Complex structures, T-duality and worldsheet instantons in Born sigma models

We investigate doubled (generalized) complex structures in 2D-dimensional Born geometries where T-duality symmetry is manifestly realized. We show that Kähler, hyperkähler, bi-hermitian and bi-hypercomplex structures of spacetime are implemented in Born geometries as doubled structures. We find that the Born structures and the generalized Kähler (hyperkähler) structures appear as subalgebras of bi-quaternions and split-tetra-quaternions. We find parts of these structures are classified by Clifford algebras. We then study the T-duality nature of the worldsheet instantons in Born sigma models. We show that the instantons in Kähler geometries are related to those in bi-hermitian geometries in a non-trivial way.


Introduction
One of the important features that characterizes string theories is duality [1].T-duality, that distinguishes string theory from theories based on point particles, is the most distinctive feature to understand stringy nature of spacetime.
T-duality among spacetime geometries is studied in various contexts.For example, the famous Buscher rule of T-duality [2] is derived in the two-dimensional string sigma model as a target space transformation.In supersymmetric theories, a duality symmetry between chiral and twisted chiral multiplets of two-dimensional N = (2, 2) sigma models is interpreted as T-duality [3][4][5].In general, an N = (2, 2) theory only with chiral multiplets requires that the target space geometry is Kähler [6,7].In particular, the presence of the twisted chiral multiplets requires that the target space is the bi-hermitian geometry admitting two independent complex structures (J + , J − ) that commute with each other and are compatible with the target space metric [8,9].A pair of noncommuting complex structures in N = (2, 2) models with semi-chiral multiplets is also studied [10].It is shown in the sigma model language that Kähler and bihermitian geometries are T-dual with each other [11][12][13][14].Similarly, N = (4, 4) supersymmetry requires that the target space is generically a bi-hypercomplex geometry which admits three sets of complex structures (J a,+ , J a,− ) (with a = 1, 2, 3) satisfying J a,+ J b,− = J b,− J a,+ .
On the other hand, geometric realization of T-duality symmetry is developed in the context of generalized geometry [15,16].The generalized tangent bundle TM over a D-dimensional spacetime manifold M is defined by the formal sum of the tangent and the cotangent bundles The spacetime geometry M is encoded by 2D-dimensional generalized structures on TM in the O(D, D) covariant fashion.For example, the Kähler structure on spacetime M is realized as the generalized Kähler structure (J J , J ω ) on TM , via so-called the Gualtieri map [16].It is also shown that the bi-hermitian structure on M is realized by the generalized Kähler structure (J + , J − ).The physical origin of this correspondence is studied in supersymmetric sigma models [17][18][19][20][21][22].This also holds for the hyperkähler and the bi-hypercomplex cases.They are realized as the generalized hyperkähler structures on TM .
Generalized geometry is closely related to doubled formalism [23][24][25].The idea of T-duality symmetric geometries is further sophisticated in the study of double field theory (DFT) [26].DFT is developed on the basis of the doubled formalism in which the spacetime metric g µν , the NSNS B-field and the dilaton ϕ are organized into the 2D × 2D generalized metric H M N and the generalized dilaton d.They are defined in the 2D-dimensional doubled space M where T-duality symmetry is manifestly and geometrically realized.T-duality symmetries of geometric quantities are implemented as global O(D, D) transformations in the doubled space.For example, the Buscher rule of g µν , B µν and ϕ is reproduced by an O(D, D) rotation of the generalized metric H M N and the generalized dilaton d.The general T-duality transformation law of Kähler, hyperkähler, bi-hermitian and bi-hypercomplex structures of spacetime geometries are also discussed in the doubled formalism [27].The geometry of the doubled space M is implemented by the Born structures [28][29][30][31][32].The Born geometry is endowed with the doubled foliations, the O(D, D) structure, the natural inner product by the O(D, D) invariant metric η M N , the generalized metric H M N and a unique connection.Furthermore, it is shown that the tangent bundle of the doubled space T M is identified with the generalized tangent bundle TM through a natural isomorphism.
The purpose of this paper is to study the T-duality nature of the spacetime structures of Kähler, hyperkähler, bi-hermitian and bi-hypercomplex geometries by embedding them into extensions of the generalized (hyper)Kähler structures and the Born structures.We call these doubled structures in general.In particular, we study compatibility of the doubled structures with the Born geometry on which DFT is naturally defined.We will show that the Born geometry is compatible with the generalized (hyper)kähler structures by encoding the complex structures of spacetime into doubled structures in an appropriate way.Along the way, we will encounter interesting connections between the doubled structures and certain algebras of hypercomplex numbers.We will analyze the algebras that the doubled structures obey.With these results, in the latter half of this paper, we study the T-duality covariant expression of the worldsheet instantons.The existence of the generalized complex structures in the doubled space leads us to the notion of doubled worldsheet instantons.We study the doubled worldsheet instantons in the Born sigma model which is a sigma model whose target space is the 2Ddimensional Born geometry [33].This provides us a T-duality covariant way of string worldsheet theory.
The organization of this paper is as follows.In the next section, we introduce the Born geometry on which DFT is defined.The relation between generalized geometry and doubled geometry is discussed.In section 3, we study the compatibility of the Born and the generalized Kähler structures.We study algebras of hypercomplex numbers that these structures obey.We show that Kähler, hyperkähler, bi-hermitian and bi-hypercomplex structures of spacetime are embedded into doubled structures together with the Born geometry.In section 4, we study the T-duality covariant expression of the worldsheet instanton equations in the Born sigma model.We discuss T-duality property of the worldsheet instantons.Section 5 is devoted to conclusion and discussions.Appendix A and B are glossaries of mathematics on hypercomplex numbers and Clifford algebras.

