Topological-antitopological fusion equations,

pluriharmonic maps and special Kähler manifolds

Vicente Cortés^{1} and Lars Schäfer^{1,2}

1Institut Élie Cartan de Mathématiques, Université Henri Poincaré - Nancy 1, B.P. 239, F-54506 Vandœuvre-lès-Nancy Cedex, France cortes@iecn.u-nancy.fr, schafer@iecn.u-nancy.fr

2 Mathematisches Institut der Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany schaefer@math.uni-bonn.de

March 31, 2004

Abstract

We introduce the notion of a tt*-bundle. It provides a simple definition, purely in terms of real differential geometry, for the geometric structures which are solutions of a general version of the equations of topological-antitopological fusion considered by Cecotti-Vafa, Dubrovin and Hertling. Then we give a simple characterization of the tangent bundles of special complex and special Kähler manifolds as particular types of tt*-bundles. We illustrate the relation between metric tt*-bundles of rank r and pluriharmonic maps into the pseudo-Riemannian symmetric space GL(r)/O(p, q) in the case of a special Kähler manifold of signature (p, q) = (2k, 2l). It is shown that the pluriharmonic map coincides with the dual Gauß map, which is a holomorphic map into the pseudo-Hermitian symmetric space Sp(ℝ^{2n})/U(k, l) ⊂ SL(2n)/SO(p, q) ⊂ GL(2n)/O(p, q), where n = k + l.

This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche For-

schungsgemeinschaft. Research of L. S. was supported by a joint grant of the ‘Deutscher Aka-

demischer Austauschdienst’ and the CROUS Nancy-Metz.

Contents

1 tt*-equations and pluriharmonic maps 1

2 Special complex and special Kähler manifolds 7

3 Special complex and special Kähler manifolds as solutions of the tt*- equations 8

4 The pluriharmonic map in the case of a special Kähler manifold 10

4.1 The Gauß maps of a special Kähler manifold . . . . . . . . . . . . . . . . . 10

4.2 Holomorphic coordinates on the Grassmannian Gr_{0}^{k,l}(ℂ^{2n}) of complex Lagrangian subspaces of signature (k, l) . . . . . . . . . . . . . . . . . . . . . 11

4.3 The special Kähler metric in affine coordinates . . . . . . . . . . . . . . . . 13

1 tt*-equations and pluriharmonic maps

Definition 1 A tt*-bundle (E, D, S) over a complex manifold (M, J) is a real vector bundle E → M endowed with a connection D and a section S ∈ Γ(T^{∗}M ⊗ End E) which satisfy the tt*-equation

R^{θ} = 0 for all θ ∈ ℝ ,  (1.1)

where R^{θ} is the curvature tensor of the connection D^{θ} defined by

D_{X}^{θ} := D_{X} + (cos θ)S_{X} + (sin θ)S_{JX} for all X ∈ T M .  (1.2)

A metric tt*-bundle (E, D, S, g) is a tt*-bundle (E, D, S) endowed with a possibly indefinite D-parallel fibre metric g such that for all p ∈ M

g(S_{X}Y, Z) = g(Y, S_{X}Z) for all  X, Y, Z ∈ T_{p}M .  (1.3)

A unimodular metric tt*-bundle (E, D, S, g) is a metric tt*-bundle (E, D, S, g) such that tr S_{X} = 0 for all X ∈ T M. An oriented unimodular metric tt*-bundle (E, D, S, g, or) is a unimodular metric tt*-bundle endowed with an orientation or of the bundle E.

Remarks: 1) In special cases, particularly emphasized in the literature, such as the moduli spaces of topological quantum field theories [CV, D] and the moduli spaces of singularities [H], the complexified tt*-bundle E^{ℂ} is identified with T^{1,0}M and the metric g is positive definite. Here we will consider the case E = T M, and hence E^{ℂ} = T^{1,0}M + T^{0,1}M. This includes special complex and special Kähler manifolds, as we shall see.

2) If (E, D, S) is a tt*-bundle then (E, D, S^{θ}) is a tt*-bundle for all θ ∈ ℝ, where

S^{θ} := D^{θ} − D = (cos θ)S + (sin θ)S_{J} .  (1.4)

The same remark applies to metric tt*-bundles.

3) Notice that an oriented unimodular metric tt*-bundle (E, D, S, g, or) carries a canonical metric volume element ν ∈ Γ(∧^{r}E^{∗}), r = rk E, determined by g and or, which is D^{θ} parallel for all θ ∈ ℝ.

Proposition 1 Let E → M be a real vector bundle over a complex manifold (M, J) such that E is endowed with a connection D and a section S ∈ Γ(T^{∗}M ⊗ End E). Then (E, D, S) is a tt*-bundle if and only if the following equations are satisfied

(i) d^{D}S = d^{D}S_{J} = 0, where S and S_{J} are considered as one-forms with values in End E and d^{D} is the exterior covariant derivative defined by D,

(ii) [S_{X}, S_{Y}] = [S_{JX}, S_{JY}] for all X and Y,

(iii) R^{D}(X, Y) + [S_{X}, S_{Y}] = 0 for all X and Y.

Proof: Using the relations 2 cos θ sin θ = sin 2θ, 2 cos^{2} θ = 1 + cos 2θ and 2 sin^{2} θ = 1 − cos 2θ, we obtain a (finite) Fourier decomposition of R^{D^{θ}} in the variable θ. The tt*-equation R^{D^{θ}} = 0 shows that all Fourier components are zero. This yields (i-iii).

Definition 2 Let (M, J) be a complex manifold and (N, h) a pseudo-Riemannian manifold. A map f : M → N is called pluriharmonic if f|_{C} is harmonic for all complex curves C ⊂ M.

Notice that the harmonicity of f|_{C} is independent of the choice of a Riemannian metric in the conformal class of C, by conformal invariance of the harmonic map equation for (real) surfaces.

Proposition 2 Let (M, J) be a complex manifold and (N, h) a pseudo-Riemannian manifold with Levi-Civita connection ∇^{h}, D a connection on M which satisfies

D_{JY} X = JD_{Y} X  (1.5)

for all vector fields which satisfy ℒ_{X}J = 0 (i.e. for which X − iJX is holomorphic) and ∇ the connection on T^{∗}M ⊗ f^{∗}T N which is induced by D and ∇^{h}.

