aa r X i v : . [ h e p - t h ] J u l arXiv:1412.1757 Mirror Symmetry in Emergent Gravity
Hyun Seok Yang ∗ †
SchoolofPhysics,KoreaInstituteforAdvanced Study,Seoul 130-722,Korea
ABSTRACT
Given a six-dimensional symplectic manifold ( M, B ) , a nondegenerate, co-closed four-form C in-troduces a dual symplectic structure e B = ∗ C independent of B via the Hodge duality ∗ . We show thatthe doubling of symplectic structures due to the Hodge duality results in two independent classes ofnoncommutative U (1) gauge fields by considering the Seiberg-Witten map for each symplectic struc-ture. As a result, emergent gravity suggests a beautiful picture that the variety of six-dimensionalmanifolds emergent from noncommutative U (1) gauge fields is doubled. In particular, the doublingfor the variety of emergent Calabi-Yau manifolds allows us to arrange a pair of Calabi-Yau manifoldssuch that they are mirror to each other. Therefore, we argue that the mirror symmetry of Calabi-Yaumanifolds is the Hodge theory for the deformation of symplectic and dual symplectic structures.Keywords: Emergent gravity, Mirror symmetry, Calabi-Yau manifoldSeptember 11, 2018 ∗ [email protected] † Present address: Center for Quantum Spacetime, Sogang University, Seoul 04107, Korea
Introduction
Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theoremor the Moser lemma in symplectic geometry stating that the electromagnetic force can always beeliminated by a local coordinate transformation as far as spacetime admits a symplectic structure,in other words, a microscopic spacetime becomes noncommutative (NC) [1, 2, 3]. See also closelyrelated works [4, 5, 6] and reviews [7, 8]. A basic idea of emergent gravity is to isomorphically mapthe deformations of symplectic structure on a symplectic manifold ( M, B ) to those of Riemannianmetric on an emergent Riemannian manifold ( M , G ) . The deformation of symplectic structure isdescribed by considering a line bundle L → M over the symplectic manifold ( M, B ) such that thecurvature F = dA of the line bundle L defines a locally deformed symplectic two-form F = B + F on an open neighborhood U ⊂ M . The resulting symplectic structure ( M, F ) defines a dynamicalsystem and it can be quantized using the underlying Poisson structure θ ≡ B − . The quantization ofthe dynamical system results in a dynamical NC spacetime which is described by a NC U (1) gaugetheory and interpreted as the quantization of the emergent Riemannian manifold ( M , G ) [3].The dynamical variables described by the NC U (1) gauge theory form a NC algebra A θ under aquantum ⋆ -bracket [ − , − ] . Given a quantum algebra ( A θ , [ − , − ]) , there are two isomorphic represen-tations of the quantum algebra. Since the NC algebra A θ admits a separable Hilbert space H and so acountable basis, there is a Lie algebra homomorphism ρ : A θ → End( H ) which is a matrix represen-tation in the Hilbert space H . An element in End( H ) is an N × N matrix where N = dim( H ) and,in our case, N → ∞ . Another representation is given by the adjoint representation ad : A θ → D which is also a Lie algebra homomorphism. Since D is the inner derivation of the algebra A θ underthe ⋆ -bracket [ − , − ] , an element in D is given by a differential operator in the differential algebra.An interesting problem is to identify the theories described by the set of large N matrices in End( H ) and the set of differential operators in D , respectively. It turns out [2, 3] that the former is describedby a large N gauge theory in lower dimensions whose dynamical variables take values in End( H ) and the latter in a classical limit describes a higher-dimensional gravity. Since these two theories areequally derived from the NC U (1) gauge theory taking values in the NC algebra A θ , they should bephysically equivalent (or dual) to each other. The relationship between a lower-dimensional large N gauge theory and a higher-dimensional gravity is known as the gauge/gravity duality or large N duality.Emergent gravity is very similar to mirror symmetry in the sense that the deformation of symplec-tic structure is isomorphically mapped to the deformation of Riemannian metric [2]. In particular,a K¨ahler manifold carries a natural symplectic structure inherited from a K¨ahler metric. Thereforethe K¨ahler manifold is a symplectic manifold. Mirror symmetry in string theory is a correspondencebetween two topologically distinct Calabi-Yau (CY) manifolds ( X, Y ) that give rise to the exactlysame physical theory [9]. The idea is that mirror symmetry provides an isomorphism between certainaspects of complex geometry on X and certain aspects of symplectic geometry on Y . In particular, the1omological mirror symmetry [10] states that the derived category of coherent sheaves on a K¨ahlermanifold should be isomorphic to the Fukaya category of a mirror symplectic manifold. The Fukayacategory is described by the Lagrangian submanifolds of a given symplectic manifold as its objectsand the Floer homology groups as their morphisms. Since emergent gravity is aimed at construct-ing a Riemannian geometry from a symplectic geometry, it will be interesting to see how the mirrorsymmetry in string theory is realized from the emergent gravity approach.Recently we showed [11, 12] that a holomorphic line bundle with a nondegenerate curvature two-form of rank n is equivalent to a n -dimensional K¨ahler manifold. This relation was first foundand explored in the context of topological strings in [13]. In particular, CY n -folds for n = 2 and 3are emergent from the commutative limit of NC U (1) instantons in four and six dimensions, respec-tively, where quantum algebra reduces to a classical Poisson algebra. Since CY manifolds are derivedfrom the commutative limit of NC Hermitian U (1) instantons, it should be interesting to investigatehow to realize the mirror symmetry of CY manifolds from the NC U (1) gauge theory