# Seiberg-Witten Like Equations on Pseudo-RiemannianSpinc Manifolds with G2(2)∗ Structure

Authors: Nülifer Özdemir; Nedim Deǧirmenci
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

We consider 7-dimensional pseudo-Riemannianspinc manifolds with structure group G2(2)∗. On such manifolds, the space of 2-forms splits orthogonally into components Λ2M=Λ72⊕Λ142. We define self-duality of a 2-form by considering the part Λ72 as the bundle of self-dual 2-forms. We express the spinor bundle and the Dirac operator and write down Seiberg-Witten like equations on such manifolds. Finally we get explicit forms of these equations on R4,3 and give some solutions.


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## Body

## 1. Introduction

The Seiberg-Witten theory, introduced by Witten in [1], became one of the most important tools to understand the topology of smooth 4-manifolds. The Seiberg-Witten theory is based on the solution space of two equations which are called the Seiberg-Witten equations. The first one of the Seiberg-Witten equations is Dirac equation and the second one is known as curvature equation [2]. The first equation is the harmonicity condition of spinor fields; that is, the spinor field belongs to the kernel of the Dirac operator. The second equation couples the self-dual part of the curvature 2-form with a spinor field. There exist various generalizations of Seiberg-Witten equations to higher dimensional Riemannian manifolds [3–6]. All of these generalizations are done for the manifolds which have special structure groups. Also Seiberg-Witten like equations are studied over 4-dimensional Lorentzian spinc manifolds [7] and 4-dimensional pseudo-Riemannian manifolds with neutral signature [8].Parallel spinors on pseudo-Riemannianspinc manifolds are studied by Ikemakhen [9]. In the present work, we consider 7-dimensional manifolds with structure group G2(2)∗. In order to define spinors and Dirac operator, the manifold M must have a spinc-structure. We assume that 7-dimensional pseudo-Riemannian manifold M with signature (-,-,-,-,+,+,+) has spinc-structure. On the other hand, to write down curvature equation, we need a self-duality notion of a 2-form on such manifolds. In 4 dimensions, self-duality concept of 2-forms is well known. The bundle of 2-forms Λ2(M) decomposes into two parts on this manifold [10]. Then we will define self-duality of a 2-form on a 7-manifold with structure group G2(2)∗ by using decomposition of 2-forms on this manifold.


## 2. Manifolds with Structure GroupG2(2)∗

The exceptional Lie groupG2, automorphism group of octonions, is well known. There is another similar Lie group G2(2)∗ which is automorphism group of split octonions [11]. On R7, we consider the metric (1)g4,3x,y=-x1y1-x2y2-x3y3-x4y4+x5y5+x6y6+x7y7,where x=(x1,x2,…,x7) and y=(y1,y2,…,y7)∈R7. From now on, we denote the pair (R7,g4,3) by R4,3. The isometry group of this space is (2)O4,3=A∈GL7,R:g4,3Ax,Ay=g4,3x,y,∀x,y∈R7.The special orthogonal subgroup of O(4,3) is (3)SO4,3=A∈O4,3:det⁡A=1.The group G2(2)∗ is the subgroup of SO(4,3), preserving the following 3-form: (4)φ0=-e127-e135+e146+e236+e245-e347+e567,where {e1,…,e7} is the dual base of the standard basis {e1,…,e7} of R4,3, with the notation eijk=ei∧ej∧ek and with the metric g4,3=(-1,-1,-1,-1,1,1,1); that is, (5)G22∗=A∈GL7,R:A∗φ0=φ0,where φ0 is called the fundamental 3-form on R4,3 [10, 11]. The space of 2-forms Λ2R7 decomposes into two parts Λ2R7=Λ72R7⊕Λ142R7, where (6)Λ72R7=α∈Λ2R7:⋆φ0∧α=2α,Λ142R7=α∈Λ2R7:⋆φ0∧α=-α.A semi-Riemannian7-manifold M with the metric of signature (-,-,-,-,+,+,+) is called a G2(2)∗ manifold if its structure group reduces to the Lie group G2(2)∗; equivalently, there exists a nowhere vanishing 3-form on M whose local expression is of the form φ0. Such a form is called a G2(2)∗ structure on M [12]. If the structure group of M is the group G2(2)∗ then the bundle of 2-forms Λ2(M) decomposes into two parts similar to Λ2R7 and we denote it by Λ2(M)=Λ72(M)⊕Λ142(M) [10].It is known that square of the Hodge∗ operator on 2-forms over 4-dimensional Riemannian manifolds is identity and ±1 are eigenvalues of the Hodge ∗ operator. The elements of eigenspace of 1 are called self-dual 2-forms and the others are called anti-self-dual forms. But this situation does not generalize to higher dimensional manifolds directly. Self-duality of 2-form has been studied on some higher dimensions [3, 13]. In this work, we need self-duality concept of 2-forms on 7-dimensional manifolds with structure group G2(2)∗.Now we define a duality operator over bundle of 2-formΛ2(M) as (7)Tφ:Λ2M⟶Λ2M,Tφα≔⋆φ∧α.The eigenvalues of this map are 2 and -1. Note that the subbundle Λ72(M) corresponds to the eigenvalue 2 and the subbundle Λ142(M) corresponds to the eigenvalue -1. Let α be a 2-form over M. If α belongs to Λ72(M), then we call α a self-dual 2-form. If α belongs to Λ142(M), then we call α an anti-self-dual 2-form. Because of decomposition of 2-forms on M, any 2-form α on M can be written uniquely as (8)α=α++α-,where α+∈Λ72(M) and α-∈Λ142(M). Similar to the 4-dimensional case, we say that α+ is self-dual part of α and α- is anti-self-dual part of α.


