Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry

We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) 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.


Introduction
String theory predicts [1,2] that six-dimensional Riemannian manifolds have to play an important role in explaining our four-dimensional Universe. They serve as an internal geometry of string theory with 6 extra dimensions and their shapes and topology determine a detailed structure of the multiplets for elementary particles and gauge fields through the compactification, which leads to a low-energy phenomenology in four dimensions. In particular, a Calabi-Yau manifold, which is a compact, Kähler manifold with vanishing Ricci curvature and so a solution of the Einstein equations without matters, has a prominent role in superstring theory and has been a central focus in both contemporary mathematics and mathematical physics. As the holonomy group of Calabi-Yau manifolds is SU(3), a compactification onto the Calabi-Yau manifold in superstring theory preserves N = 1 supersymmetry in four dimensions. One of the most interesting features in the Calabi-Yau compactification is an equivalence between type II string theories compactified on distinct Calabi-Yau manifolds. String theories compactified on these two manifolds can lead to identical effective field theories. This suggests that Calabi-Yau manifolds exist in mirror pairs M and M where the number of vector multiplets h 1,1 (M) on a Calabi-Yau manifold M is the same as the number of hypermultiplets h 2,1 ( M) on another Calabi-Yau manifold M and vice versa. This duality between two Calabi-Yau manifolds is known as the mirror symmetry [3]. While many beautiful properties of the mirror symmetry have been discovered, it is fair to say that we do not have a deep understanding of the reason for the existence of mirror symmetry.
In order to address the mirror symmetry in a different angle, we will formulate six-dimensional Euclidean gravity as a gauge theory. In general relativity, the Lorentz group appears as the structure group acting on orthonormal frames in the tangent space of a Riemannian manifold M [4]. On a Riemannian manifold M of dimension 6, the spin connection ω is an SO(6) gauge field. To be precise, under a local Lorentz transformation which is the orthogonal rotation in SO(6), a matrixvalued spin connection ω AB = ω M AB dx M plays a role of gauge fields in SO(6) gauge theory. From the SO(6) gauge theory point of view, the Riemann curvature tensors precisely correspond to the field strengths of the SO(6) gauge fields ω AB . Since the Lie algebra of SO(6) is isomorphic to that of SU(4), six-dimensional Euclidean gravity may be formulated as an SU(4) ∼ = SO(6) Yang-Mills gauge theory. In this gauge theory approach, the SO(6) holonomy group of a six-dimensional Riemannian manifold M plays a role of gauge group.
Via the gauge theory formulation of six-dimensional Euclidean gravity, we want to find gauge theory objects corresponding to Calabi-Yau manifolds and to understand their mirror symmetry in terms of Yang-Mills gauge theory. To get some insight, it will be useful to address the same problem in four-dimensional situation, which is comparatively simple. For the four-dimensional case, see, for example, [5,6] and references therein. In four dimensions, the Euclidean gravity can be formulated as an SO(4) = SU(2) L × SU(2) R gauge theory. The four-dimensional space has mystic features. The group SO(4) for d ≥ 3 is the only non-simple Euclidean Lorentz group and one can define a of SO(4), i.e. an SU(2) R spinor space, in which a mirror Calabi-Yau 2-fold or an SU(2) Yang-Mills anti-instanton lives. In this correspondence, the SU(2) gauge group of Yang-Mills instantons is identified with the holonomy group of Calabi-Yau 2-folds. With a clever guess, an extension to six dimensions is somewhat obvious. First, 2 will be replaced by 3 because, instead of a Calabi-Yau 2-fold, there is a Calabi-Yau 3-fold whose holonomy group is SU(3) and the SU(2) Yang-Mills instanton will be replaced by an SU(3) Hermitian Yang-Mills instanton. If true, it will be worthwhile to recall that the fundamental representation of SU(3) is a complex representation and so the complex conjugate 3 of a complex representation 3 is a different and inequivalent representation. Therefore, one can embed the mirror Hermitian Yang-Mills instanton into the anti-fundamental representation 3. This structure may be summarized with a schematic form: The purpose of this paper is to understand the structure in the diagram (1.3). We will formulate six-dimensional Euclidean gravity as SU(4) ∼ = SO(6) Yang-Mills gauge theory. Since the SU(3) holonomy group of Calabi-Yau manifolds appears as a gauge group in the gauge theory formulation, it is natural to consider the SU(3) gauge group in the Hermitian Yang-Mills instanton as a subgroup of the original SU(4) ∼ = SO(6) gauge group. It is noted that, as the SO(6) Lorentz algebra is isomorphic to the Lie algebra of SU(4) and SO(6) has two irreducible spinor representations, the positiveand negative-chirality spinors of SO (6) can be identified with the fundamental representation 4 and the anti-fundamental representation 4 of SU(4), respectively. Thereby, if the structure in the diagram (1.3) is true, it implies that a mirror pair of two Calabi-Yau manifolds in different Weyl spinor representations can be understood as a mirror pair of two Hermitian Yang-Mills instantons in different fundamental representations. We will show that this inference is true. This paper is organized as follows. In Section 2, we will formulate d-dimensional Euclidean gravity as SO(d) Yang-Mills gauge theory. The explicit relation between gravity and gauge theory variables will be established.
