\)-invariant of the homogeneous structure inducing the normal metric in terms of the reductive connection \(\nabla^0\). This follows the approach of [Crowley2025?, section 1.3].
Proposition 1. Let \(\varphi\) be a homogeneous \(G_2\)-structure inducing the normal metric. Then the \(\bar{\nu}\)-invariant can be computed as: \[\bar{\nu}(\varphi) = -24I_D + 3I_B -24J_D + 3J_B.\]
Proof of Proposition 1. First we use formula ?? to rewrite the Mathai-Quillen term as:
\[2\int_M s^*\psi(\nabla^{SM},g^{SM}) = 2\int_M s^*\psi(\nabla^0,g^{SM}) + 2\int_M \widetilde{e}(\nabla^0,\nabla^{SM}).\]
Using the standard formulas for the Euler class, this becomes \[= 2\int_M s^*\psi(\nabla^0,g^{SM}) + 48 \int_M \widetilde{\widehat{\mathrm{A}}}(\nabla^0,\nabla^{SM}) - 3 \int_M \widetilde{L}(\nabla^0,\nabla^{SM}).\]
Since \(s\) is homogeneous, it is parallel with respect to \(\nabla^0\). By the lemma 2 we have \[s^*\psi(\nabla^0,g^{SM}) = 0.\]
Consequently, \[2\int_M s^*\psi(\nabla^{SM},g^{SM}) = 48 \int_M \widetilde{\widehat{\mathrm{A}}}(\nabla^0,\nabla^{SM}) - 3 \int_M \widetilde{L}(\nabla^0,\nabla^{SM}).\]
We now apply the formulas ?? and ?? for the \(\eta\)-invariants of the Dirac and odd-signature operators:
\[\begin{align} \bar{\nu}(\varphi) &= 48 \int_M \widetilde{\widehat{\mathrm{A}}}(\nabla^0,\nabla^{SM}) - 3 \int_M \widetilde{L}(\nabla^0,\nabla^{SM}) -\\ &-24I_D - 48\int_M \widetilde{\widehat{\mathrm{A}}}(\nabla^0,\nabla^{SM}) - 24J_D +\\ &+ 3I_B + 3\int_M \widetilde{L}(\nabla^0,\nabla^{SM}) + 3J_B =\\ &=-24I_D + 3I_B -24J_D + 3J_B. \end{align}\] ◻
This formula was used in Crowley2025? to compute the \(\bar{\nu}\)-invariant of the homogeneous \(G_2\) structure on the Berger space \(SO(5)/SO(3)\). It shows that, in the homogeneous case, \(\bar{\nu}\) can be expressed purely in terms of representation-theoretic data of the pair \((G,H)\).
We will compute these terms for the Aloff-Wallach spaces in section 5.
Let \((k,l)\) be the pair of integer numbers such that \(k \neq \pm l\), \(l \neq \pm(k+l)\), \(k+l \neq \pm k\), and \(k\) and \(l\) are coprime.
The Aloff-Wallach spaces are defined as quotients \(SU(3)/S^1_{k,l}\), where the subgroup \(S^1_{k,l}\) is given as \(\left\{\mathop{\mathrm{diag}}\left(e^{ikx},e^{ilx},e^{-i(k+l)x}\right)\right\}\) We will also assume that \(k,l > 0\) (other cases can be obtained from this one by the change of orientation and permutations of \((k,l,-k-l)\))
First, we describe the structure of the Aloff-Wallach spaces. Fix the metric on \(\mathfrak{su}(3)\) by \(\langle X,Y \rangle = -\mathop{\mathrm{tr}}(XY)\). Let \(\mathfrak{su}(3) = \mathfrak{m} \oplus \mathfrak{u}(1)_{k,l}\) be the orthogonal decomposition. Choose the following basis for the subspace \(\mathfrak{m}:\) \[\begin{align} e_1 &= \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, & e_5 &= \frac{i}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \notag \\[1em] e_2 &= \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}, & e_6 &= \frac{i}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \label{eq:basis} \\[1em] e_3 &= \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}, & e_7 &= \frac{i}{\sqrt{2}} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix},\notag \\[1em] e_4 &= \frac{i}{\sqrt{6}\,\sqrt{k^2 + l^2 + kl}} \begin{pmatrix} 2l + k & 0 & 0 \\ 0 & -2k - l & 0 \\ 0 & 0 & k - l \end{pmatrix}.\notag \end{align}\tag{1}\] The subspace \(\mathfrak{m}^\perp = \mathfrak{u}(1)_{k,l}\) is generated by \[e_8 = \frac{i}{\sqrt{2}\,\sqrt{k^2 + l^2 + kl}} \begin{pmatrix} k & 0 & 0 \\ 0 & l & 0 \\ 0 & 0 & -k - l \end{pmatrix}.\] Denote the adjoint action of \(S^1\) as \(\pi\). Under the action of \(\pi\) vector \(e_4\) is fixed, while the planes \(\langle e_1,e_5 \rangle\), \(\langle e_2,e_6 \rangle\), and \(\langle e_3,e_7 \rangle\) carry weights \(i(k-l)\), \(i(2l+k)\), and \(i(-2k-l)\) respectively.
The tangent bundle is given as \(TN_{k,l} = SU(3)\times_\pi \mathfrak{m}\).
Let \(s = \sqrt{6(k^2+kl+l^2)}\). The multiplication table for the commutators is given as:
\[\begin{align} [e_1,e_2]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_3,\\ [e_1,e_3]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_2,\\ [e_1,e_4]_\mathfrak{m}&=-\tfrac{3(k+l)}{s}e_5,\\ [e_1,e_5]_\mathfrak{m}&=\tfrac{3(k+l)}{s}e_4,\\ [e_1,e_6]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_7,\\ [e_1,e_7]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_6.\\ \\ [e_3,e_4]_\mathfrak{m}&=\tfrac{3l}{s}e_7,\\ [e_3,e_5]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_6,\\ [e_3,e_6]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_5,\\ [e_3,e_7]_\mathfrak{m}&=\tfrac{3l}{s}e_7.\\ \\ [e_6,e_4]_\mathfrak{m}&=-\tfrac{3k}{s}e_2,\\ [e_6,e_7]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_1. \end{align}\]
\[\begin{align} [e_2,e_3]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_1,\\ [e_2,e_4]_\mathfrak{m}&=\tfrac{3k}{s}e_6,\\ [e_2,e_5]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_7,\\ [e_2,e_6]_\mathfrak{m}&=\tfrac{-3k}{s}e_4,\\ [e_2,e_7]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_5.\\ \\ \\ [e_5,e_4]_\mathfrak{m}&=\tfrac{3(k+l)}{s}e_1,\\ [e_5,e_6]_\mathfrak{m}&=\tfrac{1}{\sqrt2}e_3,\\ [e_5,e_7]_\mathfrak{m}&=-\tfrac{1}{\sqrt2}e_2. \\ \\ \\ [e_7,e_4]_\mathfrak{m}&= -\frac{3l}{s}e_3. \end{align}\]
Let \(\widetilde{\pi}: \mathfrak{h} \to \mathop{\mathrm{End}}(S)\) denote the spin representation induced from the isotropy representation \(\pi\). Then the spinor bundle is given as \(SM = G \times_{\widetilde{\pi}} S\). The homogeneous bundle \(\Omega^{ev}M\) in dimension seven is isomorphic to the bundle \(G \times_{\widetilde{\pi} \otimes \hat{\widetilde{\pi}}} S\otimes S\), where \(\hat{\widetilde{\pi}}\) denotes the representation isomorphic to \(\widetilde{\pi}\) acting on the second factor.