(i) A map f : M → N is pluriharmonic if and only if it satisfies the following equation

∇′′∂f = 0 ,  (1.6)

where ∂f = df^{1,0} ∈ Γ(⋀^{1,0} T^{∗}M ⊗_{ℂ} (T N)^{ℂ}) is the (1, 0)-component of df and ∇′′ is the (0, 1)-component of ∇ = ∇′ + ∇′′.

(ii) Any complex manifold (M, J) admits a torsion-free complex connection, i.e. a torsionfree connection D which satisfies DJ = 0.

(iii) Any torsion-free complex connection D satisfies (1.5).

Proof: (i) The condition (1.5) means that D′′Z = 0 for all local holomorphic vector fields Z, i.e. Γ_{ᾱβ}^{γ} = Γγ̄ ; ᾱβ = 0 in terms of the Christoffel symbols of D with respect to holomorphic coordinates z^{α}. This implies that the Christoffel symbols of D do not contribute to the equation (1.6). The equation is therefore independent of the choice of connection D. In fact, it is straightforward to check that the restriction of (1.6) to every complex curve C reduces to the harmonic map equation for f|_{C} : C → N. (ii) is well known, see [KN]. (iii) The conditions T^{D} = 0 and DJ = 0 imply that

D_{JY} X − JD_{Y} X = [JY, X] + D_{X}(JY) − JD_{Y} X = [JY, X] + J[X, Y] = −(ℒ_{X}J)Y . (1.7)

The right-hand side vanishes if ℒ_{X}J = 0.

Given a Hermitian metric γ on T^{1,0}M, or, more generally, a pseudo-Hermitian metric, the Chern connection of γ is the unique Hermitian connection 𝒟 in the holomorphic bundle T^{1,0}M which satisfies 𝒟′′Z = 0 for all holomorphic local sections Z of T^{1,0}M. The last property is usually written as 𝒟′′ = ¯∂.

Proposition 3 Let (M, J) be a complex manifold and 𝒟 the Chern connection of a pseudo-Hermitian metric γ on T^{1,0}M. Then there is a unique connection D in the real tangent bundle T M such that DZ = 𝒟Z for all local sections Z of T^{1,0}M, where D has been complex bilinearly extended to a connection on the complexified tangent bundle. The connection D satisfies (1.5), DJ = 0 and Dg = 0, where g is the J-invariant pseudo-Riemannian metric defined by

g(X, X) = 2γ(X^{1,0}, X^{1,0}) , X^{1,0} := \frac{1}{2}(X − iJX) ,  (1.8)

for all X ∈ T M.

Conversely, let g be a J-invariant pseudo-Riemannian metric on a complex manifold (M, J). Then there exists a unique connection D in T M, which satisfies the conditions (1.5), DJ = 0 and Dg = 0. Moreover, D induces a connection in T^{1,0}M, which is the Chern connection of the pseudo-Hermitian metric γ on T^{1,0}M defined by (1.8).

The factor 2 is chosen such that γ coincides with the restriction to T^{1,0}M of the sesquilinear extension of g to the complexified tangent bundle.

Proof: We define a connection D in the complexified tangent bundle (T M)^{ℂ} by

D_{X}Z := 𝒟_{X}Z and D_{X} Z̄ := \overline{𝒟X̄Z}  (1.9)

for all local sections X of (T M)^{ℂ} and Z of T^{1,0}M. By construction D is real, i.e. is the complex bilinear extension of a connection in T M, which we denote by the same symbol D. Obviously, it is the only real connection such that DZ = 𝒟Z for all local sections Z of T^{1,0}M. The equation (1.5) follows from 𝒟′′ = ¯∂. By construction, D preserves the decomposition (T M)^{ℂ} = T^{1,0}M + T^{0,1}M. Therefore, DJ = 0. Finally, Dg = 0 follows from the fact that 𝒟 is Hermitian.

Conversely, let (M, J, g) be a pseudo-Hermitian manifold. Then we can define a pseudo-Hermitian metric γ in T^{1,0}M by (1.8) and consider its Chern connection 𝒟. As we know, it induces a connection D in T M which satisfies (1.5), DJ = 0 and Dg = 0. To prove the uniqueness, let D̃ be an other connection satisfying (1.5), D̃J = 0 and D̃g = 0. D̃ induces a connection 𝒟̃ in T^{1,0}M, which satisfies 𝒟̃′′ = ¯∂, due to (1.5), and which is Hermitian with respect to γ. Therefore, 𝒟̃ is the Chern connection of γ, i.e. 𝒟̃ = 𝒟. This implies D = D̃, by the first part of the proof.

Given a metric tt*-bundle (E, D, S, g), we consider the flat connection D^{θ} for θ = 0: ∇ := D^{0}. Any parallel frame s = (s_{1}, …, s_{r}) of E with respect to ∇ defines a map

G = G^{(s)} : M → Sym_{p,q}(ℝ^{r}) = {A ∈ GL(r)|A^{t} = A has signature (p, q)}
x ↦ G(x) := (g_{x}(s_{i}(x), s_{j}(x))) ,  (1.10)

where (p, q) is the signature of the metric g.