We will apply in Section 3 the gauge theory formulation of Euclidean gravity to six-dimensional Riemannian manifolds. For that purpose we will devise a six-dimensional version of the 't Hooft symbols which realizes the isomorphism between SO(6) Lorentz algebra and SU(4) Lie algebra. As the SO(6) Lorentz algebra has two irreducible spinor representations, there are accordingly two kinds of the 't Hooft symbols depending on the chirality of SO(6) Weyl representation. This leads to a topological classification of six-dimensional Riemannian manifolds according to the Euler characteristic whose sign is correlated with the six-dimensional chirality. The Kähler condition can be imposed on the 't Hooft symbols which are projected to U(3)-valued ones and results in the reduction of the gauge group from SU(4) to U(3). After imposing the Ricci-flat condition, the gauge group in the Yang-Mills gauge theory is further reduced to SU(3). Consequently, we find that six-dimensional Calabi-Yau manifolds are equivalent to Hermitian Yang-Mills instantons in SU(3) Yang-Mills gauge theory.
In Section 4, we will explore the geometrical properties of Calabi-Yau manifolds in the positiveand negative-chirality representations of SO (6). We construct cohomology classes in each chiral representation and find a mirror relation in their represent ation acting on spinor states S ± of definite chirality. We show that the Euler characteristic of Calabi-Yau manifolds in different chiral representations of SO(6) has an opposite sign consistent with the mirror symmetry.
In Section 5, we will again derive the relation between Calabi-Yau manifolds and Hermitian Yang-Mills instantons and then discuss the mirror symmetry between Calabi-Yau manifolds from a completely gauge theory setup. We find that a mirror Calabi-Yau manifold corresponds to a Hermitian Yang-Mills instanton in a different complex representation 3 or 3 of SU(3) ⊂ SU(4). As a result, the integral of the third Chern class c 3 (E) for the gauge bundle E which is equal to the Euler characteristic in the case of tangent bundle, has an inevitable sign flip between the fundamental representation 3 and the anti-fundamental representation 3.
Finally we will discuss in Section 6 the results obtained in this paper with some remarks about some generalization of mirror symmetry.
In Appendix A, we fix the basis for the chiral representation of SO(6) and the fundamental representation of SU(4) and list their structure constants. In Appendix B, we present an explicit matrix representation and their algebras of the six-dimensional 't Hooft symbols in each chiral basis.

Gravity As A Gauge Theory
On a Riemannian manifold M of dimension d, the spin connection ω is an SO(d)-valued one-form and can be identified, in general, with an SO(d) gauge field. In order to make an explicit identification between the spin connections and the corresponding gauge fields, let us first introduce the d-dimensional Clifford algebra where Γ A (A = 1, · · · , d) are Dirac matrices. Then the SO(d) Lorentz generators are given by which satisfy the following Lorentz algebra 3) The SO(d)-valued spin connection is defined by ω = 1 2 ω AB J AB where ω AB = ω M AB dx M are connection one-forms on M, which transforms in the standard way as an SO(d) gauge field under local Lorentz transformations 2 ) are connection one-forms on M and T a are Lie algebra generators of SO(d) satisfying (2.5) The identification we want to make is then given by One may adopt the identification (2.6) by applying a group homomorphism of SO(d) such that (2), and SO(6) = SU(4). 1 Then the Lorentz transformation (2.4) can be translated into a usual gauge transformation The SO(d)-valued Riemann curvature tensor is defined by or, in terms of gauge theory variables, it is given by (2.10) The above pairing leads to the relation In terms of the non-coordinate (anholonomic) basis in Γ(T M) or Γ(T * M), a Riemannian metic can be written as the field strength (2.9) of SO(d) gauge fields in the non-coordinate basis takes the form where we used the structure equation (2.14) The frame basis E A = E M A ∂ M ∈ Γ(T M) satisfies the Lie algebra under the Lie bracket are the structure functions in (2.14). One can also translate the covariant derivative defined by into the covariant derivative in gauge theory given by It is then easy to show [5] that the second Bianchi identity for curvature tensors is transformed into the Bianchi identity for SO(d) gauge fields: where the bracket [MNP ] ≡ 1 3 (MNP + NP M + P MN) denotes the cyclic permutation of indices.

