Abstract
Associated with every quaternionic representation of a compact, connected Lie group there is a Seiberg–Witten equation in dimension three. The moduli spaces of solutions to these equations are typically non-compact. We construct Kuranishi models around boundary points of a partially compactified moduli space. The Haydys correspondence identifies such boundary points with Fueter sections—solutions of a non-linear Dirac equation—of the bundle of hyperkähler quotients associated with the quaternionic representation. We discuss when such a Fueter section can be deformed to a solution of the Seiberg–Witten equation.
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Notes
The following observation is due to [12, Section 3.1] and becomes important when formulating the Seiberg–Witten equation in dimension four. Suppose there is a homomorphism \(\mathbf {Z}_2 \rightarrow Z(H)\) such that the non-unit in \(\mathbf {Z}_2\) acts through \(\rho \) as \(-1\). Set \({\hat{H}} :=(\mathrm {Sp}(1)\times H)/\mathbf {Z}_2\). All of the constructions in Sect. 2.2 go through with \({\mathfrak s}\times Q\) replaced by a \({\hat{H}}\)–principal bundle \({\hat{Q}}\). In the classical Seiberg–Witten theory, this corresponds to endowing the manifold with a \(\hbox {spin}^c\) structure rather than a spin structure and a U(1)–bundle.
If \(H = G \times K\), then the G–bundle P alluded to in this notation does exist. In general, it does not exist but traces of it do—e.g., its adjoint bundle and its gauge group.
In fact, if we allow the Lie groups and the representations to be infinite-dimensional, we can also recover (special cases of) the \(G_2\)– and \(\mathrm {Spin}(7)\)–instanton equations [11, Section 4.2].
We say that \({\mathfrak c}_0\) is irreducible if \(\Gamma _{{\mathfrak c}_0} :=\lbrace u \in \mathscr {G}(P) : u{\mathfrak c}_0 = {\mathfrak c}_0 \rbrace = \lbrace \mathrm {id} \rbrace \), see Definition 3.3. There is a natural generalization of Proposition 2.19 to the case when \({\mathfrak c}_0\) is reducible. Then \(\Gamma _{{\mathfrak c}_0}\) acts on U and O and \(\mathrm {ob}\) can be chosen to be \(\Gamma _{{\mathfrak c}_0}\)-equivariant, cf. [3, Section 4.2.5]. However, in this paper we focus on neighborhoods of infinity of the moduli space, and as we will see those contain only irreducible solutions.
In the following, we will suppress the subscript B from the notation.
The term \({\mathfrak e}_{{\mathbf {p}},0}\) vanishes for \({\mathbf {p}}= {\mathbf {p}}_0\).
Here we engage in the slight abuse of notation to use the same notation for a bilinear map and its associated quadratic form.
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Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. 1754967 and the Simons Collaboration “Special Holonomy in Geometry, Analysis and Physics”.
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Appendices
A Examples of Seiberg–Witten equations
Example A.1
Let \(G = \mathrm {U}(n)\) and \(S = \mathbf {H}\otimes _{\mathbf {C}}{\mathbf {C}}^n\), where the complex structure on \(\mathbf {H}\) is given by right-multiplication by i. Let \(\rho : \mathrm {U}(n) \rightarrow \mathrm {Sp}(\mathbf {H}\otimes _{\mathbf {C}}{\mathbf {C}}^n)\) be induced from the standard representation of \(\mathrm {U}(n)\). The corresponding Seiberg–Witten equation is the \(\mathrm {U}(n)\)–monopole equation in dimension three. The closely related \({\mathbf {P}\mathrm {U}}(2)\)–monopole equation on 4–manifolds plays a crucial role in Pidstrigach and Tyurin’s approach to proving Witten’s conjecture relating Donaldson and Seiberg–Witten invariants; see, e.g., [8, 20, 27].
In this example as well as in Example 2.15, we have \(\mu ^{-1}(0) = \lbrace 0 \rbrace \).
