Transversality for the full rank part of Vafa-Witten moduli spaces (Q2071792)

\noindent Vafa and Witten proposed a pair of gauge-theoretic equations when they studied an \(N=4\) topologically twisted supersymmetric Yang-Mills theory on 4-manifolds [\textit{C. Vafa} and \textit{E. Witten}, Nucl. Phys., B 431, No. 1--2, 3--77 (1994; Zbl 0964.81522)]. \begin{align*} &d^{\ast}_{A}B+d_{A}C=0,\\ &F^{+}_{A}+\frac{1}{8}[B\cdot B]+\frac{1}{2}[B,C]=0.\tag{1} \end{align*} Let \((X,g)\) be a closed smooth Riemannian 4-manifold. On \((X,g)\) the Vafa-Witten equations become \begin{align*} &d^{\ast}_{A}B=d_{A}C=0,\\ &F^{+}_{A}+\frac{1}{8}[B\cdot B]=[B,C]=0.\tag{2} \end{align*} In the article under review the authors carry on the study about the necessary conditions under which the space of solutions of Equations (1) is a smooth manifold, and when it is not. Let \(P\to X\) be a principal \(G\)-bundle over \(X\), where \(G\) is a Lie Group with Lie Algebra \(\mathfrak{g}\). Let \(\mathfrak{g}_{P}=P\times_{ad}\mathfrak{g}\) be the associated adjoint bundle. Along the review we use the notation \(L^{p}_{k}(S)\) for the Sobolev space structure induced on a set \(S\). Thus we have \begin{align*} \Omega^{p}(X)\, \text{(\(\mathbb{R}\) valued \(p\)-forms on \(X\))} \to \ &L^{2}_{k}(\Omega^{p}),\, (0\le p\le 4),\\ \Omega^{p}(\mathfrak{g})\, \text{(\(\mathfrak{g}\) valued \(p\)-forms on \(X\))} \to\ &L^{2}_{k}(\Omega^{p}\otimes \mathfrak{g}),\\ \mathcal{A}(P)\, \text{(connection 1-forms on \(P\))} \to\ &\mathcal{A}_{k}(P)=L^{2}_{k}(\mathcal{A}),\\ \mathcal{G}_{P}\, \text{(gauge group of \(P\))} \to\ &\mathcal{G}_{k}=L^{2}_{k}(\mathcal{G}_{P}). \end{align*} For \(k\ge 2\) the spaces above are all Banach manifolds. By fixing a connection 1-form \(A_{0}\in\mathcal{A}\) the affine space structure on \(\mathcal{A}\) is induced by the vector space \(L^{2}_{k}(\mathfrak{g}\otimes \Omega^{1})\) by the correspondence \[ L^{2}_{k}(\mathcal{A})=A_{0}+L^{2}_{k}(\Omega^{1}\otimes \mathfrak{g}) \] As usual, \(\Omega^{2}=\Omega^{2,+}\oplus \Omega^{2,-}\) is the decomposition into SD and ASD 2-forms. The Configuration space \(\mathcal{C}_{k}(P)\) and the target space \(\mathcal{C}'_{k}(P)\) are defined as \begin{align*} \mathcal{C}_{k}(P)&=\mathcal{A}_{k}(P)\times L^{2}_{k}(\mathfrak{g}_{P}\otimes\Omega^{2,+})\times L^{2}_{k}(\mathfrak{g}_{P}),\\ \mathcal{C}'(P)&=L^{2}_{k-1}(\mathfrak{g}\otimes\Omega^{1})\times L^{2}_{k-1}(\mathfrak{g}\otimes\Omega^{2,+}). \end{align*} The gauge action is defined as follows: \begin{align*} \mathcal{G}_{k}\times \mathcal{C}_{k}&\to\mathcal{C}_{k}\\ g\cdot (A,B,C)&=\left(g^{-1}d_{A}g,g^{-1}Bg,g^{1}Cg\right) \end{align*} The moduli space is \(\mathcal{B}_{k}(P)=\mathcal{C}_{k}(P)/\mathcal{G}_{k+1}\). A point \((A,B,C)\in\mathcal{B}_{k}\) is called irreducible if