## 3. Spinor Bundles overG2(2)∗ Manifolds

It is known that the groupSO(4,3) has two connected components. The connected component to the identity of SO(4,3) is denoted by SO+(4,3). In this work we deal with the group SO+(4,3). The covering space of SO(4,3) is the group Spin(4,3) which lies in Clifford algebra Cl4,3=Cl(R7,-g4,3)⊂Cl4,3 and we denoted the connected component of 1∈Spin(4,3) by Spin+(4,3). There is a covering map λ:Spin+(4,3)→SO+(4,3) which is a 2 : 1 group homomorphism given by λ(g)(x)=g·x·g-1 for x∈R4,3, g∈Spin+(4,3) [10, 11, 14].One can define another group which lies in the complex Clifford algebraCl(R4,3)≅Cl7 by (9)Spin+c4,3≔Spin+4,3×S1Z2,where the elements of Spin+c(4,3) are the equivalence classes [g,z] of pair (g,z)∈Spin+(4,3)×S1, under the equivalence relation (g,z)~(-g,-z) [9]. There exist two exact sequences as(10)1⟶Z2⟶Spin+4,3⟶λSO+4,3⟶1,1⟶Z2⟶Spin+c4,3⟶ξSO+4,3×S1⟶1,where ξ([g,z])=(λ(g),z2).Lete1,…,e7 be an orthonormal basis of R4,3; then the Lie algebras of Spin(4,3) and Spinc(4,3) are (11)spin4,3=eiej:1≤i,j≤7,spinc4,3=spin4,3⊕iR,respectively. The derivative of ξ:Spin+c(4,3)→SO+(4,3)×S1 is obtained as (12)ξ∗eiej,ir=λ∗eiej,ir=2Eij,2ir,where Eij is the 8×8-matrix whose (i,j)-entry is 1, (j,i)-entry is -1, and the other entries are zero [9]. Since the Clifford algebra Cl7 is isomorphic to the algebra C(8)⊕C(8), we can project this isomorphism onto the first component. Hence, we get spinor representation: (13)κ:Cl7⟶C8≅EndC8.By restricting κ to the group Spin+c(4,3) we get (14)κSpin+c4,3:Spin+c4,3⟶AutC8and κSpin+c(4,3) is called spinor representation of the group Spin+c(4,3); shortly we denote it by κ. The elements of C8 are called spinors and the complex vector space C8 is called the spinor space and it is denoted by Δ4,3. By using spinor representation, the Clifford multiplication of vectors with spinors is defined by (15)X·ψ≔κXψ,where X∈R4,3 and ψ∈Δ4,3. The spinor space has a nondegenerate indefinite Hermitian inner product as (16)ψ1,ψ2Δ4,3≔i44-1/2κe1e2e3e4ψ1,ψ2,where z,w=∑i=18ziw-i is the standard Hermitian inner product on C8 for z=z1,…,z8,w=(w1,…,w8)∈C8. The new inner product ,Δ4,3 is invariant with respect to the group spin+c(4,3) and satisfies the following property: (17)κZψ1,ψ2Δ4,3=-ψ1,κZψ2Δ4,3,where Z∈R4,3 and ψ1,ψ2∈Δ4,3. In this work, we use the following spinor representation κ: (18)κe1=ε⊗ε⊗δ,κe2=-δ⊗δ⊗τ,κe3=-δ⊗I⊗δ,κe4=δ⊗τ⊗τ,κe5=-I⊗ε⊗τ,κe6=-τ⊗ε⊗δ,κe7=I⊗I⊗ε,where (19)I=1001,δ=0110,τ=100-1,ε=0-110.Now, we recall the main definitions concerning spinc-structure and the spinor bundle. Let M be a 7-dimensional pseudo-Riemannian manifold with structure group G2(2)∗. Then, there is an open covering {Uα}α∈A