Example A.2
Let G be a compact Lie group, \({\mathfrak g}= {{\,\mathrm{Lie}\,}}(G)\), and fix an \({{\,\mathrm{Ad}\,}}\)–invariant inner product on \({\mathfrak g}\). \(S :=\mathbf {H}\otimes _\mathbf {R}{\mathfrak g}\) is a quaternionic Hermitian vector space, and \(\rho : G \rightarrow \mathrm {Sp}(S)\) induced by the adjoint action is a quaternionic representation. The moment map \(\mu : \mathbf {H}\otimes _\mathbf {R}{\mathfrak g}\rightarrow ({\text {Im}}\mathbf {H}\otimes {\mathfrak g})^*\) is given by
for \(\xi = \xi _0\otimes 1 + \xi _1\otimes i + \xi _2\otimes j + \xi _3\otimes k \in \mathbf {H}\otimes _\mathbf {R}{\mathfrak g}\). Set \(H :=\mathrm {Sp}(1) \times G\) and extend the above quaternionic representation of G to H by declaring that \(q \in \mathrm {Sp}(1)\) acts by right-multiplication with \(q^*\).
Taking Q to be the product of the chosen spin structure \({\mathfrak s}\) with a principal G–bundle, and choosing B such that it induces the spin connection on \({\mathfrak s}\), (2.14) becomes
for \(\xi \in \Gamma ({\mathfrak g}_P)\), \(a \in \Omega ^1(M,{\mathfrak g}_P)\) and \(A \in \mathscr {A}(P)\). If M is closed, then integration by parts shows that every solution of this equation satisfies \(\mathrm{d}_A\xi =0\) and \([\xi ,a]=0\); hence, \(A + ia\) defines a flat \(G^{\mathbf {C}}\)–connection. Here \(G^{\mathbf {C}}\) denotes the complexification of G.
In the above situation, we have \(\mu ^{-1}(0)/G \cong (\mathbf {H}\otimes {\mathfrak t})/W\) where \({\mathfrak t}\) is a the Lie algebra of a maximal torus \(T \subset G\) and \(W = N_G(T)/T\) is the Weyl group of G. However, since each \(\xi \in \mu ^{-1}(0)\) has stabilizer conjugate to T, we have \(\mu ^{-1}(0)\cap S^\mathrm {reg}= \varnothing \), and the hyperkähler quotient \(S^\mathrm {reg}{/\!\! /\!\! /}G\) is empty.
Example A.3
The motivating example for us is the (r, k) ADHM Seiberg–Witten equation, which we expect to play in important role in gauge theory on \(G_2\)–manifolds,Footnote 8 and which arises from
with
where \(\mathrm {SU}(r)\) acts on \({\mathbf {C}}^r\) in the obvious way, \(\mathrm {U}(k)\) acts on \({\mathbf {C}}^k\) in the obvious way and on \({\mathfrak u}(k)\) by the adjoint representation, and \(\mathrm {Sp}(1)\) acts on the first copy of \(\mathbf {H}\) trivially and on the second copy by right-multiplication with the conjugate. Accoding to Atiyah, Drinfeld, Hitchin, and Manin [1], if \(r \geqslant 2\), then \(S^\mathrm {reg}{/\!\! /\!\! /}G\) is the moduli space of framed \(\mathrm {SU}(r)\) ASD instantons of charge k on \(\mathbf {R}^4\), and \(\mu ^{-1}(0)/G\) is its Uhlenbeck compactification, If \(r = 1\), then \(\mu ^{-1}(0)\cap S^\mathrm {reg}= \varnothing \), and \(\mu ^{-1}(0)/G = {{\,\mathrm{Sym}\,}}^k\mathbf {H}:=\mathbf {H}^k/S_k\) by [19, Example 3.14].
B Useful identities involving \(\mu \)
This appendix summarizes and proves a few useful identities regarding \(\mu \), some of which are used in this article.
