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Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry
We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.
String theory predicts  that six-dimensional Riemannian manifolds have to play an important role in explaining our four-dimensional world. They serve as an internal geometry of string theory with six extra dimensions and their shapes and topology determine a detailed structure of the multiplets for elementary particles and gauge fields through the Kaluza-Klein compactification. This program, initiated by a classic paper , tries to make contact with a low-energy phenomenology in our four-dimensional world. In particular, a Calabi-Yau (CY) manifold is a (compact) Kähler manifold with vanishing Ricci curvature and so a vacuum solution of the Einstein equations. They have a prominent role in superstring theory and have been a central focus in both contemporary mathematics and mathematical physics. As the holonomy group of CY manifolds is , the compactification onto a CY manifold in heterotic superstring theory preserves supersymmetry in four dimensions. One of the most interesting features in the CY compactification is that type II superstring theories compactified on two distinct CY manifolds lead to an identical effective field theory in four dimensions [1, 3]. This suggests that CY manifolds exist with a mirror pair where the number of vector multiplets on is the same as the number of hypermultiplets on and vice versa. Here is the Hodge number of a CY manifold . This duality between two CY manifolds is known as the mirror symmetry . While many beautiful properties of the mirror symmetry have been discovered and it has been even proven for some cases, it is fair to say that we are still far away from a deep understanding for the origin of mirror symmetry.
Mirror symmetry is a correspondence between two topologically distinct CY manifolds that give rise to exactly the same physical theory. To recapitulate the mirror symmetry, let be a compact CY manifold. The only nontrivial cohomology of the CY manifold is contained in and besides the one-dimensional cohomologies . These cohomology classes parameterize CY moduli. It is known  that every , on one hand, is represented by a real closed -form which forms a Kähler class represented by the Kähler form of a CY manifold . The elements in infinitesimally change the Kähler structure of the CY manifold and are therefore called Kähler moduli. (In string theory, these moduli are usually complexified by including -field.) On the other hand, parameterizes the complex structure moduli of a CY manifold . It is thanks to the fact that the cohomology class of -forms is isomorphic to the cohomology class , the first Dolbeault cohomology group of with values in a holomorphic tangent bundle that characterizes infinitesimal complex structure deformations. Hence the mirror symmetry of CY manifolds is the duality between two different CY 3-fold and such that the Hodge numbers of and satisfy the relations  or in a more general form where the Hodge number of a CY manifold satisfies the relations and . As we have mentioned above, the only nontrivial deformations of a CY manifold are generated by the cohomology classes in and where is the number of possible (in general, complexified) Kähler forms and is the dimension of the complex structure moduli space of . Mirror symmetry suggests that for each CY 3-fold there exists another CY 3-fold whose Hodge numbers obey the relation (1).
From a physical point of view, two CY manifolds are related by mirror symmetry if the corresponding superconformal field theories are mirror . Two superconformal field theories are said to be mirror if they are equivalent as quantum field theories. The mirror symmetry was interpreted mathematically by Kontsevich in his 1994 ICM talk as an equivalence of derived categories, dubbed the homological mirror symmetry . The homological mirror symmetry states that the derived category of coherent sheaves on a Kähler manifold should be isomorphic to the Fukaya category of a mirror symplectic manifold . The Fukaya category is described by the Lagrangian submanifold of a given symplectic manifold as its objects and the Floer homology groups as their morphisms. Hence the homological mirror symmetry formulates the mirror symmetry as an equivalence between certain aspects of complex geometry of a CY manifold and certain aspects of symplectic geometry of a mirror CY manifold in all dimensions. The geometric approach to mirror symmetry was also unveiled in ; that mirror symmetry is a geometric version of the Fourier-Mukai transformation along a dual special Lagrangian tori fibration on a mirror CY manifold which interchanges the symplectic geometry and the complex geometry of a mirror pair.