according tothe emergent gravity picture. The basic idea is to observe that there are two independent sources ofsymplectic structures in six dimensions as was briefly explained in Ref. [12]. Since a symplecticmanifold ( M, B ) is an orientable manifold, so admits a globally defined volume form, one can in-troduce the Hodge dual operator ∗ : Ω k ( M ) → Ω − k ( M ) between k -form and (6 − k ) -form vectorspaces. In particular, this implies that the vector space of two-forms Λ M = Ω ( M ) ⊕ ∗ Ω ( M ) isdoubled since one can get a two-form e B = ∗ C ∈ Ω ( M ) by taking the Hodge-dual of a four-form C ∈ Ω ( M ) . It can be shown that the new two-form e B = ∗ C is a symplectic two-form if the four-form C is co-closed and nondegenerate. The symplectic structure ( M, e B ) is completely independentof the original one ( M, B ) . Since emergent geometry is derived from the deformation of symplecticstructures and this deformation can be represented by NC U (1) gauge fields via the Darboux theo-rem, the symplectic structure ( M, e B ) will also generate (dual) NC U (1) gauge fields and give riseto a corresponding (dual) emergent geometry, as illustrated in Eqs. (3.15) and (3.16). In this paperwe will explore the physical consequence of the doubling of sympectic structures from the emergentgeometry perspective.In this paper, we explore the relation between the NC U (1) gauge theory in six dimensions and theK¨ahler gravity on a non-compact CY threefold and identify the curvature of a holomorphic line bundlewith the K¨ahler form for a CY manifold. We observe that, due to the nontrivial four-form on a CYmanifold, one can define the second dual holomorphic line bundle whose curvature is related to theHodge-dual of the four-form and argue that the CY manifold emergent from the dual holomorphic linebundle is mirror to the CY manifold emergent from the ordinary holomorphic line bundle. In fact, thisrelation has been found and studied in [13] and the similar question has been further discussed in thepapers [14, 15, 16], although the language is a bit different, involving the counting of cohomologiesvia topological strings or the duality transformations of D-branes of different codimensions. It wasalso noted in [15, 16] that the duality in [13] follows from the S-duality of type IIB superstrings.Although the duality transformation implies essentially the same result as ours for the mirror relation2n CY manifolds, it has not been surfaced yet to explicitly show how to construct the mirror pair interms of U (1) gauge theory. Our paper fills out the gap by the explicit gauge theory construction ofthe mirror pair and our result is obtained without using the machinery of topological string theory andthe conjectured web of dualities in string theory. Since the duality found in [13] can be reformulatedfrom the emergent gravity perspective, it is hoped that our gauge theory realization of the mirror pairssheds light on the duality between Gromov-Witten invariants of a CY threefold X and a topologicallytwisted NC U (1) gauge theory on X .This paper is organized as follows. In section 2, we recapitulate the basic idea on the K¨ahler man-ifolds emergent from a holomorphic line bundle with a nondegenerate curvature two-form [11, 12].The emergent CY manifolds are derived in a more elegant way by realizing the emergent gravityusing the associative algebra of NC U (1) gauge fields. In section 3, we observe that the variety ofsix-dimensional manifolds emergent from NC U (1) gauge fields is doubled thanks to the isomor-phism between two-forms and four-forms by the Hodge duality. Therefore the emergent gravity insix dimensions suggests a beautiful picture that the doubling for the variety of emergent CY mani-folds allows us to arrange a pair of CY manifolds such that they are mirror to each other. In section4, we argue that two CY manifolds arising from a line bundle and a dual line bundle are a mirror pairand thus the mirror symmetry of CY manifolds is the Hodge theory for the deformation of symplec-tic and dual symplectic structures. In section 5, we discuss some generalization of emergent K¨ahlermanifolds. Let π : L → M be a line bundle over a six-dimensional complex manifold M whose connection isdenoted by A = A a ( x ) dx a . The curvature of the line bundle L is given by F = F ab dx a ∧ dx b = d A .Given a complex structure of the base manifold M , one can canonically decompose the curvature two-form as F = F (2 , ⊕ F (1 , ⊕ F (0 , . (2.1)A holomorphic line bundle L over M is a complex vector bundle of (complex) rank one admittingholomorphic transition functions [17]. A line bundle L becomes a holomorphic line bundle if F (2 , = F (0 , = 0 . (2.2)Accordingly, the curvature of a holomorphic line bundle L consists of a (1 , -form only, i.e., F = F (1 , .For simplicity, we will assume that M = C , whose complex coordinates are given by z i = y i − + √− y i , z ¯ i = y i − − √− y i , i, ¯ i = 1 , , . (2.3)Later we will briefly discuss a generalization to a compact complex manifold, e.g., M = T ∼ = T C , athree-dimensional complex torus. According to the complex coordinates in Eq. (2.3), the connection3f L , which is called U (1) gauge fields, also takes the decomposition given by A i = 12 (cid:0) A i − − √− A i (cid:1) , A i = 12 (cid:0) A i − + √− A i (cid:1) . (2.4)Then the field strengths of (2 , and (1 , parts in Eq. (2.1) are, respectively, given by F ij = 14 (cid:0) F i − , j − − F i, j (cid:1) − √− (cid:0) F i − , j + F i, j − (cid:1) , (2.5) F ij = 14 (cid:0) F i − , j − + F i, j (cid:1) + √− (cid:0) F i − , j − F i, j − (cid:1) . (2.6)Therefore, the curvature of a holomorphic line bundle, i.e., F ij = F ij = 0 , must obey the relation F i − , j − = F i, j , F i − , j = −F i, j − , i, j = 1 , , . (2.7)Since F ij = ∂ i A j − ∂ j A i and F ¯ i ¯ j = ∂ ¯ i A j − ∂ ¯ j A i , the condition (2.2) for the holomorphic line bundlecan be solved by A i = − √− ∂ i φ ( z, z ) , A i = √− ∂ ¯ i φ ( z, z ) (2.8)where φ ( z, z ) is a real smooth function on C . Then the field strength of a holomorphic line bundleis given by F ij = √− ∂ i ∂ ¯ j φ ( z, z ) . (2.9)Suppose that M is a six-dimensional complex manifold with the metric ds = G µν ( x ) dx µ dx ν . Ona complex manifold, the metric can also be decomposed into three types: G µν = G αβ ⊕ G αβ ⊕ G αβ , (2.10)where we have split a curved space index µ = 1 , · · · , α, ¯ α ) into a holomorphic index α = 1 , , and an anti-holomorphic one ¯ α = 1 , , , similarly a tangent space index a = 1 , · · · , i, ¯ i ) into i = 1 , , and ¯ i = 1 , , . A complex manifold M is called a Hermitian manifold [18] if G αβ = G αβ = 0 . (2.11)In terms of real components, the Hermitian condition (2.11) means that G α − , β − = G α, β , G α − , β = −G α, β − , α, β = 1 , , , (2.12)which looks similar to Eq. (2.7) although one is antisymmetric and the other is symmetric. After all,the Hermitian metric consists of (1 , -type only, i.e., ds = G αβ ( z, z ) dz α dz ¯ β . (2.13)Given a Hermitian metric, one can introduce a fundamental two-form defined by Ω = √− G αβ ( z, z ) dz α ∧ dz ¯ β . (2.14)4 K¨ahler manifold is defined as a Hermitian manifold with a closed fundamental two-form, i.e., d Ω = 0 [18]. The so-called K¨ahler condition, d Ω = 0 , can be solved by the metric given by G αβ = ∂ α ∂ ¯ β K ( z, z ) , (2.15)where K ( z, z ) is a real smooth function on a complex manifold M and is called a K¨ahler potential.Now let us look at the curvature F of a holomorphic line bundle and the K¨ahler form Ω of aK¨ahler manifold that are, respectively, given by F = √− ∂ i ∂ ¯ j φ ( z, z ) dz i ∧ dz ¯ j = √− ∂∂φ ( z, z ) , (2.16) Ω = √− ∂ α ∂ ¯ β K ( z, z ) dz α ∧ dz ¯ β = √− ∂∂K ( z, z ) . (2.17)Since φ ( z, z ) and K ( z, z ) are arbitrary smooth functions on a complex manifold in addition to astriking superficial similarity of F and Ω , an innocent question naturally arises whether it is possibleto identify them or when we can identity them. If one recalls that the K¨ahler form Ω is a sym-plectic structure, then the answer may be obvious. The curvature two-form F must be a symplecticstructure to make sense the identification. Indeed, it was shown in [11, 13] that one can identify φ ( z, z ) with K ( z, z ) if the curvature F of a holomorphic line bundle is a symplectic structure, i.e.,a nondegenerate, closed two-form. A nondegenerate two-form F = F ab ( x ) dx a ∧ dx b means that det F ab ( x ) = 0 , ∀ x ∈ M . Of course, it is not a typical situation in the Maxwell’s electromagnetismwhere F ab ( x ) | | x |→∞ → . To emphasize the nondegenerateness of the field strength, let us representit by F = B + F (2.18)where B ≡ F | | x |→∞ is a nowhere vanishing two-form of rank 6 and F = dA . The identificationof φ ( z, z ) with K ( z, z ) means that a holomorphic line bundle with a nondegenerate curvature two-form of rank 6 is equivalent to a six-dimensional K¨ahler manifold. Then the real function φ ( z, z ) and so K ( z, z ) will be determined by solving the equations of motion of U (1) gauge fields. In otherwords, (generalized) Maxwell’s equations for U (1) gauge fields on a holomorphic line bundle can betranslated into Einstein’s equations for a K¨ahler manifold. For example, one may wonder what is thegauge theory object that gives rise to a CY manifold which is a K¨ahler manifold with a vanishingfirst Chern class. It was verified in [11, 12] that CY n -folds for n = 2 and are emergent from thecommutative limit of NC U (1) instantons in four and six dimensions, respectively.Let us recapitulate why it is possible to make the identification up to holomorphic gauge transfor-5ations: φ ( z, z ) = K ( z, z ) , (2.19)if the curvature F of a holomorphic line bundle is regarded as a symplectic structure on M . Animportant fact is that a symplectic structure, for instance, the B -field in Eq. (2.18), provides a bundleisomorphism B : T M → T ∗ M by X A = ι X B where X ∈ Γ( T M ) is an arbitrary vector field,since B is a nondegenerate two-form. As a result, the field strength in Eq. (2.18) can be written as F = (1 + L X ) B ≈ e L X B, (2.20)where L X = dι X + ι X d is the Lie derivative with respect to the vector field X . Since a vectorfield is an infinitesimal generator of local coordinate transformations, in other words, a Lie algebragenerator of diffeomorphisms Diff( M ) , the result (2.20) implies [1, 2] that it is possible to find alocal coordinate transformation φ ∈ Diff( M ) eliminating dynamical U (1) gauge fields in F such that φ ∗ ( F ) = B , i.e., φ ∗ = (1 + L X ) − ≈ e −L X . This statement is known as the Darboux theorem or theMoser lemma in symplectic geometry [20]. It is arguably a novel form of the equivalence principlefor the electromagnetic force. This fact leads to a remarkable conclusion [3] that, in the presenceof B -fields, the “dynamical” symplectic manifold ( M, F ) respects a (dynamical) diffeomorphismsymmetry generated by the vector field X ∈ Γ( T M ) , so the underlying local gauge symmetry israther enhanced. Here we mean the “dynamical” for fluctuating fields around a background like Eq.