the stabilizer subgroup \(\mathrm{Stab}(A,B,C)\) of \((A,B,C)\) is the center \(Z(G)\) of \(G\); otherwise \((A,B,C)\) is reducible. Let \(\mathcal{B}^{\ast}_{k}(P)\) be the irreducible moduli space. By taking a curve \(g_{t}:(-\epsilon,\epsilon)\rightarrow\mathcal{G}_{k}(P)\), such that \(g_{0}=I\), the linearization of the action at \((A,B,C)\in\mathcal{C}_{k+1}(P)\) defines the linear operator \begin{align*} d^{0}_{(A,B,C)}:L^{2}_{k}(\mathfrak{g}_{P})&\to L^{2}_{k}(\mathfrak{g}_{P}\otimes\Omega^{1})\oplus L^{2}_{k}(\mathfrak{g}_{P}\otimes\Omega^{2,+})\oplus L^{2}_{k}(\mathfrak{g}_{P})\\ \chi&\rightarrow \big(d_{A}\chi,[B,\chi],[C,\chi]\big). \end{align*} Its \(L^{2}\)-adjoint is given by \begin{align*} d^{0,\ast}_{(A,B,C)}:&L^{2}_{k}(\mathfrak{g}_{P}\otimes\Omega^{1})\oplus L^{2}_{k}(\mathfrak{g}_{P}\otimes\Omega^{2,+})\oplus L^{2}_{k}(\mathfrak{g}_{P})\to L^{2}_{k}(\mathfrak{g}_{P})\\ &(a,b,c)\, \rightarrow \, d^{\ast}_{A}a+[b\cdot B]+[c,C]. \end{align*} \textbf{Theorem 1.} For any \((A,B,C)\in \mathcal{C}_{k}(P)\), there exists a \(\mathrm{Stab}(A,B,C)\)-invariant open neighborhood of \(0\in \mathcal{U }\subset\mathrm{Ker}\big( d^{0,\ast}_{(A,B,C)}\big)\) such that the map \[ \frac{\mathcal{G}_{k+1}(P)\times \mathcal{U}}{\mathrm{Stab}(A,B,C)}\rightarrow\mathcal{C}_{k}(P) \] is a diffeomorphism onto an open neighborhood of the orbit of \((A,B,C)\). It follows that the moduli space \(\mathcal{B}^{\ast}_{k}(P)\) is a smooth Hilbert manifold locally diffeomorphic to \(\mathcal{U}\). Motivated by the Vafa-Witten Equations (1), the Vafa-Witten map is defined by \begin{align*} \mathcal{VW}&:\mathcal{C}_{k}\rightarrow\mathcal{C}'_{k-1}\\ \mathcal{VW}(A,B,C)&=\begin{pmatrix} d^{\ast}_{A}B+d_{A}C,\\ F^{+}_{A}+\frac{1}{8}[B\cdot B]+\frac{1}{2}[B,C]. \end{pmatrix} \end{align*} Therefore, the space \(\mathcal{VW}^{-1}(0)\) is \(\mathcal{G}_{k+1}\)-invariant. The moduli space of solutions is defined as \(\mathcal{M}_{k}(P)=\mathcal{VW}^{-1}(0)\diagup\mathcal{G}_{k+1}\). The deformation complex for the \(\mathcal{VW}\) moduli space is \begin{align*} 0\to L^{2}_{k+1}(\mathfrak{g}(P))&\overset{d^{0}_{(A,B,C)}}{\rightarrow} L^{2}_{k}(\mathfrak{g}(P))\oplus L^{2}_{k}(\mathfrak{g}(P)\otimes \Omega^{2,+})\oplus L^{2}_{k} (\mathfrak{g}(P))\\ &\overset{d^{1}_{(A,B,C)}}{\rightarrow} L^{2}_{k-1}(\mathfrak{g}(P)\otimes\Omega^{1})\oplus L^{2}_{k-1}(\mathfrak{g}(P)\otimes\Omega^{2,+}) \rightarrow 0, \end{align*} where \(d^{1}_{(A,B,C)}\) is the derivative of the Vafa-Witten map; \begin{align*} d^{1}_{(A,B,C)}(a,b,c)=d\big(\mathcal{VW}\big)_{(A,B,C)}.