In this paper we will explore the gauge theory formulation of six-dimensional Riemannian manifolds to address the issue why CY manifolds exist with a mirror pair. In order to simplify an underlying argumentation, we will focus on orientable six-dimensional manifolds with spin structure. In general relativity, the Lorentz group appears as the structure group acting on orthonormal frames of the tangent bundle of a Riemannian manifold . On the frame bundle, a Riemannian metric on spacetime manifold is replaced by a local orthonormal basis of the tangent bundle . Then Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. On a six-dimensional Riemannian manifold , for example, local Lorentz transformations are orthogonal rotations in , and spin connections are the -valued gauge fields from the gauge theory point of view (we will use large letters to indicate a Lie group and small letters for its Lie algebra ). Then the Riemann curvature tensor precisely corresponds to the field strength of gauge fields in gauge theory. Since the Lie group is isomorphic to , the six-dimensional Euclidean gravity can be formulated as an Yang-Mills gauge theory. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we want to identify gauge theory objects corresponding to CY manifolds and address their mirror symmetry from the perspective of Yang-Mills gauge theory. To understand why there exists a mirror pair of CY manifolds, in particular, we will employ the following well-known propositions for a -dimensional Riemannian manifold :(A)The Riemann curvature tensors are -valued two forms in .(B)There exists a global isomorphism between -dimensional Lorentz groups and classical Lie groups: (C)There is an isomorphism between the Clifford algebra in -dimensions and the exterior algebra of cotangent bundle over [9, 10] (the space of the Clifford algebra is isomorphic, as a vector space, to the vector space of the exterior algebra . This is not, however, an isomorphism of associative algebras because the product in is anticommutative while that in is not due to the central term in the Dirac algebra (5)):where .
For the isomorphism (C) between the vector spaces, the “volume operator” in the Clifford algebra corresponds to the Hodge-dual operator in the exterior algebra where are -dimensional Dirac matrices obeying the Dirac algebra It is amusing to note that the Clifford algebra from a modern viewpoint can be thought of as a quantization of the exterior algebra , in the same way that the Weyl algebra is a quantization of the symmetric algebra. In particular, the Clifford map (4) implies that the Lorentz generators in are in one-to-one correspondence with two forms in the space . And the representation space of the Clifford algebra is a spinor vector space whose elements are called fermions and essential ingredients in Standard Model. It may also be worthwhile to remark that any physical force is represented by two forms in the exterior algebra taking values in a classical Lie algebra. In addition recall that the representation of Clifford algebra in even dimensions is reducible and its irreducible representations are given by chiral fermions. Then the isomorphism (C) implies that there must be a corresponding irreducible decomposition of two forms in . This fact, in our case, has a nontrivial consequence for the Riemann curvature tensors since the Lie algebra indices and the form indices must have an identical structure in a representation space of the Lorentz symmetry according to the isomorphism (C). Our principal concern is then to pin down a geometrical consequence of the rudimentary fact (A) after implementing the isomorphisms (B) and (C) to six-dimensional CY manifolds.
Let us briefly state the result summarized in the Figure 1 in advance. Compared to the four-dimensional case [11–14], some acute changes arise. First of all, there are two sources of two forms on an orientable six-dimensional manifold . One is of course usual two forms in and the other is the Hodge duality of four forms in . Therefore, the vector space of two forms is doubled in six dimensions: The doubling of two forms is resonant with the fact that the irreducible representation of Lorentz symmetry is given by the chiral Lorentz generators . Definitely it corresponds to the mixture of two forms and four forms in according to the correspondence . Since we need to take an irreducible representation of Lorentz symmetry, this demands us to think of the irreducible components of Riemann curvature tensors as a sum of the usual curvature tensors and dual curvature tensors defined by , where is a 4-form tensor taking values in Lie algebra . Moreover, it is necessary to impose the torsion-free condition for both spin connections, and , which leads to the symmetry property of the curvature tensors: and . This is another reason why two kinds of indices must be treated symmetrically although they belong to different vector spaces. To summarize, the Hodge duality admits two independent types of curvature tensors and they have to be decomposed according to the irreducible representation of Lie algebra. In the end, the duplication of curvature tensors leads to the doubling for the variety of six-dimensional spin manifolds.