Similarly, for an oriented unimodular metric tt*-bundle (E, D, S, g, ν) with canonical volume element ν and a ∇-parallel frame s = (s_{1}, …, s_{r}) such that ν(s_{1}, s_{2}, …, s_{r}) = 1 we have a map

G = G^{(s)} : M → Sym_{p,q}^{1}(ℝ^{r}) = {A ∈ Sym_{p,q}(ℝ^{r})| det A = (−1)^{q}} .  (1.11)

By Sylvester’s Theorem, the general linear group GL(r) acts transitively on the manifold Sym_{p,q}(ℝ^{r}), which we can identify with the pseudo-Riemannian symmetric space

S(p, q) := GL(r)/O(p, q) .  (1.12)

The subgroup O(p, q) ⊂ GL(r) is the stabilizer of the matrix I_{p,q} = diag(𝟙_{p}, −𝟙_{q}). We shall identify the tangent space of the coset space S(p, q) at the canonical base point o = eO(p, q) with the vector space

sym(p, q) := {A ∈ 𝔤𝔩(r)|η(A·, ·) = η(·, A·)}  (1.13)

of symmetric endomorphisms of ℝ^{r} with respect to the standard scalar product η = η_{p,q} of signature (p, q), which is represented by the matrix I_{p,q}. The structure of a symmetric space is defined by the symmetric decomposition

𝔤𝔩(r) = 𝔬(p, q) + sym(p, q) .  (1.14)

The pseudo-Riemannian metric is defined by an O(p, q)-invariant pseudo-Euclidean scalar product on sym(p, q). For instance, we may choose the metric induced by the trace form:

𝔤𝔩(r) ∋ (X, Y) ↦ tr XY .  (1.15)

Similarly, SL(r) acts transitively on the manifold Sym_{p,q}^{1}(ℝ^{r}), which we can identify with the pseudo-Riemannian symmetric space

S_{1}(p, q) := SL(r)/SO(p, q) .  (1.16)

We have the de Rham decomposition

S(p, q) = ℝ × S_{1}(p, q) ,  (1.17)

where the flat factor corresponds to the connected central subgroup

ℝ^{>0} = {λId|λ > 0} ⊂ GL(r)  (1.18)

and the other factor is always indecomposable and even irreducible if (p, q) ≠ (1, 1). The tangent space of SL(r)/SO(p, q) at the canonical base point o = eSO(p, q) is identified with the trace-free η-symmetric matrices:

sym_{0}(p, q) := {A ∈ sym(p, q)|tr A = 0} .  (1.19)

Under a change of parallel (respectively, parallel unimodular) frame s → su, u ∈ GL(r) (respectively, u ∈ SL(r)), the map G = G^{(s)} transforms as

G^{(su)} = u^{−1} · G^{(s)} = u^{t}G^{(s)}u ,  (1.20)

where the dot stands for the action of GL(r) on Sym_{p,q}(ℝ^{r}).

The following theorem is proven in [S2], cf. [S1]. In the case where E^{ℂ} = T^{1,0}M and the metric g is positive definite it is due to Dubrovin [D].

Theorem 1 Let (E, D, S, g) be a metric tt*-bundle over a simply connected complex manifold M. Then the map

G^{(s)} = (g(s_{i}, s_{j})) : M → Sym_{p,q}(ℝ^{r}) ≅ GL(r)/O(p, q) = S(p, q)  (1.21)

associated to a parallel frame s = (s_{1}, …, s_{r}) of E with respect to the flat connection ∇ = D^{0} is pluriharmonic. Moreover, for all x ∈ M, the image of T_{x}^{1,0}M ⊂ (T_{x}M) ⊗ ℂ under the complex linear extension of dL_{u}^{−1}dG_{x} : T_{x}M → T_{o}S(p, q) = sym(p, q) consists of commuting matrices, where u ∈ GL(r) is any element such that G(x) = u · o and L_{u} : S(p, q) → S(p, q) is the isometry of S(p, q) induced by the left-multiplication by u in GL(r).

Conversely, let M be a simply connected complex manifold and f : M → Sym_{p,q}(ℝ^{r}) ≅ S(p, q) a pluriharmonic map such that, for all x ∈ M, the image of T_{x}^{1,0}M under the complex linear extension of dL_{u}^{−1}df_{x} : T_{x}M → T_{o}S(p, q) = sym(p, q) consists of commuting matrices, where u ∈ GL(r) is any element such that f(x) = u · o. Then there exists a metric tt*-bundle (E, D, S, g) over M and a parallel frame s such that f = G^{(s)}. The condition on the image of T_{x}^{1,0}M is automatically satisfied if pq = 0, which corresponds to a positive or negative definite metric g.

The same correspondence holds for oriented unimodular tt*-bundles and pluriharmonic maps into Sym_{p,q}^{1}(ℝ^{r}) ≅ SL(r)/SO(p, q) = S_{1}(p, q).

Now we shall explain in more detail the condition on the image of T^{1,0}M under the differential of f in the theorem. Above we have always identified Sym_{p,q}(ℝ^{r}) with S(p, q). Let us denote by

φ : Sym_{p,q}(ℝ^{r}) → S(p, q) , S ↦ S̃ = φ(S) ,  (1.22)

that identification, which is GL(r)-equivariant and maps I = I_{p,q} to the canonical base point o. We can identify the tangent space T_{S}Sym_{p,q}(ℝ^{r}) at S ∈ Sym_{p,q}(ℝ^{r}) with the (ambient) vector space of symmetric matrices:

T_{S}Sym_{p,q}(ℝ^{r}) = Sym(ℝ^{r}) := {A ∈ Mat(r, ℝ)|A^{t} = A} .  (1.23)

As above for S = I, the tangent space TS̃S(p, q) is canonically identified with the vector space of S-symmetric matrices:

TS̃S(p, q) = sym(S) := {A ∈ 𝔤𝔩(r)|A^{t}S = SA} .  (1.24)

Note that sym(I_{p,q}) = sym(p, q).

Proposition 4 The differential of φ at S ∈ Sym_{p,q}(ℝ^{r}) is given by

Sym(ℝ^{r}) ∋ X ↦ −\frac{1}{2}S^{−1}X ∈ S^{−1}Sym(ℝ^{r}) = sym(S) .  (1.25)

Let us now consider the differential

df_{x} : T_{x}M → Sym(ℝ^{r})  (1.26)

of f : M → Sym_{p,q}(ℝ^{r}) at x ∈ M and the differential

df̃_{x} : T_{x}M → sym(f(x))  (1.27)

of f̃ = φ ∘ f : M → S(p, q). Then the condition on the image of the differential of f in the theorem is that

dL_{u}^{−1}df̃(T_{x}^{1,0}M) ⊂ sym(p, q) ⊗ ℂ consists of commuting matrices ,  (1.28)

where f̃(x) = uo. This is equivalent to the condition that df̃(T_{x}^{1,0}M) ⊂ sym(f̃(x)) ⊗ ℂ consists of commuting matrices. This follows from the fact that

dL_{u} : T_{o}S(p, q) → T_{uo}S(p, q) = T_{f̃(x)}S(p, q)  (1.29)

corresponds to

Ad_{u} : sym(p, q) = sym(I) → sym(u · I) = sym(f̃(x))  (1.30)

and that the adjoint representation preserves the Lie bracket.