of M and transition functions gαβ:Uα∩Uβ→G2(2)∗⊂SO+(4,3) for TM.If there exists another collection of transition functions(20)g~αβ:Uα∩Uβ⟶Spin+c4,3such that the following diagram commutes (i.e., ξ∘g~αβ=gαβ and the cocycle condition g~αβg~βγ=g~αγ on Uα∩Uβ∩Uγ is satisfied), then M is called a spinc manifold. Then one can construct a principal Spin+c(4,3)-bundle PSpin+c(4,3) on M and a bundle map Λ:PSpin+c(4,3)→PSO+(4,3).Let(PSpin+c(4,3),Λ) be a spinc-structure on M. We can construct an associated complex vector bundle: (22)S=PSpin+c4,3×κΔ4,3,where κ:Spin+c4,3→Aut(Δ4,3) is the spinor representation of Spin+c(4,3). This complex vector bundle is called spinor bundle for a given spinc-structure on M and sections of S are called spinor fields. The Clifford multiplication given by (15) can be extended to a bundle map: (23)μ:TM⊗S⟶S.Parallel spinors on the spinor bundle S are studied in [9].SinceM is a pseudo-Riemannian spinc manifold, then by using the map (24)l:Spin+c4,3⟶S1,lg,z=z2,we can get an associated principal S1-bundle: (25)PS1=PSpin+c4,3×lS1.Also, the map l induces a bundle map: (26)L:PSpin+c4,3⟶PS1.Now, fix a connection 1-formA:TPS1→iR over the principal U(1)-bundle PS1. Let ∇ be the Levi-Civita covariant derivative associated with the metric g4,3 which determines an so(4,3)-valued connection 1-form ω on the principal bundle PSO+(4,3). The connection 1-form ω can be written locally (27)ω=∑i<jωijEij,where e1,e2,…,e7 is a local orthonormal frame on open set U⊂M and ωij=g4,3(∇ei,ej). By using the connection 1-form A and ω, one can obtain a connection 1-form on the principal bundle PSO+(4,3)×~PS1 (the fibre product bundle): (28)ω×A:TPSO+4,3×~PS1⟶SO+4,3×iR.The connection ω×A can be lift to a connection 1-form ZA on the principal bundle PSO+c(4,3) via the 2-fold covering map:(29)π≔Λ,L:PSpin+c4,3⟶PSO+4,3×~PS1and the following commutative diagram. One can obtain a covariant derivative operator ∇A on the spinor bundle S by using the connection 1-form ZA. The local form of the covariant derivative ∇A is (31)∇AΨ=dΨ+12∑i<jεiεjωijκeiejΨ+12AΨ,where e1,…,e7 is a orthonormal frame on open set U⊂M. We note that some authors use the term AΨ instead of 1/2AΨ in the local formula of ∇AΨ. The covariant derivative ∇A is compatible with the metric ,Δ4,3(32)Xψ1,ψ2Δ4,3=∇XAψ1,ψ2Δ4,3+ψ1,∇XAψ2Δ4,3and the Clifford multiplication (33)∇XAY·ψ=Y·∇XAψ+∇XY·ψ,where ψ,ψ1,ψ2 are spinor fields and sections of S, X, and Y are vector fields on M. We can define the Dirac operator DA as the following composition: (34)DA≔μ∘∇A:ΓS⟶∇AΓTM∗⊗S≃g4,3TM⊗S⟶μΓS,which can be written locally as (35)DAψ=∑i=17εiκei∇eiAψ,where {e1,e2,…,e7} is any oriented local orthonormal frame of TM.