It might be stressed that the doubling of six-dimensional spin manifolds is an inevitable consequence of the elementary facts (A, B, C). It should be instructive to apply the foregoing propositions (A, B, C) to four manifolds to grasp their significance [12–14] although the four-dimensional situation is in stark contrast to the six-dimensional case. In four dimensions, the Lorentz group is isomorphic to whose Lie algebras consist of chiral Lorentz generators with for chiral and antichiral representations. The splitting of the Lie algebra, , is precisely isomorphic to the canonical decomposition of the vector space of two forms: where and . That is, the six-dimensional vector space of two forms splits canonically into the sum of three-dimensional vector spaces of self-dual and anti-self-dual two forms. One can apply the canonical splitting of the two vector spaces to Riemann curvature tensors simultaneously according to (4). It results in the well-known decomposition of the curvature tensor into irreducible components [15, 16], schematically given by where is the scalar curvature, is the traceless Ricci tensor, and are the (anti-)self-dual Weyl tensors. An important lesson from the four-dimensional example is that the irreducible (chiral) representation of Lorentz symmetry corresponds to the canonical split (7) of two forms with the projection operator . We observe that the same analysis in six dimensions brings about a more dramatic result due to the fact that . The doubling of six-dimensional spin manifolds will be important to understand why CY manifolds arise with a mirror pair.
The gauge theory formulation of six-dimensional spin manifolds also leads to a valuable perspective for the doubling. The first useful access is to identify a gauge theory object corresponding to a CY 3-fold in the same sense that a gravitational instanton (or a hyper-Kähler manifold) can be identified with an Yang-Mills instanton in four dimensions [11–13]. An obvious guess goes toward a six-dimensional generalization of the four-dimensional Yang-Mills instantons known as Hermitian Yang-Mills (HYM) instantons. Indeed this relationship has been well-known to string theorists and mathematicians under the name of the Donaldson-Uhlenbeck-Yau (DUY) theorem [17, 18]. We quote a paragraph in  (Page 221) to clearly summarize this picture.
The point of intersection between the Calabi conjecture and the DUY theorem is the tangent bundle. And here’s why: once you have proved the existence of CY manifolds, you have not only those manifolds but also their tangent bundles, because every manifold has one. Since the tangent bundle is defined by the CY manifold, it inherits its metric from the parent manifold (in this case, the CY). The metric for the tangent bundle, in other words, must satisfy the CY equations. It turns out, however, that, for the tangent bundle, the Hermitian Yang-Mills equations are the same as the CY equations, provided the background metric you have selected is the CY. Consequently, the tangent bundle, by virtue of satisfying the CY equations, automatically satisfies the Hermitian Yang-Mills equations, too.
If a CY manifold can be related to a HYM instanton, a natural question immediately arises. Since a CY manifold has a mirror manifold, there will be a mirror CY manifold obeying the mirror relation (1). This in turn implies that there must be a mirror HYM instanton which can be derived from the mirror CY manifold . Thus we want to understand the relation between the HYM instanton and its mirror instanton from the gauge theory perspective. Since the Lorentz group is isomorphic to , the chiral and antichiral representations and of are equivalent to the fundamental and antifundamental representations and of . Recall that the fundamental representation of is a complex representation and so its complex conjugate is an inequivalent representation different from . Therefore, given a CY manifold , one can embed the HYM instanton inherited from into two different representations. But this situation is equally true for the mirror CY manifold . Thus there is a similar doubling for the variety of HYM instantons as occurred to CY manifolds, as summarized in Figure 1.
It may be interesting to compare this situation with the four-dimensional case [13, 14]. In four dimensions, the positive and negative chirality spinors of are given by and spinors, and , respectively. In this case, it is necessary to have two independent factors to be compatible with the splitting (7) because the irreducible representation of is real. It is interesting to see how (A, B, C) take part in the conspiracy. First, a CY 2-fold can be mapped to a self-dual or instanton which lives in the chiral representation , while a mirror CY 2-fold is isomorphically related to an anti-self-dual or instanton in the antichiral representation . For this correspondence, the gauge group of Yang-Mills instantons is identified with the holonomy group of CY 2-fold. This picture is generalized to six dimensions in an interesting way. In six dimensions, the canonical splitting (7) is applied to the enlarged vector space (6) as where the decomposition is dictated by the chiral splitting according to the isomorphism (C). From the gauge theory perspective, the splitting (9) is also compatible with the fundamental and antifundamental representations of the gauge group because the chiral representation of is identified with the fundamental representation of . After all, we will get the picture that the HYM instanton on embedded in the fundamental representation is mirror to the HYM instanton on in the antifundamental representation . This structure is summed up in Figure 1, where refers to a CY 3-fold and its mirror . And denotes a HYM instanton on in the complex representation either or of and its mirror on in the opposite complex representation.
