nLab Seiberg-Witten moduli space

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Context

Quantum field theory

Super-geometry

Contents

Idea

The Seiberg-Witten moduli space (short SW moduli space, also monopole moduli space) is the moduli space of the Seiberg-Witten equations, hence the space of its solutions up to gauge. It is used to defined the Seiberg-Witten invariants used to study 4-manifolds. A very useful property of the Seiberg-Witten moduli space is that it is always compact, which is an improvement over the previously used Yang-Mills moduli space and allowed to simplify the derivation of many results from Donaldson theory. The Seiberg-Witten moduli space is named after Nathan Seiberg and Edward Witten, who introduced the underlying Seiberg-Witten equations in 1994.

Basics

Let MM be a compact orientable Riemannian 4-manifold with Riemannian metric gΓ (S 2T *M)g\in\Gamma^\infty(S^2T^*M) and spinᶜ structure 𝔰:MBSpin c(4)\mathfrak{s}\colon M\rightarrow BSpin^\mathrm{c}(4). Because of the exceptional isomorphism: (Perutz 2002, p. 2)

Spin c(4)U(2)× U(1)U(2)={A ±U(2)|det(A )=det(A +)} Spin^\mathrm{c}(4) \cong U(2)\times_{U(1)}U(2) =\{A^\pm\in U(2)|det(A^-)=det(A^+)\}

the spinᶜ structure 𝔰\mathfrak{s} consists of two complex plane bundles W ±MW^\pm\twoheadrightarrow M, called associated spinor bundles (whose sections are called (anti) self-dual spinors), with same determinant line bundle L=det(W ±)L=det(W^\pm). Since the determinant line bundle preserves the first Chern class, one has c 1(𝔰)c 1(L)=c 1(W ±)c_1(\mathfrak{s})\coloneqq c_1(L)=c_1(W^\pm) with c 1(𝔰)mod2=w 2(M)c_1(\mathfrak{s})mod 2=w_2(M). (Perutz 2002, p. 2) Given a fundamental class [M] H 4(M,)[M]_\mathbb{Z}\in H^4(M,\mathbb{Z})\cong\mathbb{Z} and its reduction [M] 2:=[M] mod2H 4(M, 2) 2 [M]_{\mathbb{Z}_2}:=[M]_\mathbb{Z}\operatorname{mod}2\in H^4(M,\mathbb{Z}_2)\cong\mathbb{Z}_2, one therefore has:

c 1 2(𝔰)[M] mod2=w 2 2(M)[M] 2=σ(M)mod2, c_1^2(\mathfrak{s})[M]_\mathbb{Z} mod 2 =w_2^2(M)[M]_{\mathbb{Z}_2} =\sigma(M) mod 2,

hence c 1 2(𝔰)[M] σ(M) c_1^2(\mathfrak{s})[M]_\mathbb{Z}-\sigma(M) is always even.

Let Ω + 2(M)\Omega_+^2(M) be the space of self-dual forms fulfilling η=η\star\eta=\eta and let + 2(M)\mathcal{H}_+^2(M) be the vector subspace of additionally harmonic forms fulfilling Δη=0\Delta\eta=0. Let b +(M)=dim + 2(M)b_+(M)=dim\mathcal{H}_+^2(M) be the self-dual Betti number, then there is a vector subspace Π<Ω + 2(M)\Pi\lt\Omega_+^2(M) with codimΠ=b +(M)codim\Pi=b_+(M) (using the Hodge decomposition), so that a self-dual form ηΩ + 2(M)\eta\in\Omega_+^2(M) is the self-dual part of the curvature form of a connection AΩ 1(M)A\in\Omega^1(M), hence:

η=F A +=12(dA+dA), \eta =F_A^+ =\frac{1}{2}(\star d A+d A),

if and only if ηΠ\eta\in\Pi. Both the vector subspace and this result play an essential role as they are the reason the Seiberg-Witten equations are perturbed with a self-dual form before considering their moduli space. Both also lead to topological obstructions since for b +(X)=0b_+(X)=0 there is no complement and for b +(X)=1b_+(X)=1 the complement is not connected. Let Ω 2(X)\Omega_-^2(X) be the analogous space of anti self-dual forms with η=η\star\eta=-\eta and 2(M)\mathcal{H}_-^2(M) be the analogous vector subspace of additionally harmonic forms fulfilling Δη=0\Delta\eta=0. Let b (M)=dim 2(M)b_-(M)=dim\mathcal{H}_-^2(M) be the anti self-dual Betti number, then the second Betti number and signature can be expressed as:

b 2(M)=b +(M)+b (M); b_2(M) =b_+(M) +b_-(M);
σ(M)=b +(M)b (M). \sigma(M) =b_+(M) -b_-(M).

Both formulas, which are later used to calculate the dimension of the moduli space, can also be reversed:

b +(M)=12(b 2(M)+σ(M)); b_+(M) =\frac{1}{2}(b_2(M)+\sigma(M));
b (M)=12(b 2(M)σ(M)). b_-(M) =\frac{1}{2}(b_2(M)-\sigma(M)).

Configuration space

It is helpful to first consider the space of all possible solutions. Since the space of connections on the complex line bundle LL is an affine vector space, it is helpful to chose a single such connection d Ad_A and express every other as being shifted from it by a form iaΩ 1(M,𝔲(1))ia\in\Omega^1(M,\mathfrak{u}(1)) with aΩ 1(M)a\in\Omega^1(M) using 𝔲(1)i\mathfrak{u}(1)\cong i\mathbb{R}. Self-dual spinors also form a vector space with the zero section providing a canonical center. Let the configuration space and reduced configuration space be: (Moore 2010, p. 77-79)

𝒜{(d Aia,ψ)|aΩ 1(M),ψΓ (W +)}; \mathcal{A} \coloneqq\{(d_A-ia,\psi)|a\in\Omega^1(M),\psi\in\Gamma^\infty(W^+)\};
𝒜 *{(d Aia,ψ)𝒜|ψ0}. \mathcal{A}^* \coloneqq\{(d_A-ia,\psi)\in\mathcal{A}|\psi\neq 0\}.

Since the reduced configuration space 𝒜 *\mathcal{A}^* is an infinite dimensional vector space without a single point and hence homotopy equivalent to the infinite-dimensional sphere, it is contractible.

Although the definition of the reduced configuration space 𝒜 *\mathcal{A}^* is mainly motivated by the action of the gauge group below, its excluded cases are already important in the Seiberg-Witten equations themselves, which then reduced to the self-dual Yang-Mills equations.

Gauge group

Smooth maps g:MU(1)g\colon M\rightarrow U(1) act on the elements of the configuration space by: (Perutz 2002, p. 6, Moore 2010, p. 77-79)

g(d Aia,ψ)=(d Aia+gd(g 1),gψ). g\cdot(d_A-ia,\psi) =(d_A-ia+g d(g^{-1}),g\psi).

Hence the gauge group can be taken as: (Perutz 2002, p. 6, Moore 2010, p. 77-79)

𝒢C (M,U(1)). \mathcal{G} \coloneqq C^\infty(M,U(1)).

Using that the first unitary group U(1)U(1) is the Eilenberg-MacLane space K(,1)K(\mathbb{Z},1), which classifies singular cohomology, as well as the universal coefficient theorem and the Hurewicz theorem yields:

[M,U(1)]=[M,K(,1]=H 1(M,)Hom(H 1(M,),)=Hom(π 1(M) ab,). [M,U(1)] =[M,K(\mathbb{Z},1] =H^1(M,\mathbb{Z}) \cong\Hom(H_1(M,\mathbb{Z}),\mathbb{Z}) =\Hom(\pi_1(M)^\mathrm{ab},\mathbb{Z}).