Finally, df̃_{x} = dφ df_{x} = −\frac{1}{2}f(x)^{−1}df_{x} and, therefore,

df̃(T_{x}^{1,0}M) = f(x)^{−1}df_{x}(T_{x}^{1,0}M) .  (1.31)

This shows that f satisfies the condition (1.28) if and only if the matrices f(x)^{−1}df_{x}(Z) and f(x)^{−1}df_{x}(W) commute for all Z, W ∈ T_{x}^{1,0}M. This is equivalent to

[f(x)^{−1}df_{x}(JX), f(x)^{−1}df_{x}(JY)] = [f(x)^{−1}df_{x}(X), f(x)^{−1}df_{x}(Y)]  (1.32)

for all X, Y ∈ T_{x}M.

2 Special complex and special Kähler manifolds

In this section we recall some basic results on special complex manifolds and special Kähler manifolds. For more detailed information the reader is referred to [ACD], see also [F].

Definition 3 A special complex manifold (M, J, ∇) is a complex manifold (M, J) endowed with a flat torsion-free connection ∇ (on the real tangent-bundle) such that ∇J is symmetric. A special Kähler manifold (M, J, ∇, ω) is a special complex manifold (M, J, ∇) endowed with a J-invariant and ∇-parallel symplectic form ω. The (pseudo)-Kähler-metric g(·, ·) = ω(J·, ·) is called the special Kähler metric of the special Kähler manifold (M, J, ∇, ω).

Given a complex manifold (M, J) with a flat connection ∇, we define its conjugate connection by

∇_{X}^{J} = ∇_{X} − J∇_{X}J with X ∈ T M.  (2.1)

On a special complex manifold (M, J, ∇) the connection ∇^{J} is torsion-free. In addition, one can introduce a torsion-free connection

D := \frac{1}{2}(∇ + ∇^{J}) = ∇ − S, where S := \frac{1}{2}J∇J,  (2.2)

which satifies DJ = 0, as follows from a short calculation. In the case of a special Kähler manifold (M, J, ∇, ω) the connection D is the Levi-Civita connection of the special Kähler metric g and the endomorphism-field S anticommutes with the complex structure J, i.e. :

JS_{X} = −S_{X}J for all X ∈ T M .  (2.3)

Now we explain part of the extrinsic construction of special Kähler-manifolds given in [ACD]. In order to do this, we consider the complex vector space V = T^{∗}ℂ^{n} = ℂ^{2n} with canonical coordinates (z^{1}, …, z^{n}, w_{1}, …, w_{n}) endowed with the standard complex symplectic form Ω = ∑_{i=1}^{n} dz^{i} ∧dw_{i} and the standard real structure τ : V → V with fixed points V^{τ} = T^{∗}ℝ^{n}. These define a Hermitian form γ := iΩ(·, τ·). Let (M, J) be a complex manifold (M, J) of complex dimension n. We call a holomorphic immersion ϕ : M → V nondegenerate (respectively Lagrangian) if ϕ^{∗}γ is nondegenerate (respectively, if ϕ^{∗}Ω = 0). If ϕ is nondegenerate it defines a, possibly indefinite, Kähler metric g = Re ϕ^{∗}γ on the complex manifold (M, J) and the corresponding Kähler form g(·, J·) is a J−invariant symplectic form.

The following theorem gives a description of simply connected special Kähler-manifolds in terms of the above data:

Theorem 2 [ACD] Let (M, J, ∇, ω) be a simply connected special Kähler manifold of complex dimension n, then there exists a holomorphic nondegenerate Lagrangian immersion ϕ : M → V = T^{∗}ℂ^{n} inducing the Kähler metric g, the connection ∇ and the symplectic form ω = 2ϕ^{∗} (∑_{i=1}^{n} dx^{i} ∧ dy_{i}) = g(·, J·) on M. Moreover, ϕ is unique up to an affine transformation of V preserving the complex symplectic form Ω and the real structure τ. The flat connection ∇ is completely determined by the condition ∇ϕ^{∗}dx^{i} = ∇ϕ^{∗}dy_{i} = 0, i = 1, …, n, where x^{i} = Re z^{i} and y_{i} = Re w_{i}.

3 Special complex and special Kähler manifolds as solutions of the tt*-equations

Let (E, D, S) be a tt*-bundle over a complex manifold (M, J). We are now interested in the case E = T M. In that case it is natural to consider tt*-bundles for which the connection D^{θ} = D + (cos θ)S + (sin θ)S_{J} is torsion-free.

Definition 4 A tt*-bundle (T M, D, S) over a complex manifold (M, J) is called special if D^{θ} is torsion-free and special, i.e. D^{θ}J is symmetric for all θ.

Proposition 5 A tt*-bundle (T M, D, S) is special if and only if D is torsion-free and DJ, S and S_{J} are symmetric.

Proof: The torsion T^{θ} of D^{θ} is given by

T^{θ}(X, Y) = T (X, Y) + cos θ(S_{X}Y − S_{Y} X) + sin θ(S_{JX}Y − S_{JY} X) ,  (3.1)

where T is the torsion of D. This shows that T^{θ} = 0 for all θ if and only if T = 0 and S and S_{J} are symmetric. The equation

(D_{X}^{θ}J)Y = (D_{X}J)Y + cos θ[S_{X}, J]Y + sin θ[S_{JX}, J]Y  (3.2)

shows that D^{θ}J is symmetric if DJ, S and S_{J} are symmetric. Conversely, if T^{θ} = 0 and D^{θ}J is symmetric, then, by the first part of the proof, S and S_{J} are symmetric and equation (3.2) shows that DJ is symmetric.