Double field theory and Born geometry
In this section, we clarify the relations among double field theory, the Born geometry and generalized geometry.

Double field theory
We start by introducing double field theory (DFT) [26].DFT is a formulation of supergravities for which T-duality is manifestly realized.The fundamental fields of DFT are the generalised metric H M N and the generalized dilaton d.They are defined in the 2D-dimensional doubled space M. The doubled coordinate X M , (M = 1, . . ., 2D) on M is decomposed as where X µ and Xµ are the Kaluza-Klein (KK) and the winding coordinates respectively.The action of DFT is given by where R(H, d) is the generalized Ricci scalar defined by where * are any quantities in DFT.The first equation in (2.4) is the level-matching condition of closed strings while the second one is specific to DFT.This is called the strong constraint.
The physical spacetime M is defined by a D-dimensional slice in the doubled space M.This is defined by a solution to the constraints (2.4).For example, when all the components in H M N , d and gauge parameters depend only on X µ , the constraints (2.4) are trivially satisfied.In this case, a slice Xµ = const., parameterized by X µ , is chosen as the D-dimensional physical spacetime.The components g µν (X), B µν (X) and ϕ(X) are identified with the spacetime metric, the NSNS B-field and the dilaton, respectively.Then, the action (2.1) reduces to that of the NSNS sector of type II supergravities where R is the Ricci scalar defined by the spacetime metric g µν and H (3) = dB is the field strength of the B-field.In this sense, DFT is a T-duality covariant formulation of supergravity.
We next move to a more sophisticated treatment of the doubled space M and discuss its geometric structures.

Born geometry
The structures of the doubled space M in the previous subsection are furnished in the Born manifold [28][29][30][31].Before introducing the Born manifold, we start from an almost para-complex manifold.Given a 2D-dimensional differential manifold M, an endomorphism K : T M → T M that satisfies K 2 = 1 2D and whose ±1-eigenbundle has the same rank is called an almost paracomplex structure.Then the pair (M, K) gives an almost para-complex manifold.Since K is a real analogue of the almost complex structure J 2 = −1, we call this kind endomorphism the almost real structure.We also call endomorphisms on T M the doubled structures in general.Due to the property K 2 = 1 2D , we have two eigenbundles (distributions) L and L in T M associated with two eigenvalues K = ±1.They are defined by the projection operators Then the tangent space of the almost para-complex manifold M is decomposed as T M = L ⊕ L.
We then introduce the notion of integrability of doubled structures.For a distribution D ⊂ T M and vector fields X, Y ∈ Γ(D), when the Lie bracket •, • evaluated on X and Y becomes again a vector field in D, namely X, Y ∈ Γ(D), then D is called involutive.By the Frobenius theorem, a distribution D is integrable if and only if it is involutive.With this definition, we consider the projected Lie brackets, Apparently N P = 0 implies the involutivity and hence the integrability of L. The same is true for N P = 0 and the integrability of L. The Nijenhuis tensor of the endomorphism K is defined by where X, Y ∈ Γ(T M).Since N K = 0 means that K is integrable, it is obvious that the integrabilities of L and L imply that of K.An almost para-complex manifold (M, K) is said to be a para-complex manifold when K is integrable.On the other hand, when only L( L) is integrable, (M, K) is an L( L)-para-complex manifold.We next introduce a metric in the almost para-complex manifold (M, K).A neutral metric η of signature (D, D) is defined by a map η : is called a para-hermitian metric.The triple (M, K, η) is called an almost parahermitian manifold.Although, the para-hermitian metric η M N is not necessarily flat, we always take it to be flat in this paper.In this case, the metric η M N is identified with the O(D, D) invariant metric in DFT.By the neutral metric η M N and K M N , we define the fundamental two-form (ω K ) M N = η M L K L N which is not closed in general.When ω K is non-degenerate, it defines an almost symplectic structure on M. Then the para-hermitian structure defines an almost symplectic manifold (M, ω K ).When dω K = 0, then ω K is a symplectic structure on M In this case M is an almost para-Kähler manifold.When K is integrable, (M, ω K ) is a para-Kähler manifold.This is also known as a bi-Lagrangian manifold.The almost para-hermitian manifolds are classified by the integrability of K and the closedness of ω K .See Table 1.
The last quantity we introduce is the metric H M N of signature (2D, 0).Let (M, K, η) be a para-hermitian manifold.We define H as a Riemannian metric of signature (2D, 0) that satisfies Then (η, ω K , H) is called the Born structure and the quadruple (M, η, ω K , H) is the Born manifold.In DFT language, H M N is identified with the generalized metric.The condition (2.8) is rewritten as This means that the quantity J = η −1 H defines an almost real structure J 2 = 1 2D on T M and the compatibility condition becomes (2.10) The pair (η, J ) is called the chiral structure on M. On the other hand, the quantity I = H −1 ω K defines an almost complex structure I 2 = −1 2D on T M and the condition becomes The pair (H, I) is called an almost hermitian structure on M. The equations (2.8) are compatibility conditions on η, H and ω In summary, a 2D-dimensional Born manifold M is equipped with the neutral metric η of signature (D, D), the fundamental two-form ω K and the Riemannian metric H of signature (2D, 0).The pair (ω K , K) is the para-hermitian structure, (η, J ) is the chiral structure, (H, I) is the almost hermitian structure on M. The triple (η, ω K , H) is the Born structure which The doubled structure (I, J , K) is called an almost para-quaternionic structure on M and satisfies Here {•, •} is the anti-commutator of the doubled structures.The consistency conditions for Born structure are summarized as follows; Altogether we call these structures the Born geometry.