Spinor Representation of Six-dimensional Riemannian Manifolds
From now on, we will apply the gauge theory formulation in the previous section to six-dimensional Riemannian manifolds. For this purpose, the SO(6) ∼ = SU(4)/Z 2 Lorentz group for Euclidean gravity will be identified with the SU(4) gauge group in Yang-Mills gauge theory. Via the gauge theory formulation of six-dimensional Euclidean gravity, we want to find gauge theory objects corresponding to Calabi-Yau manifolds and to understand their mirror symmetry in terms of Yang-Mills gauge theory. Because our formulation of six-dimensional gravity in terms of Yang-Mills gauge theory is based on the identification (2.6) (i.e., the spin connection instead of the Levi-Civita connection), it is essential to consider a spinor representation of SO(6) to realize the relationship. Let us start with the Clifford algebra Cl(6) whose generators are given by where Γ A (A = 1, · · · , 6) are six-dimensional Dirac matrices satisfying the algebra (2.1), with the complete antisymmetrization of indices, Γ 7 = iΓ 1 · · · Γ 6 is the chiral matrix given by (A.3) and Γ ABC ± = 1 2 (I 8 ± Γ 7 )Γ ABC . It will be useful to notice [2,7] that the Clifford algebra (3.1) can be identified with the exterior algebra of a cotangent bundle T * M → M where the chirality Γ 7 corresponds to the Hodge operator * : The spinor representation of SO(6) can be constructed by 3 fermion creation operators a * i (i = 1, 2, 3) and the corresponding annihilation operators a j (j = 1, 2, 3) (see Appendix 5.A in [1]). This fermionic system can be represented in a Hilbert space V of dimension 8 whose states are obtained by acting on a Fock vacuum |Ω , annihilated by all the annihilation operators, by the product of k creation operators a * i 1 · · · a * i k , i.e. 3) The spinor representation in the Hilbert space V is reducible, i.e. V = S + ⊕ S − , and there are two irreducible spinor representations S ± each of dimension 4, namely the spinors of positive and negative chirality. If the Fock vacuum |Ω has positive chirality, the positive chirality spinors of SO(6) are states given by while the negative chirality spinors of SO(6) are those obtained by As the SO(6) Lorentz algebra is isomorphic to the Lie algebra of SU(4) and SO(6) has two irreducible spinor representations, the positive-and negative-chirality spinors of SO(6) are the fundamental representations 4 and the anti-fundamental representation 4 of SU(4), respectively [1]. In other words, since SO(6) has two inequivalent spinor representations and the 4 and 4 of SU(4) are inequivalent, the Weyl spinor representations on S ± can be identified with the (anti-)fundamental SU(4) representations on C 4 . One can form a direct product of the fundamental representations 4 and 4 of SU(4) in order to classify the Clifford generators in Eq. (3.1): where Γ ± ≡ 1 2 (I 8 ± Γ 7 ), Γ A ± ≡ Γ ± Γ A and Γ AB ± ≡ Γ ± Γ AB . Note that 15 in Eqs. (3.6) and (3.7) is the adjoint representation of SU(4) and 6 in Eqs. (3.8) and (3.9) is the antisymmetric second-rank tensor of SU(4) while 10 is the symmetric second-rank tensor of SU(4). See Appendix A for the chiral representation of SO(6) and the fundamental representation of SU(4).
According to the identification (2.6), we have the following relation On the right hand side, the doubling of SU(4) algebra in four-dimensional representations R 1 and R 2 was considered because the SO(6) spinor representation on the left hand side is eight-dimensional. We will regard the Riemann tensor R AB as a linear map acting on the Hilbert space V in Eq. (3.3). As R AB contains two gamma matrices, it does not change the chirality of the vector space V . Therefore, we can represent it in a subspace of definite chirality as either R AB : S + → S + or R AB : S − → S − . The former case R AB : S + → S + will take values in 4 ⊗ 4 in (3.6) with a singlet being removed while the latter case R AB : S − → S − will take values in 4 ⊗ 4 in (3.7) with no singlet. Therefore, there exist two independent identifications defined by where we distinguish the two classes A and B depending on the six-dimensional chirality. See Appendix A for explicit chiral representations J AB ± of SO (6). Because the classes A and B are now represented by 4 × 4 matrices on both sides, we can take a trace operation for the matrices which leads to the following relations 2 (3.14) Here we have introduced the six-dimensional analogue of the 't Hooft symbols defined by As was pointed out in Eq. (3.2), the Clifford algebra (3.1) can be identified with the exterior algebra Λ * M and so the 't Hooft symbols in Eq. (3.15) have a one to one correspondence with the basis of two-forms in Λ 2 T * M for a given orientation. Consequently, we expand the six-dimensional Riemann 2 Since the distinction of the class A and the class B is meaningful only in the eight-dimensional spinor space (3.3), the curvature tensor R ABCD in the classes A and B should be understood as the ones defined by Eq. (3.10) obeying the chirality condition Γ + R AB = R AB and Γ − R AB = R AB , respectively, as is obvious from the definition (3.11) and (3.12). Actually, the Riemann curvature tensor R ABCD in a usual SO(6)-frame bundle with Levi-Civita connections can be equally expanded in either the basis A or B because, in this case, the curvature tensor is immune from the chirality. In our case, we will always assume a spinor bundle lifted from the SO(6)-frame bundle where the structure group of its fiber is Spin (6). See the footnote 1.
curvature tensors according to their six-dimensional chirality class into two different ways: Of course, the index pairs [AB] and [CD] in the curvature tensor R ABCD should have the same chirality structure because the class A lives in the vector space of positive-chirality spinors defined by (3.4) while the class B lives in the vector space of negative-chirality spinors defined by (3.5).
Note that the Riemann curvature tensor in 6 dimensions has 225 = 15 × 15 components in total which is the number of expansion coefficients in each class. Because the torsion free condition has been assumed for the curvature tensor, the first Bianchi identity R A[BCD] = 0 should be imposed which leads to 120 constraints. After all, the curvature tensor has 105 = 225 − 120 independent components which must be equal to the number of remaining expansion coefficients in the classes A and B after solving the 120 constraints It is worthwhile to notice that the curvature tensor automatically satisfies the symmetry property R ABCD = R CDAB after dictating the first Bianchi identity (3.20). Therefore, one can split the 120 constraints in Eq. 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 f ab (±±) ∈ 15 ⊗ 15. The 84 components in the symmetric part is the number of Weyl tensors in six dimensions and the 21 = 20 + 1 components are coming from Ricci tensors. The remaining 15 components will be further removed by the first Bianchi identity (3.20) after expelling the antisymmetric components in Eq. (3.21).
