Seiberg-Witten Like Equations on Pseudo-Riemannian Manifolds with Structure
We consider 7-dimensional pseudo-Riemannian manifolds with structure group . On such manifolds, the space of 2-forms splits orthogonally into components . We define self-duality of a 2-form by considering the part 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 and give some solutions.
The Seiberg-Witten theory, introduced by Witten in , became one of the most important tools to understand the topology of smooth -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 [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 manifolds  and 4-dimensional pseudo-Riemannian manifolds with neutral signature .
Parallel spinors on pseudo-Riemannian manifolds are studied by Ikemakhen . In the present work, we consider -dimensional manifolds with structure group . In order to define spinors and Dirac operator, the manifold must have a -structure. We assume that 7-dimensional pseudo-Riemannian manifold with signature has -structure. On the other hand, to write down curvature equation, we need a self-duality notion of a -form on such manifolds. In dimensions, self-duality concept of -forms is well known. The bundle of 2-forms decomposes into two parts on this manifold . Then we will define self-duality of a -form on a -manifold with structure group by using decomposition of -forms on this manifold.
2. Manifolds with Structure Group
The exceptional Lie group , automorphism group of octonions, is well known. There is another similar Lie group which is automorphism group of split octonions . On , we consider the metric where and . From now on, we denote the pair by . The isometry group of this space is The special orthogonal subgroup of is The group is the subgroup of , preserving the following 3-form: where is the dual base of the standard basis of , with the notation and with the metric ; that is, where is called the fundamental 3-form on [10, 11]. The space of 2-forms decomposes into two parts , where
A semi-Riemannian -manifold with the metric of signature is called a manifold if its structure group reduces to the Lie group ; equivalently, there exists a nowhere vanishing 3-form on whose local expression is of the form . Such a form is called a structure on . If the structure group of is the group then the bundle of -forms decomposes into two parts similar to and we denote it by .
It is known that square of the Hodge operator on 2-forms over -dimensional Riemannian manifolds is identity and are eigenvalues of the Hodge operator. The elements of eigenspace of 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 -form has been studied on some higher dimensions [3, 13]. In this work, we need self-duality concept of -forms on -dimensional manifolds with structure group .
Now we define a duality operator over bundle of 2-form as The eigenvalues of this map are and . Note that the subbundle corresponds to the eigenvalue and the subbundle corresponds to the eigenvalue . Let be a 2-form over . If belongs to , then we call a self-dual 2-form. If belongs to , then we call an anti-self-dual 2-form. Because of decomposition of 2-forms on , any 2-form on can be written uniquely as where and . Similar to the 4-dimensional case, we say that is self-dual part of and is anti-self-dual part of .
3. Spinor Bundles over Manifolds
It is known that the group has two connected components. The connected component to the identity of is denoted by . In this work we deal with the group . The covering space of is the group which lies in Clifford algebra and we denoted the connected component of by . There is a covering map which is a 2 : 1 group homomorphism given by for , [10, 11, 14].
One can define another group which lies in the complex Clifford algebra by where the elements of are the equivalence classes of pair , under the equivalence relation . There exist two exact sequences aswhere .
Let be an orthonormal basis of ; then the Lie algebras of and are respectively. The derivative of is obtained as where is the -matrix whose -entry is , -entry is , and the other entries are zero . Since the Clifford algebra is isomorphic to the algebra , we can project this isomorphism onto the first component. Hence, we get spinor representation: By restricting to the group we get and is called spinor representation of the group ; shortly we denote it by . The elements of are called spinors and the complex vector space is called the spinor space and it is denoted by . By using spinor representation, the Clifford multiplication of vectors with spinors is defined by where and . The spinor space has a nondegenerate indefinite Hermitian inner product as where is the standard Hermitian inner product on for . The new inner product is invariant with respect to the group and satisfies the following property: where and . In this work, we use the following spinor representation : where Now, we recall the main definitions concerning -structure and the spinor bundle. Let be a -dimensional pseudo-Riemannian manifold with structure group . Then, there is an open covering of and transition functions for .