## 4. Seiberg-Witten Like Equations onG2(2)∗ Manifolds

LetM be a spinc manifold with structure group G2(2)∗. Fix a spinc-structure and a connection A in the principal U(1)-bundle PS1 associated with the spinc-structure. Note that the curvature FA of the connection A is iR-valued 2-form. The curvature 2-form FA on the PS1 determines an iR-valued 2-form on M uniquely (see [15]) and we denote it again by FA.We can define a map(36)σψX,Y=X·Y·ψ,ψΔ4,3+g4,3X,Yψ2,where X,Y∈Γ(TM). Note that the map σ(ψ) satisfies the following properties: (37)σψX,Y=-σψY,X,σψX,Y¯=-σψX,Y.Hence, the mapσ associates an iR-valued 2-form with each spinor field ψ∈Γ(S), so we can write (38)σ:ΓS⟶Ω2M,iR.In local frame {e1,e2,…,e7} on U⊂M, the map σ can be expressed as (39)σψ=-14∑i<jκeiejψ,ψΔ4,3ei∧ej.Now we are ready to express the Seiberg-Witten equations. LetM be a spinc manifold with structure group G2(2)∗. Fix a Spin+c(4,3) structure and take a connection 1-form A on the principal bundle PS1 and a spinor field ψ∈Γ(S). We write the Seiberg-Witten like equations as (40)DAψ=0,FA+=-14σψ+,where FA+ is the self-dual part of the curvature FA and σ(ψ)+ is the self-dual part of the 2-form σ(ψ) corresponding to the spinor ψ∈Γ(S).