(a,b,c)= \begin{pmatrix} d^{\ast}_{A}b+d_{A}c-[B\cdot a]-[C\cdot a]\\ d^{+}_{A}a +\frac{1}{4}[B\cdot b]+\frac{1}{2}[b\cdot C]+\frac{1}{2}[B,C] \end{pmatrix}. \end{align*} If \([A,B,C]\in \mathcal{M}_{k}(P)\), then \(d^{1}_{(A,B,C)}\circ d^{0}_{(A,B,C)}=0\). It follows that the deformation complex for the Vafa-Witten moduli space is an elliptic complex, with cohomology groups \[ H^{0}_{(A,B,C)}=\ker \big(d^{0}_{(A,B,C)}\big),\, H^{1}_{(A,B,C)}=\frac{\ker \big(d^{1}_{(A,B,C)}\big)}{\mathrm{Im} \big(d^{0}_{(A,B,C)}\big)},\, H^{2}_{(A,B,C)}=\mathrm{coker} \big(d^{1}_{(A,B,C)}\big). \] Therefore, \(\mathcal{M}_{k}(P)\) is a smooth manifold whenever we have \(H^{0}_{(A,B,C)}=H^{2}_{(A,B,C)}=0\). There exists also a regularity up to Gauge equivalence for the solutions of the VW-Equations (1); \textbf{Theorem} ([\textit{B. A. Mares}, Some analytic aspects of reduced Vafa-Witten twisted \(N=4\) supersymmetric Yang-Mills theory. Massachusetts Institute of Technology (PhD Thesis) (2010), Theorem 3.3.2], cf. [\textit{P. M. N. Feehan} and \textit{T. G. Leness}, J. Differ. Geom. 49, No. 2, 265--410 (1998; Zbl 0998.57057), Proposition 3.7]) With notation as above, for any \(L^{2}_{k}\)-solution \((A,B,C)\) to the Vafa-Witten Equations (1), there is a gauge transformation \(\psi\in \mathcal{G}_{k}(P)\) such that \(\psi.(A,B,C)\) is a smooth solution on \(X\). \textbf{Main Theorem.} Let \((X,g)\) be a closed, oriented, smooth Riemannian 4-manifold and let \(P\to X\) be a principal \(\mathrm{SU}(2)\)-bundle or an \(\mathrm{SO}(3)\)-bundle. Then the full rank part of the Vafa-Witten moduli space for such data is a smooth, zero-dimensional manifold. The rank of a section \(B\in\Omega^{2,+}(\mathfrak{g}_{P})\) is defined as follows: choose local frames for \(\mathfrak{g}_{P}\) and \(\Omega^{2,+}(\mathfrak{g}_{P})\), then the section \(B\) is represented by a \(d\times 3\) matrix-valued function with respect to the local frames, where \(d=\dim(G)\). The rank of \(B\) at a point of \(X\) is just the rank of the matrix at that point, and \(\mathrm{rank}(B)\) is the maximum of the pointwise rank over \(X\). The pointwise rank of \(B\) also gives a stratification of the manifold \(X\), namely, \[ X^{(i)}=\{x\in X\mid \, \mathrm{rank}\big(B(x)\big)=i\},\, 0\le i\le \mathrm{rank}(B). \] The top rank stratum is a nonempty open subset of \(X\). By Theorem 3, we may assume that such solutions are smooth. Aronszajn's theorem (cf. [\textit{N. Aronszajn}, J. Math. Pures Appl. (9) 36, 235--249 (1957; Zbl 0084.30402), Remark 3], [\textit{J. L. Kazdan}, Commun. Pure Appl. Math. 41, No. 5, 667--681 (1988; Zbl 0632.35015), Theorem 1.8]) implies that the zero locus of \(B\) is either the whole manifold \(X\), or a nowhere-dense closed subset of \(X\). Thus there are the following cases: (i) \(\mathrm{rank}(B)=0\). So \(B=0\) and the VW-Equations (1) reduce to \begin{align*} d_{A}C&=0,\\ F^{+}_{A}&=0. \end{align*} \(F^{+}_{A}=0\) implies that \(A\) is an ASD-connection. When \(A\) is irreducible we have \(d_{A}C=0\) \(\Leftrightarrow\) \(C=0\). If \(A\) is reducible, then \(d_{A}C=0\) means that \(C\) is a parallel section of \(\mathfrak{g}(P)\) with respect to \(A\). By the transversality theorem of [\textit{D. S. Freed} and \textit{K. K. Uhlenbeck}, Instantons and four-manifolds. Cambridge University Press, Cambridge; Mathematical Sciences Research Institute, Berkeley, CA (1984; Zbl 0559.57001), \S3]), if \(b^{+}_{2}(X)\ge 2\), and \(g\) is a generic metric, then the moduli space of irreducible ASD connections modulo gauge transformations is a finite dimensional smooth manifold. (ii) \(\mathrm{rank}(B)=1\). Then \([B\cdot B]=0\) (cf. [\textit{B. A. Mares}, loc. cit., \S4.1]). The Vafa-Witten equations (1) are reduced to \begin{align*} d^{\ast}B&=d_{A}C=0,\\ F^{+}_{A}&=[B,C]. \end{align*} The two equations for \(C\) imply that \(C\in \mathrm{Stab}(A,B)\), the Lie algebra of the stabilizer of the gauge group action on the pair \((A,B)\). From the above discussion the article obtains the following conclusions: \(\bullet\) The irreducible portion of the rank one part of the Vafa-Witten moduli space consists of equivalence classes of solutions \([A,B,C]\), where \(A\) is an irreducible ASD connection, \(B=\chi\otimes\omega\) with \(\omega\in\Omega^{2,+}(X)\) a harmonic 2-form on \(X\) and \(\chi\) a parallel section of \(\mathfrak{g}_{P}\) on \(X\diagdown\omega^{-1}(0)\), and \(C=0\). \(\bullet\) The reducible portion of the rank one part of the moduli space consists of equivalence classes of solutions \([A,B,C]\), where \(A\) is a reducible ASD connection, \(B=\eta\otimes\omega\) with a harmonic 2-form on \(X\) and \(\eta\) a parallel section of \(\mathfrak{g}_{P}\) on \(X\diagdown\omega^{-1}(0)\), and \(C=\eta\). According to a result of [\textit{S. K. Donaldson} and \textit{P. B. Kronheimer}, The geometry of four-manifolds. Oxford: Clarendon Press (1990; Zbl 0820.57002), Lemma 4.3.21], if \(X\) is simply connected and \(A\) is an irreducible ASD connection, then the restriction of \(A\) to any nonempty open subset is also irreducible. This implies that there are no irreducible rank one solutions if \(X\) is simply connected. Assuming that the structure group \(G\) of the principal bundle \(P\to X\) is either \(\mathrm{SU}(2)\) or \(\mathrm{SO}(3)\) whose centers are \(\mathbb{Z}_{2}=\{\pm I\}\) and \(I\), respectively, then the following lemma implies that if \((A,B,C)\) is a smooth solution to the Vafa-Witten equations with \(\mathrm{rank}(B)\ge 2\), then such solutions are irreducible. \textbf{Lemma.} Suppose \((A,B,C)\) is a smooth solution to the Vafa-Witten equations (1) with structure group \(\mathrm{SU}(2)\) or \(\mathrm{SO}(3)\), and \(\mathrm{rank}(B)\ge 2\). Then \(C=0\) and \(\mathrm{Stab}(A, B,C) = Z(G)\). Following these lines the authors prove the main theorem.