It is easy to check that in the case of Aloff-Wallach spaces the weights of \(\widetilde{\pi}\) and \(\hat{\widetilde{\pi}}\) are \((0,0, \pm i(k-l), \pm i(2k+l), \pm i(2l+k))\).
Under our assumptions on \(k,l\) the most general homogeneous metric is given by choosing the orthonormal basis of the form \[(ae_1, ae_5, be_2, be_6, ce_3, ce_7, de_4).\]
The associated \(G_2\) structure is then chosen by identifying \(\mathfrak{m}\) with \(\mathop{\mathrm{Im}}\mathbb{O}\) as follows: \(ae_1\) is identified with \(i\), \(ae_5\) with \(ie\), \(be_2\) with \(j\), \(be_6\) with \(je\), \(ce_3\) with \(c'k + s'ke\), \(ce_7\) with \(-s'k + c'ke\) (\(c' = \cos x, s' = \sin x\) for some \(x\)), \(de_4\) with \(e\). According to Cabrera1996? every homogeneous \(G_2\)-structure on \(N_{k,l}\) arises in this way.
The \(G_2\) 3-form is given as: \[\begin{align} \varphi &= abc \, c' e_1 \wedge e_2 \wedge e_3 - abc \, s' e_1 \wedge e_2 \wedge e_7 + (a^2 d) e_1 \wedge e_4 \wedge e_5 \\ &\quad - abc \, c' e_1 \wedge e_6 \wedge e_7 + abc \, s' e_1 \wedge e_3 \wedge e_6 + (b^2 d) e_2 \wedge e_4 \wedge e_6 \\ &\quad + abc \, c' e_2 \wedge e_5 \wedge e_7 - abc \, s' e_2 \wedge e_3 \wedge e_5 + (c^2 d) e_3 \wedge e_4 \wedge e_7 \\ &\quad - abc \, c' e_3 \wedge e_5 \wedge e_6 + abc \, s' e_5 \wedge e_6 \wedge e_7. \end{align}\]
The coclosed homogeneous \(G_2\)-structures are given precisely by the condition \(s = 0\). The space of such structures admits an obvious \(\mathbb{Z}_2\times \mathbb{Z}_2\) symmetry. As shown in Ball2019? the space of such \(G_2\)-structures can therefore be identified with \[(\mathbb{R}^+)^2 \times (\mathbb{R}\backslash \{0\})\] by fixing the signs of \(a,b\) to be positive.
Up to scaling there are two non-equivalent nearly parallel \(G_2\)-structures. They correspond to \(x = 0\), \(c < 0\) and have the opposite signs of \(d\), see Ball2019?. We denote these structures \(\varphi_+\) and \(\varphi_-\), where \(\varphi_+\) is the structure given by \(d>0\) and \(\varphi_-\) is given by \(d < 0\).
Lemma 5. For the homogeneous nearly-parallel \(G_2\)-structures \(\varphi_\pm\) we have \[\bar{\nu}(\varphi_+) = -\bar{\nu}(\varphi_-).\]
In this section we will discuss the scalar curvature of homogeneous metrics on \(N_{k,l}\).
Let \(\{f_i\}\) be the dual basis to \(\{e_i\}\). Recall that the most general homogeneous metric on \(N_{k,l}\) is given by \[g = a^2\left({f_1}^2 + f_5^2\right) + b^2\left(f_2^2 + f_6^2\right) + c^2\left(f_3^2 + f_7^2\right) + d^2f_4^2.\]
We are interested in the sign of the scalar curvature, which is preserved under rescaling, so we consider the rescaled metric \[g/d^2 = \lambda_1\left({f_1}^2 + f_5^2\right) + \lambda_2\left(f_2^2 + f_6^2\right) + \lambda_3\left(f_3^2 + f_7^2\right) + f_4^2.\] The scalar curvature of such metric can by computed using the following result:
Theorem 3 (Park2013?). \[S_{(\lambda_1,\lambda_2,\lambda_3)} = \frac{-(\lambda_1^2 + \lambda_2^2 + \lambda_3^2) + 6(\lambda_1\lambda_2 + \lambda_2\lambda_3 + \lambda_3\lambda_1)}{6\,\lambda_1\lambda_2\lambda_3} - \frac{1}{8q}\left(\frac{(k+l)^2}{\lambda_1^2} + \frac{l^2}{\lambda_2^2} + \frac{k^2}{\lambda_3^2}\right),\] where \(q = k^2 + kl + l^2\).
We write this as \[S(\lambda) = f(\lambda) - g(\lambda),\] where \(f\) is homogeneous of degree \(-1\) and \(g\) is homogeneous of degree \(-2\). Consequently, for any \(t > 0\)
\[S(t\lambda) = \frac{1}{t}f(\lambda) - \frac{1}{t^2}g(\lambda).\]
Lemma 6. The space of homogeneous metrics with positive scalar curvature is connected.
Proof of Lemma 6. Note that if for some \(\lambda \in \mathbb{R}_+^3\) \(f(\lambda) > 0\) then \(S(t\lambda) > 0\) for all sufficiently large \(t > 0\). Hence, the ray \(\{t\lambda, t > 0\}\) intersects the set \(\{S > 0\}\) in an unbounded interval. In particular, the set \(\{S > 0\}\) is a cone over \(\{S = 0\}\). Now, we project this set to the plane \(\lambda_3 = 1\). The projection will be the same as projection of \(\{f > 0\}\), which is connected. Since \(S = 0\) is a smooth surface in \(\mathbb{R}^3_+\) and its projection to the plane \(\lambda_3 = 1\) is connected and one to one, the set {S = 0} is also connected. Since \(\{S > 0\}\) is a cone over {S = 0}, it is also connected. ◻
Proof of Lemma 5. Let \(\alpha\) be a homogeneous coclosed \(G_2\) structure with \(a = b = -c = d = 1\), \(\beta\) be a homogeneous coclosed \(G_2\) structure with \(a = b = -c = -d = 1\). Then \(\alpha = -\beta\). Both of these structures induce the normal metric, which has the positive scalar curvature.
Since the space of metrics with positive scalar curvature is connected, there exist paths in the parameter space \((a,b,c,d)\) connecting \(\varphi_+\) to \(\alpha\) and \(\varphi\) to \(\beta\), such that induced metrics along these paths have positive scalar curvature. Hence, \[\bar{\nu}(\varphi_+) = \bar{\nu}(\alpha) = -\bar{\nu}(\beta) = -\bar{\nu}(\varphi_-).\] ◻
Remark 2. From the proof of Lemma 5 we can see that to compute \(\bar{\nu}\)-invariants of \(\varphi_\pm\) it is enough to compute them for some homogeneous \(G_2\)-structure inducing the normal metric.
Remark 3. From the proof of Lemmas 5 and 6 we can see that structures \(-\varphi_-\) and \(\varphi_+\) are homotopic through the path of \(G_2\)-structures inducing metrics with positive scalar curvatures.
In this section we compute the \(I\) and \(J\) terms appearing in the Goette’s formulas for \(\eta\)-invariants in the case of the Aloff-Wallach spaces.
First we state the general formulas.