The purpose of this paper is to understand the structure in Figure 1. To the best of our knowledge, there is no concrete work to address the mirror symmetry based on the picture in Figure 1 although the mirror symmetry has been extensively studied so far. We will show that CY manifolds and HYM instantons exist with mirror pairs as a consequence of the doubling (6) of two forms in six dimensions. It is arguably a remarkable consequence of the mysterious Clifford isomorphism (C).
This paper is organized as follows. In Section 2, we formulate -dimensional Euclidean gravity as a Yang-Mills gauge theory. The explicit relations between gravity and gauge theory variables are established. In particular, we construct the dual curvature tensors that are necessary for an irreducible representation of Lorentz symmetry. We observe that the geometric structure described by dual spin connections and curvature tensors is exactly parallel to the usual one described by and so clarify why the variety of orientable spin manifolds is doubled.
We apply in Section 3 the gauge theory formulation to six-dimensional Riemannian manifolds. For that purpose we devise a six-dimensional version of the ’t Hooft symbols which realizes the isomorphism between Lorentz algebra and Lie algebra. As the Lorentz algebra has two irreducible spinor representations, there are accordingly two kinds of the ’t Hooft symbols depending on the chirality of irreducible representations. Our construction of six-dimensional ’t Hooft symbols is new to the best of our knowledge. Using this construction, we impose the Kähler condition on the ’t Hooft symbols. This is done by projecting the ’t Hooft symbols to -valued ones and so results in the reduction of the gauge group from to . After imposing the Ricci-flat condition, the gauge group in Yang-Mills gauge theory is further reduced to . This result is utilized to show that six-dimensional CY manifolds can be recast as HYM instantons in Yang-Mills gauge theory. We elucidate why the canonical splitting (9) of six-dimensional spin manifolds corresponds to the chiral representation of . It turns out that this splitting is equally applied to CY manifolds as well as HYM instantons.
In Section 4, we apply the results in Section 3 to CY manifolds to see how the mirror symmetry between them can be explained by the doubling of six-dimensional spin manifolds. We observe that it is always possible to find a pair of CY manifolds such that their Euler characteristics in different chiral representations obey the mirror relation (1). This implies that a pair of CY manifolds in the opposite chiral representation are mirror to each other as indicated by the arrow in Figure 1.
In Section 5, we revisit the relation between CY manifolds and HYM instantons to discuss the mirror symmetry from a completely gauge theory perspective. We show that a pair of HYM instantons embedded in different complex representations and correspond to a mirror pair of CY manifolds as summarized in Figure 1. This result is consistent with the mirror symmetry because the integral of the third Chern class for a vector bundle is equal to the Euler characteristic of tangent bundle when and the third Chern class has a desired sign flip between a complex vector bundle in the fundamental representation and its conjugate bundle in the antifundamental representation. Therefore, we confirm the picture in Figure 1 that the mirror symmetry between CY manifolds can be understood as a mirror pair of HYM instantons in holomorphic vector bundles.
Finally we recapitulate in Section 6 the results obtained in this paper and conclude the paper with a few speculative remarks.
In Appendix A, we fix the basis for the chiral representation of and the fundamental representation of and list their structure constants. In Appendix B, we present an explicit construction of the six-dimensional ’t Hooft symbols and their algebraic properties in each chiral basis.
