Conjectural Existence of the Categorial Complex Branes for Generalized Calabi-Yau.

Geometric Langlands Duality can be formulated as follows: Let C be a Riemann surface (compact, without boundary), G be a compact reductive Lie group, GC be its complexification, and Mflat(G, C) be the moduli space of stable flat GC-connections on C. The Langlands dual of G is another compact reductive Lie group LG defined by the condition that its weight and coweight lattices are exchanged relative to G. Let Bun(LG, C) be the moduli stack of holomorphic LG-bundles on C. One of the statements of Geometric Langlands Duality is that the derived category of coherent sheaves on Mflat(G, C) is equivalent to the derived category of D-modules over Bun(LG, C).

Mflat(G, C) is mirror to another moduli space which, roughly speaking, can be described as the cotangent bundle to Bun(LG, C). The category of A-branes on T Bun(LG, C) (with the canonical symplectic form) is equivalent to the category of B-branes on a noncommutative deformation of T Bun(LG, C). The latter is the same as the category of (analytic) D-modules on Bun(LG, C).

So, what exactly is, the relationship between A-branes and noncommutative B-branes. This relationship arises whenever the target space X is the total space of the cotangent bundle to a complex manifold Y. It is understood that the  symplectic form ω is proportional to the canonical symplectic form on T Y. With the B-field vanishing, and Y as a complex, we regard ω as the real part of a holomorphic symplectic form Ω. If qi are holomorphic coordinates on Y, and pi are dual coordinates on the fibers of T Y,  Ω can be written as

Ω = 1/ħdpi ∧ dqi = dΘ

Since ω (as well as Ω) is exact, the closed A-model of X is rather trivial: there are no nontrivial instantons, and the quantum cohomology ring is isomorphic to the classical one.

We would like to understand the category of A-branes on X = T Y. The key observation is that ∃ a natural coisotropic A-brane on X well-defined up to tensoring with a flat line bundle on X. Its curvature 2-form is exact and given by

F = Im Ω

If we denote by I the natural almost complex structure on X coming from the complex structure on Y , we have F = ωI, and therefore the endomorphism ω−1F = I squares to −1. Therefore any unitary connection on a trivial line bundle over X whose curvature is F defines a coisotropic A-brane. 

Now, what about the endomorphisms of the canonical coisotropic A-brane, i.e., the algebra of BRST-closed open string vertex operators? This is easy if Y is an affine space. If one covers Y with charts each of which is an open subset of Cn, and then argues that the computation can be performed locally on each chart and the results “glued together”, one gets closer to the fact that the algebra in question is the cohomology of a certain sheaf of algebras, whose local structure is the same as for Y = Cn. In general, the path integral defining the correlators of vertex operators does not have any locality properties in the target space. Each term in perturbation theory depends only on the infinitesimal neighbourhood of a point. This shows that the algebra of open-string vertex operators, regarded as a formal power series in ħ, is the cohomology of a sheaf of algebras, which is locally isomorphic to a similar sheaf for X = Cn × Cn.

Let us apply these observations to the canonical coisotropic A-brane on X = T Y. Locally, we can identify Y with a region in Cn by means of holomorphic coordinate functions q1, . . . , qn. Up to BRST-exact terms, the action of the A-model on a disc Σ 􏰠takes the form

S = 1/ħ ∫∂Σ φ (pidqi)

where φ is a map from Σ to X. This action is identical to the action of a particle on Y with zero Hamiltonian, except that qi are holomorphic coordinates on Y rather than ordinary coordinates. The BRST-invariant open-string vertex operators can be taken to be holomorphic functions of p, q. Therefore quantization is locally straightforward and gives a noncommutative deformation of the algebra of holomorphic functions on T Y corresponding to a holomorphic Poisson bivector

P = ħ∂/∂pi ∧ ∂/∂qi

One can write an explicit formula for the deformed product:

􏰋(f ⋆ g)(p, q) = exp(􏰋ħ/2(∂2/∂pi∂q̃i  −  ∂2/∂qi∂p̃i )) f(p, q) g (p̃, q̃)|p̃ = p, q̃ = q

This product is known as the Moyal-Wigner product, which is a formal power series in ħ that may have zero radius of convergence. To rectify the situation, one can restrict to functions which are polynomial in the fiber coordinates pi. Such locally-defined functions on T Y can be thought of as symbols of differential operators; the Moyal-Wigner product in this case reduces to the product of symbols and is a polynomial in ħ. Thus locally the sheaf of open-string vertex operators is modelled on the sheaf of holomorphic differential operators on Y (provided we restrict to operators polynomial in pi).