Born geometry and generalized geometry
The para-hermitian structure (ω K , K) of a Born manifold defines the eigenbundles L and L. By the Frobenius theorem, the involutive bundle L defines a foliation structure in M that allows L = T F. The physical spacetime in DFT is identified with a leaf of the foliation.When we write the basis of L as ∂ µ = ∂ ∂X µ , then the local coordinate of the base leaves F is X µ .The same is true for L. Since L is integrable in the para-hermitian manifold, there is a foliation structure that defines leaves whose coordinate is Xµ , and the basis of L is given by ∂µ = ∂ ∂ Xµ .
The pair (F, F) defines a doubled foliation in M and we find the natural coordinate system X M = (X µ , Xµ ) in M.This is identified with the KK and the winding coordinates.The D-dimensional physical spacetime M in a 2D-dimensional Born manifold M is a leaf in F or F. A physical field Φ on M is given by an (anti)para-holomorphic quantity dΦ = 0 (dΦ = 0) defined by K [34].This is a trivial solution to the constraints (2.4).
We now examine the relation between the doubled space and the generalized tangent bundle on a D-dimensional physical space M .The generalized tangent bundle of M is the formal sum of the tangent and the cotangent bundles TM = T M ⊕ T * M .Since we have the neutral metric η in M, there is a map T M = L ⊕ L → L * ⊕ L * .This induces the following isomorphisms; Then L is identified with the dual vector space L * of L. This defines natural isomorphisms [28,30,31]; The distribution L = T F is the tangent bundle of the leaves F and L * is its dual T * F. Therefore T M is identified with the generalized tangent bundle T F ⊕ T * F over F, or T F ⊕ T * F over F.
In the following, we choose M ⊂ F without loss of generality and identify doubled structures on T M with generalized structures on TM through the natural isomorphism (2.16).In this case, doubled vectors and generalized vectors are identified as follows; (2.17) For later convenience, we introduce a particular representation of the Born structure; Note that when we impose the constraints (2.4) in DFT, g µν and B are the spacetime metric and the B-field on a leaf in F. The other structures are parameterized as (

2.19)
With this parameterization, we find that I, J and K anti-commute with each other and they indeed satisfy We call the equations (2.18), (2.19) the standard representation.We note that the neutral metric η defines a natural inner product on the doubled or generalized vectors U and V ; where we have used the expansions 3 Born structure and generalized complex structures In this section, we study the compatibility conditions for the Born structure on M and the Kähler structure on M .The Kähler structure on spacetime M is embedded into the generalized Kähler structure on TM via the Gualtieri map.This is identified with a doubled structure on T M via the natural isomorphism.We analyze algebras that govern the Born geometry and the generalized Kähler structures.We also examine how they are combined into the doubled space M.