2. Gravity as a Gauge Theory
In this section we consider the gauge theory formulation of Riemannian manifolds taking values in an irreducible spinor representation of the Lorentz group [13, 14]. This section is to establish the notation for the doubled variety of Riemannian manifolds, but more detailed exposition will be deferred to the next section. On a Riemannian manifold of dimension , the spin connection is a -valued one form and can be identified, in general, with a gauge field. In order to make an explicit identification between the spin connections and the corresponding gauge fields, let us first consider the -dimensional Dirac algebra (5) where are Dirac matrices. Then the Lorentz generators are given by which satisfy the following Lorentz algebra: The spin connection is defined by , which transforms in the standard way as a gauge field under local Lorentz transformations where .
In even dimensions, the spinor representation is reducible and its irreducible representations are given by positive and negative chiral representations. In next section we will provide an explicit chiral representation for the six-dimensional case. The Lorentz generators for the chiral representation are given by where and . Therefore, the spin connection in the chiral representation takes the form Here we used a sloppy notation for which must be understood as , where , and . For a notational simplicity we will use this notation since it will not introduce too much confusion. Note that the spin connections and are considered as independent since they resulted from the doubling of one form due to the Hodge duality, as will be shown later.
Now we introduce a gauge field defined by where are one-form connections on . We will take definition (15) by adopting the group isomorphism (3). The Lie algebra generators are matrices obeying the commutation relation where and are generators in a representation and its conjugate representation , respectively. The identification we want to make is then given by Then the Lorentz transformation (12) can be interpreted as a usual gauge transformation where . The Riemann curvature tensor is defined by  where . Or, in terms of gauge theory variables, it is given by where .
As we outlined in Section 1, in addition to the usual curvature tensor , we need to introduce the dual curvature tensor defined by where is a -form tensor taking values in Lie algebra. One may consider the dual spin connection as the Hodge dual of a -form in Lie algebra. It is useful to introduce the adjoint exterior differential operator defined by where the Hodge-dual operator obeys the well-known relation for . Using the adjoint differential operator , the -valued -form in (21) can be written as where we devised a simplifying notation for and . Using the nilpotency of the adjoint differential operator , that is , one can derive the (second) Bianchi identity: It may be compared with the ordinary Bianchi identity in general relativity written as
Let us also introduce dual vielbeins , where , in addition to the usual vielbeins which independently form a local orthonormal coframe at each spacetime point in . We combine the dual one-form with the usual coframe to define a matrix of vielbeins: where The coframe basis defines dual vectors by a natural pairing: The above pairing leads to the relation . Since we regard the spin connections and as independent, let us consider two kinds of geometrical data on a spin manifold , dubbed and classes: We emphasize that the geometric structure of a -dimensional spin manifold can be described by either type or type but they should be regarded as independent even topologically. In other words, we can separately consider a Riemannian metric for each class given by or
In order to recover general relativity from the gauge theory formulation of gravity, it is necessary to impose the torsion-free condition; that is, As a result, the spin connections are determined by vielbeins, that is, , from which one can deduce the first Bianchi identity where the curvature tensors are defined by (19). It is not difficult to see that (35) leads to the symmetry property for the Riemann curvature tensors ; . It may be convenient to introduce the torsion matrix defined by where we have defined the inverted spin connection . It is straightforward to show that and so the first Bianchi identity (35) is automatic because of the torsion-free condition, . Similarly, using definition (19), it is easy to derive the second Bianchi identity, , whose matrix form reads as where In terms of gauge theory variables, it can be stated as or .
To sum up, a -dimensional orientable Riemannian manifold admits a globally defined volume form which leads to the isomorphism between and . In particular, it doubles the two-form vector space which leads to the enlargement for the geometric structure of Riemannian manifolds. The Hodge duality is thus the origin of the doubling for the variety of Riemannian manifolds. One is described by and the other independent construction is given by . According to the isomorphism (C), they are decomposed into two irreducible representations of Lorentz symmetry. In the next section we will apply the irreducible decomposition to six-dimensional spin manifolds to see why the variety of Riemannian manifolds is doubled.