Hence for MM simply connected or more generally if its fundamental group is perfect, every gauge is nullhomotopic and therefore has a global logarithm, meaning that for every smooth map g:MU(1)g\colon M\rightarrow U(1) there exists a smooth map u:Mu\colon M\rightarrow\mathbb{R} with g=e iug=e^{iu}. In this case, the action on the configuration space simplifies to: (Moore 2010, p. 77-79)

e iu(d Aia,ψ)=(d Ai(a+du),e iuψ). e^{iu}\cdot(d_A-ia,\psi) =(d_A-i(a+d u),e^{iu}\psi).

For a base point x 0Mx_0\in M, the gauge group can be separated as a product using the based gauge group:

𝒢 0{g𝒢|g(x 0)=1}; \mathcal{G}_0 \coloneqq\{g\in\mathcal{G}|g(x_0)=1\};
𝒢=𝒢 0×U(1). \mathcal{G} =\mathcal{G}_0\times U(1).

As the product shows, the gauge group 𝒢\mathcal{G} is not contractible. But as the argument above shows, for MM simply connected, the based gauge group 𝒢 0\mathcal{G}_0 is contractible.

Moduli space

Since both the gauge group 𝒢\mathcal{G} and its subgroup, the based gauge group 𝒢 0\mathcal{G}_0, act on the configuration space 𝒜\mathcal{A} and its subspace, the reduced configuration space 𝒜 *\mathcal{A}^*, there are quotient spaces: (Nicolaescu 2000, p. 89, Kronheimer & Mrowka 2007, Def. 1.3.1. & Eq. (1.16), Moore 2010, p. 77-79)

𝒜/𝒢; \mathcal{B} \coloneqq\mathcal{A}/\mathcal{G};
˜𝒜/𝒢 0; \widetilde\mathcal{B} \coloneqq\mathcal{A}/\mathcal{G}_0;
*𝒜 */𝒢; \mathcal{B}^* \coloneqq\mathcal{A}^*/\mathcal{G};
˜ *𝒜 */𝒢 0. \widetilde\mathcal{B}^* \coloneqq\mathcal{A}^*/\mathcal{G}_0.

As the formula of the action above shows, the gauge group 𝒢\mathcal{G} doesn’t act free on the configuration space 𝒜\mathcal{A}, since the points with vanishing self-dual spinor field ψ=0\psi=0 are invariant under all constant gauges g=constg=const, but it therefore does act free on the reduced configuration space 𝒜 *\mathcal{A}^* and the based gauge group 𝒢 0\mathcal{G}_0 even acts free on both. \mathcal{B} therefore has singularities, while the other spaces don’t. If MM is simply connected, then ˜\widetilde\mathcal{B} can furthermore be identified with a linear subspace of the configuration space 𝒜\mathcal{A} by: (Moore 2010, p. 77-79)

˜{(d Aia,ψ)𝒜|δa=0}. \widetilde\mathcal{B} \cong\{(d_A-ia,\psi)\in\mathcal{A}|\delta a=0\}.

Equivalently, for every aΩ 1(M)a\in\Omega^1(M), there is a unique smooth map u:Mu\colon M\rightarrow\mathbb{R} with δ(a+du)=δa+Δu=0\delta(a+\mathrm{d}u)=\delta a+\Delta u=0, which can be shown again using the Hodge decomposition C (M,)= 0(M)δΩ 1(M)C^\infty(M,\mathbb{R})=\mathcal{H}_0(M)\oplus\delta\Omega^1(M) with 0(M)\mathcal{H}_0(M) just being the constant functions for MM connected. (Moore 2010, p. 77-79)

