_{1}

^{*}

Some derived categories and their deformed versions are used to develop a theory of the ramifications of field studied in the geometrical Langlands program to obtain the correspondences between moduli stacks and solution classes represented cohomologically under the study of the kernels of the differential operators studied in their classification of the corresponding field equations. The corresponding D-modules in this case may be viewed as sheaves of conformal blocks (or co-invariants) (images under a version of the Penrose transform) naturally arising in the framework of conformal field theory. Inside the geometrical Langlands correspondence and in their cohomological context of strings can be established a framework of the space-time through the different versions of the Penrose transforms and their relation between them by intertwining operators (integral transforms that are isomorphisms between cohomological spaces of orbital spaces of the space-time), obtaining the functors that give equivalences of their corresponding categories.(For more information,please refer to the PDF version.)

The extensions given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic ^{1}.

In more general sense, the conjectured to the group

First, the election of the derived sheaves to one theory of sheaf cohomology on ^{2}

Let

considering the moduli space as base,

If we consider

where

From the perspective of the Zuckerman functors produced from the Penrose transform, the kernels associated with the

Proposition 2. 1. (F. Recillas). The equations with non-flat differential operators can be solved by the corresponding Szegö kernels associated with Harish-Chandra modules [

Proof. Some results of representation theory obtained by the seminar of representation theory of real reductive Lie groups IM/UNAM (2000-2007) [

One geometrical argument is the condition established in the kernel of equivalences inside the moduli space^{3} associated to

Theorem (F. Bulnes). 2. 1 [

After we generalize the functors

with the correspondence rule given as:

We can enounce the following theorem explained and proved in [

Theorem 3.2 (F. Bulnes). The derived category of quasi-

Proof. [

Then as example of some consequences that derive of the classification of differential operators proposed by the theorem 3.2, to the solution of the corresponding field equations (using the philosophy of Langlands program to field ramifications) is the following short ^{4} to ramifications:

If we use the topological gauge theory through of a scheme on Stein manifolds of a complex Riemannian manifold of the space-time, using the generalization given by Gindikin conjectures formulated in the Section 7 in [

Theorem 4.1 (F. Bulnes). Consider the classes of Hecke category

. One short table of some Penrose transforms and their geometrical picture

# | |||
---|---|---|---|

Penrose Transform | Geometrical Langlands Correspondences (Cycles and Co-cycles) | Geometrical Picture | |

1 | Penrose-Radon Transform | CY-Manifolds | |

2 | Penrose-Ward Transform^{a} | Hyper-plane Twistor Space | |

3 | Classic Penrose Transform | Celestial Spheres | |

4 | Penrose-Schmid Transform | ||

5 | Classic Penrose Transform | Space-Time (Minkowski Space) | |

5 | Radon Transform | Hyperbolic Disk |

a. The Penrose-Ward transform for this case maps very naturally 1-gerbes over the hyper-plane twistor space to solutions to the self-dual string equation.

Langlands correspondences given by

(as was established in the Theorem 4.1 [

a) Orbi-folds or,

b) Strings as twisted hyperlines and twisted hyper-planes or,

c) Super-twistor surfaces (from complex 2- and 3-dimension spaces).

Proof. The functor given in the theorem 3.2,

where ^{6}. To the twisted Hecke category and electing a correct character, these centers are Drinfeld centers in derived algebraic geometry [

Finally, using the proper to generalized flag manifolds that appear in the ^{7} (which to certain conditions can be Grothendieck alteration [^{8}

where these images are co-cycles of the ramifications of the category

The obtaining of a first approaching through the integral geometry methods of the different geometrical pictures that defines the different actions of loop group

through the equivalences of categories

equivariant

“The true source of the transformations and determination of all field interactions in the space-time born from a field that can be ramified under the same scheme of connections that involves the Deligne connection adding other connection on singularities (that is to say, of other secondary sources) to certain

I grateful the invitation offered for the “Sophus Lie” Conference Center, in Norway and the Engineering Information Institute of China to give a cycle of talks related with the contents of this mathematical research paper. Also I grateful the financing and moral supports to realize the present research work to the director of TESCHA, Demetrio Moreno Arcega, M. in L.