3. Spinor Representation of Six-Dimensional Riemannian Manifolds
We will apply the gauge theory formulation in the previous section to six-dimensional Riemannian manifolds. For this purpose, the Lorentz group for Euclidean gravity will be identified with the gauge group in Yang-Mills gauge theory. A motivation for the gauge theory formulation of six-dimensional Euclidean gravity is to identify a gauge theory object corresponding to a CY manifold and to understand the mirror symmetry of CY manifolds in terms of Yang-Mills gauge theory. Because our gauge theory formulation is based on identification (17), we will restrict ourselves to orientable six-dimensional manifolds with spin structure and consider a spinor representation of in order to scrutinize the relationship.
Let us start with the Clifford algebra whose generators are given by where are six-dimensional Dirac matrices satisfying the algebra (5) and assumes the complete antisymmetrization of indices. is the chiral matrix given by (A.6). According to the isomorphism (4), the Clifford algebra (40) can be isomorphically mapped to the exterior algebra of a cotangent bundle where the chirality operator corresponds to the Hodge-dual operator .
The spinor representation of can be constructed by 3 fermion creation operators and the corresponding annihilation operators (see Appendix 5.A in ). This fermionic system can be represented in a Hilbert space of dimension 8 with a Fock vacuum , annihilated by all the annihilation operators. The states in are obtained by acting the product of creation operators on the vacuum ; that is, The spinor representation of the algebra (40) is reducible and has two irreducible spinor representations. Indeed the Hilbert space splits into the spinors of positive and negative chirality; that is, , each of dimension 4. If the Fock vacuum has positive chirality, the positive chirality spinors of are states given by while the negative chirality spinors are those obtained by As Lorentz algebra is isomorphic to Lie algebra, the positive and negative chirality spinors of can be identified with the fundamental representation and the antifundamental representation of , respectively . As a result, the chiral spinor representations and of are identified with the fundamental representations and of .
One can form a direct product of the fundamental representations and in order to classify the Clifford generators in (40): where are the projection operators onto the space of definite chirality and . Note that in (45) and (46) is the adjoint representation of and and in (47) and (48) are the antisymmetric and symmetric representations of , respectively. See Appendix A for the Lie algebra generators in the chiral representation of and the fundamental representation of . It is important to notice that and are independent of each other; that is, , and this doubling of the Clifford basis is parallel to the doubling of two forms according to the Clifford isomorphism (41) as will be clarified below.
We want to find the irreducible decomposition of Riemann curvature tensors under the Lorentz symmetry as the six-dimensional version of (8). As we noticed before, there are two kinds of Lorentz generators given by the irreducible components which correspond to the chiral and antichiral representations of Lorentz algebra . Recall that (19) takes the following split of curvature tensors : where and both and obey the Lie algebra defined by (16). The doubling of Lie algebra in four-dimensional representations and on the right-hand side was considered in parallel to the spinor representation on the left-hand side. Since the Lorentz generators are in one-to-one correspondence with two forms in the vector space (9), we identify the following map: Since the role of the chiral operator is parallel with the Hodge-dual operator , the chiral Lorentz generators correspond to the canonical split of the enlarged vector space (6). Therefore, the two forms in (49) must be understood as the element of the irreducible vector space in (9); that is, As a result, on the left-hand side of (49) has twice as many components as the usual Riemann curvature tensor.
Let us summarize the gauge theory formulation in Section 2. Suppose that and are Lie algebra generators in an irreducible representation of and , respectively. First consider an gauge field in the representation obtained by taking the Hodge duality of a four-form and make the following identification: where is the dual spin connection and . Then the dual curvature tensors (21) and (24) are, respectively, written as where is a four-form field strength whose Hodge duality is the field strength in gauge theory. The nilpotency of exterior differentials, , immediately leads to the Bianchi identity Hence the geometric structure described by the dual variables will be exactly parallel to the usual one described by .