If there exists another collection of transition functions such that the following diagram commutes (i.e., and the cocycle condition on is satisfied), then is called a manifold. Then one can construct a principal -bundle on and a bundle map .
Let be a -structure on . We can construct an associated complex vector bundle: where is the spinor representation of . This complex vector bundle is called spinor bundle for a given -structure on and sections of are called spinor fields. The Clifford multiplication given by (15) can be extended to a bundle map: Parallel spinors on the spinor bundle are studied in .
Since is a pseudo-Riemannian manifold, then by using the map we can get an associated principal -bundle: Also, the map induces a bundle map:
Now, fix a connection 1-form over the principal -bundle . Let be the Levi-Civita covariant derivative associated with the metric which determines an -valued connection 1-form on the principal bundle . The connection 1-form can be written locally where is a local orthonormal frame on open set and . By using the connection -form and , one can obtain a connection 1-form on the principal bundle (the fibre product bundle): The connection can be lift to a connection 1-form on the principal bundle via the 2-fold covering map:and the following commutative diagram. One can obtain a covariant derivative operator on the spinor bundle by using the connection 1-form . The local form of the covariant derivative is where is a orthonormal frame on open set . We note that some authors use the term instead of in the local formula of . The covariant derivative is compatible with the metric and the Clifford multiplication where are spinor fields and sections of , , and are vector fields on . We can define the Dirac operator as the following composition: which can be written locally as where is any oriented local orthonormal frame of .
4. Seiberg-Witten Like Equations on Manifolds
Let be a manifold with structure group . Fix a -structure and a connection in the principal -bundle associated with the -structure. Note that the curvature of the connection is -valued 2-form. The curvature 2-form on the determines an -valued 2-form on uniquely (see ) and we denote it again by .
We can define a map where . Note that the map satisfies the following properties:
Hence, the map associates an -valued 2-form with each spinor field , so we can write In local frame on , the map can be expressed as
Now we are ready to express the Seiberg-Witten equations. Let be a manifold with structure group . Fix a structure and take a connection 1-form on the principal bundle and a spinor field . We write the Seiberg-Witten like equations as where is the self-dual part of the curvature and is the self-dual part of the -form corresponding to the spinor .
5. Seiberg-Witten Like Equations on
Let us consider these equations on the flat space with the structure given by . We use the standard orthonormal frame on and the spinor representation in (18). The connection on is given by where and are smooth maps. Then, the associated connection on the line bundle is the connection -form and its curvature -form is given by where for . Now we can write the Dirac operator on with respect to a given -structure and -connection .
We denote the dual basis of by . Now one can give a frame for the space of self-dual -forms on as
Let be the curvature form of the -valued connection 1-form and let be its self-dual part. Then, Now we calculate the -form , for a spinor . Then can be written in the following way: The projection onto the subspace is given by If is calculated explicitly, then we obtain the following identity: Hence, the curvature equation can be written explicitly as Dirac equation can be expressed as follows:
These equations admit nontrivial solutions. For example, direct calculation shows that the spinor field with and the connection -form satisfy the above equations.
Now we consider the space where is the space of connection 1-forms on the principle bundle and is the space of spinor fields. The space is called the configuration space. There is an action of the gauge group on the configuration space by where and . The action of the gauge group enjoys the following equalities: Hence, if the pair is a solution to the Seiberg-Witten equations, then the pair is also a solution to the Seiberg-Witten equations.
One can obtain infinitely many solutions for the Seiberg-Witten equations on : Consider the spinor and the connection 1-form Since the pair is a solution on , the pair is also a solution, where and is a smooth real valued function on .
The moduli space of Seiberg-Witten equations on the manifold with structure group is
Whether the moduli space has similar properties of moduli space of Seiberg-Witten equations on a -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.
This study was supported by Anadolu University Scientific Research Projects Commission under Grant no. 1501F017.
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