Embedding Kähler structures
It is widely known that an (almost) complex structure J in spacetime M is embedded into a generalized almost complex structure J on TM [16].A generalized almost complex structure is an endomorphism J : TM → TM that preserves the inner product ⟨J •, J •⟩ = ⟨•, •⟩ and squares to minus identity J 2 = −1 2D .Since J 2 = −1 2D , the generalized almost complex structure defines ±i-eigenbundles on the complexified generalized tangent bundle; The eigenbundles l ± have the complex rank D and are maximally isotropic and l + ∩ l − = 0.The integrability of the generalized almost complex structures are defined through the Courant involutivity of the eigenbundles [16].When this is the case, J becomes a generalized complex structure.A generalized Kähler structure is defined by a pair of two commuting generalized complex structures (J 1 , J 2 ) whose product G = J 1 J 2 defines a positive-definite metric on TM .With these definitions, we exhibit an explicit example of the generalized Kähler structure.Given a complex structure J in a Kähler manifold M , we have generalized complex structures of the form; where J * is the adjoint of J and ω = −gJ is the Kähler two-form associated with the complex structure J.It is shown that J J and J ω are Courant involutive when J is integrable and dω = 0 which holds for any Kähler manifold M .We find that J J and J ω commute with each other and their product becomes a positive-definite metric on TM1 .Then the pair (J J , J ω ) defines a generalized Kähler structure.
We examine the compatibility of the generalized Kähler structure (J J , J ω ) and the Born structure on M. Since the physical spacetime M ⊂ F admits the metric g µν and the Kähler structure, namely, an integrable complex structure J and a symplectic form ω = −gJ, they satisfy J 2 = −1 and ωJ = −J * ω on T F = L.We also assume B = 0 for the time being.In the standard representation, the building blocks of the doubled structure I, J and K in the Born manifold are given by They obey the algebra of the almost para-quaternionic structure (2.13).In other words, this is the algebra of the split-quaternions involving two real and one imaginary units (see Appendix A).On the other hand, the generalized Kähler structure (J J , J ω , G) obeys the algebra Here [•, •] is the commutator of the doubled structures.This is the algebra of the bi-complex numbers C 2 (see Appendix A).We examine algebraic structures that incorporate the splitquaternions and the bi-complex numbers as subalgebras.By the explicit calculation, it is obvious that the almost product structure G in the generalized Kähler structure gives the chiral structure in the Born structure, G = J .Hereafter we denote J instead of G.The products of the generalized Kähler structure (J J , J ω ) and the doubled structure (I, J , K) introduce the additional structures on T M; This means that the algebra is not closed by (J J , J ω , I, J , K).The newly appeared structures P and Q satisfy They play as real and complex structures on T M. By evaluating all the products involving P and Q, we find Table 2: The product table including the Born and the generalized complex structures.Left × right products are shown.
Here we have used the relation ω = −gJ in the evaluation of the products.From (3.8), we find that no additional structures appear.Then the algebra is closed by the basis (1 2D , J J , J ω , I, J , K, P, Q). (3.9) In this algebra, we have four complex and four real structures on T M; The basis (1 2D , J J , J ω , I, J , K, P, Q) defines an eight-dimensional algebra whose product table is given in Table 2.We find that this is the algebra of the bi-quaternions.It is known that algebras of some hypercomplex numbers are isomorphic to Clifford algebras.Indeed, the biquaternion algebras are equivalent to the Clifford algebras As we show in Appendix A, there are commutative and anti-commutative bases which form subalgebras in the bi-quaternion algebra.The subalgebras include bi-complex numbers C 2 , split-quaternions SpH and quaternions H.We find all the subalgebras of the bi-quaternion algebra; (1).
We then examine the compatibility conditions of the structures (J J , J ω , I, J , K, P, Q) and the metrics η, H on M. The building blocks of the Born structures I, J and K satisfy Since η defines the natural inner product on T M ≃ TM , the generalized Kähler structure (J J , J ω ) satisfies Note that this together with J = J J J ω implies the second condition in (3.11).Furthermore, since P = KJ ω and Q = −J ω I, we have This means that η is anti-hermitian with respect to P and Q.
We have the compatibility conditions for the Born structures; The metric of the Kähler spacetime M satisfies the hermitian condition g(J•, J•) = g(•, •).This implies the following properties; Then, the compatibility conditions for P and Q on the generalized metric H is found to be We here comment on the effects of the B-field on the above structures.Until now, we have not cared about the B-field.For a given (almost) real or complex structure A on T M, the B-field is introduced by an O(D, D) transformation on T M, known as the B-transformation Indeed, the equation (3.4) becomes the standard representation (2.19) by the B-transformation.
The same is true for the representation (3.6).Since the B-transformation is a similarity transformation, the algebra closes even in the presence of the B-field.
In summary, the Kähler structure (J, ω) of spacetime M is embedded into the doubled structures on T M satisfying the algebra of bi-quaternions.The compatibility of the generalized Kähler and the Born structures requires the bi-quaternion algebra that encompasses four real and four complex structures (1 2D , J J , J ω , I, J , K, P, Q).The algebra has substructures given by bi-complex numbers, split-quaternions and quaternions.

Embedding bi-hermitian structures
We next consider embeddings of the bi-hermitian structure (J ± , ω ± ) of spacetime M .Note that the metric g µν is hermitian with respect to J ± , and ω ± = −gJ ± are the fundamental two-forms.The bi-hermitian structures are embedded into the generalized complex structures J ± as [16]; where the matrices of J J ± and J ω ± are given by (3.2).We find that J + and J − commute with each other and they give the chiral structure J = J + J − .The triples (J J ± , J ω ± , J ) form the algebras of two bi-complex numbers sharing the common real (chiral) structure J .Since the algebra of the bi-quaternions does not support such subalgebras, we need to enlarge the algebra to incorporate the bi-hermitian structure (J ± , ω ± ) into the Born geometry.We find that an algebra that allows subalgebras of two bi-complex numbers sharing one real structure J is the bi-quaternions over the field C.This is a 16-dimensional algebra and schematically written as C ⊗ C ⊗ H, which is isomorphic to the Clifford algebra Cl 3 (C).This algebra contains 8 real and 8 imaginary units and the bi-complex numbers are subalgebras generated by the bases (see Appendix A); (e 0 i 1, e i 1 1, e i i 1), (e 0 1 î, e i i î, e i i 1), (i = 1, 2, 3).(3.19)Here e µ , (1, i) and ( 1, î) are bases of H, C and C, respectively.The bases (3.19) share the real unit e i i 1.For example, if we assign we find that they obey the algebras of two bi-complex numbers.The Born structure (I, J , K) is represented by a split-quaternion subalgebra of C ⊗ C ⊗ H.This is given by the basis (e 1 i 1, e 2 i 1, e 3 1 1). (3.21) Therefore we employ the following assignment; We have five imaginary units J J ± , J ω ± , I. The other three imaginary units P ′ , Q ′ , R ′ are represented as P ′ , Q ′ , R ′ are obtained as (without the B-field) (3.25) We easily confirm that Here we have used the relations of the bi-hermitian structure J * ± = −gJ ± g −1 and the fact that J + and J − commute with each other.