(2.18). Therefore, we fall into a situation similar to general relativity that the dynamical symplecticmanifold ( M, F ) can be locally trivialized by a coordinate transformation φ ∈ Diff( M ) .In terms of local coordinates, the coordinate transformation φ ∈ Diff( M ) may be represented by φ : y a x a ( y ) = y a + θ ab a b ( y ) (2.21)where θ ≡ B − and the dynamical coordinates a b ( y ) will be called symplectic gauge fields. By usingthe above coordinates, the Darboux transformation obeying φ ∗ ( F ) = B is explicitly written as (cid:0) B ab + F ab ( x ) (cid:1) ∂x a ∂y µ ∂x b ∂y ν = B µν , (2.22)where B is assumed to be constant without loss of generality. Since both sides of Eq. (2.22) areinvertible, one can deduce [21, 22, 23] that Θ ab ( x ) ≡ ( F − ) ab ( x ) = { x a ( y ) , x b ( y ) } θ = (cid:0) θ ( B − f ) θ (cid:1) ab ( y ) , (2.23) Note that both φ ( z, z ) and K ( z, z ) are locally defined. They may not fit together on the overlap U i ∩ U j to give aglobally defined function on a complex manifold M where S i ( U i , z ( i ) ) is a holomorphic atlas of M [17]. However, thecurvature F and the K¨ahler form Ω can be globally defined. For example, one can use a U (1) gauge transformation, A → A + df where f ∈ C ∞ ( M ) , to glue locally defined functions φ ( i ) on each coordinate patch U i . On the overlap U i ∩ U j of two coordinate patches, the U (1) gauge transformation reads as φ ( i ) = φ ( j ) + f ( ij ) ( z ) + f ( ij ) ( z ) where tworeal functions φ ( i ) and φ ( j ) are defined on U i and U j , respectively [19]. This gluing of U (1) gauge fields can be translatedinto that of K¨ahler potentials according to the identification (2.19). { ψ ( y ) , ϕ ( y ) } θ = θ µν ∂ψ ( y ) ∂y µ ∂ϕ ( y ) ∂y ν (2.24)for ψ, ϕ ∈ C ∞ ( M ) and the field strength of symplectic gauge fields is given by f ab ( y ) = ∂ a a b ( y ) − ∂ b a a ( y ) + { a a ( y ) , a b ( y ) } θ . (2.25)The identification (2.19) suggests a fascinating path for the quantization of K¨ahler manifolds.Note that the symplectic manifold ( M, F ) is a dynamical system since it can be understood as thedeformation of a symplectic manifold ( M, B ) by the electromagnetic force F = dA . Thus onemay quantize the dynamical system of the symplectic manifold ( M, F ) rather than trying to quantizea K¨ahler manifold directly [3]. The quantization Q is straightforward as the dynamical systemequips with the intrinsic Poisson structure (2.24) like as quantum mechanics. An underlying math isessentially the same as quantum mechanics. It results in a NC U (1) gauge theory [24] on a quantizedor NC space, denoted by R θ , whose coordinate generators satisfy the commutation relation [ y a , y b ] = iθ ab . (2.26)The NC ⋆ -algebra generated by the Moyal-Heisenberg algebra (2.26) will be denoted by A θ [25].The NC U (1) gauge theory is constructed by lifting the coordinate transformation (2.21) to a localautomorphism of A θ defined by Q : φ
7→ D A which acts on the NC coordinates y a as [22, 26] D A ( y a ) ≡ b X a ( y ) = y a + θ ab b A b ( y ) ∈ A θ . (2.27)It ascertains that NC U (1) gauge fields are obtained by quantizing symplectic gauge fields, i.e., b A a = Q ( a a ) . Upon quantization, the Poisson bracket is similarly lifted to a NC bracket in A θ . For example,the Poisson bracket relation (2.23) is now defined by the commutation relation [ b X a , b X b ] ⋆ = i (cid:0) θ ( B − b F ) θ (cid:1) ab , (2.28)where the field strength of NC U (1) gauge fields b A a is given by b F ab = ∂ a b A b − ∂ b b A a − i [ b A a , b A b ] ⋆ . (2.29)Here we observe [3] that NC U (1) gauge fields describe a dynamical NC spacetime (2.28) which is adeformation of the vacuum NC spacetime (2.26). To sum up, a dynamical NC spacetime is defined bythe quantization of a line bundle L over a symplectic manifold ( M, B ) and described by a NC U (1) gauge theory. We have come to a notice that the basic idea on the emergent K¨ahler manifold in this paper is essentially the same asthe realization of K¨ahler gravity in terms of U (1) gauge theory presented in a beautiful paper [13]. The authors in [13]conclude that for topological strings the U (1) gauge theory is the fundamental description of gravity at all scales includingthe Planck scale, where it leads to a quantum gravitational foam. p det( g + F ) = p det( G + B ) (2.30) = g s G s q det( G + b F − B ) , (2.31)where the flat metrics ( g, G ) are the K¨ahler metric of C and B its K¨ahler form. The identity (2.30)clearly verifies that U (1) gauge fields on the left-hand side must be a connection of a holomorphicline bundle obeying F ij = F ij = 0 to give rise to a K¨ahler metric G on the right-hand side and viceversa. Thus it establishes the identification (2.19). This demonstration is also true for the identity(2.31); a connection on a NC holomorphic line bundle obeying b F (2 , = b F (0 , = 0 gives rise to aK¨ahler metric G [11, 12].Although the identities in Eqs. (2.30) and (2.31) clearly illustrate how to realize the gauge/gravityduality using a NC U (1) gauge theory based on an associative algebra A θ , the ⋆ -algebra A θ providesa more elegant approach for the gauge/gravity duality. A preliminary step to derive gravitationalvariables from NC U (1) gauge fields [2, 3, 8] is to note that the NC ⋆ -algebra A θ admits a nontrivialinner automorphism A defined by O 7→ O ′ = U ⋆ O ⋆U − where U ∈ A and O ∈ A θ . Its infinitesimalgenerators consist of an inner derivation D . Then the inner derivation D manifests a well-known Liealgebra homomorphism defined by the map A θ → D : O 7→ ad O = − i [ O , · ] ⋆ (2.32)for any O ∈ A θ . Using the Jacobi identity of the NC ⋆ -algebra A θ , it is easy to verify the Lie algebrahomomorphism: [ad O , ad O ] = − i ad [ O , O ] ⋆ (2.33)for any O , O ∈ A θ . In particular, we define the set of NC vector fields given by { b V a ≡ ad b D a ∈ D | b D a ( y ) = p a + b A a ( y ) ∈ A θ , a = 1 , · · · , } (2.34)where p a = B ab y b and b D a ( y ) ≡ D A ( p a ) = B ab b X b ( y ) . One can apply the Lie algebra homomorphism(2.33) to the commutation relation − i [ b D a , b D b ] ⋆ = − B ab + b F ab (2.35) Since U (1) instantons on a commutative space M are singular and this singularity is resolved in the NC descriptionof U (1) instantons [27, 28], it is necessary to allow singular U (1) gauge fields on M to make sense the identification(2.19) in a general context. To admit such a singular gauge field, we need to relax the notion of the line bundle L . Thenatural replacement for the holomorphic line bundle L is the rank one torsion free sheaf with the same first Chern class[13]. We will assume the generalization of the line bundle by allowing singular U (1) gauge fields.