Let \(W_G\) denote the Weyl group of \(G\), \(\Delta^+_G\) the set of positive roots of \(G\), \(\mathfrak{t}, \mathfrak{s}\) be the maximal Cartan subalgebras inside \(\mathfrak{g},\mathfrak{h}\). Let \(\rho_G\) and \(\rho_H\) be the half-sums of positive roots of \(G\) and \(H\). Let \(\widehat{A}(z) = \tfrac{z/2}{\sinh(z/2)}\).
Take \(E\in \mathfrak{s}^\perp \subset \mathfrak{h}\) be the positive unit vector, let \(\delta \in -i\mathfrak{t}^*\) be the unique weight such that \(-i\delta(E) > 0\) and \(\delta(X) \in 2\pi i\mathbb{Z} \Leftrightarrow e^X \in S.\)
Then, by Goette2009? the first term for the \(\eta\)-invariant of the Dirac operator is given as: \[\begin{align} I_D = 2 \sum_{w \in W_G} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)} \biggl( &\prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX)) e^{- \frac{\delta}{2}(wX)} -\\ &- \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX|_\mathfrak{s}))e^{-\rho_H(wX|_{\mathfrak{s}})}\biggr) \cdot \prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0}. \end{align}\]
Let \(\hat{\widetilde{\pi}}: H \to \mathop{\mathrm{End}}(S)\) denote the action of \(H\) inducing the \(\Lambda^{ev}\) bundle on \(G/H\). Let \(\{\kappa_j\}\) be the weights of \(\hat{\widetilde{\pi}}\). Take \(\{\alpha_j\}\) to be unique weights in \(i\mathfrak{t}^*\) such that \({\alpha_j}|_\mathfrak{s} = \kappa_j + \rho_H\) and \(-i(\alpha_j-\delta)(E)< 0 \leqslant -i\alpha_j(E)\). We will call weight \(\alpha_j\) a lift of the weight \(\kappa_j\).
Then, by Goette2009? the first term for the \(\eta\)-invariant of the odd-signature operator is given as: \[\begin{align} I_B = 2 \sum_j\sum_{w \in W_G} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)} \biggl( &\prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX)) e^{-\left(\alpha_j + \frac{\delta}{2}\right)(wX)} -\\ &- \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX|_\mathfrak{s})) e^{-(\kappa_j + \rho_H)(wX|_\mathfrak{s})}\biggr)\cdot \prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0}. \end{align}\]
Consider two paths of \(G\)-equivariant Dirac operators \(D^\lambda\) and \(B^{\lambda,\mu}\) connecting the Dirac operator \(D\) on \(\Gamma(S)\) and the odd-signature operator \(B\) \(\Omega^{ev}(M)\) induced by the Levi-Civita connection to their reductive counterparts \(\widetilde{D}\) and \(\widetilde{B}\) in the terminology of Goette1999? and Goette2009?. Using Frobenius reciprocity and Peter-Weyl theorem we write: \[\Gamma(S) = \overline{\bigoplus\limits_{\gamma \in \widehat{G}} V^\gamma \otimes \mathop{\mathrm{Hom}}_H(V^\gamma, S)},\] \[\Omega^{ev}(M) = \overline{\bigoplus\limits_{\gamma \in \widehat{G}} V^\gamma \otimes \mathop{\mathrm{Hom}}_H(V^\gamma,S\otimes S)}.\]
For each summand above, we may write \[\begin{align} \tag{2} D^\lambda|_{V^\gamma \otimes \mathop{\mathrm{Hom}}_H(V^\gamma, S)} &= \mathop{\mathrm{id}}_{V^\gamma} \otimes {}^\gamma D^\lambda, \\ \tag{3} B^{\lambda,\mu}|_{V^\gamma \otimes \mathop{\mathrm{Hom}}_H(V^\gamma,S\otimes S)} &= \mathop{\mathrm{id}}|_{V^\gamma} \otimes {}^\gamma{B}^{\lambda,\mu}. \end{align}\]
The explicit formulas for \({}^\gamma D^\lambda\) and \({}^\gamma{B}^{\lambda,\mu}\) are given in the section 5.2.
Then, \(J\) terms are the spectral flow terms given as: \[\begin{align} \tag{4} J_D &= \sum\limits_{\gamma \in \widehat{G}} \chi_G^\gamma \cdot (\eta({}^\gamma D) - (\eta+h)( {}^\gamma\widetilde{D})),\\ \tag{5} J_B &= \sum\limits_{\gamma \in \widehat{G}} \chi_G^\gamma \cdot (\eta({}^\gamma B) - (\eta+h)({}^\gamma \widetilde{B})). \end{align}\]
Here, \(h\) denotes the dimension of the kernel of the corresponding operator.
We now consider the case of \(G = SU(3)\), \(H = S^1\) and define Cartan subalgebras as follows: \[\begin{align} \mathfrak{t} &= \{i\mathop{\mathrm{diag}}(x_1,x_2,x_3)\;| \;x_1 + x_2 + x_3 = 0\},\\ \mathfrak{s} &= \{i\mathop{\mathrm{diag}}(kt,lt,-(k+l)t)\;| \;t \in \mathbb{R}\}. \end{align}\]
Let \(L_j \in i\mathfrak{t}^*\) be given by \(L_j(i\mathop{\mathrm{diag}}(x_1,x_2,x_3)) = ix_j.\)
The Weyl group \(W_{SU(3)}\) is the symmetric group \(S_3\). We pick the Weyl chamber \(P_{SU(3)}= \{x_1 > x_2 > x_3\}\). With this choice the positive roots are: \[\beta_1 = L_1 - L_2, \beta_2 = L_2 - L_3, \beta_3 = L_1 - L_3.\]
The half-sum of the weights is \(\rho_G = \frac{1}{2}(\beta_1 + \beta_2 + \beta_3) = \beta_3.\) For \(S^1\) we pick the Weyl chamber \(P_{S^1} = \{t>0\}\). Since \(H = S^1\), we have that \(\rho_H = 0\).
Following the orientation conventions from Goette2009? and the orientation chosen for \(\mathfrak{m}\), the space \(\mathfrak{s}^\perp\) is oriented by taking \(-e_4\) to be the positive vector. Thus, \[E = -e_4.\]
\[\delta = \frac{1}{\varepsilon(k,l)}\left((-2l-k)L_1 + (2k+l)L_2 + (l-k)L_3\right).\]
\[\varepsilon(k,l) = \gcd(2k+l, 2l+k) = \begin{cases} 3 \text{ if } k \equiv l \mod 3 \text{ and } 3 \not | k, \\ 1 \text{ otherwise }. \end{cases}\]
Then the first term for the \(\eta\)-invariant of the Dirac operator can be expressed as: \[I_D = 2 \sum_{w \in W_G} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)} \left( \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX)) e^{- \frac{\delta}{2}(wX)} - \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX|_\mathfrak{s}))\right)\cdot \prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0}.\]
Let \(\{\kappa_j\}\) be the weights of the representation of \(\hat{\widetilde{\pi}}\) and \(\{\alpha_j\}\) be their lifts. Then the first term of the \(\eta\)-invariant of the odd signature operator can be expressed as:
\[\begin{align} I_B &= 2 \sum_j\sum_{w \in W_G} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)} \biggl( \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX)) e^{-\left(\alpha_j + \frac{\delta}{2}\right)(wX)} -\\ &- \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX|_\mathfrak{s})) e^{-\kappa_j(wX|_\mathfrak{s})}\biggr)\cdot \prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0}. \end{align}\]
Where the weights of \(\hat{\widetilde{\pi}}\) are \((0,0, \pm i(k-l), \pm i(2k+l), \pm i(2l+k))\).