Theorem 3

(i) Let (M, J, ∇) be a special complex manifold. Put S := \frac{1}{2}J∇J and D := ∇ − S. Then (T M, D, S) is a special tt*-bundle, which satisfies the additional conditions:

a) S_{X}J = −JS_{X} for all X ∈ T M and

b) DJ = 0.

This defines a map Φ from special complex manifolds to special tt*-bundles.

(ii) Let (T M, D, S) be a special tt*-bundle over a complex manifold (M, J). Then (M, J, ∇ := D+S) is a special complex manifold. This defines a map Ψ from special tt*-bundles to special complex manifolds such that Ψ ∘ Φ = Id. If (T M, D, S) is a special tt*-bundle satisfying the conditions a) and b) in (i), then Φ(Ψ(T M, D, S)) = (T M, D, S).

(iii) Let (M, J, g, ∇) be a special Kähler manifold with S and D defined as in (i). Then (T M, D, S, g) is a special metric tt*-bundle. This defines a map Φ from special Kähler manifolds to special metric tt*-bundles.

(iv) Let (T M, D, S, g) be a special metric tt*-bundle over a pseudo-Hermitian manifold (M, J, g) satisfying the conditions a) and b) in (i). Then (M, J, ∇ := D + S, g) is a special Kähler manifold. In particular, we have a map Ψ from special metric tt*-bundles over pseudo-Hermitian manifolds satisfying a) and b) to special Kähler manifolds. Moreover, Ψ is a bijection and Ψ^{−1} = Φ.

(v) Let (T M, D, S, g) be a metric tt*-bundle over a pseudo-Hermitian manifold (M, J, g) satisfying a) and b) in (i) and such that D is torsion-free. Then it is special if and only if (M, J, ∇ := D + S, g) is a special Kähler manifold.

Proof: (i) Let (M, J, ∇) be a special complex manifold with S and D defined as above. Then

∇^{θ} = e^{θJ} ∘ ∇ ∘ e^{−θJ}  (3.3)

is a family of flat torsion-free special connections. Using ∇ = D + S and (2.3) we can write

∇_{X}^{θ} = D_{X} + e^{2θJ}S_{X} .  (3.4)

The following calculation shows that ∇^{θ} = D^{−2θ}, where D^{θ} is defined in (1.2):

∇_{X}^{θ} − D_{X} = e^{2θJ}S_{X} = cos(2θ)S_{X} + sin(2θ)JS_{X}
=^{(∗)} cos(−2θ)S_{X} + sin(−2θ)S_{JX} = D_{X}^{−2θ} − D_{X} , X ∈ T M .

At (∗) we have used that JS_{X} = −S_{JX}, which follows from

JS_{X}Y = JS_{Y} X = −S_{Y} JX = −S_{JX}Y , X, Y ∈ T_{p}M .  (3.5)

Here we used the symmetry of S and (2.3). This shows that (T M, D, S) is a special tt*-bundle.

(ii) Let (T M, D, S) be a special tt*-bundle. This means that D^{θ} is flat, torsion-free and special. In particular, ∇ = D + S = D^{0} is flat, torsion-free and special and (M, J, ∇) is a special complex manifold. It is clear that Ψ ∘ Φ = Id.

Conversely, let (T M, D, S) be a special tt*-bundle such that DJ = 0 and S_{X}J = −JS_{X} for all X. Then we can recover D and S from ∇ = D + S by the formulas S = \frac{1}{2}J∇J and D = ∇ − S. In fact, Let (T M, D′, S′) be an other special tt*-bundle over (M, J) such that D′J = 0 and S′_{X}J = −JS′_{X} for all X ∈ T M and ∇ = D + S = D′ + S′. Then

0 = D′_{X}J = ∇_{X}J − [S′_{X}, J] = ∇_{X}J + 2JS′_{X}  (3.6)

for all X ∈ T M. This shows that S′_{X} = \frac{1}{2}J∇_{X}J = S_{X} and D′ = ∇ − S′ = ∇ − S = D.

(iii) Let (M, J, g, ∇) be a special Kähler manifold with S and D defined as in (i). Then, by (i), (T M, D, S) is a special tt*-bundle and satifies a) and b). To prove that it is a metric tt*-bundle we have to check that Dg = 0 and that (1.3) is satisfied. Since DJ = 0, by b), the equation Dg = 0 is equivalent to the following claim:

Claim: The Kähler form ω is D-parallel; Dω = 0.

In fact ∇ω = 0 and S_{X} = \frac{1}{2}J∇_{X}J, X ∈ T M, is the product of two anticommuting ω-skew-symmetric endomorphisms A = \frac{1}{2}J and B = ∇_{X}J. This implies that S_{X} is ω-skew-symmetric and, thus, Dω = 0.

The endomorphism S_{X} is ω-skew-symmetric and anticommutes with J, by a). Therefore S_{X} is symmetric with respect to g = ω(J·, ·).

(iv) Let (T M, D, S, g) be a special metric tt*-bundle over a pseudo-Hermitian manifold (M, J, g) satisfying a) and b) in (i). Thanks to (ii), we know already that (M, J, ∇ := D + S) is a special complex manifold. Therefore it suffices to prove that ∇ω = 0. The assumption Dg = 0 and property b) imply that Dω = 0. Now it is sufficient to observe that the endomorphisms S_{X}, X ∈ T M, are ω-skew-symmetric. In fact, S_{X} is g-symmetric in virtue of (1.3) and anticommutes with J, by a). This shows that (M, J, ∇, g) is a special Kähler manifold. The remaining statements follow from (ii).

(v) Let (T M, D, S, g) be a metric tt*-bundle over a pseudo-Hermitian manifold (M, J, g) such that (M, J, ∇ := D + S, g) is a special Kähler manifold. If D is torsion-free, then it is the Levi-Civita connection of g and, thus, D = ∇ − \frac{1}{2}J∇J, see section 2. Now we can conclude that Φ(M, J, ∇, g) = (T M, D, S, g). This shows that (T M, D, S, g) is a special metric tt*-bundle. The converse follows from (iv).

Corollary 1 Any special metric tt*-bundle (T M, D, S, g) over a pseudo-Hermitian manifold (M, J, g) which satisfies a) and b) in Theorem 3 (i) is oriented and unimodular.

