AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2016/2173214 2173214 Research Article Seiberg-Witten Like Equations on Pseudo-Riemannian Spinc Manifolds with G2(2) Structure Özdemir Nülifer Deǧirmenci Nedim Tsimpis Dimitrios Department of Mathematics Anadolu University Eskisehir Turkey anadolu.edu.tr 2016 612016 2016 26 08 2015 28 09 2015 612016 2016 Copyright © 2016 Nülifer Özdemir and Nedim Deǧirmenci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider 7-dimensional pseudo-Riemannian spinc manifolds with structure group G2(2). On such manifolds, the space of 2-forms splits orthogonally into components Λ2M=Λ72Λ142. 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.

1. Introduction

The Seiberg-Witten theory, introduced by Witten in , 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 . 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 . 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  and 4-dimensional pseudo-Riemannian manifolds with neutral signature .

Parallel spinors on pseudo-Riemannian spinc manifolds are studied by Ikemakhen . 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 . 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 Group <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M26"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo mathvariant="bold">(</mml:mo><mml:mn>2</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

The exceptional Lie group G2, automorphism group of octonions, is well known. There is another similar Lie group G2(2) which is automorphism group of split octonions . 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=AGL7,R:g4,3Ax,Ay=g4,3x,y,x,yR7.The special orthogonal subgroup of O(4,3) is (3)SO4,3=AO4,3:detA=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=eiejek and with the metric g4,3=(-1,-1,-1,-1,1,1,1); that is, (5)G22=AGL7,R:Aφ0=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=Λ72R7Λ142R7, where (6)Λ72R7=αΛ2R7:φ0α=2α,Λ142R7=αΛ2R7:φ0α=-α.

A semi-Riemannian 7-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 . 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) .

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-form Λ2(M) as (7)Tφ:Λ2MΛ2M,Tφαφα.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 over <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M102"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo mathvariant="bold">(</mml:mo><mml:mn>2</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> Manifolds

It is known that the group SO(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 1Spin(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 xR4,3, gSpin+(4,3) [10, 11, 14].

One can define another group which lies in the complex Clifford algebra Cl(R4,3)Cl7 by (9)Spin+c4,3Spin+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) . There exist two exact sequences as(10)1Z2Spin+4,3λSO+4,31,1Z2Spin+c4,3ξSO+4,3×S11,where ξ([g,z])=(λ(g),z2).

Let e1,,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:1i,j7,spinc4,3=spin4,3iR,respectively. The derivative of ξ:Spin+c(4,3)SO+(4,3)×S1 is obtained as (12)ξeiej,ir=λeiej,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 . 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:Cl7C8EndC8.By restricting κ to the group Spin+c(4,3) we get (14)κSpin+c4,3:Spin+c4,3AutC8and 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·ψκXψ,where XR4,3 and ψΔ4,3. The spinor space has a nondegenerate indefinite Hermitian inner product as (16)ψ1,ψ2Δ4,3i44-1/2κe1e2e3e4ψ1,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 ZR4,3 and ψ1,ψ2Δ4,3. In this work, we use the following spinor representation κ: (18)κe1=εεδ,κe2=-δδτ,κe3=-δIδ,κe4=δττ,κe5=-Iετ,κe6=-τεδ,κe7=IIε,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}αA of M and transition functions gαβ:UαUβG2(2)SO+(4,3) for TM.

If there exists another collection of transition functions (20)g~αβ:UαUβSpin+c4,3such that the following diagram commutes (i.e., ξg~αβ=gαβ and the cocycle condition g~αβg~βγ=g~αγ on UαUβUγ 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+c4,3Aut(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:TMSS.Parallel spinors on the spinor bundle S are studied in .

Since M is a pseudo-Riemannian spinc manifold, then by using the map (24)l:Spin+c4,3S1,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+c4,3PS1.

Now, fix a connection 1-form A:TPS1iR 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 UM 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×~PS1SO+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)πΛ,L:PSpin+c4,3PSO+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+12i<jεiεjωijκeiejΨ+12AΨ,where e1,,e7 is a orthonormal frame on open set UM. 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)DAμA:ΓSAΓTMSg4,3TMSμΓS,which can be written locally as (35)DAψ=i=17εiκeieiAψ,where {e1,e2,,e7} is any oriented local orthonormal frame of TM.

4. Seiberg-Witten Like Equations on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M248"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo mathvariant="bold">(</mml:mo><mml:mn>2</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:mrow><mml:mrow><mml:mi>∗</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> Manifolds

Let M 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 ) 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,YΓ(TM). Note that the map σ(ψ) satisfies the following properties: (37)σψX,Y=-σψY,X,σψX,Y¯=-σψX,Y.

Hence, the map σ associates an iR-valued 2-form with each spinor field ψΓ(S), so we can write (38)σ:ΓSΩ2M,iR.In local frame {e1,e2,,e7} on UM, the map σ can be expressed as (39)σψ=-14i<jκeiejψ,ψΔ4,3eiej.

Now we are ready to express the Seiberg-Witten equations. Let M 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 ψΓ(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 ψΓ(S).