One can easily solve the symmetry property R ABCD = R CDAB with the coefficients obeying which results in 120 components and belongs to the symmetric part in Eq. (3.21). Now the remaining 15 conditions can be imposed by the equations It is obvious that Eq. (3.23) gives rise to nontrivial relations only for the coefficients satisfying Eq.
In the end, there are 105 remaining components for f ab (±±) which precisely match with the independent components of Riemann curvature tensor in the classes A and B.
Let us introduce the following projection operator acting on 6 × 6 antisymmetric matrices defined by where I ≡ iσ 2 ⊗ I 3 . Because any 6 × 6 antisymmetric matrix of rank 4 spans a four-dimensional subspace R 4 ⊂ R 6 , the operator (3.25) in this case can be written as and so it is a projection operator for such a rank 4 matrix, i.e., Note that some combination of antisymmetric rank 4 matrices can give rise to a 6 × 6 antisymmetric matrix of rank 6 which can be transformed into the canonical form I AB by an SO(6) rotation. In this case, the operator (3.25) is not a projection operator for the rank 6 matrix I AB but it acts as In general, one can deduce by a straightforward calculation the following properties Therefore, one can decompose the 't Hooft symbols in Eq. (3.15) into eigenspaces of the operator (3.25): whereâ,b = 1, · · · , 8 andȧ,ḃ = 1, · · · , 6 are SU(4) indices in the entries of l (+)â AB and m (+)ȧ AB , respectively. They obey the following relations which can be summarized as The exactly same properties such as Eqs. (3.33) and (3.34) hold for the above 't Hooft symbols. The geometrical meaning of the projection operators in Eq. (3.25) can be understood as follows. Consider an arbitrary two-form and introduce the 15-dimensional basis of two-forms in Λ 2 T * M for each chirality of SO(6) Lorentz algebra It is easy to derive the following useful identity from Eqs. (B.9) and (B.10) where vol(g) = √ gd 6 x. The Hodge star is an isomorphism of vector bundle * : Λ k T * M → Λ 6−k T * M which depends upon a metric g and the orientation of M. Consider a nondegenerate 2-form on M Ω = 1 2 This two-form can be wedged with the Hodge star to construct a diagonalizable operator on Λ 2 T * M as follows: The 15 × 15 matrix representing * Ω is found to have eigenvalues 2, 1 and −1 with eigenspaces of dimension 1, 6 and 8, respectively. On any six-dimensional Riemannian manifold M, the space of 2-forms Λ 2 T * M can thus be decomposed into three subspaces where Λ 2 1 and Λ 2 6 are locally spanned by and Λ 2 8 by A similar decomposition can be done with the negative chirality basis J a − . Later we will explain why there is such a decomposition of the 15-dimensional vector space Λ 2 T * M.
Note that the entries of Λ 2 1 , Λ 2 6 and Λ 2 8 coincide with those of n (±)0 AB , m (±)ȧ AB and l (±)â AB , respectively. One can quickly see that this coincidence is not an accident. Consider the action of the projection operator (3.25) on the two-form (3.38) given by or in terms of form notation Hence we see that, if F ∈ Λ 2 8 = {F | P + F = 0}, it satisfies the Ω-anti-self-duality equation whereas F ∈ Λ 2 6 = {F | P − F = 0} satisfies the Ω-self-duality equation It is not difficult to show [2] that the set {l AB } can be identified with U(3) generators which are embedded in SO (6). In general, an element of U(3) group can be represented as where λ 0 = I 3 is a unit matrix of rank 3, λâ (â = 1, · · · , 8) are the SU (3) Gell-Mann matrices and θ a 's are real parameters for U to be unitary. Now the 3 × 3 anti-Hermitian matrix Θ with matrix elements which are complex numbers Θ ij (i, j = 1, 2, 3) can easily be embedded into a 6 × 6 real matrix in SO(6) by replacing Θ ij = ReΘ ij + iImΘ ij by the 2 × 2 real matrix Θ ij = I 2 · ReΘ ij + iσ 2 · ImΘ ij . A straightforward calculation shows that the resulting 6 × 6 antisymmetric real matrix Θ AB can be written as Note that U (3) is the holonomy group of a Kähler manifold. That is, the projection operators in Eq. (3.25) can serve to project a Riemannian manifold whose holonomy group is SO(6) ∼ = SU (4) to a Kähler manifold with U(3) holonomy. Now we will show that it is indeed the case. Suppose that M is a complex manifold and let us introduce local complex coordinates z α = {x 1 + ix 2 , x 3 + ix 4 , x 5 + ix 6 }, α = 1, 2, 3 and their complex conjugateszᾱ,ᾱ = 1, 2, 3, in which an almost complex structure J takes the form J α β = iδ α β , Jᾱβ = −iδᾱβ [2]. Note that, relative to the real basis x M , M = 1, · · · , 6, the almost complex structure is given by J = I = iσ 2 ⊗ I 3 which was already introduced in Eq. (3.25). We further impose a Hermitian condition on the complex manifold M defined by g(X, Y ) = g(JX, JY ) for any X, Y ∈ T M. This means that a Riemannian metric g on a complex manifold M is a Hermitian metric, i.e., g αβ = gᾱβ = 0, g αβ = gβ α . The Hermitian condition can be solved by taking the vielbeins as where a tangent space index A = 1, · · · , 6 has been split into a holomorphic index i = 1, 2, 3 and an anti-holomorphic indexī = 1, 2, 3. This in turn means that J i j = iδ i j , Jīj = −iδ¯ij. Then one can see that the two-form Ω in Eq. (3.41) is a Kähler form, i.e., Ω(X, Y ) = g(JX, Y ) and it is given by where E i = E i α dz α is a holomorphic one-form and E¯i = E¯iᾱdzᾱ is an anti-holomorphic one-form. It is then easy to see that the condition that a Hermitian manifold (M, g) is a Kähler manifold, i.e. dΩ = 0, is equivalent to the one that the spin connection ω A B is U(3)-valued, i.e., Therefore, the spin connections after the Kähler condition (3.55) can be written as the form (3.52).