Embedding hyperkähler and bi-hypercomplex structures
The hyperkähler structure (J a , ω a ) (a = 1, 2, 3) on M is embedded into the generalized hyperkähler structure on T M; For later convenience, we denote J Ja = J a,+ and J ωa = J a,− .These structures satisfy the algebra Here ϵ abc is the Levi-Civita symbol and J is the chiral structure satisfying J 2 = 1 2D .In fact, the algebra (3.29) is the definition of the generalized hyperkähler structure [35] and it is the algebra of the split-bi-quaternions or Cl 0,3 (R) in disguise [27].
An algebra that incorporates the split-bi-quaternions and the algebra of the Born structure (split-quaternions) is split-tetra-quaternions.This is a hypercomplex number generating a 16-dimensional algebra and isomorphic to Cl 4,0 (R), Cl 1,3 (R) and Cl 0,4 (R).The split-tetraquaternions contain the bases of 10 imaginary and 6 real units.They are represented by (see Appendix A and B) 1e 0 e 0 , 1e 0 e a , 1e 1 e 0 , 1e 1 e a , ie 2 e 2 , ie 2 e a , ie 3 e 0 , ie 3 e a , (a = 1, 2, 3), (3.31)where e µ and e µ are two commuting quaternions and (1, i) is the basis of C. Hereafter we omit the "1" in the products.We find that the triples (J a,+ , J a,− , J ) (a = 1, 2, 3) form three independent bi-complex numbers sharing the common real structure J = J a,+ J a,− (a : no sum).Since the bi-complex numbers are realized as subalgebras of split-tetra-quaternions as (ie 2 e 0 , e 0 e a , ie 2 e a ), (ie 3 e 0 , e 0 e a , ie 3 e a ), (a = 1, 2, 3), (3.32) we make an assignment2 ; J = ie 2 e 0 , J a,+ = e 0 e a , J a,− = −ie 2 e a , (a = 1, 2, 3). (3.33) The Born structure (I, J , K) of M obeys the algebra of split-quaternions.Since the splitquaternion that contains J is represented by the basis (e 1 e 0 , ie 2 e 0 , ie 3 e 0 ), (3.34) we find the assignment The other three imaginary units are given by ie 3 e a , (a = 1, 2, 3) (3.36) which are decomposed like ie 3 e a = (e 1 1)(ie 2 e a ).Then by assigning the remaining three complex structures P ′′ , Q ′′ and R ′′ as we find Indeed, the direct calculations reveal that they are expressed by satisfying the desired properties . Similarly, we find that the remaining real structures S ′′ , T ′′ , U ′′ , V ′′ are given by We finally consider the embedding of the bi-hypercomplex structure (J a,± , ω a,± ) on M into the doubled structures of M. The bi-hypercomplex structure (I a,± , ω a,± ) is embedded into the generalized hyperkähler structure Note that the Kähler case corresponds to J a,+ = J a,− = J a and hence J a,+ = J Ja , J a,− = J ωa .The generalized hyperkähler structure (J a,+ , J a,− ) forms the algebra of the split-bi-quaternions.This contains 6 imaginary units (J a,± ) 2 = −1 2D (a : no sum).Furthermore, this contains 6 generalized Kähler structures (J J a,± , J ω a,± , J ) that share the common real structure J .Each forms the algebra of the bi-complex numbers.Then we need 6 bi-complex subalgebras to incorporate these structures.This is not possible by the split-tetra-quaternions and we therefore enlarge this algebra.An appropriate algebra is the split-tetra-quaternions over H (see Appendix A).This is a 64-dimensional algebra on M. Indeed, if we assign the bases J = ie 2 e 0 ê0 , J J a,+ = e 0 e a ê0 , J ω a,+ = −ie 2 e a ê0 , J J a,− = e 0 e 0 êa , J ω a,− = −ie 2 e 0 êa , then we find that the triples obey the algebra of bi-complex numbers.In this basis, J a,± are represented by Here we have used the fact that e µ , e µ , êµ are commuting quaternions and denoted e 0 e 0 ê0 = 1 2D .Then we confirm the algebra of the split-bi-quaternions for the generalized hyperkähler structure.
In summary, the hyperkähler structure (J a , ω a ) on M is embedded into the doubled structure on M that obey the 16-dimensional algebra of the split-tetra-quaternions.The bi-hypercomplex structure (J a,± , ω a,± ) is embedded into the doubled structure that obeys the 64-dimensional algebra of the split-tetra-quaternions over H.The algebras of doubled structures on T M ≃ T M ⊕ T * M are summarized in Table 3.

Worldsheet instantons in Born sigma models
We have established the T-duality covariant embeddings of complex structures of spacetime.
One of the notions that has deep connections with the spacetime complex structures is the worldsheet instantons [36,37].In [27], we studied the T-duality relation between the instantons in Kähler and bi-hermitian geometries.Due to the fact that there are complex structures J and J ± in the Kähler and the bi-hermitian geometries respectively, we find a one-to-two correspondence between the instantons in each geometry.We here elucidate this relation within the T-duality covariant formulation.
In this section, we study the worldsheet instantons in a T-duality covariant doubled formalism.The doubled formalism of string sigma models that makes T-duality be manifest has been studied in various viewpoints [38][39][40][41].Among other things, more direct connections to DFT and the doubled space appear in the Born sigma model [33] 3 .In the following, we show that the worldsheet instanton equations respecting T-duality symmetry are obtained in the Born sigma model by utilizing the doubled structures discussed in the previous sections.