Proposition B.1
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\), \(a \in \Omega ^1(M,{\mathfrak g}_P)\), and \(\phi ,\psi \in \Gamma ({\mathfrak S})\), we have
and for \(a \in \Omega ^1(M,{\mathfrak g}_P)\) and \(\phi ,\psi \in \Gamma ({\mathfrak S})\), we have
Proof
For all \(a \in \Omega ^1(M,{\mathfrak g}_P)\), we have
This proves the first identity. To prove the second identity, note that, for all \(\eta \in \Omega ^0(M,{\mathfrak g}_P)\), we have
\(\square \)
Proposition B.4
For all \(A \in \mathscr {A}(Q)\) and \(\phi , \psi \in \Gamma ({\mathfrak S})\) we have
and
Proof
Fix a point \(x \in M\), a positive local orthonormal frame \((e_i)\) around x with \((\nabla e_i)(x) = 0\), and let \(\xi \) be a local section of \({\mathfrak g}_P\) defined in a neighborhood of x satisfying \((\nabla \xi )(x) = 0\). We set \(\nabla ^A_i :=\nabla ^A_{e_i}\).
At the point \(x \in M\), we compute with
This proves the first identity. To prove the second identity, we compute
\(\square \)
Proposition B.7
If \((\varepsilon ,\Phi ,A) \in (0,\infty )\times \Gamma ({\mathfrak S})\times \mathscr {A}_B(Q)\) is a solution of (2.21) and \(R_\Phi (\xi ) = \rho (\xi )\Phi \), then
Here \((e_1,e_2,e_3)\) is local orthonormal frame, \((e^1,e^2,e^3)\) is the dual coframe, \(F^B_{ij} :=F_B(e_i,e_j)\), \(F^{\mathfrak s}_{ij} :=F_{\mathfrak s}(e_i,e_j)\) with \(F_{\mathfrak s}\) denoting the curvature of the spin connection on \({\mathfrak s}\), and \(e^{ij} :=e^i \wedge e^j\).
Proof
We compute
Since
the result now follows from Proposition B.4. \(\square \)
C Proof of Proposition 3.2
For the reader’s convenience, we recall the definitions of the graded vector space \(L^\bullet \),
the graded Lie bracket \(\llbracket \cdot ,\cdot \rrbracket \),
and the graded differential \(\delta _{{\mathfrak c}}\),
We proceed in four steps.
Step 1\((L^\bullet ,\llbracket \cdot ,\cdot \rrbracket )\) is a graded Lie algebra.
We need to verify the graded Jacobi identity, that is, for every three homogeneous elements \(x,y,z \in L^\bullet \) we need to show that
vanishes. Here \(\deg {x}\) denotes the degree of x.
For degree reasons \(J(x,y,z) = 0\), unless \(\deg x + \deg y + \deg z \leqslant 3\). \((\Omega ^\bullet (M,{\mathfrak g}_P),{[\cdot \wedge \cdot ]})\) is a graded Lie algebra. Since J(x, y, z) is invariant under permutations of x, y, and z, we can assume that \(z \in \Gamma ({\mathfrak S})\) in degree 1 or 2. Hence, only the following five cases remain:
-
For \(\xi ,\eta \in \Omega ^0(M,{\mathfrak g}_P)\), and \(\phi \in \Gamma ({\mathfrak S})\) in degree 1 or 2, we have
$$\begin{aligned} J(\xi ,\eta ,\phi )&= \llbracket \xi ,\llbracket \eta ,\phi \rrbracket \rrbracket + \llbracket \eta ,\llbracket \phi ,\xi \rrbracket \rrbracket + \llbracket \phi ,\llbracket \xi ,\eta \rrbracket \rrbracket \\&= \rho (\xi )\rho (\eta )\phi - \rho (\eta )\rho (\xi )\phi - \rho ([\xi ,\eta ])\phi = 0. \end{aligned}$$ -
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\), and \(\phi ,\psi \in \Gamma ({\mathfrak S})\) in degree 1, we have
$$\begin{aligned} J(\xi ,\phi ,\psi )&= \llbracket \xi ,\llbracket \phi ,\psi \rrbracket \rrbracket + \llbracket \phi ,\llbracket \psi ,\xi \rrbracket \rrbracket - \llbracket \psi ,\llbracket \xi ,\phi \rrbracket \rrbracket \\&= -2[\xi ,\mu (\phi ,\psi )] + 2\mu (\phi ,\rho (\xi )\psi ) + 2\mu (\psi ,\rho (\xi )\phi ) = 0 \end{aligned}$$by Proposition B.1.