Lemma 7. We have: \[-24I_D + 3I_B = 1.\]
Proof sketch of Lemma 7. The expressions for \(I_D\) and \(I_B\) admit apparent singularities of the order \(|\Delta_G^+| + 1\), which cancel after symmetrization over the Weyl group \(W_G\). In the case \(G = SU(3)\), there are three positive roots, and hence the singularities are of the fourth order. Consequently, to evaluate the limit as \(X \to 0\), it is enough to compute the fourth power term in the Taylor expansion of the expressions involved in the formulas for \(I_D\) and \(I_B\). A direct but lengthy computation shows that most of the terms in the expression \(3I_B - 24I_D\) cancel, yielding a constant value independent of the parameters \(k,l\).
The detailed computation is provided in the appendix [appx]. ◻
In this section we compute the spectral flow terms appearing in Goette’s formulas for \(\eta\)-invariants. This calculation is similar to the computations of the \(\eta\) invariants of the Berger space \(SO(5)/SO(3)\) in Goette2004?. The main result of this section is:
Lemma 8. The spectral flow terms for the Dirac operator and odd signature operator are \[J_D = 0,\] \[J_B = -14.\]
Proof of Lemma 8. By Goette1999?, we have \(\eta(D) = \eta(\widetilde{D})\), and by Goette1999? the kernel of \(\widetilde{D}\) is trivial for \(SU(3)/S^1\). Consequently, the spectral flow term from equation 4 is zero for the Dirac operator: \[J_D = 0.\]
We now focus on the odd signature operator. Let \(e_1,\ldots,e_7\) be the orthonormal basis of \(\mathfrak{m}\) as in 1 . Denote by \(c_i\), \(\hat{c}_i\) the Clifford multiplication by \(e_i\) on the first and second factor of \(\Lambda^{ev}\mathfrak{m} \cong S\otimes S\), respectively.
We define two maps \(\widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}}\) and \(\widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m}}: \mathfrak{g} \to \mathop{\mathrm{End}}(\Lambda^{ev}\mathfrak{m})\) by
\[\widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}}(X) = \frac{1}{4}\sum_{i,j = 1}^7 \langle [X, e_i], e_j\rangle c_ic_j \text{ and } \widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m}}(X) = \frac{1}{4}\sum_{i,j = 1}^7 \langle [X, e_i], e_j\rangle \hat{c}_i\hat{c}_j,\] and set:
\[\widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, i} = \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}}(e_i) \text{ and } \widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m},i} = \widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m}}(e_i).\] Then \(\widetilde{\pi} = \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}}|_{\mathfrak{h}}\) and \(\hat{\widetilde{\pi}} = \widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m}}|_{\mathfrak{h}}\) are the differentials of the representation of \(H\) on the two factors of \(S\otimes S\) that induce the bundle \(\Lambda^{ev}\mathfrak{m}\).
Let \(\gamma_i\) denote the action of \(e_i\) on the dual of the representation space \(V^\gamma\). Then the operators from equations 2 and 3 are defined in the following way:
\[{}^\gamma D^\lambda = \sum_{i=1}^7 c_i(\gamma_i + \lambda \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m},i})\]
\[{}^\gamma {B}^{\lambda,\mu} = \sum_{i=1}^7 c_i(\gamma_i + \lambda\widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m},i} + \mu\widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m},i}).\]
Note that \(D^{\tfrac{1}{2}} = D\) and \(B^{\tfrac{1}{2},\tfrac{1}{2}}\) are respectively the Dirac operator and the odd signature operator associated to the Levi-Civita connection on \(M\), while \(D^{\frac{1}{3}} = \widetilde{D}\) and \(B^{\frac{1}{3},0}\) are respectively the reductive Dirac operator and odd signature operators from the Goette1999?, Goette2009?.
Now, consider the one-parameter family \(B^{\lambda,3\lambda-1}\) for \(\lambda \in [\frac{1}{3}, \frac{1}{2}]\):
\[{}^\gamma {B}^{\lambda,3\lambda - 1} = {}^\gamma {\widetilde{B}} + \mu B_0.\]
Where \(B_0 = \sum_{i=1}^7 c_i\left(\frac{1}{3}\widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m},i} + \widehat{\widetilde{\mathop{\mathrm{ad}}}}_{\mathfrak{m},i}\right),\) and \(\mu = 3\lambda-1\).
The square of \({}^\gamma {\widetilde{B}}\) has been computed in Goette1999?:
\[{}^\gamma {\widetilde{B}}^2 = ||\gamma + \rho_G||^2 - c_H^{\hat{\widetilde{\pi}}} - ||\rho_H||^2\]
Since in our case \(\rho_H = 0\), this simplifies to: \[{}^\gamma {\widetilde{B}}^2 = ||\gamma + \rho_G||^2 - c_H^{\hat{\widetilde{\pi}}}\]
We now compute \(c_H^{\hat{\widetilde{\pi}}}\). On the weight space \(V_\mu\) of \(S^1\) the Casimir operator is given as \(||\mu(h)||^2\) where \(h\) is the unit generator of \(\mathfrak{s}\) with respect to the norm induced from the embedding \(\iota_{k,l}: S^1 \to SU(3)\). Hence, on the weight space \(V_m\) the Casimir operator is \(\frac{m^2}{2(k^2+kl+l^2)}\). In our case the weights of \(\hat{\widetilde{\pi}}\) are \(0, 0, \pm i(k-l), \pm i(2l+k), \pm i(2k+l)\). Thus, \[c_H^{\hat{\widetilde{\pi}}} \in \left\{0, \frac{(k-l)^2}{2(k^2+kl+l^2)}, \frac{(2k+l)^2}{2(k^2+kl+l^2)}, \frac{(2l+k)^2}{2(k^2+kl+l^2)} \right\}.\] In particular, \[c_H^{\hat{\widetilde{\pi}}} \leqslant 2.\]
Next we compute the \(||\gamma + \rho_G||^2\) terms. Let \(\gamma_{(p,q)}\) denote the irreducible representation of \(SU(3)\) with the highest weight \(pL_1 - qL_3\), then \[||\gamma_{(p,q)} + \rho_G||^2 = ||(p+1)L_1 - (q+1)L_3||^2 = \frac{2}{3}(p^2 + q^2 + pq) + 2(p+q) + 2.\] Note that \(||\gamma + \rho_G||^2 - c_H^{\hat{\widetilde{\pi}}}\) is always non-negative.
We now compute the eigenvalues of \(B_0\) using the following model for \(S\). We identify \(S\) with \(\Lambda^* V\) where \(V\) is 3-dimensional totally isotropic subspace with basis \[f_1 = \frac{1}{\sqrt2}(e_1 + ie_5), f_2 = \frac{1}{\sqrt2}(e_2 + ie_6), f_3 = \frac{1}{\sqrt2}(e_3 + ie_7).\]
The Clifford multiplication is given as:
\[\begin{align} c_1 &= \varepsilon_1 + \iota_1\\ c_2 &= \varepsilon_2 + \iota_2\\ c_3 &= \varepsilon_3 + \iota_3\\ c_4 &= \pm i(-1)^{\deg}\\ c_5 &= i(\iota_1 - \varepsilon_1)\\ c_6 &= i(\iota_2 - \varepsilon_2)\\ c_7 &= i(\iota_3 - \varepsilon_3).\\ \end{align}\]
The choice of the sign for \(c_4\) is given by the choice of orientation on \(N_{k,l}\). We want the volume element \(c_1c_5c_2c_6c_3c_7c_4\) to act as \(\mathop{\mathrm{id}}\) on \(S\). Since \(c_1c_5c_2c_6c_3c_7\) acts as \(i(-1)^{\deg}\), we have to choose \(c_4 = -i(-1)^{\deg}\).