Born sigma models
The Born sigma model is a sigma model whose target space is the Born manifold M.This is closely related to the doubled sigma model of a string introduced in [39,40].The action of the Born sigma model in the Minkowski signature is given by Here Σ is the two-dimensional worldsheet, X M = (X µ , Xµ ) is the local coordinate of the Born manifold M, * is the Hodge star operator in Σ, H M N is the generalized metric in the Born manifold M and Ω M N = −Ω N M is an anti-symmetric constant matrix.We have neglected the Fradkin-Tseytlin term that involves the dilaton [33] which is not relevant in our discussion.The action (4.1) is invariant under the O(D, D) rotation In the following, we use the standard parameterization of the generalized metric (2.3).The second term in the action (4.1) is topological but plays an important role in the instanton equations.Following [39,40], we employ the expression of the topological term Using the standard parameterization, the term involving H M N in the action (4.1) is expanded as where h αβ , (α, β = 0, 1) is the metric of the two-dimensional worldsheet Σ and the topological term is Here ε αβ and ϵ αβ are the totally anti-symmetric tensor and the Levi-Civita symbol in Σ, respectively.
Since the Born sigma model (4.1) contains double degrees of freedom, we impose constraints on the quantities.The physical (non-doubled) sigma model is obtained by imposing the DFT constraints (2.4) on the background fields g µν , B µν , ϕ and also by introducing the self-duality condition; This is rewritten by the chiral structure Then we can remove the winding degrees of freedom d Xµ from the action.Note that we have solved the DFT constrains (2.4) by making all the background fields depend only on X µ .
Plugging (4.7) back into the action (4.1), we find the H M N part vanishes.On the other hand, the topological term becomes This precisely reproduces the action of the ordinary string sigma model.

Instantons in Born sigma models
We then consider the instantons in the Born sigma model.In the following, spacetime and the worldsheet have the Euclidean signature and * 2 = −1.We first consider the term that depends on H M N in the action (4.1).Since the metric H M N is positive-definite in the Euclidean space, we have the Bogomol'nyi bound of the action; where A is a doubled structure satisfying A 2 = −1 2D in the Born manifold M and ω A = HA is the fundamental two-form associated with A. The bound is saturated when the map X : or equivalently, We call these the doubled instanton equations.By the chirality condition (4.6)4 and the doubled instanton equations, we have This means that we need (AJ ) 2 = −1 2D to have non-trivial solutions for instantons, otherwise , we obtain (AJ ) 2 = −1 2D if A commutes with J (i.e., [A, J ] = 0).On the other hand, if A anti-commutes with J (i.e., {A, J } = 0), then we find (AJ ) 2 = 1 2D .Therefore only A that commutes with the chiral structure J is allowed for the doubled instantons.
In the following, we consider Σ = S 2 and the image of the map X is a two-cycle C 2 in M. Then the map X : Σ → M is classified by the homotopy class π 2 (S 2 ).The action bound is given by Here we have restored the topological term Ω.We assume that the two-cycle C 2 lie in the physical spacetime M .Note that the topological term is written in the T-duality covariant form; In the following, we first study the doubled instanton equations for the Kähler geometry and then move to the bi-hermitian geometry.For simplicity we start by the Kähler geometry with trivial B-field B = 0.There is a doubled structure (J J , J ω , I, Q) in the doubled space whose squares are −1 2D .As we have shown, I and Q anti-commute with the chiral structure J and they give trivial solutions to the equations (4.11).On the other hand we have [J J , J ] = [J ω , J ] = 0 and the doubled instantons defined by J J , J ω provide meaningful solutions.We clarify them explicitly.A = J ω case.In the case of A = J ω , the doubled instanton equations are which in components are written as Here ω = −gJ is the Kähler form in M .By the chirality condition (4.6), we solve the winding coordinate as Plugging this into the first line in (4.16), we obtain In general, the almost complex structure J of the spacetime manifold M decomposes the complexified tangent space T M C = T M ⊗ C into T M ± such as Here T M ± are eigenbundles of the complex structure JX ± = ±iX ± and X + , X − are (anti)holomorphic vectors.When J is integrable, the Lie bracket of the (anti)holomorphic vectors become the (anti)holomorphic vectors.Therefore the doubled instanton equations by J ω restrict dX µ to (anti)holomorphic vectors by J.
The fundamental two-form ω Jω becomes Then the term | C 2 ω Jω | in the action bound (4.13) becomes trivial; Here we have used the relation gω The non-trivial action bound comes from the topological term Ω and it is nothing but the (Euclideanized) string sigma model action; A = J J case.When A = J J , the doubled instanton equations become In components, these are given by The first equation reproduces the ordinary worldsheet instanton equations [36].The second equation provides us the T-dual of the first equation.We find that the chirality condition ∂ α Xµ = −iε αβ g µν ∂ β X ν applying to the second equation gives the first one.In this case, the fundamental two-form ω J J is evaluated as Therefore we have By using the chirality condition and eliminating Xµ sector, we have Here we have used the relations ω = −gJ and ω −1 = −J −1 g −1 = Jg −1 .Then, we find the action bound coming from the H part is This is anticipated because the chirality condition makes the H M N part be trivial.On the other hand, the topological term in the action is evaluated as This reproduces the action bound of the ordinary worldsheet instantons.
For later convenience, we here introduce the B-field by the B-transformation.For a doubled structure A, the action bound is found to be where the doubled structure A is replaced by A B = e B Ae −B and ω A B is the fundamental two-form associated with A B .The general doubled instanton equations are then given by dX M ± (A B ) M N * dX N = 0. (4.30) For the generalized complex structure A = J ω , we have The doubled instanton equations in components become Using the chiral structure the chirality condition is solved by By substituting this into the first line in (4.32), we find the condition (4.17) obtained in the case of the trivial B-field.In the second line, we have This again implies the condition (4.17).Therefore dX µ is a (anti)holomorphic vector even in the presence of the B-field.The action bound is similarly obtained.For the case A = J J , we have The doubled instanton equations are then In components, we have The first line gives the worldsheet instanton equation.Under the chirality condition, the second line in (4.38) becomes Therefore (4.38) consistently reproduces the instanton equation dX µ ± J µ ν * dX ν = 0 even in the presence of the B-field.In this case, the action bound is given by Then the topological θ-term for the instanton bound is precisely obtained by the B-field.
Bi-hermitian geometry.We next consider the bi-hermitian geometry characterized by (J + , J − ).It is known that Kähler and bi-hermitian geometries are T-dual with each other.This becomes apparent when these structures are embedded into generalized Kähler structures in the doubled space [27].Indeed, the doubled instanton equations in the bi-hermitian geometry are obtained by the T-duality transformation of (4.22).The bi-hermitian structure (J + , J − ) on spacetime M is expressed by the generalized complex structures as The doubled instanton equations are given by In the following, we focus on J B + without loss of generality.The equations (4.42) are decomposed as Then the equations (4.42) are linear combinations of the doubled instanton equations defined by J B J ± and J B ω ± .As we have clarified, under the chirality condition, we have the following equations from the doubled instanton equations; This means that the solutions are restricted to the (anti)holomorphic vectors defined by J ± and they are also instantons with respect to J ± .This is possible since the bi-hermitian structure satisfies [J + , J − ] = 0 and the common eigenvectors of J ± are allowed.This also implies J ± * have common eigenvectors (instantons).The action bound in this case is This is nothing but the bound for the ordinary worldsheet instantons defined by J ± .