Thus it is natural to put the two geometric structures on an equal footing. Moreover, the irreducible representation of the Clifford algebra suggests that the curvature tensors in (51) are given by the combination One may note that, on an orientable (spin) manifold, the duplication of curvature tensors always happens by the Hodge duality. The combination (55) can be understood as follows. One may regard the Riemann tensor as a linear operator acting on the Hilbert space in (42). As contains two gamma matrices, it does not change the chirality of the vector space . Therefore, we can represent it in a subspace of definite chirality as either or . The former case takes values in in (45) with a singlet being removed while the latter case takes values in in (46) with no singlet. This implies two independent identifications defined by where class acts on the subspace of positive (negative) chirality. See Appendix A for the irreducible representation of and . Because classes and in (56) are now represented by matrices on both sides, we can take a trace operation for the matrices which leads to the following relations: Here we have introduced a six-dimensional analogue of the ’t Hooft symbols defined by where we used a bookkeeping notation, , , and , . They serve as a complete basis of the vector space in (45) and (46). An explicit expression of the six-dimensional ’t Hooft symbols and their algebra are presented in Appendix B.
Note that in (56) are the field strengths of gauge fields. Thus we introduce a pair of gauge fields whose field strengths are given by The gauge field is nothing but the spin connection resident in the vector space of positive (negative) chirality; that is, Using (B.7), the field strengths can be written as . One can apply again the same expansion to the index pair of the Riemann tensor . That is, one can expand the field strengths in terms of the chiral bases in (59): As was pointed out in (41), the Clifford algebra (40) is isomorphic to the exterior algebra as vector spaces, so the ’t Hooft symbol in (59) has a one-to-one correspondence with the basis of two forms in depending on the chirality for a given orientation. Consequently, the six-dimensional Riemann curvature tensors can be expanded as follows: Note that the index pairs and in the curvature tensor have the same chirality structure because of the symmetry property .
The Riemann curvature tensor in six dimensions has components in total which is the number of the expansion coefficients in each class. Because the torsion-free condition has been assumed for the curvature tensors, the first Bianchi identity should be imposed which leads to 120 constraints for each class. After all, the curvature tensor has independent components which must be equal to the number of remaining expansion coefficients in class or after solving the 120 constraints: It is worthwhile to notice that the curvature tensor automatically satisfies the symmetry property after dictating the first Bianchi identity (66). Therefore, one can split the 120 constraints in (66) into the conditions imposing the symmetry and the extra 15 conditions. These extra conditions can be manifest by considering the tensor product of : where the first part with 120 components is symmetric and the second part with 105 components is antisymmetric. It is obvious from our construction that . The 84 components in the symmetric part is the number of Weyl tensors in six dimensions and the components refer to Ricci tensors. The remaining 15 components in the symmetric part are removed by the first Bianchi identity (66) after expelling the antisymmetric components in (67).
One can easily solve the symmetry property with the coefficients satisfying which results in 120 components for each chirality belonging to the symmetric part in (67). Now the remaining 15 conditions can be reduced to the equations It is obvious that (69) gives rise to a nontrivial relation only for the coefficients satisfying (68). Finally, using (B.9) and (B.10), (69) can be reduced to the 15 constraints for each sector. In the end, have 105 independent components for each chirality which precisely match with the independent components of Riemann curvature tensors in class or .(It may be worthwhile to recall the four-dimensional situation [12, 13]. In four dimensions, the first Bianchi identity gives rise to 16 constraints. Thus Riemann curvature tensors have independent components. And the 16 constraints split into 15 ones for and one more constraint which reads as . The last constraint is responsible for the equality of the Ricci scalar in the chiral and antichiral sectors in (8). The constraints in (70) correspond to the six-dimensional analogue of the last one.)
Let us introduce the following (projection) operator acting on antisymmetric matrices defined by where . Because any antisymmetric matrix of rank 4 spans a four-dimensional subspace , the operator (71) in this case can be written in the four-dimensional subspace as so it reduces to the projection operator for such rank 4 matrices; that is, Note that is a antisymmetric matrix of rank 6. In this case, the operator (71) does not act as a projection operator but acts as In general, one can deduce by a straightforward calculation the following properties:
After a little algebra, one can classify the ’t Hooft symbols in (59) into the eigenspaces of the operator (71): where and are indices in the entries of and , respectively. They obey the following relations: Thus the (projection) operators (71) decompose the vector space into their eigenspaces as .