Sunday, January 18, 2015

Shimura Varieties and Motives















Hermitian symmetric domains are complex manifolds isomorphic to bounded symmetric domains.  Every hermitian symmetric domain can be realized as a moduli space for Hodge structures plus tensors.  In general, hermitian symmetric domains are not algebraic varieties.  To obtain an algebraic variety one must pass to the quotient by an arithmetic group.  To obtain a moduli variety, one further assumes the arithmetic group is defined by congruence conditions.  The algebraic varities obtained this way are called connected Shimura varieties.

For all connected Shimura varieties except those of type E6, E7 and certain D types, the variety is a moduli variety for abelian motives with additional structure.  In the remaining cases, the connected Shimura variety is not a moduli variety for abelian motives and it is not know whether it is a moduli variety at all.

In supergravity applications, the hermitian symmetric domains of interest are E6/SO(10)xSO(2) and E7/E6xSO(2).  For real non-compact E6 and E7 quotients, such spaces are moduli for extremal black holes in D=5 and D=4 supergravity, with the real E6 and E7 groups acting as U-duality groups in compactified M-theory.  In the nonperturbative regime, due to charge quantization, it is quite natural to consider the quotients of the hermitian symmetric domains by an arithmetic group--which lives in a semisimple algebraic group over the rationals Q.

Further study of these exceptional Shimura varieties should lead to a deeper understanding of nonperturbative M-theory.

Saturday, January 03, 2015

Alain Connes: Geometry and the Quantum

 


The lecture was held within the framework of the Hausdorff Trimester Program Non-commutative Geometry and its Applications. (17.12.2014)

Currently, Connes' approach is distinct from string theory; however, for some years the geometry of the worldvolume has been known to be described by C*-algebraic spectral triples.

See:

D-branes, Matrix Theory and K-homology
T. Asakawa, S. Sugimoto, S. Terashima

Hence, from an abstract perspective, Connes' approach is dual to the matrix model approach to M-theory.

Tuesday, December 23, 2014

Motivic Dreams


















To show that all L-functions associated to Shimura varieties - thus to any motive defined by a Shimura variety - can be expressed in terms of automorphic L-functions of is weaker, even very much weaker, than to show that all motivic L-functions are equal to such L-functions.  Moreover, although the stronger statement is expected to be valid, there is, so far as I know, no very compelling reason to expect that all motivic L-functions will be attached to Shimura varieties.  - R. Langlands

Tuesday, December 09, 2014

The Father of Motives





















With the passing of Alexander Grothendieck, there is much discussion comparing him to Einstein in the mathematics world.  I think it would be more fitting to compare him to Riemann, whose differential geometry made General Relativity possible.  Of course, this is meant in a future tense as it is extremely likely Grothendieck's Motives will play a central role in the nonperturbative completion of M-theory.


Thursday, November 27, 2014

Matrix theory Revisited



















Progress has been made on matrix descriptions of the lattices for E10, E11 and K27.  It's a great time to return to Chern-Simons like matrix models with cubic actions.

Saturday, September 20, 2014

Wednesday, July 02, 2014

Unified Theory of Mathematics

 



What do the Langland's program and Monstrous Moonshine have in common?  The answer lies in M-theory.  Stay tuned.