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Calabi-Yau manifolds from noncommutative Hermitian U(1) instantons

We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.


Introduction to Emergent Gravity
The emergent gravity can be described by considering the deformation of a symplectic manifold (M, B) where B is a nondegenerate, closed two-form on M [1,2,3,4]. It may be emphasized that a symplectic manifold (M, B) is necessarily an even-dimensional orientable manifold since ν = 1 n! B n defines a nowhere vanishing volume form where dim(M) = 2n. This fact will be important to understand the mirror symmetry of Calabi-Yau (CY) manifolds emergent from the deformation complex on a symplectic manifold, as will be discussed in a separate paper [5]. Let us consider a line bundle L over a symplectic manifold (M, B) whose connection one-form is denoted by A = A µ (x)dx µ . The curvature F of a line bundle is a closed two-form, i.e., dF = 0 and so locally given by F = dA. Suppose that the line bundle L over (M, B) admits a local gauge symmetry B L which acts on the connection A as well as the symplectic structure B on the base manifold M: with Λ an arbitrary one-form on M. The local gauge symmetry B L is known as the Λ-symmetry or B-field transformation in string theory. This symmetry then dictates that the curvature F = dA of L appears only with the combination F ≡ B + F since the two-form F is a gauge invariant quantity under the Λ-symmetry. Since dF = 0, the line bundle L over (M, B) results in a "dynamical" symplectic manifold (M, F ) if det(1 + F θ) = 0 where θ ≡ B −1 [4]. Here we mean the "dynamical" for a fluctuating field around a background. Therefore the electromagnetic force F = dA manifests itself as the deformation of a symplectic manifold (M, B). 1 Since B is a symplectic structure on M, it defines a bundle isomorphism B : T M → T * M by X → Λ = −ι X B where X ∈ Γ(T M) is an arbitrary vector field. As a result, the B-field transformation (1.1) can be written as where L X = dι X + ι X d is the Lie derivative with respect to the vector field X. Note that the ordinary U(1) gauge symmetry, A → A + dλ, is a particular case of the Λ-symmetry (1.1) for Λ = dλ = −ι X λ B. In this case, the vector field X λ = −θ(dλ) is called a Hamiltonian vector field. Since a vector field is an infinitesimal generator of local coordinate transformations, in other words, a Lie algebra generator of Diff(M), the B-field transformation (1.2) can be identified with a local coordinate transformation generated by the vector field X ∈ Γ(T M). Consequently the 1 It may be instructive to conceive an analogue in general relativity. According to the general theory of relativity, the gravitational force corresponds to the deformation of a given Euclidean space (M, g), which results in a "dynamical" Riemannian manifold (M, G) where G = g +h. The equivalence principle then implies that the "dynamical" Riemannian manifold (M, G) can always be trivialized in a locally inertial frame where the metric G recovers the original unperturbed one g. It may be worthwhile to remark that the electromagnetic force is to the deformation of a symplectic manifold what the gravitational force is to the deformation of a Riemannian manifold. The emergent gravity picture implies [4] that these two deformations are isomorphic to each other.
Λ-symmetry (1.1) can be considered on par with the (dynamical) diffeomorphism symmetry. This fact leads to a remarkable conclusion [1,2] that, in the presence of B-fields, the underlying local gauge symmetry is rather enhanced. Thus we fall into a situation similar to general relativity that the dynamical symplectic manifold (M, F ) can be locally trivialized by a coordinate transformation In other words, it is always possible to find a local coordinate transformation eliminating dynamical U(1) gauge fields as far as spacetime admits a symplectic structure. This statement is known as the Darboux theorem or the Moser lemma in symplectic geometry [6]. It is arguably a novel form of the equivalence principle for the electromagnetic force [4]. It may be rewarding to revisit the footnote 1 with this insight. Let us introduce an anchor map θ = B −1 : is called a Poisson structure on M [6]. It gives the vector space C ∞ (M) a Lie algebra structure, called a Poisson bracket, that is an antisymmetric, bilinear map {−, −} θ : In terms of local coordinates on a small patch U ⊂ M, the Poisson structure θ = B −1 is given by Without loss of generality, θ ab can be chosen to be a constant skew-symmetric matrix of rank 2n, typically taking the form [θ ab ] = 1 n ⊗ √ −1θ i σ 2 with θ i := θ 2i−1,2i , i = 1, · · · , n. According to the Darboux theorem stating that φ * (B + F ) = B, it is always possible to find a locally inertial frame, namely, Darboux coordinates, to eliminate the electromagnetic force F = dA. Let us represent the local coordinate transformation φ ∈ Diff(M) as φ : y a → x a (y) = y a + θ ab a b (y). (1.4) The dynamical local coordinates a a (y) will be called symplectic gauge fields, which are introduced to compensate local deformations of an underlying symplectic structure by U(1) gauge fields. The dynamical coordinates x a (y) are covariant under a symplectic gauge transformation, i.e., δx a (y) = −X λ x a (y) = {x a , λ} θ (y) and so play an important role in emergent gravity [7]. It is convenient to introduce "covariant momenta" defined by where f ab = ∂ a a b − ∂ b a a + {a a , a b } θ is the field strength of symplectic gauge fields. One can see that symplectic gauge fields a a (y) ∈ C ∞ (M) deform the background Poisson structure specified by In the end, the dynamical symplectic manifold (M, F ) can be described by a gauge theory of symplectic gauge fields introduced via the local coordinate transformation (1.4).