Locally, there is no difference between the sheaf of holomorphic differential operators D(Y ) and the sheaf of holomorphic differential operatorsD(Y, L) on a holomorphic line bundle L over Y. Thus the sheaf of open-string vertex operators could be any of the sheaves D(Y, L). Moreover, the classical problem is symmetric under pi → −pi combined with the orientation reversal of Σ; if we require that quantization preserve this symmetry, then the algebra of open-string vertex operators must be isomorphic to its opposite algebra. It is known that the opposite of the sheaf D(Y, L) is the sheaf D(Y, L−1 ⊗ KY), so symmetry under pi → −pi requires L to be a square root of the canonical line bundle KY. It does not matter which square root one takes, since they all differ by flat line bundles on Y, and tensoring L by a flat line bundle does not affect the sheaf D(Y, L). The conclusion is that the sheaf of open-string vertex operators for the canonical coisotropic A-brane α on X = T Y is isomorphic to the sheaf of noncommutative algebras D(Y, K1/2). One can use this fact to associate Y to any A-brane β on X a twisted D-module, i.e., a sheaf of modules over D(Y, K1/2). Consider the A-model with target X on a strip Σ = I × R, where I is a unit interval, and impose boundary conditions corresponding to branes α and β on the two boundaries of Σ. Upon quantization of this model, one gets a sheaf on vector spaces on Y which is a module over the sheaf of open-string vertex operators inserted at the α boundary. A simple example is to take β to be the zero section of T Y with a trivial line bundle. Then the corresponding sheaf is simply the sheaf of sections of KY1/2, with a tautological action of D(Y, KY1/2).

One can argue that the map from A-branes to (complexes of) D-modules can be extended to an equivalence of categories of A-branes on X and the derived category of D-modules on Y. The argument relies on the conjectural existence of the category of generalized complex branes for any generalized Calabi-Yau. One can regard the Geometric Langlands Duality as a nonabelian generalization. 

Philosophical Identity of Derived Correspondences Between Smooth Varieties.

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Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γf ⊂ X × Y of f, which (as a set) is defined as

Γf = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections pX, pY from X × Y to the factors X, Y. Restricted to the subvariety Γf, pX is an isomorphism (since f is a morphism). The fibres of pY restricted to Γf are just the fibres of f; so for example f is proper iff pY | Γf is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f : H(X) → H(Y)

This map can be expressed in terms of the graph Γf and the projection maps as

f(α) = pY∗ (pX(α) ∪ [Γf]) —– (2)

where [Γf] ∈ H (X × Y) is the fundamental class of the subvariety [Γf]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that pX be an isomorphism with some weaker assumption, such as pXf, pY | Γf finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf OΓ on X × Y. Associated to the projection maps pX, pY, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ Db(X × Y) with support which is proper over both factors. A derived correspondence defines a functor ΦF by

ΦF : Db(X) → Db(Y)
(−) ↦ RpY∗(LpX(−) ⊗L F)

where (−) could refer to both objects and morphisms in Db(X). F is sometimes called the kernel of the functor ΦF.

The functor ΦF is exact, as it is defined as a composite of exact functors. Since the projection pX is flat, the derived pullback LpX is the same as ordinary pullback pX. Given derived correspondences E ∈ Db(X × Y), F ∈ Db(Y × Z), we obtain functors Φ: Db(X) → Db(Y), Φ: Db(Y) → Db(Z), which can then be composed to get a functor

ΦF ◦ Φ: Db(X) → Db(Z)

which is a two-sided identity with respect to composition of kernels.

Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.

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A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way.

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If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0

….

Lie-Dragging Sections Vectorially. Thought of the Day 149.0

Generalized vector fields over a bundle are not vector fields on the bundle in the standard sense; nevertheless, one can drag sections along them and thence define their Lie derivative. The formal Lie derivative on a bundle may be seen as a generalized vector field. Furthermore, generalized vector fields are objects suitable to describe generalized symmetries.

Let B = (B, M, π; F) be a bundle, with local fibered coordinates (xμ; yi). Let us consider the pull-back of the tangent bundle  τB: TB → B along the map πk0: JkB → B:

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A generalized vector field of order k over B is a section Ξ of the fibre bundle π: πk*TB → JkB, i.e.