The above condition in turn means that If one introduces a gauge field defined by one can show that the field strength in Eq. (3.66) is given by where it is necessary to use the fact that the U(3) structure constants f abc satisfy the following relation The relation (3.69) is easy to understand because the U(1) part among the U(3) structure constants has to vanish. This establishes the result that the Ricci-flatness is equal to the vanishing of the U(1) field strength. That is, F (+)0 = dA (+)0 ∈ Λ 2 1 has a trivial first Chern class.
Again the above result is consistent with the branching rule (3.58). In terms of complex coordinates, the U(1) gauge field in Eq. (3.67) on a Kähler manifold is given by where the exterior derivative is defined by d = ∂ +∂ = dz α ∂ α +dzᾱ∂ᾱ. It is obvious that A (+)0 cannot be written as an exact one-form, say, A (+)0 = dλ with an arbitrary real function λ(z,z), though it is closed, i.e. dA (+)0 = 0. Therefore, one can see that the U(1) gauge field A (+)0 ∈ 1 0 is a nontrivial cohomology element.
In summary, the Kähler condition (3.55) projects the 't Hooft symbols to U(3)-valued ones in 1 0 ⊕8 0 and results in the reduction of the gauge group from SU(4) to U(3). After imposing the trivial first Chern class, F (+)0 = dA (+)0 = 0 ∈ 1 0 , the gauge group is further reduced to SU(3). Remaining spin connections in 8 0 that are SU(3) gauge fields satisfy the Hermitian Yang-Mills equation (3.64). As a Kähler manifold with the trivial first Chern class is a Calabi-Yau manifold, this means that the Calabi-Yau manifold is described by the Hermitian Yang-Mills equation (3.64) whose solution is known as Hermitian Yang-Mills instantons [2]. Consequently, we find that six-dimensional Calabi-Yau manifolds are equivalent to Hermitian Yang-Mills instantons in SU(3) Yang-Mills gauge theory. This equivalence will be more clarified using the gauge theory approach in Section 5.
The same formulae can be obtained for the type B case in Eq. (3.60) where the Kähler condition (3.55) is solved by (3.71) The Ricci-flat condition R AB = f ab (−−) η a AC η b BC = 0 leads to the equation which is equal to the vanishing of U(1) field strength, i.e. F (−)0 = dA (−)0 = 0, where the U(1) gauge field is given by This can be derived by using the fact that the U(3) structure constants f abc also satisfy the following relation where a, b now runs over 3,4,5,8,9,10,13,14,15. Note that the entries of U(3) generators for the type B case are different from those in the type A case. As can be expected, Calabi-Yau manifolds for the type B case are also described by the Hermitian Yang-Mills equations

Mirror Symmetry of Calabi-Yau Manifolds
In this section we want to explore the geometrical properties of six-dimensional Riemannian manifolds in the spinor representations A and B. For that purpose, consider two generic Riemannian manifolds whose metrics are given by where A and B will eventually refer to the chirality classes. Each of the metrics will define their own connections ω A B through the torsion-free conditions [4], The spin connections can take arbitrary values as far as they satisfy the integrability condition (3.20). Their symmetry properties can be characterized by decomposing them into irreducible subspaces under SO(6) group: where = 20 is a completely antisymmetric part of spin connections defined by ω [ABC] = 1 3 (ω ABC + ω BCA + ω CAB ). In six dimensions, the spin connections ω [ABC] may be further decomposed as (imaginary) self-dual and anti-self-dual parts, i.e., The above decomposition may be shaky because = 20 is already an irreducible representation of SO (6). It is just for a heuristic comparison with irreducible SU(4) representations. Note that 6 is coming from the antisymmetric tensor in 4 × 4 in Eq. So notice that the irreducible representation of SO(6) for spin connections is different from that of SU(4) which is identified with the irreducible spinor representation of SO (6). Introduce a three-form defined by where we used the definition in Eqs. (4.6) One can go further with the identity (4.6). Using the definition F (±)a = dA (±)a − 1 2 f bc a A (±)b ∧ A (±)c , the following relation can be derived from Eq. (4.6) where D (±) J a ± = dJ a ± − f bc a A (±)b ∧ J c ± and the definitions in Eqs. (3.16) and (3.17) are used. By writing ω AB = ω CAB E C , one can see that where f ABC are structure constants satisfying the Lie algebra (2.15). Therefore, the 3-form Ψ in Eq. (4.8) can be written as where we used the structure equation (2.14), which leads to the result Suppose that (M, g) is a Kähler manifold, i.e. dΩ = dJ 0 ± = 0. On a Kähler manifold M, the gauge fields A (±)a take values in U(3) Lie algebra, namely satisfying Eq. (3.61) or (3.71), and so the three-form Ψ can be expanded as (4.13) Consequently the spin connections on the Kähler manifold (M, g) can be decomposed as (4.14) The tensor products in Eq. (4.14) could be further decomposed according to the branching rule under SU(3) × U(1) [10] but it is not necessary for our purpose. If the three-form Ψ is defined on a Calabi-Yau manifold M, our previous result implies that Ψ 0 ≡ 4 3 P + Ψ = A (±)0 ∧ Ω is a closed 3-form, i.e. dΨ 0 = 0. 