Since the symplectic manifold (M, F ) is a dynamical system, one may quantize the system like as quantum mechanics [4]. The quantization is straightforward as the dynamical system equips with an intrinsic Poisson structure given by (1.3). An underlying math is essentially the same as quantum mechanics. It results in a quantized (or NC) line bundle L over a NC space [8], denoted by R 2n θ , whose coordinate generators satisfy the commutation relation [y a , y b ] = iθ ab . (1.7) The NC ⋆-algebra generated by the Moyal-Heisenberg algebra (1.7) will be denoted by A θ [9]. The quantization Q also lifts the coordinate transformation (1.4) to a local automorphism of A θ defined by Q : φ → D A which acts on the NC coordinates y a as [10] One can see [9] that NC U(1) gauge fields are obtained by quantizing symplectic gauge fields, i.e., A a = Q(a a ). Let us define dynamical momentum variables D a (y) ≡ B ab X b (y) = p a + A a (y). Upon quantization, the Poisson bracket is similarly lifted to a NC bracket in A θ . For example, the Poisson bracket relation (1.6) is now defined by the commutation relation where the field strength of NC U(1) gauge fields A a is given by (1.10) Hence we observe that NC U(1) gauge fields describe a dynamical NC spacetime (1.9) which is a deformation of the background NC spacetime (1.7). To sum up, a dynamical NC spacetime is defined by the quantization of a line bundle L over a symplectic manifold (M, B) and described by a NC U(1) gauge theory [4]. An important point [9] is that a NC space such as the Moyal-Heisenberg algebra (1.7) always admits a nontrivial inner automorphism A defined by O → O ′ = U ⋆ O ⋆ U −1 where U ∈ A and O ∈ A θ . Its infinitesimal generators consist of an inner derivation D. Then there is a well-known Lie algebra homomorphism between the NC ⋆-algebra A θ and the inner derivation D, defined by the map [2,3,4,7] A θ → D : for any O ∈ A θ . Using the Jacobi identity of the NC ⋆-algebra A θ , it is easy to verify the Lie algebra homomorphism: In particular, we define the set of NC vector fields given by where D a = p a + A a are previously introduced dynamical NC momenta. One can apply the Lie algebra homomorphism (1.12) to the commutation relation (1.9) to yield (1.14) A basic idea of emergent gravity is to realize the gauge/gravity duality using the Lie algebra homomorphism (1.11) [2,3,4,7]. The gauge theory side of the duality is defined by a NC U(1) gauge theory based on an associative algebra A θ and its gravity side is defined by associating the derivation D of the algebra A θ with a (quantized) frame bundle of an emergent spacetime manifold M. But, in order to identify global quantities such as vielbeins in gravity with elements of D, it is necessary to glue the local data of the derivation D which are derived from NC gauge fields defined on Darboux charts. A detailed exposition for the globalization was recently given in Ref. [4]. See also [11]. For our purpose, it is enough to consider the globalization for the set of vector fields defined by Eq. (1.13). In particular, we are interested in the commutative limit, i.e. |θ| → 0, of the derivation algebra D. In this limit, the NC vector fields in Eq. (1.13) reduce to ordinary vector fields A 2n-dimensional manifold emergent from NC U(1) gauge fields will be denoted by M. The global vector fields V a = V µ a (y)∂ µ ∈ Γ(T M) are related to inverse vielbeins E a = E µ a (y)∂ µ ∈ Γ(T M) in general relativity as [2,3,4,7] V a = λE a (1. 16) where λ is to be determined by a volume-preserving condition. We fix the conformal factor λ by imposing the condition that the vector fields V a preserve a volume form This means that the vector fields V a obey the condition Note that any symplectic manifold always admits such volume-preserving vector fields. See the appendix B in Ref. [4] for the discussion of modular vector fields on a symplectic manifold. In the end the dynamical metric emergent from NC gauge fields is given by where e a = e a µ (y)dy µ = λv a ∈ Γ(T * M) are orthonormal one-forms on M and Since the gravitational metric (1.19) is completely determined by NC U(1) gauge fields, a spacetime geometry described by the metric (1.19) will be determined by the dynamical law of NC U(1) gauge fields. In particular, Einstein gravity emerges from the commutative limit of NC U(1) gauge fields [2,3,4,7]. This emergent gravity picture may be strengthened by the fact that a theory of NC U(1) gauge fields respects the diffeomorphism symmetry (1.2) and the Lie algebra homomorphism (1.11) realizes a duality between algebraic objects in A θ and geometric objects in D. For instance, it was recently shown [12] that a holomorphic line bundle with a nondegenerate curvature two-form of rank 2n is equivalent to a 2n-dimensional Kähler manifold and, in particular, CY n-folds for n = 2 and 3 are emergent from the commutative limit of NC U(1) instantons in four and six dimensions, respectively. In this paper we will further elaborate the emergent gravity from NC U(1) gauge fields and give a more elegant verification for emergent CY manifolds.
