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## Sunday, January 18, 2015

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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.

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.

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## 13 comments:

This looks suspiciously like some sort of system similar to Bott periodicity. I am not sure where to look, but I wonder if there something of this form in the literature. If not then maybe this is either new or crazy.

LC

The exceptional cases are still open game, mathematically.

There are ways of thinking of Joradan matrices as E9 and E10, and maybe this continues to E∞. This does not seem clear to me right now. Then of course one would have to us Morse theory or Floer cohomology to get some form of homotopy periodicity.

The homotopy periodicity is "mod 8," which in the Connes' lecture and paper you find as well. The question might be whether this has something to do with OP^2.

LC

The hermitian symmetric domain E6/SO(10)xSO(2) = complexified OP^2.

The Wikipedia article on Hermitean symmetric spaces draws this out. E_6/SO(6)xSO(2) is equivalent to OP^2. The question is then what would be equivalent to a Grassmannian space. The quotient spin(8)/U(4) = D_4/D_2xU(1) is a 12-dim (set of roots) space of structures on R^8. The quotient is then ~ G(2, O) or an octonion Grassmannian space. There is a sort of triality of SO(8)’s in E8 and this might segue into Grassmanian structures.

The approach with heterotic groups though probably should employ γ-matrices, such as with work done by Dray and Monongue. Curiously the lecture by Connes centers around γ-matrices. I am wondering if there is something similar to π_iG(n,2n) ~ π _{i+1}SU(2n) in Bott periodicity that would hold with exceptional even sporadic groups.

LC

Yes, the bigger goal is to extend this to generalized twistor scattering amplitudes. This is where the Shimura varieties come in to play.

I have been pondering a couple of things. The first is whether the Hermitian symmetric manifolds have some relationship with Cartan decompositions and the Kostant-Sekiguchi theorem. This gives adjoint orbits that on g, the Lie algebra of G with H = G/K, so that adjoint orbits of G on elements of g that are nilpotent are diffoe to nilpotent orbits of H. In this way there is some connection between Hermitian manifolds and SLOCC. E7/E6xSO(2) is then a form of the moduli space of attractors for such orbits in N = 8, or 1/8 SUSY in BPS, O = E_{7(7)}/E_{6(6)}.

The other is whether there is some role for gerbes or sheaves on U_a∩U_b∩U_c, with a WZW type of action. I have been pondering this with respect to E8s or O so that O^3 can be mapped to the Leech lattice.

Cheers LC

Yes, your intuition serves you reliably. F4 and E6 give and LOCC and SLOCC for octonionic qutrits.

The leech lattice indeed lives in O^3, and better yet as Baez has shown, it lives in J(3,O). There is a 27 dimensional lattice that extends the E11 root lattice and in turn can be constructed using J(3,O) over Q. Such lattices should be exact descriptions in the full nonperturbative bosonic M-theory. Take a look at Baez's category theory blog to see the progress along these lines. I am commenting as Metatron there.

Thanks for the response. It seems that we are in the middle of the whole monster group realization. It seems as if one has to look at Conway groups that describe the symmetries of the Leech lattice. The J3O acts on the Monster group as an automorphism, as I recall, and one would need to look at the embedding of the Co group as a way of projecting onto the Leech lattice in J3 or O^3. Some sort of creative work like this is needed.

LC

The automorphism group of the leech lattice fits inside the exceptional group F4.

I was meaning that I think the Leech lattice is the automorphsm group for one of the Conway groups or the Fischer-Griess “monster” group.

The F_4 is the automorphism of the Leach Λ_{24}. The G_2 is the automorphism of E8. I also think that G_2 and F_4 are stabilizer groups, or have zero commutation in E8. The question I have had is whether there is some significance to this. The two are automorphisms, but F_4 is over a sort of triality of what G_2 is an automorphism on.

I have Conway and Sloane, and back around 2009 I did piles of calculations with Jacobi θ-functions for E8 and the Leech lattice. Maybe revisiting that would be worthwhile.

Cheers LC

There exists a lattice in 27-dimensions that F_4 acts on, more generally. The Conway group Co_0 are those restricted F_4 transformations that close as the automorphism group of the Leech in 24-dimensions.

The theta function for E8 has a well known description of heterotic string states in one copy of the E8xE8 lattice. Going up, there is also the theta function for E8xE8 that counts the full heterotic states on the 16D self-dual lattice.

The more interesting theta function is that for the lattice in 27-dimensions, K27=(E8+E8+E8)+++, which is related to bosonic string theory, or perhaps even better, bosonic M-theory.

Up to now, it has already been shown that E10, E11 and K27 and their symmetries, can be constructed with the exceptional Jordan algebra and its symmetries. The bigger question is what is the form of the (generalized) Kac-Moody algebra with K27 as its root lattice.

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