8o yield the relation [2, 8, 29] ad b F ab = [ b V a , b V b ] ∈ D . (2.36)The identification (2.19) may be confirmed by using the Lie algebra of derivations in Eqs. (2.34)and (2.36). To be precise, the derivation D of the associative algebra A θ defined by a NC U (1) gaugetheory is associated with a (quantized) frame bundle of an emergent spacetime manifold M [3]. Forexample, we recently verified a particular case of the identity (2.19) [11, 12] that the commutativelimit of six-dimensional NC Hermitian U (1) instantons obeying the Hermitian Yang-Mills equation[30, 31, 32] b F = − ∗ ( b F ∧ B ) , (2.37)where B = I µν dy µ ∧ dy ν is the K¨ahler form of C , is equivalent to CY manifolds obeying the (local)Einstein equation, det G αβ = 1 . We observed in the previous section that emergent gravity is defined by considering the deformationof a symplectic manifold ( M, B ) by a line bundle L → M . The line bundle L results in a dynamicalsymplectic manifold ( M, F ) by introducing a new symplectic structure F = B + F where F = dA is identified with the curvature of the line bundle [3]. It is important to note [20] that a symplecticmanifold ( M, B ) is necessarily an orientable manifold since the symplectic structure B admits anowhere vanishing volume form ν = B . Then a globally defined volume form introduces theHodge-dual operation ∗ : Ω k ( M ) → Ω − k ( M ) between vector spaces of k -forms and (6 − k ) -forms.This implies that the vector space Λ M of two-forms is enlarged twice: Λ M = Ω ( M ) ⊕ ∗ Ω ( M ) , (3.1)since there are additional two-forms from the Hodge-dual of four-forms in Ω ( M ) in addition to theoriginal two-forms in Ω ( M ) . Let C be a nondegenerate four-form that is co-closed, i.e., δC = 0 where δ = − ∗ d ∗ : Ω k ( M ) → Ω k − ( M ) (3.2)is the adjoint exterior differential operator [17, 18]. Define a two-form e B ≡ ∗ C . Then we have therelation δC = 0 ⇔ d e B = 0 . (3.3)Therefore e B defines another symplectic structure independent of B . Hence it should be possibleto consider the deformation of the dual symplectic structure e B by incorporating a dual line bundle e L → M . Then an interesting question is what is a physical consequence of the doubling of symplecticstructures in Eq. (3.1) due to the Hodge duality.Let e A be a U (1) connection of the dual line bundle e L and e F = d e A its curvature. According tothe vector space structure in Eq. (3.1), we identify the curvature e F = d e A with the Hodge-dual of9 four-form G , i.e., e F = ∗ G . The Bianchi identity for e L is then equal to the co-closedness of thefour-form G : d e F = 0 ⇔ δG = 0 . (3.4)Using the nilpotency δ = 0 [18], the so-called co-Bianchi identity, δG = 0 , can locally be solvedby G = δD and the connection e A of the dual line bundle e L can be identified with the Hodge-dual ofthe five-form connection D , viz., e A = − ∗ D . As the usual line bundle L over a symplectic manifold ( M, B ) , the dual line bundle e L will similarly deform the dual symplectic structure e B of the basemanifold M , leading to a new symplectic structure e F ≡ e B + e F = ∗ ( C + G ) . (3.5)Hence the dual line bundle e L also results in a dynamical symplectic manifold ( M, e F ) . Recall thatthe symplectic gauge fields in Eq. (2.21) have been introduced via a Darboux transformation φ ∈ Diff( M ) such that φ ∗ ( F ) = B . Similarly, one can consider a local coordinate transformation e φ ∈ Diff( M ) such that e φ ∗ ( e F ) = e B . Let us introduce Darboux coordinates u a ( a = 1 , · · · , so that thecoordinate transformation e φ ∈ Diff( M ) is given by e φ : u a w a ( u ) = u a + e θ ab c b ( u ) (3.6)where e θ ≡ e B − and the dynamical coordinates c b ( u ) will be called dual symplectic gauge fields.By using the Darboux coordinates, the coordinate transformation obeying e φ ∗ ( e F ) = e B is explicitlywritten as (cid:0) e B ab + e F ab ( w ) (cid:1) ∂w a ∂u µ ∂w b ∂u ν = e B µν . (3.7)It should be emphasized that the dual symplectic gauge fields in Eq. (3.6) are completely independentof the symplectic gauge fields in Eq. (2.21) to be compatible with the doubling of the vector space inEq. (3.1).The dual Poisson structure e θ = e B − defines a new Poisson bracket given by { ψ ( u ) , ϕ ( u ) } e θ = e θ µν ∂ψ ( u ) ∂u µ ∂ϕ ( u ) ∂u ν (3.8)for ψ, ϕ ∈ C ∞ ( M ) . From the Darboux transformation (3.7), one can then deduce the Poisson bracketrelation ( e F − ) ab ( w ) = { w a ( u ) , w b ( u ) } e θ = (cid:0)e θ ( e B − e f ) e θ (cid:1) ab ( u ) , (3.9)where the field strength of dual symplectic gauge fields is defined by e f ab ( u ) = ∂ a c b ( u ) − ∂ b c a ( u ) + { c a ( u ) , c b ( u ) } e θ (3.10)10ith ∂ a := ∂∂u a . The quantization Q of the dynamical symplectic manifold ( M, e F ) is defined bycanonically quantizing a Poisson algebra ( C ∞ ( M ) , {− , −} e θ ) [3]. It leads to another NC ⋆ -algebra A e θ generated by the Moyal-Heisenberg algebra satisfying the commutation relation [ u a , u b ] = i e θ ab . (3.11)The NC ⋆ -algebra A e θ is independent of the previous one A θ by our construction. For example, a localautomorphism of A e θ defined by Q : e φ