This paper is organized as follows. In Sect. 2, we derive the Hermitian Yang-Mills equations for NC U(1) gauge fields [13]. In Sect. 3, we show that the commutative limit of NC Hermitian U(1) instantons is isomorphically mapped to the Hermitian Yang-Mills equations for spin connections of an emergent CY manifold. In Sect. 4, we argue that the NC Hermitian U(1) instantons should correspond to quantized CY manifolds which are describe by a matrix model or large N gauge theory [4]. We also briefly discuss the mirror symmetry of CY manifolds from the emergent gravity picture that will be further illuminated in a separate paper [5]. In Appendix A, we present a calculational detail to verify that the self-duality equations for NC U(1) instantons in four and six dimensions are transformed to geometrical equations for spin connections of an emergent CY manifold.

NC Hermitian U (1) Instantons
Let π : L → R 6 θ be a NC line bundle over a NC space R 6 θ whose coordinates obey the commutation relation (1.7). Denote the connection of the NC line bundle L by A = A a (y)dy a and its curvature The structure group U(1) ⋆ of L acts on the connection as where g ∈ U(1) ⋆ . In order to substantiate the idea that Riemannian manifolds emerge from NC U(1) gauge fields obeying some equations, let us consider the six-dimensional NC U(1) gauge theory whose action is given by We will assume that the multiplication between NC fields is always the star product if it is not explicitly indicated for a notational simplicity. For example, f (y) g(y) := f (y) ⋆ g(y) for f , g ∈ A θ . One can show [14] that the action (2.3) can be written as the Bogomol'nyi form where the constant symplectic matrix I ab is given by The above action may be written in a more compact form as where Ω = 1 2 I ab dy a ∧ dy b is the two-form of rank 6 and will be identified with the Kähler form of R 6 ∼ = C 3 , i.e., dΩ = 0. We assume that the wedge product between forms is defined under the star product, e.g., F ∧ F = 1 4 F ab ⋆ F cd dy a ∧ dy b ∧ dy c ∧ dy d . Then, using the Bianchi identity where K is a NC Chern-Simons term defined by and Note that the second term in Eq. (2.7) can be written as and thus it vanishes under the integral thanks to the property [9] d 6 y[ f, g] ⋆ = 0 (2.11) for f , g ∈ A θ . After all, the last term in Eq. (2.6) can be written as a boundary term on ∂R 6 ∼ = S 5 : 2 (2.12) As was shown in Eq. (2.12), the last term in Eq. (2.6) is a topological term which depends only on the topological class of the vector bundle L over R 6 θ . Then we can show that the minimum of the action (2.6) is achieved at the configuration obeying the equations or, equivalently, using the fact * 2 α = α for any even form α, * F = ∓ F ∧ Ω. (2.14) Note that the condition I ab F ab = 0 needs not be imposed separately because it can be derived from Eq. (2.13) by using the identity 1 8 ε ab cdef I cd I ef = I ab . Therefore the term I ab F ab in Eq. (2.6) identically vanishes as long as Eq. (2.13) is satisfied. It may be convenient to write the six-dimensional version of the self-duality equation (2.13) in terms of component notation: where T ab cd ≡ 1 2 ε ab cdef I ef . We will call Eq. (2.15) the Hermitian Yang-Mills (HYM) equations [13].