3 This means that Ψ 0 is a nontrivial element of the third cohomology group H 3 (M), because A (±)0 cannot be written as an exact oneform as was pointed out in Eq. (3.70) and Ω is the Kähler form in the second cohomology group H 2 (M). It is obvious from our construction that Ψ 0 ∈ 3 2 3 ⊕ 3 − 2 3 in Eq. (4.14) is coming from 10 = 1 2 ⊕ 3 2 3 ⊕ 6 − 2 3 and its conjugate 10 in Eq. (4.8) and it consists of (2,1)-and (1,2)-forms, i.e., Ψ 0 ∈ H 2,1 (M) ⊕ H 1,2 (M) in the Dolbeault cohomology. Now let us summarize where nontrivial classes of Dolbeault cohomology on a Calabi-Yau manifold come from. We observed that a part of them is coming from the metrics in (4.1) and the other is coming from the spin connections in (4.4). It is well-known [2] that a compact Kähler manifold has a nontrivial second cohomology group H 2 (M) which gives rise to a positive Betti number b 2 > 0. This second cohomology H 2 (M) is coming from the Kähler class Ω of a Kähler metric defined by Eq. (3.54). Another nontrivial cohomology class coming from the metric on a Calabi-Yau 3-fold is the holomorphic 3-form Φ. A Calabi-Yau 3-fold always admits a globally defined and nowhere vanishing holomorphic volume-form Φ satisfying the property [3] Ω ∧ Ω ∧ Ω = iΦ ∧ Φ. The nontrivial cohomology class on a Calabi-Yau manifold M may be represented as where E i = E i α dz α is a holomorphic one-form and E¯i = E¯iᾱdzᾱ is an anti-holomorphic one-form. According to the correspondence (3.2), we make the following identification [2] where a i and a * i are annihilation and creation operators, respectively, acting on the Hilbert space V in Eq. (3.3). Therefore, we identify the cohomology classes in Eqs. (4.16)-(4.18) with the following fermion operators with a symmetric ordering prescription where the factor i was absorbed in the definition such that Ω becomes a Hermitian operator. Given a metric ds 2 = E A ⊗ E A , one can determine the spin connection ω AB via the torsion free condition, Because we are taking an irreducible spinor representation of SO(6) for the identification (2.6), it is necessary to specify which representation is chosen to embed the spin connection ω AB . As we remarked in the footnote 2, one can equally choose either the positive chirality representation or the negative chirality representation. Or one may consider the case where a Dirac operator D M = ∂ M + ω M acts on a chiral spinor η which obeys a well-known commutation relation [D M , D N ]η = R M N P Q Γ P Q η. In this case the representation has been fixed from the outset by the chiral spinor η. At any rate, the identification of SU(4) gauge fields, according to the definition (3.59) and (3.60), depends on which representation has been chosen. When a specific chirality representation is chosen for a given metric, one can determine the coefficients f ab (++) in Eq.
where we used the fact that the creation and annihilation operators change the chirality, i.e. (a * i , a i ) : S ± → S ∓ . It might be remarked that the ± 1 2 shift in the Kähler operator ( Ω ± 1 2 ) is to correct the U(1) charge difference between Ω and Ψ ± 0 , Ψ ± 0 operators. We will see that the relation in Eqs.   Every complex vector bundle E of rank n has an underlying real vector bundle E R of rank 2n, obtained by discarding the complex structure on each fiber. Then the top Chern class of a complex vector bundle E is the Euler class of its realization [11] c n (E) = e(E R ) (4.28) where n = rank E. Therefore, the Euler characteristic χ(M) of M for a tangent bundle E R = T M is given by the integral of the top Chern class Recall that if E is a complex vector bundle, then there exists a dual or conjugate bundle E with an opposite complex structure whose j-th Chern class is given by [7,11] c j (E) = (−1) j c j (E). (4.30) The Euler characteristic χ(M) for a six-dimensional Riemannian manifold M is given by On one hand, for the type A in Eq. (3.13) where R AB = F (+)a η a AB , it is given by where Eq. (B.9) was used. On the other hand, for the type B in Eq. (3.14) where R AB = F (−)a η a AB , the Euler characteristic in Eq. (4.31) can be written as where Eq. (B.10) was used. Note that two irreducible spinor representations of SO(6) can be identified with the fundamental and anti-fundamental representations of SU(4). By choosing a complex structure, e.g., Eq. (3.53), the SO(6) tangent bundle T M reduces to a U(3) vector bundle E. In order to utilize the relation (4.29), let us consider a U(3) ⊂ SU(4) sub-bundle E such that T M ⊗ C = E ⊕ E. Note that the U(3) = SU(3) × U(1) subgroup among the Lorentz group SO(6) does not mix the creation and annihilation operators in the Hilbert space (3.3). Thus we will consider the class A in Eq. (3.11) as a U(3) vector bundle E and the class B in Eq. (3.12) as its dual (conjugate) bundle E. According to the definition (4.29), one can see that the sign difference (4.30) between a vector bundle E and its conjugate bundle E originates from different chiral representations in our case. That is, the sign difference (4.30) for the third Chern class of U(3) vector bundle E and its dual bundle E has been attributed to the opposite chirality (or parity).