-
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\), \(\phi \in \Gamma ({\mathfrak S})\) in degree 1 and \(\psi \in \Gamma ({\mathfrak S})\) in degree 2, we have
$$\begin{aligned} J(\xi ,\phi ,\psi )&= \llbracket \xi ,\llbracket \phi ,\psi \rrbracket \rrbracket + \llbracket \phi ,\llbracket \psi ,\xi \rrbracket \rrbracket + \llbracket \psi ,\llbracket \xi ,\phi \rrbracket \rrbracket \\&= -([\xi ,*\rho ^*\left( \phi \psi ^*\right) ] - *\rho ^*\left( \phi (\rho (\xi )\psi )^*\right) + *\rho ^*\left( \psi (\rho (\xi )\phi )^*\right) ) \\&= -*\rho ^*\left( [\rho (\xi ),\phi \psi ^*] + \phi \psi ^*\rho (\xi ) - \rho (\xi )\phi \psi ^*\right) = 0. \end{aligned}$$ -
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\), \(a \in \Omega ^1(M,{\mathfrak g}_P)\), and \(\phi \in \Gamma ({\mathfrak S})\) in degree 1, we have
$$\begin{aligned} J(\xi ,a,\phi )&= \llbracket \xi ,\llbracket a,\phi \rrbracket \rrbracket + \llbracket a,\llbracket \phi ,\xi \rrbracket \rrbracket - \llbracket \phi ,\llbracket \xi ,a \rrbracket \rrbracket ] \\&= -\rho (\xi ){\bar{\gamma }}(a)\phi + {\bar{\gamma }}(a)\rho (\xi )\phi + {\bar{\gamma }}({[\xi ,a]})\phi = 0. \end{aligned}$$ -
For \(a \in \Omega ^1(M,{\mathfrak g}_P)\) and \(\phi ,\psi \in \Gamma ({\mathfrak S})\) in degree 1, we have
$$\begin{aligned} J(a,\phi ,\psi )&= - \llbracket a,\llbracket \phi ,\psi \rrbracket \rrbracket - \llbracket \phi ,\llbracket \psi ,a \rrbracket \rrbracket - \llbracket \psi ,\llbracket a,\phi \rrbracket \rrbracket \\&= 2{[a\wedge \mu (\phi ,\psi )]} + *\rho ^*\left( ({\bar{\gamma }}(a)\psi )\phi ^*\right) + *\rho ^*\left( ({\bar{\gamma }}(a)\phi )\psi ^*\right) = 0 \end{aligned}$$by Proposition B.1.
Step 2\((L^\bullet ,\delta _{\mathfrak c}^\bullet )\) is a DGA.
We need to show that \(\delta _{\mathfrak c}\circ \delta _{\mathfrak c}= 0\). Using Proposition B.1, we compute that
and, using Proposition B.4 and Proposition B.1, we compute that
Step 3\((L^\bullet ,\llbracket \cdot ,\cdot \rrbracket ,\delta _{\mathfrak c}^\bullet )\) is a DGLA.
We need to verify that \(\delta _{\mathfrak c}^\bullet \) satisfies the graded Leibniz rule with respect to \(\llbracket \cdot ,\cdot \rrbracket \), that is for every two homogeneous elements \(x,y \in L^\bullet \) we need to show that
vanishes.
For degree reasons, \(D(x,y) = 0\) unless \(\deg x + \deg y \leqslant 2\); hence, only the following eight cases remain:
-
For \(\xi ,\eta \in \Omega ^0(M,{\mathfrak g}_P)\), we have
$$\begin{aligned} D(\xi ,\eta ) = \begin{pmatrix} -\rho ([\xi ,\eta ])\Phi \\ \mathrm{d}_A[\xi ,\eta ] \end{pmatrix} - \llbracket \begin{pmatrix} -\rho (\xi )\Phi \\ \mathrm{d}_A\xi \end{pmatrix} , \eta \rrbracket - \llbracket \xi , \begin{pmatrix} -\rho (\eta )\Phi \\ \mathrm{d}_A\eta \end{pmatrix} \rrbracket = 0. \end{aligned}$$ -
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\) and \(\phi \in \Gamma ({\mathfrak S})\) in degree 1, we have
by Proposition B.1.