Although the canonical projection ˜\widetilde\mathcal{B}\rightarrow\mathcal{B} might not be even be a fiber bundle due to the singularities, the canonical projection ˜ * *\widetilde\mathcal{B}^*\rightarrow\mathcal{B}^*, after a suitable Sobolev completion, is a principal U(1)-bundle. For MM simply connected, ˜ *\widetilde\mathcal{B}^* is contractible, since 𝒜 *\mathcal{A}^* always is and 𝒢 0\mathcal{G}_0 is in this case as argued before. It then follows from the long exact sequence of homotopy groups of the principal U(1)-bundle ˜ * *\widetilde\mathcal{B}^*\rightarrow\mathcal{B}^*, that *\mathcal{B}^* is an Eilenberg-MacLane space K(,2)K(\mathbb{Z},2) (as S 1S^1 is a K(,1)K(\mathbb{Z},1)) and since the infinite complex projective space P \mathbb{C}P^\infty is as well, there is a weak homotopy equivalence *P \mathcal{B}^*\rightarrow\mathbb{C}P^\infty. (Moore 2010, p. 81) Homotopy classes of such maps are classified by [K(,2),K(,2)][K(\mathbb{Z},2),K(\mathbb{Z},2)]\cong\mathbb{Z} and the weak homotopy equivalence must correspond to a generator ±1\pm 1\in\mathbb{Z}. But the principal U(1)-bundle also bijectively corresponds to the homotopy class of a classifying map f: *BU(1)P f\colon\mathcal{B}^*\rightarrow BU(1)\cong\mathbb{C}P^\infty with ˜ *f *EU(1)f *S \widetilde\mathcal{B}^*\cong f^*EU(1)\cong f^*S^\infty, which falls under the exact same classification, but doesn’t necessarily correspond to a generator. It is exactly the first Chern class c 1()H 2( *,)c_1(\mathcal{B})\in H^2(\mathcal{B}^*,\mathbb{Z})\in\mathbb{Z} of the principal U(1)-bundle, but the perturbed Seiberg-Witten equations need to enter for it to be of use for the Seiberg-Witten invariants. Its moduli spaces are then given by the subspaces of its solutions: (Perutz 2002, p. 6, Moore 2010, p. 81)

{[d Aia,ψ]|(d Aia,ψ)fulfills SW eq.}; \mathcal{M} \coloneqq\{[d_A-ia,\psi]\in\mathcal{B}|(d_A-ia,\psi)\text{fulfills SW eq.}\};
η{(d Aia,ψ)˜|(d Aia,ψ)fulfills pSW eq. with per.η}; \mathcal{M}_\eta \coloneqq\{(d_A-ia,\psi)\in\widetilde\mathcal{B}|(d_A-ia,\psi)\text{fulfills pSW eq. with per.}\eta\};
˜ η{(d Aia,ψ)˜|(d Aia,ψ)fulfills pSW eq. with per.η}. \widetilde\mathcal{M}_\eta \coloneqq\{(d_A-ia,\psi)\in\widetilde\mathcal{B}|(d_A-ia,\psi)\text{fulfills pSW eq. with per.}\eta\}.

With the canonical projection ˜\widetilde\mathcal{B}\rightarrow\mathcal{B}, there is a canonical projection ˜ η η\widetilde\mathcal{M}_\eta\rightarrow\mathcal{M}_\eta. Since the former isn’t a fiber bundle, it seems that the latter isn’t as well. But this isn’t necessarily the case and exactly the reason why the Seiberg-Witten equations are considered with a perturbation. For this, the self-dual Betti number is important:

  • If b +(M)1b_+(M)\geq 1, then a perturbation ηΩ + 2(M)Π\eta\in\Omega_+^2(M)\setminus\Pi\neq\emptyset forces solutions of the perturbed Seiberg-Witten equations with ψ=0\psi=0 to fulfill F A +=ηF_A^+=\eta, which is not possible due to ηΠ\eta\notin\Pi. Hence both perturbed moduli spaces avoid all singularities in this case and ˜ η η\widetilde\mathcal{M}_\eta\rightarrow\mathcal{M}_\eta becomes a principal U(1)-subbundle of ˜ * *\widetilde\mathcal{B}^*\rightarrow\mathcal{B}^* with first Chern classc 1(˜ η)H 2( η,)c_1(\widetilde\mathcal{M}_\eta)\in H^2(\mathcal{M}_\eta,\mathbb{Z}), which is then used to define the Seiberg-Witten invariants.