Proof: T M is canonically oriented by the complex structure J. By Theorem 3, (M, J, g, ∇ = D + S) is a special Kähler manifold. Its Kähler form is parallel with respect to D and ∇ and hence invariant under S_{X} = ∇_{X} −D_{X} for all X ∈ T M. This shows that tr S_{X} = 0.

In [H] special complex and special Kähler geometry is interpreted in terms of variations of Hodge structure of weight 1 on the complexified tangent bundle. From this interpretation and his discussion of tt*-geometry, it follows that any special complex (respectively, special Kähler) manifold defines a tt*-bundle (respectively, a metric tt*-bundle) in the sense of our definition.

4 The pluriharmonic map in the case of a special Kähler manifold

4.1 The Gauß maps of a special Kähler manifold

Let (M, J, g, ∇) be a special Kähler manifold of complex dimension n = k + l and of Hermitian signature (k, l), i.e. g has signature (2k, 2l). Let (M̃, J, g, ∇) be its universal covering with the pullback special Kähler structure, which is again denoted by (J, g, ∇). According to Theorem 2, there exists a (holomorphic) Kählerian Lagrangian immersion ϕ : M̃ → V = T^{∗}ℂ^{n} = ℂ^{2n}, which is unique up to a complex affine transformation of V with linear part in Sp(ℝ^{2n}). We consider the dual Gauß map of ϕ

L : M̃ → Gr_{0}^{k,l}(ℂ^{2n}) , p ↦ L(p) := T_{ϕ(p)}M̃:= dϕ_{p}T_{p}M̃ ⊂ V  (4.1)

into the Grassmannian of complex Lagrangian subspaces W ⊂ V of signature (k,l), i.e. such that the restriction of γ to W is a Hermitian form of signature (k, l). The map L : M̃ → Gr_{0}^{k,l}(ℂ^{2n}) is in fact the dual of the Gauß map

L^{⊥} : M̃ → Gr_{0}^{l,k}(ℂ^{2n}) , p ↦ L(p)^{⊥} = \overline{L(p)} ≅ L(p)^{∗} .  (4.2)

Here L(p)^{⊥} stands for the γ-orthogonal complement of L(p) and the isomorphism \overline{L(p)} ≅ L(p)^{∗} is induced by the symplectic form Ω on V = L(p) ⊕ \overline{L(p)}.

The Grassmannian Gr_{0}^{k,l}(ℂ^{2n}) is an open subset of the complex Grassmannian Gr_{0}(ℂ^{2n}) of complex Lagrangian subspaces W ⊂ V and hence a complex submanifold.

Proposition 6 (i) The dual Gauß map L : M̃ → Gr_{0}^{k,l}(ℂ^{2n}) is holomorphic

(ii) The Gauß map L^{⊥} : M̃ → Gr_{0}^{l,k}(ℂ^{2n}) is antiholomorphic.

Proof: The holomorphicity of L follows from that of ϕ. Part (ii) follows from (i), since L^{⊥} = \overline{L} : p ↦ \overline{L(p)}.

The real symplectic group Sp(ℝ^{2n}) acts transitively on Gr_{0}^{k,l}(ℂ^{2n}) and we have the following identification:

Gr_{0}^{k,l}(ℂ^{2n}) = Sp(ℝ^{2n})/U(k, l) .  (4.3)

Here U(k, l) ⊂ Sp(ℝ^{2n}) is defined as the stabilizer of

W_{o} = span\{\frac{∂}{∂z^{1}} + i\frac{∂}{∂w_{1}} , ⋯, \frac{∂}{∂z^{k}} + i\frac{∂}{∂w_{k}} , \frac{∂}{∂z^{k+1}} − i\frac{∂}{∂w_{k+1}} , ⋯, \frac{∂}{∂z^{n}} − i\frac{∂}{∂w_{n}} \} . (4.4)

The Gauß maps L and L^{⊥} induce Gauß maps

L_{M} : M → Γ \ Gr_{0}^{k,l}(ℂ^{2n})  (4.5)
L_{M}^{⊥} : M → Γ \ Gr_{0}^{l,k}(ℂ^{2n})  (4.6)

into the quotient of the Grassmannian by the holonomy group Γ = Hol(∇) ⊂ Sp(ℝ^{2n}) of the flat symplectic connection ∇.

Corollary 2 (i) The dual Gauß map L_{M} : M → Γ \ Gr_{0}^{k,l}(ℂ^{2n}) of M is holomorphic.

(ii) The Gauß map L_{M}^{⊥} : M → Γ \ Gr_{0}^{l,k}(ℂ^{2n}) is antiholomorphic.

The Grassmannian Gr_{0}^{k,l}(ℂ^{2n}) is a pseudo-Hermitian symmetric space and, in particular, a homogeneous pseudo-Kähler manifold. If Γ ⊂ Sp(ℝ^{2n}) acts properly discontinuously on Gr_{0}^{k,l}(ℂ^{2n}) then Γ \ Gr_{0}^{k,l}(ℂ^{2n}) is a locally symmetric space of pseudo-Hermitian type.

4.2 Holomorphic coordinates on the Grassmannian Gr_{0}^{k,l}(ℂ^{2n}) of complex Lagrangian subspaces of signature (k, l)

In this section we shall introduce a local model for the Grassmannian Gr_{0}^{k,l}(ℂ^{2n}) and determine the corresponding local expression for the dual Gauß map. This model is a pseudo-Riemannian analogue of the Siegel upper half-space

Sym^{+}(ℂ^{n}) := {A ∈ Mat(n, ℂ)|A^{t} = A and Im A is positive definite} .  (4.7)

Our aim is to construct holomorphic coordinates for the complex manifold Gr_{0}^{k,l}(ℂ^{2n}) in a Zariski-open neighborhood of a point W_{0} of the Grassmannian represented by a Lagrangian subspace W_{0} ⊂ V of signature (k, l). Using a transformation from Sp(ℝ^{2n}) we can assume that W_{0} = W_{o}, see (4.4). Let U_{0} ⊂ Gr_{0}^{k,l}(ℂ^{2n}) be the open subset consisting of W ∈ Gr_{0}^{k,l}(ℂ^{2n}) such that the projection