7→ D e A acts on the NC coordinates u a as [22, 26] D e A ( u a ) ≡ c W a ( u ) = u a + e θ ab b C b ( u ) ∈ A e θ , (3.12)where b C a = Q ( c a ) are another NC U (1) gauge fields obtained by quantizing dual symplectic gaugefields c a ( u ) . The covariant dynamical coordinates c W a ( u ) satisfy the commutation relation [ c W a , c W b ] e ⋆ = i (cid:0)e θ ( e B − b H ) e θ (cid:1) ab , (3.13)where the field strength of NC U (1) gauge fields b C a is given by b H ab = ∂ a b C b − ∂ b b C a − i [ b C a , b C b ] e ⋆ . (3.14)In consequence, there exist two independent NC ⋆ -algebras to define a dynamical NC spacetime.They are separately obtained by quantizing the line bundles L and e L describing the deformation ofsymplectic structures in Ω ( M ) and ∗ Ω ( M ) , respectively. Since the two vector spaces in Eq. (3.1)are isomorphic to each other so that they should be treated on an equal footing, the exactly sameargument for the previous symplectic manifold ( M, F ) can be equally applied to the dual symplecticmanifold ( M, e F ) . It is straightforward to derive from the Darboux transformation (3.7) the followingidentity between Dirac-Born-Infeld densities (up to total derivatives) [11]: q det( e g + e F ) = q det( e G + e B ) (3.15) = e g s e G s q det( e G + b H − e B ) , (3.16)where ( e g, e G ) are the K¨ahler metric of C and e B its K¨ahler form and ( e g s , e G s ) are coupling constantsin the dual gauge theories. The identity (3.15) immediately verifies that U (1) gauge fields on theleft-hand side must be a connection of a holomorphic line bundle obeying e F ij = e F ij = 0 to giverise to a K¨ahler metric e G αβ = ∂ α ∂ ¯ β e K ( z, z ) on the right-hand side where e K ( z, z ) is the K¨ahler po-tential of a K¨ahler manifold f M . Thus the field strength of a holomorphic line bundle e L is givenby e F ij = ∂ i ∂ ¯ j e φ ( z, z ) where e φ ( z, z ) is a real smooth function on C . The identity (3.15) then de-mands to identify the real function e φ ( z, z ) with the K¨ahler potential e K ( z, z ) up to holomorphic gaugetransformations (see the footnote 1), i.e., [11, 12] e φ ( z, z ) = e K ( z, z ) . (3.17)11onversely, if the metric e G µν is K¨ahler, e F must be the curvature of a holomorphic line bundle. As wepointed out in footnote 3, it is necessary to replace holomorphic line bundles with torsion free sheavesof rank one in order to include singular U (1) gauge fields such as U (1) instantons. The torsion freesheaves fail to be a line bundle in real codimension four [13], which are nothing but the ideal sheavesin our case. The identity (3.16) similarly requires that NC U (1) gauge fields should be a connectionof a NC holomorphic line bundle satisfying b H ij = b H ij = 0 . In particular, if the NC U (1) gauge fieldsin Eq. (3.16) are NC Hermitian U (1) instantons obeying the Hermitian Yang-Mills equation b H = − ∗ ( b H ∧ e B ) , (3.18)the K¨ahler metric e G αβ in Eq. (3.15) describes a CY manifold f M [12].The emergent CY manifold f M can be demonstrated on a more concrete basis. As a counterpart of b D a ( y ) = D A ( p a ) , let us introduce covariant NC momenta defined by b K a ( u ) ≡ D e A ( e p a ) = e B ab c W b ( u ) where e p a = e B ab u b . Then they satisfy the commutation relation − i [ b K a , b K b ] e ⋆ = − e B ab + b H ab . (3.19)To bear a close parallel to Eq. (2.34), let us consider the set of NC vector fields defined by { b Z a ≡ ad b K a ∈ D | b K a ( u ) = e p a + b C a ( u ) ∈ A e θ , a = 1 , · · · , } . (3.20)One can apply the Lie algebra homomorphism (2.33) to Eq. (3.19) to yield the relation [3, 8] ad b H ab = [ b Z a , b Z b ] ∈ D . (3.21)After all, the Hermitian Yang-Mills equation (3.18) can be transformed as [12, 32] [ b Z a , b Z b ] = − T abcd [ b Z c , b Z d ] , (3.22)where T abcd = ε abcdef e B ef and e B = ⊗ √− σ . Following the exactly same calculation given inRef. [12] (see Appendix B), one can show that the commutative limit of Eq. (3.22) is equivalent togeometric equations for spin connections given by e ω ab = − T abcd e ω cd . (3.23)Note that the spin connections e ω ab are determined by solving the torsion-free conditions: e T a = d e E a + e ω ab ∧ e E b = 0 (3.24)for a six-dimensional manifold f M whose metric is given by ds = e G µν ( x ) dx µ ⊗ dx ν = e E a ⊗ e E a . (3.25)It is not difficult to show [12] that the six-dimensional manifold f M must be a CY manifold if its spinconnections satisfy the relation (3.23). 12 Mirror Symmetry of Emergent Geometry