## 5. Seiberg-Witten Like Equations onR4,3

Let us consider these equations on the flat spaceM=R4,3 with the G2(2)∗ structure given by φ0. We use the standard orthonormal frame {e1,e2,…,e7} on M=R4,3 and the spinor representation in (18). The spinc connection ∇A on R4,3 is given by (41)∇jAΨ=∂Ψ∂xj+AjΨ,where Aj:R4,3→iR and Ψ:R4,3→Δ4,3 are smooth maps. Then, the associated connection on the line bundle LΓ=R4,3×C is the connection 1-form (42)A=∑i=17Aidxi∈Ω1R4,3,iRand its curvature 2-form is given by (43)FA=dA=∑i<jFijdxi∧dxj∈Ω2R4,3,iR,where Fij=∂Aj/∂xi-∂Ai/∂xj for i,j=1,…,7. Now we can write the Dirac operator DA on R4,3 with respect to a given spinc-structure and spinc-connection ∇A.We denote the dual basis of{e1,e2,…,e7} by {e1,e2,…,e7}. Now one can give a frame for the space of self-dual 2-forms on R4,3 as(44)f1=e1∧e2+e3∧e4-e5∧e6,f2=e1∧e3-e2∧e4-e6∧e7,f3=e1∧e4+e2∧e3-e5∧e7,f4=e1∧e5-e2∧e6-e4∧e7,f5=e1∧e6+e2∧e5-e3∧e7,f6=e1∧e7+e3∧e6+e4∧e5,f7=e2∧e7+e3∧e5-e4∧e6.LetFA be the curvature form of the iR-valued connection 1-form A and let FA+ be its self-dual part. Then, (45)FA+=∑i=17FA,fififi2=13F12+F34-F56f1+F13-F24-F67f2+F14+F23-F57f3+F15-F26-F47f4+F16+F25-F37f5+F17+F36+F45f6+F27+F35-F46f7.Now we calculate the 2-form σ(ψ)+, for a spinor ψ∈S. Then σ(ψ) can be written in the following way: (46)σψ=∑i<jeiejψ,ψei∧ej.The projection onto the subspace Λ72(R4,3,iR) is given by (47)σψ+=∑i=17σψ,fififi2.If σ(ψ)+ is calculated explicitly, then we obtain the following identity: (48)3σψ+=-3ψ2ψ¯1+3ψ1ψ¯2+ψ4ψ¯3-ψ3ψ¯4-ψ6ψ¯5+ψ5ψ¯6-ψ8ψ¯7+ψ7ψ¯8f1+3ψ3ψ¯1+ψ4ψ¯2-3ψ1ψ¯3-ψ2ψ¯4+ψ7ψ¯5-ψ8ψ¯6-ψ5ψ¯7+ψ6ψ¯8f2+-3ψ4ψ¯1+ψ3ψ¯2-ψ2ψ¯3+3ψ1ψ¯4+ψ8ψ¯5+ψ7ψ¯6-ψ6ψ¯7-ψ5ψ¯8f3+-3ψ6ψ¯1+ψ5ψ¯2+ψ8ψ¯3+ψ7ψ¯4-ψ2ψ¯5+3ψ1ψ¯6-ψ4ψ¯7-ψ3ψ¯8f4+-3ψ5ψ¯1-ψ6ψ¯2-ψ7ψ¯3+ψ8ψ¯4+3ψ1ψ¯5+ψ2ψ¯6+ψ3ψ¯7-ψ4ψ¯8f5+-3ψ7ψ¯1-ψ8ψ¯2+ψ5ψ¯3-ψ6ψ¯4-ψ3ψ¯5+ψ4ψ¯6+3ψ1ψ¯7+ψ2ψ¯8f6+-3ψ8ψ¯1+ψ7ψ¯2-ψ6ψ¯3-ψ5ψ¯4+ψ4ψ¯5+ψ3ψ¯6-ψ2ψ¯7+3ψ1ψ¯8f7.Hence, the curvature equation can be written explicitly as (49)F12+F34-F56=143ψ2ψ¯1-3ψ1ψ¯2-ψ4ψ¯3+ψ3ψ¯4+ψ6ψ¯5-ψ5ψ¯6+ψ8ψ¯7-ψ7ψ¯8,F13-F24-F67=14-3ψ3ψ¯1-ψ4ψ¯2+3ψ1ψ¯3+ψ2ψ¯4-ψ7ψ¯5+ψ8ψ¯6+ψ5ψ¯7-ψ6ψ¯8,F14+F23-F57=143ψ4ψ¯1-ψ3ψ¯2+ψ2ψ¯3-3ψ1ψ¯4-ψ8ψ¯5-ψ7ψ¯6+ψ6ψ¯7+ψ5ψ¯8,F15-F26-F47=14-3ψ6ψ¯1+ψ5ψ¯2+ψ8ψ¯3+ψ7ψ¯4-ψ2ψ¯5+3ψ1ψ¯6-ψ4ψ¯7-ψ3ψ¯8,F16+F25-F37=14-3ψ5ψ¯1-ψ6ψ¯2-ψ7ψ¯3+ψ8ψ¯4+3ψ1ψ¯5+ψ2ψ¯6+ψ3ψ¯7-ψ4ψ¯8,F17+F36+F45=14-3ψ7ψ¯1-ψ8ψ¯2+ψ5ψ¯3-ψ6ψ¯4-ψ3ψ¯5+ψ4ψ¯6+3ψ1ψ¯7+ψ2ψ¯8,F27+F35-F46=14-3ψ8ψ¯1+ψ7ψ¯2-ψ6ψ¯3-ψ5ψ¯4+ψ4ψ¯5+ψ3ψ¯6-ψ2ψ¯7+3ψ1ψ¯8.Dirac