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for each section σ: M → B, one can define Ξσ = i ○ Ξ ○ jkσ: M → TB, which is a vector field over the section σ. Generalized vector fields of order k = 0 are ordinary vector fields over B. Locally, Ξ(xμ, yi, …, yiμ1,…μk) is given the form:

Ξ = ξμ(xμ, yi, …, yiμ1,…μk)∂μ + ξi(xμ, yi, …, yiμ1,…μk)∂i

which, for k ≠ 0, is not an ordinary vector field on B due to the dependence of the components (ξμ, ξi) on the derivative of fields. Once one computes it on a section σ, then the pulled-back components depend just on the basic coordinates (xμ) so that Ξσ is a vector field over the section σ, in the standard sense. Thus, generalized vector fields over B do not preserve the fiber structure of B.

A generalized projectable vector field of order k over the bundle B is a generalized vector field Ξ over B which projects on to an ordinary vector field ξ = ξμ(x)∂μ on the base. Locally, a generalized projectable vector field over B is in the form:

Ξ = ξμ(xμ)∂μ + ξi(xμ, yi, …, yiμ1,…μk)∂i

As a particular case, one can define generalized vertical vector fields (of order k) over B, which are locally of the form:

Ξ = ξi(xμ, yi, …, yiμ1,…μk)∂i

In particular, for any section σ of B and any generalized vertical vector field Ξ over B, one can define a vertical vector field over σ given by:

Ξσ = ξi(xμ, σi(x),…, ∂μ1,…, μkσi(x))∂i

If Ξ = ξμμ + ξii is a generalized projectable vector field, then Ξ(v) = (ξi – yiμξμ)∂i = ξi(v)i is a generalized vertical vector field, where Ξ(v) is called the vertical part of Ξ.

If σ’: ℜ x M → B is a smooth map such that for any fixed s ∈ ℜ σs(x) = σ'(s, x): M → B is a global section of B. The map σ’ as well as the family {σs}, is then called a 1-parameter family of sections. In other words, a suitable restriction of the family σs, is a homotopic deformation with s ∈ ℜ of the central section σ = σ0. Often one restricts it to a finite (open) interval, conventionally (- 1, 1) (or (-ε, ε) if “small” deformations are considered). Analogous definitions are given for the homotopic families of sections over a fixed open subset U ⊆ M or on some domain D ⊂ M (possibly with values fixed at the boundary ∂D, together with any number of their derivatives).

A 1-parameter family of sections σs is Lie-dragged along a generalized projectable vector field Ξ iff

(v))σs = d/ds σs

thus dragging the section.

Principal Bundles Preserve Structures…

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A bundle P = (P, M ,π; G) is a principal bundle if the standard fiber is a Lie group G and ∃ (at least) one trivialization the transition functions of which act on G by left translations Lg : G → G : h ↦ f  g . h (where . denotes here the group multiplication).

The principal bundles are slightly different from affine bundles and vector bundles. In fact, while in affine bundles the fibers π-1(x) have a canonical structure of affine spaces and in vector bundles the fibers π-1(x) have a canonical structure of vector spaces, in principal bundles the fibers have no canonical Lie group structure. This is due to the fact that, while in affine bundles transition functions act by means of affine transformations and in vector bundles transition functions act by means of linear transformations, in principal bundles transition functions act by means of left translations which are not group automorphisms. Thus the fibers of a principal bundle do not carry a canonical group structure, but rather many non-canonical (trivialization-depending) group structures. In the fibers of a vector bundle there exists a preferred element (the “zero”) the definition of which does not depend on the local trivialization. On the contrary, in the fibers of a principal bundle there is no preferred point which is fixed by transition functions to be selected as an identity. Thus, while in affine bundles affine morphisms are those which preserve the affine structure of the fibers and in vector bundles linear morphisms are the ones which preserve the linear structure of the fibers, in a principal bundle P = (P, M, π; G) principal morphisms preserve instead a structure, the right action of G on P.

Let P = (P, M, π; G) be a principal bundle and {(Uα, t(α)}α∈I a trivialization. We can locally consider the maps

R(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, h . g](α) —– (1)

∃ a (global) right action Rg of G on P which is free, vertical and transitive on fibers; the local expression in the given trivialization of this action is given by R(α)g .