-
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\) and \(a \in \Omega ^1(M,{\mathfrak g}_P)\), we have
$$\begin{aligned} D(\xi ,a)&= \begin{pmatrix} -{\bar{\gamma }}([\xi ,a])\Phi \\ \mathrm{d}_A [\xi ,a] \end{pmatrix} - \llbracket \begin{pmatrix} -\rho (\xi )\Phi \\ \mathrm{d}_A\xi \end{pmatrix} , a \rrbracket - \llbracket \xi , \begin{pmatrix} -{\bar{\gamma }}(a)\Phi \\ \mathrm{d}_A a \end{pmatrix} \rrbracket \\&= \begin{pmatrix} -{\bar{\gamma }}([\xi ,a])\Phi - {\bar{\gamma }}(a)\rho (\xi )\Phi + \rho (\xi ){\bar{\gamma }}(a)\Phi \\ \mathrm{d}_A [\xi ,a] - [\mathrm{d}_A\xi \wedge a] - [\xi ,\mathrm{d}_A] \end{pmatrix} = 0. \end{aligned}$$ -
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\) and \(\phi \in \Gamma ({\mathfrak S})\) in degree 2, we have
$$\begin{aligned} D(\xi ,\phi )&= *\rho ^*(\rho (\xi )\phi \Phi ^*) - \llbracket \rho (\xi )\Phi ,\phi \rrbracket - \llbracket \mathrm{d}_A\xi ,\phi \rrbracket - \llbracket \xi , *\rho ^*(\phi \Phi ^*) \rrbracket \\&= *\rho ^*(\rho (\xi )\phi \Phi ^*) - * \rho ^*( \phi \Phi ^* \rho (\xi )) - [\xi ,*\rho ^*(\phi \Phi ^*)] = 0. \end{aligned}$$ -
For \(\xi \in \Omega ^0(M,{\mathfrak g}_P)\) and \(b \in \Omega ^2(M,{\mathfrak g}_P)\), we have
$$\begin{aligned} D(\xi ,b) = \mathrm{d}_A [\xi ,b] - [\mathrm{d}_A\xi ,b] - [\xi ,\mathrm{d}_Ab] = 0. \end{aligned}$$ -
For \(\phi ,\psi \in \Gamma ({\mathfrak S})\) in degree 1, we have
by Proposition B.4.
-
For \(a \in \Omega ^1(M,{\mathfrak g}_P)\) and \(\phi \in \Gamma ({\mathfrak S})\) in degree 1, we have
by Proposition B.1.
-
For \(a,b \in \Omega ^1(M,{\mathfrak g}_P)\), we have
$$\begin{aligned} D(a,b) = \begin{pmatrix} - {\bar{\gamma }}{ {[a\wedge b]}} \Phi \\ \mathrm{d}_A {[a\wedge b]} \end{pmatrix} - \llbracket \begin{pmatrix} {\bar{\gamma }}(a)\Phi \\ \mathrm{d}_A a \end{pmatrix} , b \rrbracket + \llbracket a , \begin{pmatrix} {\bar{\gamma }}(b)\Phi \\ \mathrm{d}_A b \end{pmatrix} \rrbracket = 0. \end{aligned}$$
Step 3 For every \({\hat{{\mathfrak c}}} :=(a,\phi ) \in L^1\), \((A+a,\Phi +\phi )\) solves (2.14) if and only if \(\delta _{\mathfrak c}{\hat{{\mathfrak c}}} + \frac{1}{2}\llbracket {\hat{{\mathfrak c}}},{\hat{{\mathfrak c}}} \rrbracket = 0\).
For \({\hat{{\mathfrak c}}} :=(a,\phi ) \in L^1\), we have
which vanishes if and only if \((A+a,\Phi +a)\) solves (2.14). \(\square \)
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Doan, A., Walpuski, T. Deformation theory of the blown-up Seiberg–Witten equation in dimension three. Sel. Math. New Ser. 26, 48 (2020). https://doi.org/10.1007/s00029-020-00574-6
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DOI: https://doi.org/10.1007/s00029-020-00574-6