It is obvious that a solution of the HYM equations is automatically a solution of the equations of motion, D b F ab = 0, due to the Bianchi identity, D a F bc + D b F ca + D c F ab = 0.
In six dimensions, the four-form tensor T abcd in Eq. (2.15) breaks the Lorentz symmetry SO(6) ∼ = SU(4)/Z 2 to U(3) = SU(3) × U(1). Thus it is useful to decompose the 15-dimensional vector space of two-forms Λ 2 T * M under the unbroken symmetry group U(3) into three subspaces [14]: 3 In general, the property (2.11) holds up to total derivative terms. Hence one may worry that the second term (2.10) may generate an additional boundary term on S 5 . The asymptotic boundary condition for NC U (1) gauge fields is that F | |y|→∞ → 0 and so A| |y|→∞ → i g ⋆ d g −1 where g ∈ U (1) ⋆ . Then the topological invariant (2.12) is coming from the second term in Eq. (2.8) which behaves like ∼ S 5 ( g ⋆ d g −1 ) 3 ∧ Ω. Note that the commutator term (2.10) contains more derivatives and thus more rapidly decays at the asymptotic boundary compared to the Chern-Simons term (2.8). As a result, total derivative terms in Eq. (2.10) do not contribute any nontrivial boundary term. 3 This decomposition can easily be understood by the Clifford isomorphism Cl(d) = where Λ 2 1 , Λ 2 6 , and Λ 2 8 are one-dimensional (singlet), six-dimensional and eight-dimensional vector spaces taking values in U(1) ⊂ U(3), SU(4)/U(3) = CP 3 , and SU(3) ⊂ U(3), respectively. One can show [14] that, under the choice (2.5) for the Kähler form Ω of R 6 , the NC U(1) gauge fields in Λ 2 6 , i.e. F = 1 2 F ab dy a ∧ dy b ∈ Λ 2 6 , obey the (+)-equations while, if F ∈ Λ 2 8 , they obey the (−)-equations Explicitly, the (−)-equations (2.18), for example, are given by The symplectic structure (2.5) provides a natural pairing between coordinates which picks up a particular complex structure on R 6 . The complex coordinates specified by the symplectic matrix (2.5) are given by 4 and the corresponding NC U(1) gauge fields take the combination Then the field strengths of (2, 0) and (1, 1) parts are, respectively, given by

Emergent Calabi-Yau Manifolds
Using the Lie algebra homomorphism (1.14), one can translate the HYM equations (2.15) into some geometric equations for the vector fields determined by NC U(1) gauge fields in Eq. (1.13) [2,3,4,7]. For instance, the (−)-equations (2.18) are equivalently stated as (3.1) In the commutative limit (1.15), NC vector fields V a reduce to ordinary vector fields and Eq. (3.1) in this limit is defined by the Lie bracket. Hence we introduce the Lie bracket of the vector fields The Lie algebra (3.2) means that the dual covectors v a ∈ Γ(T * M) obey the structure equations dv a = 1 2 g bc a v b ∧ v c . Using Eq. (3.2), the commutative limit of Eq. (3.1) can be written as In Appendix A, we show that, for a particular choice of volume form (1.17), Eq. (3.3) can be transformed into simple equations for spin connections given by Now we will show that a six-dimensional manifold obeying Eq. (3.4) must be a Ricci-flat, Kähler manifold which is called a CY manifold [15,16]. Therefore we come to a conclusion that CY manifolds are emergent from the commutative limit of six-dimensional NC Hermitian U(1) instantons.

(3.8)
It is well-known [15,16] that the Ricci tensor of a Kähler manifold is the field strength of the U(1) part of U(3) = SU(3) × U(1) spin connections and the U(1) gauge field is given by the trace part of U(3) spin connections, i.e., A Ricci-flat and Kähler manifold is known as a CY manifold which plays an important role in string theory compactification [15]. Consequently a CY manifold with SU(3) holonomy is characterized, up to a gauge choice, by Eqs. (3.7) and (3.10). In terms of real coordinates, they are succinctly summarized by Eq. (3.4). In Appendix A, we show that the (generalized) self-duality equation (3.4) for spin connections is equivalent to the commutative limit of NC Hermitian U(1) instantons defined by the Hermitian Yang-Mills equation (2.18) in six dimensions. Therefore we see that the commutative limit of six-dimensional NC Hermitian U(1) instantons can be rephrased into the Ricciflat and Kähler condition for Calabi-Yau manifolds.