We showed that the doubling of symplectic structures due to the Hodge duality results in two indepen-dent classes of NC U (1) gauge fields by considering the Seiberg-Witten map [24] for each symplecticstructure. It may be emphasized that this result is a direct consequence of the well-known Hodgeduality stating the doubling of two-form vector spaces in Eq. (3.1). As a result, emergent gravityleads to an intriguing conclusion that the variety of six-dimensional manifolds emergent from NC U (1) gauge fields is doubled. Note that a CY manifold X always arises with a mirror pair Y obeyingthe mirror relation [9] h , ( X ) = h , ( Y ) , h , ( X ) = h , ( Y ) (4.1)where h p,q ( M ) = dim H p,q ( M ) ≥ is a Hodge number of a CY manifold M . When we conceive theemergent CY manifolds from the mirror symmetry perspective, we cannot help investigating how thedoubling for the variety of emergent geometry is related to the mirror symmetry of CY manifolds.Suppose that M is a six-dimensional orientable manifold to equip a globally defined volume form.This volume form allows us to define the Hodge dual operator ∗ : Ω k ( M ) → Ω − k ( M ) on a vectorspace Λ ∗ M = M k =0 Ω k ( M ) , (4.2)where Ω k ( M ) is the space of k -forms on T ∗ M . Consider a subspace of nondegenerate, closed two-forms and co-closed four-forms in Λ ∗ M denoted by S ( M ) and S ( M ) , respectively. Let us take adirect sum S ( M ) ≡ S ( M ) ⊕ ∗ S ( M ) . (4.3)If ω ∈ S ( M ) , then ω is a closed, dω = 0 , and nondegenerate two-form. Therefore, ω is a symplecticstructure on M . According to the Hodge decomposition theorem [18], any two-form ω ∈ Λ M in Eq.(3.1) is decomposed as ω = ω H + dα + δβ, (4.4)and thus the decomposition for a general symplectic two-form ω ∈ S ( M ) is given by ω = ω H + dα, (4.5)where ω H is a harmonic two-form and α ∈ Ω ( M ) , β ∈ Ω ( M ) . A harmonic k -form ω H ∈ Ω k ( M ) is defined by ∆ ω H = 0 where the Laplace-Beltrami operator ∆ : Ω k ( M ) → Ω k ( M ) is defined by ∆ = dδ + δd. (4.6)A k -form ω H is harmonic if and only if dω H = 0 and δω H = 0 . Then ω H is a unique harmonicrepresentative in the k -th de Rham cohomology H k ( M ) [18]. Note that the harmonic two-form ω H = ω H ⊕ ∗ ω H in Eq. (4.5) in general consists of harmonic forms in H ( M ) and H ( M ) . Similarlythe one-form α ∈ Ω ( M ) in Eq. (4.5) contains α = − ∗ γ with γ ∈ Ω ( M ) as well as α = a in13 ( M ) , that means dα = da ⊕ ∗ δγ . We remark that the Hodge decomposition on the exterior algebra(4.2) is a canonical decomposition given a globally defined volume form from which an orientedinner product is defined. Hence it is necessary to consider the direct sum (4.3) to realize the Hodgedecomposition (4.5) for a general symplectic structure since ω H + δγ ∈ S ( M ) .Since emergent gravity is based on the symplectic geometry or more generally a Poisson geometry[3], it is necessary to exhaust, at least, all possible symplectic structures to realize a complete emergentgeometry. Therefore, it is demanded to consider the direct sum (4.3) to exhaust all possible symplecticstructures. For instance, F = B + F in Eq. (2.18) and e F = e B + e F = ∗ ( C + G ) in Eq. (3.5) belong tothe vector space S ( M ) . In general, as we have shown before, the vector space (4 . can be understoodas a deformation space of primitive symplectic and dual symplectic structures ( B, e B ) which is locallydescribed by a line bundle L over ( M, B ) and a dual line bundle e L over ( M, e B ) . We verified howthe doubling of symplectic structures in Eq. (4.3) due to the Hodge duality leads to two independentclasses of NC U (1) gauge fields and results in the doubling of emergent geometry. For example, weshowed in Sect. 3 that NC Hermitian U (1) instantons arise as a solution of the Hermitian Yang-Millsequation (3.18) defined by dual NC U (1) gauge fields b C a ( u ) and give rise to CY manifolds in thecommutative limit, which are independent of CY manifolds emergent from the line bundle L over asymplectic manifold ( M, B ) . In other words, the variety of emergent CY manifolds is doubled thanksto the Hodge duality ∗ : S ( M ) → S ( M ) .Note that the Euler characteristic of a CY manifold M is given by [9] χ ( M ) = 2 (cid:0) h , ( M ) − h , ( M ) (cid:1) . (4.7)Since two classes of emergent CY manifolds are completely independent of each other, it shouldbe possible to arrange a pair of CY manifolds ( X, Y ) such that χ ( X ) = − χ ( Y ) . (A very similardoubling for the variety of CY manifolds was observed in [33].) Because of the fact h p,q ( M ) =dim H p,q ( M ) ≥ , χ ( X ) = − χ ( Y ) necessarily implies the mirror relation (4.1). Consequently,the emergent gravity suggests a beautiful picture that the mirror symmetry of CY manifolds simplyoriginates from the doubling of symplectic structures in Eq. (4.3). Furthermore, according to theHodge decomposition theorem, generic deformations of a symplectic structure can be written as theform (4.5), in which ω H = ω H ⊕ ∗ ω H is a sum of harmonic forms in H ( M ) and H ( M ) and dα = da ⊕ ∗ δγ with a ∈ Ω ( M ) and γ ∈ Ω ( M ) .In summary, the generic deformation of a symplectic two-form can be written as the form ω = ( ω H + da ) + ∗ ( ω H + δγ ) . (4.8)Note that ω belongs