Monday, April 28, 2014

T-Duality, Mirror Symmetry, Dedekind-Weil-Langlands .Rosetta Stone, Ising Model, Automorphy, Galois, Automorphic Representations

If we look at current work on Mirror Symmetry in the context of quantum field theory and string theory, and in the context of mathematics, there are a number of suggestive analogies with the solutions to the two-dimensional Ising model. There are very many approaches, and they reveal different aspects of the Ising system.

1. If T-Duality is taken as (m,n) and R becoming (n,m) and 1/R, it may be useful to look at duality in the Ising lattice: (K,K') and k go to (K'*, K*) and 1/k, connecting high and low temperature lattices, rotating the lattice pi/2.  k=1/sinh2Ksinh2K' and K'* is defined by sinh2K'sinh2K'*=1.

2. The Dedekind-Weil-Langlands triplet of Riemannian (analysis, automorphy), Italian (function theory), and German (arithmetic number theory) nicely classifies the various ways of solving the Ising problem: Yang-Baxter, functional equations and automorphy, and adding up by means of transfer matrices or graph counting. As for the Langlands program connecting Galois groups to automorphic representations, perhaps think in terms of the physicists' partition function and its also being automorphic, and that the transfer matrix might well be a group representation (I am unsure of just what), its characters or traces, being the actual partition function (the thermodynamic free energy is the -1/kBT ln Partition-Function = 1/kBT ln tr Transfer-Matrix = 1/kBT ln Lamda-max, where Lambda is the eigenvalue of the Tranfer-Matrix. Note as well that the Ising pseudo-Hamiltonian, in Onsager, A+B/k, where if k is small (low T, B dominates and is a potential energy, and if k is large A dominates and is a kinetic energy, much as we might have for the harmonic oscillator, as is the case in the T-duality system.

3. None of this denies the value of the electromagnetic duality that inspires much of Mirror Symmetry, but provides a rather simpler model.

4. The Big Questions are:
Why is there regularity here, here being crystalline solids, quadratic reciprocity, in the congruences and the distribution of primes, in the number of solutions to an equation modulo p, How does counting lead to scaling? In effect, what makes harmonic analysis so powerful.

How does the regularity encoded in the Galois group, permutations essentially, become the regularity encoded in automorphic forms? We do know that in the asymptotic realm, sums of reasonably nice random variables have asymptotic form (the Gaussian) and scale nicely at the square root of N.

We know, by the way, that the counting of eigenvalues or the spectrum of a differential operator gives information about areas, volumes, kinetic energy sums. Weyl asymptotics

And we know that particles and fields are intimately connected.
Why are group representations so powerful? For physicists, they allow calculation of numbers!

There are two sorts of non-answers to these questions. Kant would say that we cannot have knowledge of the ultimate source, but we surely can go deeper and deeper. Husserl would say that what we are learning are various aspects of a phenomenon, an identity in a manifold presentation of profiles, when we have many way of getting at something. In any case, questions about why mathematics and physics are so useful to each other, demand historical answers, not in the end something mysterious.

Having many layers and aspects will often lead us to understand something and see it at self-evident, as described by Gian-Carlo Rota in his discussions of mathematics and beauty and evidence.

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