5. Seiberg-Witten Like Equations on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M291"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>4,3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>

Let us consider these equations on the flat space M=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=Ψxj+AjΨ,where Aj:R4,3iR and Ψ:R4,3Δ4,3 are smooth maps. Then, the associated connection on the line bundle LΓ=R4,3×C is the connection 1-form (42)A=i=17AidxiΩ1R4,3,iRand its curvature 2-form is given by (43)FA=dA=i<jFijdxidxjΩ2R4,3,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=e1e2+e3e4-e5e6,f2=e1e3-e2e4-e6e7,f3=e1e4+e2e3-e5e7,f4=e1e5-e2e6-e4e7,f5=e1e6+e2e5-e3e7,f6=e1e7+e3e6+e4e5,f7=e2e7+e3e5-e4e6.

Let FA 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 ψS. Then σ(ψ) can be written in the following way: (46)σψ=i<jeiejψ,ψeiej.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)ψ8x1-ψ7x2-ψ6x3+ψ5x4-ψ3x5-ψ4x6+ψ2x7=-A1ψ8+A2ψ7+A3ψ6-A4ψ5+A5ψ3+A6ψ4-A7ψ2,ψ7x1+ψ8x2-ψ5x3-ψ6x4+ψ4x5-ψ3x6-ψ1x7=-A1ψ7-A2ψ8+A3ψ5+A4ψ6-A5ψ4+A6ψ3+A7ψ1,-ψ6x1-ψ5x2-ψ8x3-ψ7x4+ψ1x5+ψ2x6+ψ4x7=A1ψ6+A2ψ5+A3ψ8+A4ψ7-A5ψ1-A6ψ2-A7ψ4,-ψ5x1+ψ6x2-ψ7x3+ψ8x4-ψ2x5+ψ1x6+ψ3x7=A1ψ5-A2ψ6+A3ψ7-A4ψ8+A5ψ2-A6ψ1-A7ψ3,-ψ4x1-ψ3x2-ψ2x3+ψ1x4-ψ7x5+ψ8x6+ψ6x7=A1ψ4+A2ψ3+A3ψ2-A4ψ1+A5ψ7-A6ψ8-A7ψ6,-ψ3x1+ψ4x2-ψ1x3-ψ2x4+ψ8x5+ψ7x6-ψ5x7=A1ψ3-A2ψ4+A3ψ1+A4ψ2-A5ψ8-A6ψ7+A7ψ5,ψ2x1-ψ1x2-ψ4x3-ψ3x4+ψ5x5-ψ6x6+ψ8x7=-A1ψ2+A2ψ1+A3ψ4+A4ψ3-A5ψ5+A6ψ6-A7ψ8,ψ1x1+ψ2x2-ψ3x3+ψ4x4-ψ6x5-ψ5x6-ψ7x7=-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 GMap(X,S1) on the configuration space by (54)u·A,ψA+u-1du,u-1ψ,where uG 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 on R4,3: 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 group G2(2) is (58)M=A,ψC:DAψ=0,FA+=-1/4σψ+G.

Whether the moduli space M has similar properties of moduli space of Seiberg-Witten equations on a 4-dimensional manifold is a subject of another work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This study was supported by Anadolu University Scientific Research Projects Commission under Grant no. 1501F017.

Witten E. Monopoles and four-manifolds Mathematical Research Letters 1994 1 6 769 796 10.4310/mrl.1994.v1.n6.a13 Morgan J. W. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds 1996 Princeton, NJ, USA Princeton University Press 10.1515/9781400865161 Deǧirmenci N. Özdemir N. Seiberg-Witten-like equations on 7-manifolds with G2-structure Journal of Nonlinear Mathematical Physics 2005 12 4 457 461 10.2991/jnmp.2005.12.4.1 2-s2.0-30444439549 Değirmenci N. Özdemir N. Seiberg-Witten like equations on 8-manifolds with structure group spin(7) Journal of Dynamical Systems and Geometric Theories 2009 7 1 21 39 10.1080/1726037X.2009.10698560 Gao Y. H. Tian G. Instantons and the monopole-like equations in eight dimensions Journal of High Energy Physics 2000 5, article 036 Nitta T. Taniguchi T. Quaternionic Seiberg-Witten equation International Journal of Mathematics 1996 7 5 697 10.1142/S0129167X96000360 Değirmenci N. Özdemir N. Seiberg-Witten like equations on Lorentzian manifolds International Journal of Geometric Methods in Modern Physics 2011 8 4 Değirmenci N. Karapazar S. Seiberg-Witten like equations on Pseudo-Riemannian Spinc-manifolds with neutral signature Analele stiintifice ale Universitatii Ovidius Constanta 2012 20 1 Ikemakhen A. Parallel spinors on pseudo-Riemannian Spinc manifolds Journal of Geometry and Physics 2006 56 9 1473 1483 10.1016/j.geomphys.2005.07.005 Baum H. Kath I. Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds Annals of Global Analysis and Geometry 1999 17 1 1 17 10.1023/A:1006556630988 Harvey F. R. Spinors and Calibrations 1990 Academic Press Kath I. G 2 2 -structures on pseudo-riemannian manifolds Journal of Geometry and Physics 1998 27 3-4 155 177 10.1016/s0393-0440(97)00073-9 2-s2.0-0032164132 Corrigan E. Devchand C. Fairlie D. B. Nuyts J. First-order equations for gauge fields in spaces of dimension greater than four Nuclear Physics, Section B 1983 214 3 452 464 10.1016/0550-3213(83)90244-4 2-s2.0-33646048135 Lawson H. B. Michelsohn M. Spin Geometry 1989 Princeton, NJ, USA Princeton University Press Friedrich T. Dirac Operators in Riemannian Geometry 2000 American Mathematical Society