Discussion
Suppose that F = dA = B + F is the curvature of a holomorphic line bundle, i.e., is a real smooth function on C n . Then the curvature of a holomorphic line bundle is given by Note that the Kähler form (3.6) of a 2n-dimensional Kähler manifold is given by It was shown in the Appendix of [12] that one can identify φ(z, z) and K(z, z) if the curvature F of a holomorphic line bundle is a symplectic structure, i.e., a nondegenerate, closed two-form. This means that a holomorphic line bundle with a nondegenerate curvature two-form of rank 2n is equivalent to a 2n-dimensional Kähler manifold. A CY manifold is a Kähler manifold with a vanishing first Chern class [16]. In this paper we have verified a particular case for the equivalence between a Kähler manifold and a holomorphic line bundle.
Since we have considered a line bundle over R 6 with a symplectic structure B, the emergent CY manifolds from NC U(1) gauge fields are noncompact. But the result can be generalized to compact CY manifolds by considering a line bundle L → M over a compact Kähler manifold M with a symplectic structure B if the two-form B is a Kähler form of the base manifold M, although an explicit construction may be more difficult. Actually the argument in [12] can be generalized to a holomorphic line bundle over a compact Kähler manifold M whose Kähler structure is given by a background B-field. Then the result in [12] will be equally applied to the compact case.
We observed in Sect. 1 that a line bundle over a symplectic manifold (M, B) results in a dynamical symplectic manifold (M, F ). The quantization of the dynamical symplectic manifold (M, F ) gives rise to a dynamical NC spacetime described by a NC U(1) gauge theory [4]. Therefore NC U(1) gauge fields correspond to a quantized dynamical spacetime. In this paper we showed that CY manifolds are emergent from a semi-classical limit of NC Hermitian U(1) instantons. Note that the ordinary vector field V a ∈ Γ(T M) in Eq. (1.15) is just the leading part of the NC vector field V a ∈ D when the commutative limit is taken into account and a classical Riemannian manifold M is constructed by the set of the usual vector fields V a . Thus it is natural to think of NC Hermitian U(1) instantons as a quantized geometry of CY manifolds. Since NC gauge fields can be represented by large N matrices in End(H) where H is a Hilbert space representing the NC space (1.7) [9,17], the quantized CY manifold will be described by a matrix theory or large N gauge theory. It is well-known [18,19] that such a matrix model or large N gauge theory describes a nonperturbative formulation of string/M theories. Therefore it is reasonable [4] that NC U(1) gauge fields describe a quantum geometry.
We explained in Sect. 1 that emergent gravity is defined by considering the deformation of a symplectic manifold (M, B) by a line bundle L → M. The line bundle L manifests itself only by introducing a new symplectic structure F = B + F where F = dA is identified with the curvature of the line bundle [4]. Then symplectic or NC U(1) gauge fields are introduced via a local coordinate transformation φ ∈ Diff(M) eliminating dynamical U(1) gauge fields, i.e., φ * (F ) = B. The underlying math for this argument is the well-known theorem in symplectic geometry known as the Darboux theorem [6]. Note that a CY manifold X always arises with a mirror pair Y obeying the mirror relation [15] h 1,1 (X) = h 2,1 (Y ), where h p,q is a Hodge number of a CY manifold. Since we showed that six-dimensional CY manifolds are emergent from the commutative limit of NC Hermitian U(1) instantons which are the connections in a stable holomorphic line bundle L → M [13], an interesting question arises when we conceive the emergent CY manifolds from the mirror symmetry perspective. What is the mirror symmetry from the emergent gravity picture ? Emergent gravity seems to provide a very elegant picture for the mirror symmetry [14]. and h p,q (M) ≥ 0, χ(X) = −χ(Y ) implies the mirror relation (4.3). Consequently, the emergent gravity suggests a beautiful picture that the mirror symmetry of CY manifolds is simply the Hodge theory for the deformation of symplectic and dual symplectic structures. We will discuss the mirror symmetry in emergent gravity elsewhere [5].

A Spin connections of NC U (1) instantons
In this appendix, we will show that the (generalized) self-duality equations for NC U(1) instantons in four and six dimensions can be transformed to geometrical ones for spin connections of an emergent Riemannian manifold using the Lie algebra homomorphism (1.14). In particular, it is shown that the resulting geometric equations are equivalent to the CY condition for a Ricci-flat, Kähler manifold. Let us consider the generalized self-duality equations defined by Eq. (2.15) where T ab cd = ε ab cd , d = 4; We showed in Sect. 2 that, using the Lie algebra homomorphism (1.14), the self-duality equations (2.15) can be isomorphically mapped to those of vector fields in the commutative limit, viz., Eq.
(1.15). To be specific, they are given by