In the basis 1 :
\[\begin{align} \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 1} &= \tfrac{1}{4}\left(-\sqrt2 c_2c_3 + \sqrt2 c_6c_7 - \tfrac{6(k+l)}{r}c_4c_5\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 2} &= \tfrac{1}{4}\left(\sqrt2 c_1c_3 - \sqrt2 c_5c_7 + \tfrac{6k}{r}c_4c_6\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 3} &= \tfrac{1}{4}\left(-\sqrt2 c_1c_2 + \sqrt2c_5c_6 + \tfrac{6l}{r}c_4c_7\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 4} &= \tfrac{1}{4}\cdot \tfrac{6}{r}\left((k+l)c_1c_5 - kc_2c_6 -lc_3c_7\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 5} &= \tfrac{1}{4}\left(\sqrt2c_2c_7 - \sqrt2c_3c_6 -\tfrac{6(k+l)}{r}c_1c_4\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 6} &= \tfrac{1}{4}\left(-\sqrt2 c_1c_7 + \sqrt2c_3c_5 + \tfrac{6k}{r}c_2c_4\right)\\ \widetilde{\mathop{\mathrm{ad}}}_{\mathfrak{m}, 7} &= \tfrac{1}{4}\left(\sqrt2c_1c_6 - \sqrt2 c_2c_5 + \tfrac{6l}{r}c_3c_4\right).\\ \end{align}\]
Using the above model for spinors we compute the matrix \(B_0\) acting on the space \(S\otimes S\) and its eigenvalues in sympy. All of these computations can be found here. The maximal absolute value of the eigenvalues of \(B_0\) is \(2\sqrt2\).
As we have seen before, \[({}^\gamma {\widetilde{B}})^2 \geqslant \frac{2}{3}(p^2 + q^2 + pq) + 2(p+q) \geqslant 2 = \lambda_{\max}^2\left(\frac{1}{2}B_0\right)\] for \((p,q) \neq (0,0)\).
Consequently, the only irreducible representation \(\gamma\) for which the sign of eigenvalues can change along the path \({}^\gamma {B}^{\lambda,3\lambda - 1}\) is the trivial representation \(\gamma_0\).
For the trivial representation, we explicitly compute the matrices \({}^{\gamma_0}{B}^{\frac{1}{3},0}\) and \({}^{\gamma_0}{B}^{\frac{1}{2},\frac{1}{2}}\) and evaluate their \(\eta\)-invariants. \[\eta({}^{\gamma_0}{B}^{\frac{1}{3},0}) = 16\mathop{\mathrm{sign}}(k) + 16\mathop{\mathrm{sign}}(l) + 16\mathop{\mathrm{sign}}(-k-l) = 16,\;h({}^{\gamma_0}{B}^{\frac{1}{3},0}) = 0.\] \[\eta({}^{\gamma_0}{B}^{\frac{1}{2},\frac{1}{2}}) = 2\mathop{\mathrm{sign}}(k) + 2\mathop{\mathrm{sign}}(l) + 2\mathop{\mathrm{sign}}(-k-l) = 2,\;h({}^{\gamma_0}{B}^{\frac{1}{2},\frac{1}{2}}) = 2.\] The resulting difference is \[\eta({}^{\gamma_0}{B}^{\frac{1}{2},\frac{1}{2}}) - (\eta+h)({}^{\gamma_0}{B}^{\frac{1}{3},0}) = -14.\] Consequently, \[J_B = -14.\] ◻
Gathering results from the previous sections, we have:
Proof of Theorem 1. Let \(\alpha\) be the homogeneous \(G_2\)-structure inducing the normal metric and the same orientation as \(\varphi_+\). By the proof of Lemma 6 we know, \[\bar{\nu}(\alpha) = \bar{\nu}(\varphi_+) = \bar{\nu}(\varphi_-).\]
By Proposition 1, we know that
\[\bar{\nu}(\alpha) = -24I_D + 3I_B - 24J_D + 3J_B.\]
By Lemma 7 we have \(-24I_D + 3I_B = 1\), and by 8 we have \(-24J_D + 3J_B = -42.\)
In total, we have \[\bar{\nu}(\varphi_+) = \bar{\nu}(\alpha) = -41.\]
And, in addition,
\[\bar{\nu}(\varphi_-) = - \bar{\nu}(\varphi_+) = 41.\] ◻
Let \((M,g)\) be a 7-dimensional 3-Sasakian manifold. It is well-known that 3-Sasakian structure admits a 3-dimensional space of Killing spinors and hence a 2-sphere worth of nearly-parallel \(G_2\)-structures given by choosing a unit Killing spinor. We denote these structures by \(\varphi_{ts}(x)\) for \(x \in S^2\).
Moreover, from the data of 3-Sasakian structure one can construct a proper (in the sense that its space of Killing spinors is 1-dimensional) nearly-parallel \(G_2\) structure called squashed nearly-parallel \(G_2\) structure (cf. Friedrich1997?). We denote it by \(\varphi_{sq}\).
We begin by recalling the definition of the 3-Sasakian structure.
Definition 4. A 3-Sasakian structure on the manifold \((M,g)\) is a triple of vector fields \((V_1,V_2,V_3)\) such that the following is satisfied:
Vector \(V_i\) defines Sasakian structure for each \(i = 1,2,3\).
The frame \((V_1,V_2,V_3)\) is orthonormal.
For each permutation (i,j,k) of the sign \(\delta:\) \(\nabla_{V_i}V_j = (-1)^\delta V_k.\)
On the distribution orthogonal to \((V_1,V_2,V_2)\) the tensors \(\phi_i = -\nabla V_i\) satisfy \(\phi_i\phi_j = (-1)^\delta\phi_k.\)
A vector is called horizontal if it is orthogonal to \(V_i\) for \(i = 1,2,3\). A vector is called vertical if it lies in the span of \(V_i\).
For \(t > 0\) define the canonical variation of the metric \(g^t\):
\(g^t(X,Y) = g(X,Y)\) if \(X,Y\) are horizontal vector fields, and \(g^t(V,W) = t^2g(V,W)\) if \(V,W\) are vertical vectors.
For \(s = \frac{1}{\sqrt5}\) this metric is Einstein and admits proper nearly-parallel \(G_2\) structure \(\varphi_{sq}\).
Lemma 9. The squashed nearly-parallel \(G_2\) structure \(\varphi_{sq}\) is homotopic to \(\varphi_{ts}(x) \;\forall x\) along the path of \(G_2\)-structures inducing metrics with positive scalar curvature.
Proof of Lemma 9. All of the \(\varphi_{ts}(x)\) are homotopic since they correspond to a choice of a unit Killing spinor associated to the 3-Sasakian structure, which is connected.