  • If b +(M)2b_+(M)\geq 2, then any two perturbations η 1,η 2Ω + 2(M)Π\eta_1,\eta_2\in\Omega_+^2(M)\setminus\Pi\neq\emptyset can furthermore be connected by a path γ:[0,1]Ω + 2(M)Π\gamma\colon[0,1]\rightarrow\Omega_+^2(M)\setminus\Pi, which describes a bordism ˜ η 1˜ η 2\widetilde\mathcal{M}_{\eta_1}\rightarrow\widetilde\mathcal{M}_{\eta_2}. Hence all perturbations give the same bordism class. (Moore 2010, p. 100) (If b +(M)=1b_+(M)=1, one has to chose a connected component of Ω + 2(M)Π\Omega_+^2(M)\setminus\Pi, which might give two different bordism classes.)

For the Seiberg-Witten invariant, which is in particular Chern number, the necessary amount of cup products of the Chern class with itself is evaluated with the fundamental class of the moduli space in the Kronecker pairing. Since the Chern class has even degree, the moduli space must have even dimension for this to work and it furthermore has to be known precisely for the amount of cup products. First relating it to the index of the Dirac operator and then applying the Atiyah-Singer index theorem yields formulas containing the Euler characteristic and the signature:

Let MM be simply connected. If ind(D A +)>0ind(D_A^+)\gt 0 for b +(M)=0b_+(M)=0 or b +(M)>0b_+(M)\gt 0, then η\mathcal{M}_\eta is an oriented smooth manifold with dimension: (Nicolaescu 2000, Lem. 2.2.10, Kronheimer & Mrowka 2007, Thrm 1.4.4., Moore 2010, Transversality Theorem 2 on p. 91 )

dim η=2ind(D A +)b +(M)1=14(c 1 2(𝔰)[M]2χ(M)3σ(M)). dim\mathcal{M}_\eta =2 ind(D_A^+) -b_+(M) -1 =\frac{1}{4}(c_1^2(\mathfrak{s})[M]-2\chi(M)-3\sigma(M)).

While the first expression is obviously always an integer, it is more difficult to see for the second expression. But as shown in the basics, c 1 2(𝔰)[M]σ(M)c_1^2(\mathfrak{s})[M]-\sigma(M) is always even, which already makes the term in the brackets even as well.

Let MM be simply connected. If b +(M)>0b_+(M)\gt 0, then ˜ η\widetilde\mathcal{M}_\eta is a compact (Donaldson 1996, Eq. (7), Moore 2010, Compactness Theorem on p. 83) oriented smooth manifold with dimension: (Moore 2010, Transversality Theorem 1 on p. 86)

dim˜ η=2ind(D A +)b +(M)=14(c 1 2(𝔰)[M]2χ(M)3σ(M))+1. dim\widetilde\mathcal{M}_\eta =2 ind(D_A^+) -b_+(M) =\frac{1}{4}(c_1^2(\mathfrak{s})[M]-2\chi(M)-3\sigma(M))+1.

(Some literature uses the convention L 2=det(W ±)L^2=\det(W^\pm) since it is indeed the square of a line bundle, which makes the above formula not include a factor in front of the Chern class.) Hence for b +(M)b_+(M) even, dim ηdim\mathcal{M}_\eta is also even and the Seiberg-Witten invariants, which are independent of the Riemannian metric gg and the perturbation η\eta Kronheimer & Mrowka 2007, Thrm 1.5.2. as argued above for the latter, can then be defined as: (Donaldson 1996, Eq. (6), Nicolaescu 2000, p. 113, Kronheimer & Mrowka 2007, Def. 1.5.3. & 1.5.4.)

SW(M,𝔰)c 1( η) dim˜ η2,[˜ η]. SW(M,\mathfrak{s}) \coloneqq\langle c_1(\mathcal{M}_\eta)^{\frac{dim\widetilde\mathcal{M}_\eta}{2}},[\widetilde\mathcal{M}_\eta]\rangle.

References

See also:

Created on August 21, 2025 at 23:32:11. See the history of this page for a list of all contributions to it.