Mirror Symmetry from Gauge Theory
We showed in Section 3 that Calabi-Yau manifolds can be identified with Hermitian Yang-Mills instantons from the gauge theory point of view. And the mirror symmetry says that a Calabi-Yau manifold has a mirror pair satisfying the relation (4.25). Therefore, there must be a corresponding Hermitian Yang-Mills instanton mirror to the original Hermitian Yang-Mills instanton that obeys the mirror property (4.37). So an interesting question is what is the mirror Hermitian Yang-Mills instanton from the viewpoint of gauge theory formulation. This question is the subject of this section. In Section 3, the six-dimensional Euclidean gravity has been formulated as SU(4) ∼ = SO(6) Yang-Mills gauge theory. We found that a Kähler manifold is described by U(3) gauge theory. After imposing the Ricci-flat condition on the Kähler manifold, the gauge group in the Yang-Mills theory is further reduced to SU(3) and so a Calabi-Yau manifold is described by SU(3) Yang-Mills gauge theory. In particular, we found that six-dimensional Calabi-Yau manifolds are equivalent to Hermitian Yang-Mills instantons in the SU(3) gauge theory. Now we will derive these results again and then discuss the mirror symmetry between Calabi-Yau manifolds from a completely gauge theory setup.
Suppose that the metric of a six-dimensional Riemannian manifold M is given by Let π : E → M be an SU(4) bundle over M whose curvature is defined by where A = A a M (x)T a dx M is a connection one-form of the vector bundle E. The generators T a of SU(4) Lie algebra satisfy the commutation relation (2.5) with a normalization TrT a T b = − 1 2 δ ab . Consider SU(4) Yang-Mills theory defined on the Riemannnian manifold (5.1) whose action is given by Using the projection operator (3.25) and the identity (3.29), it is easy to derive the following formula One can rewrite the action (5.3) using the above identity as The above action can be written in a more compact form as where Ω is the two-form of rank 6 defined in Eq. (3.41).
Using the fact one can see that the last term in Eq. (5.6) is a topological term, i.e., TrF ∧ F ∧ Ω = d(K ∧ Ω) (5.8) if and only if the two-form Ω is closed, i.e. dΩ = 0. In other words, when M is a Kähler manifold, the last term in Eq. (5.6) depends only on the topological class of the Kähler-form Ω and the vector bundle E on M. Although we have originally started with the positive definite action (5.3), the term in the brace [· · · ] in Eq. (5.6) after separating a topological term is not necessarily positive definite due to the second term. In order to keep the positive definiteness of the Yang-Mills action (5.6) after separating the topological term, it is necessary to impose the following requirement We will see later the geometrical meaning and the significance of the condition (5.9). After the condition (5.9), the action (5.6) is now positive definite up to a topological term and the minimum action can be achieved in a configuration satisfying the equations are automatically satisfied. We want to solve Eqs. (5.9) and (5.10) known as the Hermitian Yang-Mills equations [2]. It was observed in Section 3 that the 't Hooft symbols in Eq. (3.15) realizes the isomorphism between SO(6) Lorentz algebra and SU(4) Lie algebra and provide a complete basis of two-forms in Λ 2 T * M. Thus one may expand the SU(4) field strength F a AB (a = 1, · · · , 15) using the basis in Eq. (3.15), for example, like either Eq. (3.16) or Eq. (3.17). But we know that there are two independent bases, η a AB and η a AB . So the question is how to distinguish the two bases from the SU(4) gauge theory approach. The crux is that the N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the anti-fundamental representation. And the complex conjugate N of a complex representation N is a different, nonequivalent representation. In particular, the positive and negative chirality representations of SO(6) ∼ = SU(4) are four dimensional which coincides with the fundamental (4) and the anti-fundamental (4) representations of SU (4). 4 This is yet another reason why there are two independent bases of twoforms.
In the anti-fundamental representation 4 of SU(4), the generators may be given by (T a ) * = − i 2 λ * a and they obey the same Lie algebra as T a : But one can see from Eq. (A.9) that the symmetric structure constants d abc have an opposite sign, i.e., 14) It turns out that this sign flip is correlated with the opposite sign in Eq. (3.40). Thereby we will expand the SU(4) field strength F a AB in the fundamental representation 4 as the type A in Eq. Although it is not necessary to impose the symmetry property (3.22) for a general vector bundle π : E → M, we will take a symmetric prescription shows that the space CP 3 = SO(6)/U (3) can also be identified with the space of complex structure deformations [7]. This coincidence might presage the mirror symmetry.
