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.
- 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.
- 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.
- 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.
- 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.
- Langlands geometric duality becomes actually a trinity: number theorygeometryphysics. The number theorygeometry part of this trinity duality corresponds to Langlands geometric duality. The geometryphysics 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.
Source: https://matpitka.blogspot.com/2024/09/extension-of-langlands-geometric.html
Anyone can join.
Anyone can contribute.
Anyone can become informed about their world.
"United We Stand" Click Here To Create Your Personal Citizen Journalist Account Today, Be Sure To Invite Your Friends.
Humic & Fulvic Liquid Trace Mineral Complex
HerbAnomic’s Humic and Fulvic Liquid Trace Mineral Complex is a revolutionary new Humic and Fulvic Acid Complex designed to support your body at the cellular level. Our product has been thoroughly tested by an ISO/IEC Certified Lab for toxins and Heavy metals as well as for trace mineral content. We KNOW we have NO lead, arsenic, mercury, aluminum etc. in our Formula.
This Humic & Fulvic Liquid Trace Mineral complex has high trace levels of naturally occurring Humic and Fulvic Acids as well as high trace levels of Zinc, Iron, Magnesium, Molybdenum, Potassium and more. There is a wide range of up to 70 trace minerals which occur naturally in our Complex at varying levels. We Choose to list the 8 substances which occur in higher trace levels on our supplement panel. We don’t claim a high number of minerals as other Humic and Fulvic Supplements do and leave you to guess which elements you’ll be getting.
Order Your Humic Fulvic for Your Family by Clicking on this Link, or the Banner Below.
Our Formula is an exceptional value compared to other Humic Fulvic Minerals because...
It’s OXYGENATED
It Always Tests at 9.5+ pH
Preservative and Chemical Free
Allergen Free
Comes From a Pure, Unpolluted, Organic Source
Is an Excellent Source for Trace Minerals
Is From Whole, Prehisoric Plant Based Origin Material With Ionic Minerals and Constituents
Highly Conductive/Full of Extra Electrons
Is a Full Spectrum Complex
Our Humic and Fulvic Liquid Trace Mineral Complex has Minerals, Amino Acids, Poly Electrolytes, Phytochemicals, Polyphenols, Bioflavonoids and Trace Vitamins included with the Humic and Fulvic Acid. Our Source material is high in these constituents, where other manufacturers use inferior materials.
Try Our Humic and Fulvic Liquid Trace Mineral Complex today. Be 100% Satisfied or Receive a Full Money Back Guarantee. Order Yours Today by Following This Link.
![Report abuse](/img/report_abuse.png)