Fix an orthonormal frame of the horizontal distribution \(X_1,X_2,X_3,X_4\) and define \(Z_a := V_a/t\). Following Friedrich1997? we define the path of \(G_2\) 3-forms in the following way:
\[\varphi_s = F_1 + F_2,\] where \[F_1 = Z_1 \wedge Z_2 \wedge Z_3,\] \[F_2 = \sum_a Z_a \wedge \omega_a, \text{ and } \omega_a = \frac{1}{2} \sum_i X_i \wedge \nabla_{X_i}V_a.\]
The form \(\varphi_t\) induces precisely the metric \(g_t\) and gives the path between \(\varphi_1\) and \(\varphi_{\frac{1}{\sqrt5}} = \varphi_{sq}\). We also note that the scalar curvature stays positive along this path: according to Boyer2007b? the scalar curvature of the metric \(g^t\) is \[s_t = 48 + \frac{6}{t^2} - 12t^2.\] Which is positive for \(t \in \left[\tfrac{1}{\sqrt5}, 1\right]\).
Note that \(\varphi_1\) is not the one of nearly-parallel \(G_2\)-structures induces from 3-Sasakian structure.
According to Agricola2010? the \(G_2\) structure \(\varphi_1\) (called the canonical \(G_2\) structure associated to 3-Sasakian structure) admits a spinor field \(\Psi_0\), which generates Killing spinors by taking the Clifford product with the horizontal vectors \(V_a\). In particular, the space of Killing spinors is generated by \(V_a \cdot \Psi_0\) for \(a = 1,2,3\).
But since \(X\cdot \Psi_0 \perp \Psi_0\) for any vector field \(X\), these spinors can be continuously rotated one into another via \[\Psi_t = \Psi_0\cos t + V_a \cdot \Psi_0\sin t.\] Hence, the corresponding \(G_2\)-structures are homotopic, completing the proof. ◻
Example 1. In particular, we can compute the \(\bar{\nu}\)-invariant of the squashed metric on \(S^7\): \[\label{eq:nuS7} \bar{\nu}(\varphi_{sq}(S^7)) = \bar{\nu}(\varphi_{std}(S^7)) = 1.\qquad{(4)}\]
The last equality is due to Crowley2025?.
In this section we would like to gather some results regarding the first Pontryagin class of homogeneous nearly-parallel \(G_2\) manifolds.
The main statement of this section is:
Proposition 2. Let \(M\) be a 7-dimensional homogeneous space admitting the homogeneous nearly parallel \(G_2\) structure. Then the first Pontryagin class \(p_1(M)\) is a torsion class.
Proof of Proposition 2. The homogeneous nearly parallel \(G_2\) manifolds were classified by Friedrich1997?. The only proper homogeneous examples are the squashed 7-sphere, Aloff-Wallach spaces and the Berger space.
Other homogeneous nearly parallel \(G_2\) manifolds are not proper, hence they are Sasaki-Einstein and according to LeBrun2025? any Sasaki-Einstein 7-manifolds has \(p_1(M)\) a torsion class.
Next we compute \(p_1\) explicitly for the proper cases.
In the case of \(S^7\), \(p_1\) is trivially zero.
According to Kruggel1997?, the fourth cohomology group of \(N_{k,l}\) is \(H^4(N_k,l) \cong \mathbb{Z}_{2(k^2 + kl + l^2)}\), and the first Pontryagin class \(p_1(N_{k,l})\) is zero.
Let \(\xi_{m,n}\) be the vector bundle of the rank \(4\) over \(S^4\) with the Euler class \(e(\xi_{m,n}) = n\) and first Pontryagin class \(p_1(\xi_{m,n}) = 2(n+2m)\). Let \(M_{m,n}\) be the corresponding \(S^3\) bundle over \(S^4\). Then, according to Goette2004? the Berger space \(SO(5)/SO(3)\) is diffeomorphic to \(M_{\mp 1, \pm 10}\). The algebraic topological invariants of \(M_{m,n}\) are computed in Crowley2003?: \[H^4(M_{m,n}) \cong \mathbb{Z}_{n}, \;p_1(M_{m,n}) = 4m \in \mathbb{Z}_{n}.\] Hence, \[p_1(SO(5)/SO(3)) = -4 \in \mathbb{Z}_{10}.\]
◻
In fact, as we can see, all of the known examples (including nonhomogeneous ones obtained from Sasaki-Einstein or 3-Sasakian structures) of nearly parallel \(G_2\) manifolds have torsion \(p_1\) class (although, sometimes for trivial reasons).
In the appendix we carry out explicit computations required to prove Lemma 7. In the first part we discuss the problem of lifting weights of \(\hat{\widetilde{\pi}}\) from \(i\mathfrak{s}\) to \(i\mathfrak{t}\), which is required to compute \(\eta\) of the odd signature operator \(B\). In the second part we explicitly compute the sum \(-24I_D + 3I_B\).
As we have seen in the section 5 the weights of \(\widetilde{\pi}\) are \(0, \pm i(k-l), \pm i(2l+k), \pm i(2k+l)\).
In this section we compute lifts of the weights of \(\widetilde{\pi}\).
By the lift of the weight \(\kappa \in i\mathfrak{s}^*\) to the weight \(\alpha \in i\mathfrak{t}^*\) we understand the unique weight such that \({\alpha}|_\mathfrak{s} = \kappa + \rho_H\) and \(-i(\alpha-\delta)(E)< 0 \leqslant -i\alpha(E)\) for \(\delta, E\) as in 5.
Obviously, by this definition \(0\) lifts to \(0\).
For the moment we assume that \(\gcd(2k+l, 2l+k) = 1\), and deal with the other case below. Let \(s:= ||(-2l-k,2k+l,l-k)|| = \sqrt{6(k^2 +kl + l^2)}.\)
consider the weight \(-i(k-l)\). Then the lift should be of the form \(-\beta_1 + m\delta\) for some \(m \in \mathbb{z}\). \[0 \leqslant i\beta_1(E) = \frac{3(k+l)}{s} < \delta(E) = s.\] hence the lift is \[\alpha = -\beta_1.\]
consider the weight \(i(2k+l)\). Then the lift should be of the form \(\beta_2 + m\delta\) for some \(m \in \mathbb{z}\). \[0 \leqslant -i\beta_2(E) = \frac{3k}{s} < \delta(E) = s.\] hence the lift is \[\alpha = \beta_2.\]
consider the weight \(-i(2l+k)\). Then the lift should be of the form \(-\beta_3 + m\delta\) for some \(m \in \mathbb{Z}\). \[0 \leqslant i\beta_3(E) = \frac{3l}{s} < \delta(E) = s.\] hence, the lift is \[\alpha = -\beta_3.\]
Remark 4. Now, assume that weight \(\kappa\) lifts to \(\alpha\), i.e. \[0 \leqslant -i\alpha(E) < -i\delta(E).\]
Consider the weight \(-\kappa\), then \[-i\delta(E) > -i(\delta(E) - \alpha(E)) > 0.\]
Thus, the weight \(-\kappa\) lifts to \(\delta - \alpha\). In particular \(i(k-l)\) lifts to \(\delta + \beta_1\), \(-i(2k+l)\) lifts to \(\delta - \beta_2\), and \(i(2l+k)\) lifts to \(\delta + \beta_3\).
Now, assume that we are in the case, when \(\gcd(2k+l,2l+k) = 3\), then \(\delta(E) = s/3\). First, we assume that \(k = 3m+1, l = 3n+1\). Under our assumptions on \(k,l\) from section 4, we have \(m\neq n\) and \(m,n > 0\).
Then \[\begin{align} s\delta(E)-3(k+l) &= \frac{1}{3}\cdot 6(9m^2 + 9m + 9mn + 9n^2 + 9n + 3) -3(3m+3n+2) =\\ &= 2(9m^2 + 9m + 9mn + 9n^2 + 9n + 3) -3(3m+3n+2)\geqslant 0. \end{align}\] It is easy to check that the same is true for other weights and for the case \(k \equiv l \equiv -1 \mod 3\).