equation DAΨ=0 can be expressed as follows: (50)∂ψ8∂x1-∂ψ7∂x2-∂ψ6∂x3+∂ψ5∂x4-∂ψ3∂x5-∂ψ4∂x6+∂ψ2∂x7=-A1ψ8+A2ψ7+A3ψ6-A4ψ5+A5ψ3+A6ψ4-A7ψ2,∂ψ7∂x1+∂ψ8∂x2-∂ψ5∂x3-∂ψ6∂x4+∂ψ4∂x5-∂ψ3∂x6-∂ψ1∂x7=-A1ψ7-A2ψ8+A3ψ5+A4ψ6-A5ψ4+A6ψ3+A7ψ1,-∂ψ6∂x1-∂ψ5∂x2-∂ψ8∂x3-∂ψ7∂x4+∂ψ1∂x5+∂ψ2∂x6+∂ψ4∂x7=A1ψ6+A2ψ5+A3ψ8+A4ψ7-A5ψ1-A6ψ2-A7ψ4,-∂ψ5∂x1+∂ψ6∂x2-∂ψ7∂x3+∂ψ8∂x4-∂ψ2∂x5+∂ψ1∂x6+∂ψ3∂x7=A1ψ5-A2ψ6+A3ψ7-A4ψ8+A5ψ2-A6ψ1-A7ψ3,-∂ψ4∂x1-∂ψ3∂x2-∂ψ2∂x3+∂ψ1∂x4-∂ψ7∂x5+∂ψ8∂x6+∂ψ6∂x7=A1ψ4+A2ψ3+A3ψ2-A4ψ1+A5ψ7-A6ψ8-A7ψ6,-∂ψ3∂x1+∂ψ4∂x2-∂ψ1∂x3-∂ψ2∂x4+∂ψ8∂x5+∂ψ7∂x6-∂ψ5∂x7=A1ψ3-A2ψ4+A3ψ1+A4ψ2-A5ψ8-A6ψ7+A7ψ5,∂ψ2∂x1-∂ψ1∂x2-∂ψ4∂x3-∂ψ3∂x4+∂ψ5∂x5-∂ψ6∂x6+∂ψ8∂x7=-A1ψ2+A2ψ1+A3ψ4+A4ψ3-A5ψ5+A6ψ6-A7ψ8,∂ψ1∂x1+∂ψ2∂x2-∂ψ3∂x3+∂ψ4∂x4-∂ψ6∂x5-∂ψ5∂x6-∂ψ7∂x7=-A1ψ1-A2ψ2+A3ψ3-A4ψ4+A5ψ6+A6ψ5+A7ψ7.These equations admit nontrivial solutions. For example, direct calculation shows that the spinor field(51)ψ=0,0,ψ3,iψ3,ψ3,iψ3,0,0with ψ3(x1,x2,…,x7)=e-i/2x12x2 and the connection 1-form (52)Ax1,x2,…,x7=ix1x2dx1+i2x12dx2satisfy the above equations.Now we consider the space(53)C=A×ΓS,where A is the space of connection 1-forms on the principle bundle PS1 and Γ(S) is the space of spinor fields. The space C is called the configuration space. There is an action of the gauge group G≔Map(X,S1) on the configuration space by (54)u·A,ψ≔A+u-1du,u-1ψ,where u∈G and (A,ψ)∈C. The action of the gauge group enjoys the following equalities: (55)FA+u-1du=FA,DAu-1ψ=u-1DAψ.Hence, if the pair (A,ψ) is a solution to the Seiberg-Witten equations, then the pair A+u-1du,u-1ψ is also a solution to the Seiberg-Witten equations.One can obtain infinitely many solutions for the Seiberg-Witten equations onR4,3: Consider the spinor (56)ψ=0,0,ψ3,iψ3,ψ3,iψ3,0,0,ψ3x1,x2,…,x7=e-i/2x12x2and the connection 1-form (57)Ax1,x2,…,x7=ix1x2dx1+i2x12dx2.Since the pair (A,ψ) is a solution on R4,3, the pair A+idf,e-ifψ is also a solution, where u=eif and f is a smooth real valued function on R4,3.The moduli space of Seiberg-Witten equations on the manifold with structure groupG2(2)∗ is (58)M=A,ψ∈C:DAψ=0,FA+=-1/4σψ+G.Whether the moduli spaceM has similar properties of moduli space of Seiberg-Witten equations on a 4-dimensional manifold is a subject of another work.



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