Using the local trivialization, we set p = [x, h](α) = [x, g(βα)(x) . h]β following diagram commutes:

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which clearly shows that the local expressions agree on the overlaps Uαβ, to define a right action. This is obviously a vertical action; it is free because of the following:

Rgp = p => [x, h . g](α) = [x, h](α) => h · g = h => g = e —– (2)

Finally, if p = [x, h1](α) and q = [x, h2](α) are two points in the same fiber of p, one can choose g = h2-1 . h1 (where · denotes the group multiplication) so that p = Rgq. This shows that the right action is also transitive on the fibers.

On the contrary, that a global left action cannot be defined by using the local maps

L(α)g : π-1(Uα) → π-1(Uα) : [x, h](α) ↦ [x, g . h](α) —– (3)

since these local maps do not satisfy a compatibility condition analogous to the condition of the commuting diagram.

let P = (P, M, π; G) and P’ = (P’, M’, π’ ; G’ ) be two principal bundles and θ : G → G’ be a homomorphism of Lie groups. A bundle morphism Φ = (Φ, φ) : P → P’ is a principal morphism with respect to θ if the following diagram is commutative:

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When G = G’ and θ = idG we just say that Φ is a principal morphism.

A trivial principal bundle (M x G, M, π; G) naturally admits the global unity section I ∈ Γ(M x G), defined with respect to a global trivialization, I : x ↦ (x, e), e being the unit element of G. Also, principal bundles allow global sections iff they are trivial. In fact, on principal bundles there is a canonical correspondence between local sections and local trivializations, due to the presence of the global right action.

Local Lifts into Period Domains: Holonomies: Philosophies of Conjugacy. Part 2.

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Let F = GC/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ gC and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.

Every flag manifold fibres over an inner symmetric space. Conversely, every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us now see how this story relates to the geometry of J(N).

So let p : F → N be a canonical fibration. By construction, the fibres of p are complex submanifolds of F and this allows us to define a fibre map ip : F → J(N) as follows: at f ∈ F we have an orthogonal splitting of TfF into horizontal and vertical subspaces both of which are invariant under the complex structure of F. Then dp restricts to give an isomorphism of the horizontal part with Tp(f)N and therefore induces an almost Hermitian structure on Tp(f)N : this is ip(f) ∈ Jp(f)N. Such a construction is possible whenever we have a Riemannian submersion of a Hermitian manifold with complex submanifolds as fibres.

ip : F → J(N) is a G-equivariant holomorphic embedding. This implies that ip (F) is an almost complex submanifold of J(N) on which J is integrable.

If j ∈ Z ⊂ J(N) then G · j is a flag manifold canonically fibred over N. In fact, G · j = ip(F ) for some canonical fibration p : F → N of a flag manifold F .

For this, the main observation is the following: at π(j), we have the symmetric decomposition g = k ⊕ q

with q ≅ Tπ(j)N. If q is the (0,1)-space for j then [q, q] ⊕ q

is the nilradical of a parabolic subalgebra p, where G · j is equivariantly biholomorphic to the corresponding flag manifold GC/P. Each canonical fibration of a flag manifold gives rise to a G-orbit in Z for some inner symmetric G-space N and that all such orbits arise in this way. But, for fixed G, there are only a finite number of biholomorphism types of flag manifold (they are in bijective correspondence with the conjugacy classes of parabolic subalgebras of gC) and each flag manifold admits but a finite number of canonical fibrations. Thus Z is composed of a finite number of G-orbits all of which are closed. In this way, we obtain a geometric interpretation of the purely algebraic construction of the canonical fibrations: they are just the restrictions of the projection π : J(N) → N to the various realisations of F as an orbit in Z.

For each non-compact real form GR of a complex semisimple group Lie group GC, there is a unique Riemannian symmetric space GR/K of non-compact type. The corresponding involution is called the Cartan involution of GR. Consider now the orbits of such a GR on the various flag manifolds F = GC/P. Those orbits which are open subsets of F are called flag domains: an orbit is a flag domain precisely when the stabilisers contain a compact Cartan subgroup of GR. It turns out that the presence of this compact Cartan subgroup is precisely what we need to define a canonical element of gR and thus an involution of gR just as in the compact case. However the involution is not necessarily a Cartan involution (i.e. the associated symmetric space need not be Riemmanian). In case that the involution is a Cartan involution, the flag domain is a canonical flag domain which is then exponentiated such that the involution gets to a Riemannian symmetric space of non-compact type and a canonical fibration of canonical flag domain over it.