Conclusion and discussions
In this paper, we studied doubled structures that encode Kähler, hyperkähler, bi-hermitian and bi-hypercomplex geometries.
The spacetime metric g µν and the NSNS B-field are organized into the generalized metric H M N and the natural O(D, D) structures of DFT are implemented by the Born structure on the doubled space M. Due to the natural isomorphism emerged from the Born structure, the tangent bundle of the doubled space M and the generalized tangent bundle TM over the physical spacetime M is identified.On the other hand, the Kähler structure on the physical spacetime M is embedded into the generalized Kähler structure (J J , J ω ) on T M. We analyzed compatibility of the doubled and the Born structures in the doubled space.We found that the algebraic structures require the extra doubled structures P and Q in the Born geometry.Altogether we showed that they form the algebra of the bi-quaternions.The Born and the generalized complex structures appear as subalgebras of split-quaternions and the bi-complex numbers, respectively.Utilizing this fact, we extended the discussion to the bi-hermitian case.We found that the desired algebra that encodes the bi-hermitian structure on spacetime is the algebra of bi-quaternions over C.This is a 16-dimensional algebra that contains appropriate subalgebras.By using the basis of the algebra, we write down all the real and imaginary units in their explicit forms.For the hyperkähler structure on spacetime, it is represented by the generalized hyperkähler structure on T M. This satisfies the algebra of split-bi-quaternions.This together with the Born structure leads us to the algebra of split-tetra-quaternions.We exhibited the explicit representations of the doubled structures that form this algebra.We further extended the results to the bi-hypercomplex case.We found that the structure is realized as the algebra of the split-tetra-quaternions over H in the doubled space.These results provide us deep connections among the algebras of the hypercomplex numbers, the complex structures of spacetime, the doubled structures and T-duality.We also showed that some of the algebras of the hypercomplex numbers also expressed by Clifford algebras.
In the latter part of this paper, we studied the doubled worldsheet instantons in the Born sigma model.The Born sigma model is a sigma model whose target space is the Born geometry.The model keeps manifest T-duality and is governed by the generalized metric H M N and the topological term.The ordinary string sigma model is reproduced by the DFT constraints and the chirality condition defined by J .We derived the Bogomol'nyi equations defined by the doubled complex structures on T M. We clarify that appropriate doubled complex structures reproduced the ordinary worldsheet instanton equations.We then discussed the T-duality transformation of the worldsheet instantons.We in particular focused on the T-duality between Kähler and bi-hermitian geometries.The one-to-two correspondence of the worldsheet instantons discussed in [27] is naturally interpreted in the Born sigma model.We showed that the instantons in the bi-hermitian geometries are represented by a linear combination of individual instantons defined by the structures (J ± , ω ± ).The analysis can be extended to the hyperkähler and bi-hypercomplex cases.The Bogomol'nyi equations in the Born sigma models are interpreted as the T-duality covariant realization of the worldsheet instanton equations.
As we discussed, the doubled space plays an important role in revealing the T-duality among geometric structures.It has been discussed that solutions to supergravities that are related by T-duality transformations are given by a solution to DFT.This means that the spacetime geometries of various supergravity solutions are described by doubled geometry in a T-duality unified manner.For example, the H-monopole (smeared NS5-brane) and the KKmonopole (KK5-brane) in type II string theories are unified into an O(D, D) covariant solution to DFT [44].The worldsheet instanton effects in the H-monopole geometry, the KK-monopole geometry and their relations to T-duality are studied in various perspectives [45][46][47][48].Among other things, the instantons break the isometry of the H-monopole geometry and recover that of the NS5-brane which is a genuine solution in string theory.Things get more interesting when we consider this phenomenon in the T-dual side.It has been shown that instantons in the KK-monopole geometry breaks the isometry not along the KK direction, but of the winding space [47,48].The modified geometry is characterized not only by the physical coordinate x µ but also by the winding coordinate xµ .These fact mean that instantons reveal the more stringy nature of spacetime.It would be interesting to study this winding geometry in the context of the Born sigma model.The notion of T-duality covariant instantons helps us to understand geometries that are not fully captured in supergravities.They are known as nongeometries.An example of this kind of non-geometry is the T-fold [39,40] whose explicit realization includes exotic branes in string theories [49].The exotic 5 2  2 -brane in type II string theories is a typical example studied intensively.Indeed, the exotic 5 2  2 -brane obtained by the T-duality transformation of the hyperkähler (Taub-NUT) geometry is realized by a solution to DFT in a specific frame [50].It has been shown that the 5 2  2 -brane geometry is expected to admit the bi-hypercomplex structures [27] and the worldsheet instantons in the 5 2  2 -brane geometry is studied [51][52][53].It would be interesting to study the instantons and bi-hypercomplex structures in the doubled setup.We will come back to these issues in future researches.
Table 4: The product tables of the bases of quaternions (left) and split-quaternions (right).For quaternions i 2 = j 2 = k 2 = ijk = −1 and they anti-commute.For split-quaternions i 2 = −1, j 2 = k 2 = ijk = 1 and they anti-commute.The split-quaternions are obtained by replacing j → ij, k → ik in the quaternions.Here i is an auxiliary imaginary unit i 2 = −1.