Thus, \(\delta(E) \geqslant \beta_i(E)\) for all \(i\) and the lifts of the weights of \(\hat{\widetilde{\pi}}\) are the same as before.
In this section we give the complete proof of the Lemma 7 by examining the terms constituting expressions for \(I_D\) and \(I_B\).
The expressions for the \(I_D\) and \(I_B\) have an apparent singularity of the fourth order, which cancels out. Since the expressions are analytic it is enough to compute fourth power terms in the Taylor series of corresponding functions.
We denote by \(\widetilde{I}_D\) the fourth power term of \[\left( \prod_{\beta \in \Delta_G^+}\hat{A}(\beta(wX)) e^{- \frac{\delta}{2}(wX)} - \prod_{\beta \in \Delta_G^+}\hat{A}(\beta(wX|_\mathfrak{s}))\right)\] and by \(\widetilde{I}_B\) the fourth power term of
\[\sum_j\biggl( \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX)) e^{-\left(\alpha_j + \frac{\delta}{2}\right)(wX)} - \prod_{\beta \in \Delta_G^+}\widehat{A}(\beta(wX|_\mathfrak{s})) e^{-\kappa_j(wX|_\mathfrak{s})}\biggr)\]
In particular \[I_D = \sum_{w \in W_{SU(3)}} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)}\widetilde{I}_D(wX)\prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0},\] and \[I_B = \sum_{w \in W_{SU(3)}} \frac{\mathop{\mathrm{sign}}(w)}{\delta(wX)}\widetilde{I}_B(wX)\prod\limits_{\beta \in \Delta^+_G}\frac{-1}{\beta(X)}\bigg|_{X = 0}.\]
Note that \[\widehat{A}(z) = \frac{z/2}{\sinh{z/2}} = 1 - \frac{1}{24}z^2 + \frac{7}{5760}z^4 + \ldots\]
Denote \(a = -\frac{1}{24}, b = \frac{7}{5760}\).
Since we have three positive roots in the case of \(SU(3)/S^1\), the terms that we need to compute have the following expression:
\[\widehat{A}(\beta_1)\widehat{A}(\beta_2)\widehat{A}(\beta_3)\widehat{A}(\delta)e^{-\left(\alpha-\frac{\delta}{2}\right)} - \widehat{A}(\widetilde{\beta_1})\widehat{A}(\widetilde{\beta_2})\widehat{A}(\widetilde{\beta_3})e^{-\kappa}.\]
Fourth power term is given as:
\[\begin{align} &b\left(\beta_1^4 + \beta_2^4 + \beta_3^4 - \widetilde{\beta}_1^4 - \widetilde{\beta}_2^4 - \widetilde{\beta}_3^4 + \delta^4\right) + \\ +& a^2\left(\beta_1^2\beta_2^2 + \beta_2^2\beta_3^2 + \beta_3^2\beta_1^2 - \widetilde{\beta}_1^2\widetilde{\beta}_2^2 - \widetilde{\beta}_2^2\widetilde{\beta}_3^2 - \widetilde{\beta}_3^2\widetilde{\beta}_1^2\right)+\\ +& a^2\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\delta^2 + \frac{a}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2 +\delta^2\right)\left(\alpha - \frac{\delta}{2}\right)^2 -\\ -& \frac{a}{2}\left(\widetilde{\beta}_1^2 + \widetilde{\beta}_2^2 + \widetilde{\beta}_3^2\right)\kappa^2 + \frac{1}{24}\left(\alpha - \frac{\delta}{2}\right)^4 - \frac{1}{24}\kappa^4 = \\ =& b\left(\beta_1^4 + \beta_2^4 + \beta_3^4 - \widetilde{\beta}_1^4 - \widetilde{\beta}_2^4 - \widetilde{\beta}_3^4 + \delta^4\right) +\\ +& a^2\left(\beta_1^2\beta_2^2 + \beta_2^2\beta_3^2 + \beta_3^2\beta_1^2 - \widetilde{\beta}_1^2\widetilde{\beta}_2^2 - \widetilde{\beta}_2^2\widetilde{\beta}_3^2 - \widetilde{\beta}_3^2\widetilde{\beta}_1^2\right) + \\ +& a^2\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\delta^2 + \frac{a}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2 +\delta^2\right)\left(\alpha^2 - \alpha\delta + \frac{\delta^2}{4}\right) +\\ +& \frac{1}{24}\left(\alpha^4 - 2\alpha^3 \delta + \frac{3}{2}\alpha^2\delta^2 -\frac{1}{2}\alpha\delta^3 + \frac{1}{16}\delta^4\right)-\\ -& \frac{a}{2}\left(\widetilde{\beta}_1^2 + \widetilde{\beta}_2^2 + \widetilde{\beta}_3^2\right)\kappa^2 - \frac{1}{24}\kappa^4. \end{align} \label{eq:4term}\tag{6}\]
Denote \[U := \beta_1^4 + \beta_2^4 + \beta_3^4 - \widetilde{\beta}_1^4 - \widetilde{\beta}_2^4 - \widetilde{\beta}_3^4,\] \[V := \beta_1^2\beta_2^2 + \beta_2^2\beta_3^2 + \beta_3^2\beta_1^2 - \widetilde{\beta}_1^2\widetilde{\beta}_2^2 - \widetilde{\beta}_2^2\widetilde{\beta}_3^2 - \widetilde{\beta}_3^2\widetilde{\beta}_1^2.\]
In particular, since for the standard Dirac operator \(\kappa, \alpha = 0\), we have:
\[\widetilde{I}_D = bU + a^2V +\left(a^2 + \frac{a}{8}\right)\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\delta^2 + \left(b + \frac{a}{8} + \frac{1}{24\cdot 16}\right)\delta^4.\]
Now, consider \(\widetilde{I}_B\). Recall that \(\hat{\widetilde{\pi}}\) has eight weights. The two zero weights contribute two terms \(\widetilde{I}_D\), since they lift to zeroes. The remaining non-zero weights occur in pairs \((\kappa_i, -\kappa_i)\). By remark 4, if \(\kappa_i\) lifts to \(\alpha\), \(-\kappa_i\) lifts to \(\delta - \alpha\).