π_{(z)} : V = T^{∗}ℂ^{n} = ℂ^{n} ⊕ (ℂ^{n})^{∗} → ℂ^{n}  (4.8)

onto the first summand (z-space) induces an isomorphism

π_{(z)}|_{W} : W →^{∼} ℂ^{n} .  (4.9)

Notice that U_{0} ⊂ Gr_{0}^{k,l}(ℂ^{2n}) is an open neighborhood of the base point W_{0}. For elements W ∈ U_{0} we can express w_{i} as a function of z = (z^{1}, …, z^{n}). In fact,

w_{i} = ∑ C_{ij}z^{j} ,  (4.10)

where

(C_{ij}) ∈ Sym_{k,l}(ℂ^{n}) = {A ∈ Mat(n, ℂ)|A^{t} = A and Im A has signature (k, l)} . (4.11)

Proposition 7 The map

C : U_{0} → Sym_{k,l}(ℂ^{n}) , W ↦ C(W) := (C_{ij})  (4.12)

is a local holomorphic chart for the Grassmannian Gr_{0}^{k,l}(ℂ^{2n}).

Remark: The open subset Sym_{k,l}(ℂ^{n}) ⊂ Sym(ℂ^{n}) = {A ∈ Mat(n, ℂ)|A^{t} = A} is a generalization of the famous Siegel upper half-space Sym_{n,0}(ℂ^{n}) = Sym^{+}(ℂ^{n}), which is a Siegel domain of type I. In the latter case, we have U_{0} = Sp(ℝ^{2n})/U(n) and a global coordinate chart

C : Gr_{0}^{n,0}(ℂ^{2n}) = Sp(ℝ^{2n})/U(n) →^{∼} Sym_{n,0}(ℂ^{n}) .  (4.13)

We shall now describe the dual Gauß map L in local holomorphic coordinates in neighborhoods of p_{0} ∈ M̃ and L(p_{0}) ∈ Gr_{0}^{k,l}(ℂ^{2n}). Applying a transformation from Sp(ℝ^{2n}), if necessary, we can assume that L(p_{0}) ∈ U_{0}. We put U := L^{−1}(U_{0}). The open subset U ⊂ M̃ is a neighborhood of p_{0}.

Let ϕ : M̃ → T^{∗}ℂ^{n} be the Kählerian Lagrangian immersion. It defines a system of local (special) holomorphic coordinates

φ := π_{(z)} ∘ ϕ|_{U} : U →^{∼} U′ ⊂ ℂ^{n} , p ↦ (z^{1}(ϕ(p)), ⋯, z^{n}(ϕ(p)))  (4.14)

and we have the following commutative diagram

\begin{matrix}U & ⟶^{L} & U_{0} \\ φ ↓ & & ↓ C \\ U′ & ⟶^{L_{U}} & Sym_{k,l}(ℂ^{n}) , \end{matrix}  (4.15)

where the vertical arrows are holomorphic diffeomorphisms and L_{U} at z = (z^{1}, … z^{n}) is given by

L_{U}(z) = (F_{ij}(z)) := (\frac{∂^{2}F (z)}{∂z^{i}∂z^{j}}) .  (4.16)

Here F = F (z) is a holomorphic function on U′ ⊂ ℂ^{n} determined, up to a constant, by the equations

w_{j}(ϕ(p)) = \frac{∂F}{∂z^{j}} |_{z(ϕ(p))} .  (4.17)

Summarizing, we obtain the following proposition.

Proposition 8 The dual Gauß map L has the following coordinate expression

L_{U} = C ∘ L ∘ φ^{−1} = (F_{ij}) ,  (4.18)

where φ : U → ℂ^{n} is the (special) holomorphic chart of M̃ associated to the Kählerian Lagrangian immersion ϕ, see (4.14), and C : U_{0} → Sym(ℂ^{n}) is the holomorphic chart of Gr_{0}^{k,l}(ℂ^{2n}) constructed in (4.12).

4.3 The special Kähler metric in affine coordinates

As before, let (M, J, g, ∇) be a special Kähler manifold of Hermitian signature (k, l), k + l = n = dim_{ℂ} M, and (M̃, J, g, ∇) its universal covering. As in section 1, we shall now consider the metric g in a ∇-parallel frame. Such a frame is provided by the Kählerian Lagrangian immersion ϕ : M̃ → V. In fact, any point p ∈ M̃ has a neighborhood in which the functions x̃^{i} := Re z^{i} ∘ ϕ, ỹ_{i} := Re w_{i} ∘ ϕ, i = 1, …, n, form a system of local ∇-affine coordinates. We recall that the ∇-parallel Kähler form is given by ω = 2 ∑ dx̃^{i} ∧ dỹ_{i}. This implies that the globally defined one-forms \sqrt{2}dx̃^{i}, \sqrt{2}dỹ_{i} constitute a ∇-parallel unimodular frame

(e^{a})_{a=1,...,2n} = (e^{1}, …, e^{2n}) := (\sqrt{2}dx̃^{1}, …, \sqrt{2}dx̃^{n}, \sqrt{2}dỹ_{1}, …, \sqrt{2}dỹ_{n})  (4.19)

of T^{∗}M̃ with respect to the metric volume form ν = (−1)^{n+1} \frac{ω^{n}}{n!} = 2^{n}dx̃^{1} ∧ … ∧ dỹ_{n}. The dual frame (e_{a}) of T M̃ is also ∇-parallel and unimodular. The metric defines a smooth map

G : M̃ → Sym_{2k,2l}^{1}(ℝ^{2n}) = {A ∈ Mat(2n, ℝ)|A^{t} = A , det A = 1 has signature (2k, 2l)}

(4.20)

by

p ↦ G(p) := (g_{ab}(p)) := (g_{p}(e_{a}, e_{b})) .  (4.21)

We will call G = (g_{ab}) the fundamental matrix of ϕ. As before, we identify

Sym_{2k,2l}^{1}(ℝ^{2n}) = SL(2n, ℝ)/SO(2k, 2l) .  (4.22)

This is a pseudo-Riemannian symmetric space. For conventional reasons, in this section, SO(2k, 2l) ⊂ SL(2n, ℝ) is defined as the stabilizer of the symmetric matrix

E_{o} := diag(𝟙_{k}, −𝟙_{l}, 𝟙_{k}, −𝟙_{l}) .  (4.23)

The fundamental matrix induces a map

G_{M} : M → Γ \ Sym_{2k,2l}^{1}(ℝ^{2n})  (4.24)

into the quotient of Sym_{2k,2l}^{1}(ℝ^{2n}) by the action of the holonomy group Γ = Hol(∇) ⊂ Sp(ℝ^{2n}) ⊂ SL(2n, ℝ). The target Γ \ Sym_{2k,2l}^{1}(ℝ^{2n}) is a pseudo-Riemannian locally symmetric space, provided that Γ acts properly discontinuously.