Kähler classes in the representations 4 and 4 are coming from different Kähler manifolds. (So to say, it is very unnatural that the same Kähler class simultaneously belongs to two different, inequivalent representations.) Thus we consider two Calabi-Yau manifolds M and M whose background Kähler classes in each representation are given by After all, the Hermitian Yang-Mills equations, Eqs. (5.9) and (5.10), can be defined in the U(3) gauge theory. The so-called stability equation (5.9) for each representation is then reduced to the following equations By applying exactly the same argument as Eq. In conclusion, we have confirmed that the mirror pair of Calabi-Yau manifolds can be understood as the pair of Hermitian Yang-Mills instantons in the fundamental representation 3 and the antifundamental representation 3 from the gauge theory point of view.

Discussion
Mirror symmetry of Calabi-Yau manifolds is a crucial ingredient for various string dualities. Strominger, Yau and Zaslow proposed [15] that the mirror symmetry is a T-duality transformation along dual special Lagrangian tori fibrations on mirror Calabi-Yau manifolds. It is known [15] that the Tduality transformation along the dual three-tori introduces a sign flip in the Euler characteristic as even and odd forms exchange their role. Since the odd number of T-duality operations transforms type IIB string theory to type IIA string theory and vice versa, the six-dimensional chirality of two Calabi-Yau manifolds being mirror each other will be flipped after the T-duality because the ten-dimensional chirality is correlated with the six-dimensional one. Maybe our result confirms in a different context that mirror symmetry originates from the two different chiral representations of Calabi-Yau manifolds.
Via the gauge theory formulation of six-dimensional Euclidean gravity, we could show that there are two kinds of Hermitian Yang-Mills instantons in the fundamental and anti-fundamental representations of SU(4) ∼ = SO(6). Since a Calabi-Yau manifold is equivalent to a Hermitian Yang-Mills instanton from gauge theory point of view (see the quotation in the Introduction) and the chiral spinor representation of SO(6) can be identified with the (anti-)fundamental representation of SU(4), the structure in the diagram (1.3) has been naturally anticipated. In this correspondence, the SU(3) holonomy group of a Calabi-Yau manifold is realized as the SU(3) gauge group in Yang-Mills gauge theory. Therefore, the mirror symmetry of Calabi-Yau manifolds could be understood as the existence of the mirror pair of Hermitian Yang-Mills instantons embedded in the fundamental and anti-fundamental representations of SU(3) ⊂ SU(4).
Our gauge theory formulation of six-dimensional Euclidean gravity suggests that the existence of mirror pairs for Calabi-Yau manifolds may be generalized to general six-dimensional Riemannian manifolds like as the four-dimensional case [8]. One can consider two general Riemannian manifolds M and M , for example, described by the Strominger system [18] for non-Kähler complex manifolds. The existence of two chirality classes A and B in Section 3 is simply a consequence of the fact that the six-dimensional Lorentz group SO(6) has two (chiral and anti-chiral) irreducible spinor representations and the SO(6) group is isomorphic to SU(4)/Z 2 . Therefore, one may consider a pair of metrics, for instance, given by Eq. (4.1) in different chiral representations of SO (6). If the pairs are properly chosen such that their curvature tensors in Eqs. (3.18) and (3.19) satisfy the relation f ab the sign flip in the Euler characteristics (4.34) and (4.35) may be true even for general Riemannian manifolds. Furthermore, one may similarly formulate the pairing between general Riemannian manifolds in different chiral representations using a purely gauge theory approach as in Section 5. One can consider an SU(4) vector bundle E over a Riemannian manifold M and Hermitian Yang-Mills equations on (E, M) with a general Hermitian form Ω, as constructed in [13,19]. As might be indicated by Eqs. (5.26) and (5.27), it may be possible to find a pair of Hermitian Yang-Mills instantons satisfying χ(E) = −χ(E) in SU(4) Yang-Mills gauge theory. This sign flip can also be consistent with the general result (4.39). Therefore, it will be very interesting to investigate a generalization of mirror symmetry beyond Calabi-Yau manifolds. Some progress along this line will be reported elsewhere.
A SO(6) and SU (4) We will take an irreducible (Weyl) spinor representation of SO(6) whose Lorentz generators are defined by where Γ 7 = iΓ 1 · · · Γ 6 . Note that J AB + and J AB − independently satisfy the Lorentz algebra (2.3) and commute each other, i.e., [J AB + , J CD − ] = 0. They also satisfy the anti-commutation relation Because the chiral matrix Γ 7 is given by where I 4 is a 4 × 4 identity matrix, the Weyl spinor representation of the generators in Eq. (A.1) will be given by 4 × 4 matrices. We will use the following representation of Dirac matrices [16]  One can verify that the generators J AB + and J AB − separately satisfy the Lorentz algebra (2.3). One can exchange the positive-chirality, (A.6), and negative-chirality, (A.7), representations of SO(6) by a parity transformation, a reflection x M → −x M of any one element of the fundamental six-dimensional representation of SO(6) [1]; in our case, x 6 → −x 6 . But they cannot be connected by any SO(6) rotations.
The anti-Hermitian 4 × 4 matrices T a = i 2 λ a , a = 1, · · · , 15 with vanishing traces constitute the basis of SU(4) Lie algebra. The Hermitian 4 × 4 matrices λ a are given by The generators T a = i 2 λ a satisfy the following relation where the structure constants f abc are completely antisymmetric while d abc are symmetric with respect to all of their indices. Their values are shown in the tables 1 and 2. (We get these tables from [17].)

B Six-dimensional 't Hooft symbols
The matrix representation of the six-dimensional 't Hooft symbol η a AB = −Tr (T a J AB + ) is given by