to the vector space in Eq. (4.3), i.e., ω ∈ S ( M ) ⊕ ∗ S ( M ) because dω = d ( ω H + da ) + ∗ δ ( ω H + δγ ) = 0 . We showed that the deformations in S ( M ) are locally describedby a line bundle L → M while those in ∗ S ( M ) are modeled by a dual line bundle e L over M .Those two deformations are independent of each other and result in two independent classes of CYmanifolds. Therefore we can derive two independent classes of CY manifolds from the deformations14n Eq. (4.8) and classify them according to their topological invariants. Since the Euler characteristic χ ( M ) of a CY 3-fold M can have an arbitrary sign unlike the four-dimensional case in which theEuler characteristic must be positive semi-definite, we may arrange a pair of CY manifolds ( X, Y ) such that χ ( X ) = − χ ( Y ) in which CY manifolds X and Y are emergent from the classes S ( M ) and S ( M ) , respectively. The formula (4.7) indicates that the only solution for χ ( X ) = − χ ( Y ) is tosatisfy the mirror relation (4.1). (Note that the Hodge diamond for a CY 3-fold is determined by onlytwo independent Hodge numbers h , and h , besides fixed ones h , = h , = h , = h , = 1 .)Therefore, the emergent gravity picture implies that the mirror symmetry of CY manifolds can beunderstood as the Hodge theory for the deformations of symplectic and dual symplectic structurescharacterized by Eq. (4.8). The identification (2.19) implies a general result [11, 12] that a holomorphic line bundle with a nonde-generate curvature two-form is equivalent to a six-dimensional K¨ahler manifold. A generalization totorsion free sheaves or ideal sheaves must be implemented to incorporate singular U (1) gauge fieldssuch as U (1) instantons. Since the real function φ ( z, z ) will be determined by solving the equationsof motion of U (1) gauge fields, it means that (generalized) Maxwell’s equations for U (1) gauge fieldson a holomorphic line bundle can be translated into Einstein’s equations for a K¨ahler manifold. Aparticular case was verified in [11, 12] that the Einstein equations for CY n -folds for n = 2 and areequivalent to the equations of motion for the commutative limit of NC U (1) instantons in four and sixdimensions, respectively. Recall that the metric for a K¨ahler manifold is basically determined by asingle function, the so-called K¨ahler potential, although the gluings described in the footnote 1 mustbe implemented to have a globally defined metric. As a result, the Ricci tensor of a K¨ahler manifoldis extremely simple and it is given by [18] R αβ = − ∂ ln det G γδ ∂z α ∂z ¯ β . (5.1)Using the identity (2.19), it must be possible to relate the Ricci tensor (5.1) to some equations of U (1) gauge fields on a holomorphic line bundle. It will be interesting to find an explicit form of theequations for holomorphic U (1) gauge fields.So far we have assumed that a complex manifold M is noncompact, e.g., C . It is desirable togeneralize the results in this paper to compact complex manifolds, e.g., T , T × K , and CP , whichare all compact symplectic (i.e., K¨ahler) manifolds. We can put a holomorphic line bundle on sucha compact K¨ahler manifold. Then, similarly to the noncompact case, the line bundle will deform anunderlying symplectic (i.e., K¨ahler) structure of the base manifold and end in a dynamical symplecticmanifold. The resulting symplectic structure can be identified with the K¨ahler form of a K¨ahler man-ifold emergent from the holomorphic line bundle over a compact complex manifold. Therefore, we15till have the local identification (2.19) even for a compact manifold. However, an explicit construc-tion of Poisson algebras and covariant connections on a compact K¨ahler manifold will be much moredifficult than a noncompact case. In particular, the gluing of coordinate patches for a holomorphicatlas of a compact manifold, described in the footnote 1, will be more nontrivial compared to, e.g., C . The quantization of a compact K¨ahler manifold will also be a more challenging issue. Thusa sophisticated mathematical tool for emergent geometry would be requested for the compact case.Nevertheless, the conclusion for the noncompact case will be true even for compact K¨ahler manifoldsbecause main features such as Eqs. (4.3) and (4.5) invariably hold for any symplectic manifold.In four dimensions, it has been possible to accomplish an explicit test of emergent gravity withknown solutions in gravity and gauge theory [34, 35]. In higher dimensions, it becomes more difficultto obtain an explicit solution in gravity as well as gauge theory. Fortunately, some solutions in sixdimensions are explicitly known for Ricci-flat K¨ahler manifolds [36] and NC Hermitian U (1) instan-tons [31, 37]. Therefore, it will be interesting to examine an explicit test of six-dimensional emergentgravity for the known solutions in both gravity and gauge theory. Acknowledgments
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