Observe that these weights appear in 6 , only as \(\kappa_i^{\text{even}}\) and \((\alpha - \delta/2)^{\text{even}}\). Since \((\delta - \alpha - \delta/2) = -(\alpha-\delta/2)\), the even powers ensure that \(\kappa_i\) and \(-\kappa_i\) make identical contributions to \(\widetilde{I}_B\). Recall that nontrivial weights of \(\hat{\widetilde{\pi}}\) lift to \((\pm \beta_i, \delta\mp \beta_i)\). Taking the sum over the weights of \(\hat{\widetilde{\pi}}\) and substituting the weights in the formula 6 we obtain:
\[\begin{align} \widetilde{I}_B &= 2 \widetilde{I}_D + 2\biggl(3bU + 3a^2V + 3b\delta^4 + 3a^2\delta^2\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right)+\\ &+ \frac{a}{2}\left(\beta_1^2 + \beta_2^2 +\beta_3^2+\delta^2\right)\left(\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right) - \left(\beta_1 - \beta_2 + \beta_3\right)\delta + \frac{3\delta^2}{4}\right)+\\ &+ \frac{1}{24}\biggr(\left(\beta_1^4 + \beta_2^4 + \beta_3^4\right) - 2\left(\beta_1^3 - \beta_2^3 + \beta_3^3\right)\delta + \frac{3}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\delta^2 - \\ &- \frac{1}{2}\left(\beta_1 - \beta_2 + \beta_3\right)\delta^3 + \frac{3}{16}\delta^4\biggr) -\\ &- \frac{a}{2}\left(\widetilde{\beta}_1^2 + \widetilde{\beta}_2^2 + \widetilde{\beta}_3^2\right)\left(\widetilde{\beta}_1^2 + \widetilde{\beta}_2^2 + \widetilde{\beta}_3^2\right) - \frac{1}{24}\left(\left(\widetilde{\beta}_1^4 + \widetilde{\beta}_2^4 + \widetilde{\beta}_3^2\right)\right) = \\ &=2\widetilde{I}_D + 2\biggl( \left(3b +\frac{a}{2} + \frac{1}{24}\right)U + \left(3a^2 + a\right)V + \left(3b +\frac{3a}{8} + \frac{3}{16\cdot 24}\right)\delta^4 +\\ &+ \left(3a^2 + \frac{3a}{8} + \frac{3}{48}\right)\delta^2\left(\beta_1^2+\beta_2^2+\beta_3^2\right) \\ &- \frac{a}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta\\ &- \frac{2}{24}\left(\beta_1^3 - \beta_2^3 + \beta_3^3\right)\delta + \left(-\frac{1}{48} -\frac{a}{2}\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta^3\biggr). \end{align}\]
\[\begin{align} -24\widetilde{I}_D + 3\widetilde{I}_B & = -24\left(bU +a^2V +a^2\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right)\delta^2 + \left(b + \frac{a}{8} + \frac{1}{16\cdot 24}\right)\delta^4\right) +\\ &+ 3\cdot 2 \left(bU + a^2V + a^2\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right)\delta^2 +\left(b + \frac{a}{8} + \frac{1}{24\cdot 16}\right)\delta^4\right) +\\ &+ 3\cdot2\biggl[ \left(3b +\frac{a}{2} + \frac{1}{24}\right)U + \left(3a^2 + a\right)V + \left(3b +\frac{3a}{8} + \frac{3}{16\cdot 24}\right)\delta^4 +\\ &+ \left(3a^2 + \frac{3a}{8} + \frac{3}{48}\right)\delta^2\left(\beta_1^2+\beta_2^2+\beta_3^2\right) - \frac{a}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta - \\ &- \frac{1}{12}\left(\beta_1^3 - \beta_2^3 + \beta_3^3\right)\delta + \left(-\frac{1}{48} -\frac{a}{2}\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta^3\biggr] =\\ &= 6\left(\frac{a}{2}+\frac{1}{24}\right)U + 6aV + 6\cdot\left(\frac{3a}{8} + \frac{3}{48}\right)\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right)\delta^2 \\ &-\frac{a}{2}\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta -\\ &- \frac{1}{12}\left(\beta_1^3 - \beta_2^3 + \beta_3^3\right)\delta + \left(-\frac{1}{48} -\frac{a}{2}\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta^3 = \\ &= \frac{1}{8}\left(U - 2V\right) - \frac{9}{32}\left(\beta_1^2 + \beta_2^2 +\beta_3^2\right)\delta^2 +\\ &+ \frac{1}{48}\left(\beta_1^2 + \beta_2^2 + \beta_3^2\right)\left(\beta_1 - \beta_2 + \beta_3\right)\delta - \frac{1}{12}\left(\beta_1^3 - \beta_2^3 + \beta_3^3\right)\delta. \end{align}\]
Lemma 10. We have that \[U = 2V.\]
Proof of Lemma 10. Let \[z = \frac{kx_1 + lx_2 +(k+l)(x_1+x_2)}{\sqrt{2}\sqrt{k^2+kl+l^2}},\] so that \[X|_{\mathfrak{s}} = ze_8.\]
\[\begin{align} U &= \beta_1^4 + \beta_2^4 + \beta_3^4 - \widetilde{\beta}_1^4 - \widetilde{\beta}_2^4 - \widetilde{\beta}_3^4 =\\ &= (x_1 - x_2)^4 + (2x_2 + x_1)^4 + (2x_1+x_2)^4 - z^4((k-l)^4 + (2l+k)^4 + (2k+l)^4) =\\ &= (18x_1^4 + 36x_1^3x_2 + 54x_1^2x_2^2 + 36x_1x_2^3 + 18x_2^4) - z^4(18k^4 + 36k^3l + 54k^2l^2 + 36kl^3 + 18l^4). \end{align}\]
\[\begin{align} (x_1 - x_2)^4 + (2x_2 + x_1)^4 + (2x_1+x_2)^4 &= 18x_1^4 + 36x_1^3x_2 + 54x_1^2x_2^2 + 36x_1x_2^3 + 18x_2^4,\\ (k-l)^4 + (2l+k)^4 + (2k+l)^4) &= 18k^4 + 36k^3l + 54k^2l^2 + 36kl^3 + 18l^4. \end{align}\]
\[\begin{align} V =& \beta_1^2\beta_2^2 + \beta_2^2\beta_3^2 + \beta_3^2\beta_1^2 - \widetilde{\beta}_1^2\widetilde{\beta}_2^2 - \widetilde{\beta}_2^2\widetilde{\beta}_3^2 - \widetilde{\beta}_3^2\widetilde{\beta}_1^2 = \\ =& (x_1-x_2)^2(2x_2+x_1)^2 + (2x_2+x_1)^2(2x_1+x_2)^2 + (2x_1+x_2)^2(x_1-x_2)^2 -\\ -& z^4((k-l)^2(2l+k)^2 + (2l+k)^2(2k+l)^2 + (2k+l)^2(k-l)^2) =\\ =& (9x_1^4 + 18x_1^3x_2 + 27x_1^2x_2^2 + 18x_1x_2^3 + 9x_2^4) - s^4(9k^4 + 18k^3l + 27k^2l^2 + 18kl^3 + 9l^4). \end{align}\]
Thus, we can see that \(U = 2V\). ◻
Hence,
\[\begin{align} \frac{1}{\delta}\left(-24\widetilde{I}_D + 3\widetilde{I}_B\right) = -\frac{9}{32}(\beta_1^2 + \beta_2^2 +\beta_3^2)\delta &+ \frac{1}{48}(\beta_1^2 + \beta_2^2 + \beta_3^2)(\beta_1 - \beta_2 + \beta_3) - \\ &- \frac{1}{12}(\beta_1^3 - \beta_2^3 + \beta_3^3). \end{align}\]
It is easy to see via direct computation that after symmetrization over \(W_{SU(3)} = S_3\) only the \((\beta_1^3 - \beta_2^3 + \beta_3^3)\) term gives nonzero value \(6\), so:
\[\begin{align} -24I_D + 3I_B &= 2\cdot \sum_{w \in W_G} \mathop{\mathrm{sign}}(w)\frac{-24\widetilde{I}_D(wX) + 3\widetilde{I}_B(wX)}{\delta(wX)} \cdot \prod_{\beta \in \Delta_+} \frac{-1}{\beta(wX)} =\\ &= 2 \cdot \left(-\frac{1}{2}\right)(-1)^3 = 1. \end{align}\]
The author would like to thank his advisor C. LeBrun for guidance and support, and S. Goette for answering questions about his papers.