https://matpitka.blogspot.com/2024/09/extension-of-langlands-geometric.html

## Tuesday, September 03, 2024

### Extension of Langlands geometric duality to trinity involving also physics-geometry correspondence

The master formula for TGD allowing construction of quantum states using the interpretation of space-time surfaces as numbers realizes the analog of geometric Langlands duality and generalizes it to a trinity. Geometric Langlands correspondence assigns to a pair of elements of a function field, which is a number theoretic object, a geometric object as algebraic surface having interpretation also as a Riemann surface with K\"ahler structure, twistor structure and spinor structure. This extends the number-theory-algebraic geometry duality to trinity and physics becomes the third part of a trinity.
1. The most high level form of number theory corresponds to function fields, which are infinite-D structured. In TGD, the pairs (f1,f2) of two functions of generalized complex coordinates of H=M4×CP2 define a linear space and the functions fi are elements of a function field. This is the number theoretic side of the Langlands geometric duality.
2. A function pair, whose root (f1,f2)=(0,0) defines a space-time surface in H and induces the number field structure of the function field to the space of space-time surfaces, "world of classical worlds" (WCW). Basic arithmetic operations of the number field apply to the component functions fi and induce corresponding operations for space-time surfaces in WCW. The notion of induction, which is the basic principle of TGD, is central also here. It is missing from standard physics and also string models.
3. The root as a space-time surface obeys holography =holomorphy principle and is a minimal surface (as classical representation of generalized massless particle and massless field equations) and represents the geometry side of the geometric Langlands duality. This connection represents geometric Langlands duality in TGD. Riemannian geometries restricted to algebraic geometries is what makes the geometric Langlands duality possible.

It is still unclear whether the choice of the classical action defining space-time surfaces and producing, apart from singularities, a minimal surface as an outcome, is only analogous to a choice of the coordinates and whether the recent choice (volume action + Kaehler action) is only the most convenient choice. If so, the laws of physics boil down to a completely action independent form, that is to the construction of quantum states induced by the products for space-time surfaces regarded as generalized numbers.

4. Space-time surfaces as minimal surfaces with generalized complex structure and are extremals for any variational principle constructible in terms of the induced geometry since extremal property reduces to the generalized complex structure. The action makes itself visible only at the singularities.
5. Langlands geometric duality becomes actually a trinity: number theory<-->geometry<--->physics. The number theory<-->geometry part of this trinity duality corresponds to Langlands geometric duality. The geometry<--->physics part is the TGD counterpart of Einstein's equations identifying geometry and physics.

See the article About Langlands correspondence in the TGD framework or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.