Theorem 4 The fundamental matrix G : M̃ → Sym_{2k,2l}^{1}(ℝ^{2n}) = SL(2n, ℝ)/SO(2k, 2l) takes values in the totally geodesic submanifold

ι : Gr_{0}^{k,l}(ℂ^{2n}) = Sp(ℝ^{2n})/U(k, l) ↪ SL(2n, ℝ)/SO(2k, 2l)  (4.25)

and coincides with the dual Gauß map L : M̃ → Gr_{0}^{k,l}(ℂ^{2n}): G = ι ∘ L.

Proof: The proof will follow from a geometric description of the inclusion ι. To any Lagrangian subspace W ∈ Gr_{0}^{k,l}(ℂ^{2n}) we can associate the scalar product g^{W} := Re γ|_{W} of signature (2k, 2l) on W ⊂ V. The projection onto the real points

Re : V = T^{∗}ℂ^{n} → T^{∗}ℝ^{n} = ℝ^{2n} , v ↦ Re v = \frac{1}{2}(v + \overline{v})  (4.26)

induces an isomorphism of real vector spaces W →^{∼} ℝ^{2n} the inverse of which we denote by ψ = ψ_{W}. We claim that

ι(W) = ψ^{∗}g^{W} =: (g_{ab}^{W}) =: G^{W} .  (4.27)

To check this, it is sufficient to prove that the map

Gr_{0}^{k,l}(ℂ^{2n}) ∋ W ↦ G^{W} ∈ Sym_{2k,2l}^{1}(ℝ^{2n})  (4.28)

is Sp(ℝ^{2n})-equivariant and maps the base point W_{o} with stabilizer U(k, l), see (4.4), to the base point E_{o} with stabilizer SO(2k, 2l), see (4.23). Let us verify that indeed G^{W_{o}} = E_{o}.

Using the definition of γ, one finds for the basis

(e_{j}^{±}) := (\frac{∂}{∂z^{j}} ± i\frac{∂}{∂w_{j}})  (4.29)

of V that the only non-vanishing components of γ are γ(e_{j}^{±}, e_{j}^{±}) = ±2. This shows that g^{W_{o}} = Re γ|_{W_{o}} is represented by the matrix 2E_{o} with respect to the basis

(e_{1}^{+}, …, e_{k}^{+}, e_{1}^{−}, …, e_{l}^{−}, ie_{1}^{+}, …, ie_{k}^{+}, ie_{1}^{−}, …, ie_{l}^{−}) .  (4.30)

In order to calculate G^{W_{o}} = (g_{ab}^{W_{o}}) = (g(ψe_{a}, ψe_{b})), we need to pass from the real basis (4.30) of W_{o} to the real basis (ψe_{a}).

Recall that the real structure τ is complex conjugation with respect to the coordinates (z^{i}, w_{i}). This implies that

ψ^{−1}(e_{j}^{+}) = \frac{∂}{∂x^{j}} = \sqrt{2}e_{j} , ψ^{−1}(ie_{j}^{+}) = −\frac{∂}{∂y_{j}} = −\sqrt{2}e_{n+j} , j = 1, …, k, (4.31)
ψ^{−1}(e_{j}^{−}) = \frac{∂}{∂x^{j}} = \sqrt{2}e_{j} , ψ^{−1}(ie_{j}^{−}) = \frac{∂}{∂y_{j}} = \sqrt{2}e_{n+j} , j = 1, …, l . (4.32)

This shows that G^{W_{o}} = E_{o}.

It remains to check the equivariance of W ↦ G^{W} = ψ^{∗}g. Using the definition of the map ψ = ψ_{W} : ℝ^{2n} → W, one easily checks that, under the action of Λ ∈ Sp(ℝ^{2n}), ψ transforms as

ψ_{ΛW} = Λ ∘ ψ_{W} ∘ Λ^{−1}|_{ℝ^{2n}} .  (4.33)

From this we deduce the transformation law of G^{W}:

G^{ΛW} = ψ_{ΛW}^{∗} g^{ΛW} = (Λ^{−1})^{∗}ψ_{W}^{∗} Λ^{∗}g^{ΛW} = (Λ^{−1})^{∗}ψ_{W}^{∗} g^{W} = (Λ^{−1})^{∗}G^{W} = Λ · G^{W} . (4.34)

The above claim (4.27), together with the fact that

g^{L(p)} = g_{p} and G^{L(p)} = G(p)  (4.35)

for all p ∈ M̃, implies that

ι(L(p)) = G^{L(p)} = G(p) .  (4.36)

Corollary 3 The fundamental matrix G : M̃ → Sym_{2k,2l}^{1}(ℝ^{2n}) is pluriharmonic.

Proof: G = ι∘L is the composition of the holomorphic map L : M̃ → Gr_{0}^{k,l}(ℂ^{2n}) with the totally geodesic inclusion Gr_{0}^{k,l}(ℂ^{2n}) ⊂ Sym_{2k,2l}^{1}(ℝ^{2n}). The composition of a holomorphic map with a totally geodesic map is pluriharmonic.

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[S1] L. Schäfer, tt*-Geometrie und pluriharmonische Abbildungen, Diplomarbeit an der Universität Bonn, December 2002.

[S2] L. Schäfer, tt*-bundles and pluriharmonic maps, to appear.