A.1 Basic elements
Binarions.Binarions are the two-dimensional (non)associative unital algebras over the field R. Binarions are classified as the followings depending on the bases of the algebras.
Note that the complex numbers define the field C but the split-complex and dual numbers do not.This is because they have non-trivial zero divisors.Since we never treat the dual numbers in this paper, we do not care about the "dual"-hypercomplex numbers.
Quaternions.We next introduce four dimensional (non)associative unital algebras over the field R.There are two options.
(1).Quaternions are defined by a normed (associative) division algebra over the field R.This is defined by the basis (1, i, j, k) given in the product Table 4 (left).
The quaternions define a field H but the split-quaternions do not. A.
Here we have denoted e 0 e µ as e µ .The split-tetra-quaternions contain split-bi-quaternions as subalgebras.This contains algebras of the generalized hyperkähler structures and the Born structures.

B Clifford algebra and hypercomplex numbers
Some hypercomplex numbers are related to Clifford algebras.In this appendix, we present the explicit relations among them.We first define quantities A Clifford algebra Cl p,q (R) is defined by the basis 1, e i , e i ∧ e j , e i ∧ e j ∧ e k , • • • .Here 1 is the unit of the multiplication of the field R. In the following, we use e i ∧ e j = 1 2 (e i e j − e j e i ) = e i e j .
An element X in Cl p,q (R) is expanded as x ij e i e j + 1 3!
x ijk e i e j e k + • • • + 1 n! x 1,...,n e 1 • • • e n , (B.2) where x, x i , x ij , . . .∈ R are coefficients.The dimension of the algebra is dim(Cl p,q (R)) = 1 3 The algebra involves 2 n−1 imaginary and real units and Cl p,q (R) are in general non-commutative unital associative division algebras.
2-dim.The 2 1 = 2-dimensional algebras are Cl 1,0 (R) and Cl 0,1 (R).The algebra Cl 1,0 (R) is generated by 1 and e 2 1 = 1 and they commute.Then this is isomorphic to the split-complex numbers Cl 1,0 (R) ≃ SpC.The algebra Cl 0,1 (R) is generated by 1 and e 2 1 = −1 and they commute.It is obvious that this is equivalent to the complex numbers Cl 0,1 (R) ≃ C. commuting basis, cannot be written in Clifford algebras.This is not the case when the field defining Clifford algebras is changed from R to C. For example, Cl 0 (C) is a complex vector space generated by 1.This is identified with C. The complex 2-dimensional (hence the real 4-dimensional) algebra Cl Note that they all commute and define two real and two imaginary units; It is obvious that this is equivalent to the algebra of the bi-complex numbers, Cl 1 (C) ≃ C ⊗ C.
In the same way, we have isomorphisms Cl 2 (C) ≃ C ⊗ H, Cl 3 (C) ≃ C ⊗ C ⊗ H, and so on.We note that not all the hypercomplex numbers are isomorphic to Clifford algebras.For example, the tri-complex numbers by Segre C 3 = C ⊗ C ⊗ C is not obtained in this way.A summary of the algebras is found in Table 6

. 2 )
Here the derivative means ∂ M = ∂ ∂X M .The indices are raised and lowered by the O(D, D) invariant metric η M N and its inverse η M N .The action (2.1) is manifestly invariant under the global O(D, D) transformation and is invariant under the O(D, D) covariantized diffeomorphism and the B-field gauge transformation.The T-duality transformations of the spacetime fields are implemented by the O(D, D) rotation of the generalized metric H M N (X) and the generalized dilaton d(X).To see this, it is useful to employ the standard parameterization of the DFT quantities ) where g µν and B µν are D × D symmetric and anti-symmetric matrices respectively, while ϕ is a real function on M. The generalized metric H M N is an O(D, D) element and d is invariant under the O(D, D) rotation.All the quantities involving the gauge parameters in DFT are subject to the constraints

Table 1 :
The classification of almost para-hermitian manifolds (M, K).