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Locally Symmetric Spaces are the basis of the Langlands program—a set of ambitious and interconnected conjectures connecting representation theory to number theory, firstly proposed in 1967 by Robert Langlands.

These spaces have become a crossroads for many different strands of mathematical thought, and there are two topics recently an IAS 2017-2018 special program placed particular focus on. These are two of these areas:

To quote from the words of Akshay Venkatesh:

  • Analysis on locally symmetric spaces The motivation to study this comes both from the Langlands program, and from analytic number theory. The techniques are drawn from representation theory and analysis on manifolds, among others.

  • Topology of locally symmetric spaces Here the subject is guided by Langland conjecture that relates the cohomology of locally symmetric spaces to algebraic varieties. Understanding subtler features of this conjecture was a central theme of the program.

Langlands program tells a relationship between the topology of locally symmetric spaces and the theory of motive. But this is still too far from the real world (not direct physical implications.)

My questions are that have these ideas made connections to physics or physical sciences so far? What are the examples?

It looks that the crystal and lattice structure underly some algebraic structure.

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    $\begingroup$ Quantum symmetric spaces and boundary conditions via K-matrices (in spin chain sense, not abelian CS sense) $\endgroup$
    – AHusain
    Commented Nov 24, 2018 at 5:00

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There is a very interesting interaction between some aspects of the Langlands program and high-energy physics, involving elliptic polylogarithms (and other related generalisations of the classical polylogarithm functions).

On one hand, these numbers are periods of differential forms on symmetric spaces, so they clearly belong to the "Langlands world"; on the other hand, they also seem to appear very naturally in computations of scattering amplitudes in superstring theory. See for instance this paper by Broedel at al.

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Growth of heat trace coefficients for locally symmetric spaces [Journal of Mathematical Physics 53, 103506 (2012)] gives an application to quantum billiards.

We study the asymptotic behavior of the heat trace coefficients for the scalar Laplacian in the context of locally symmetric spaces. These spaces have the distinguishing property that the Huygens principle for the shifted wave equation holds. As a consequence growth estimates conjectured by Berry and Howls [“High orders of the Weyl expansion for quantum billiards: Resurgence of periodic orbits and the Stokes phenomenon”] are sharp.

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Akshay Venkatesh has some application-oriented work: On quantum unique ergodicity for locally symmetric spaces, see also:

Heat-kernel asymptotics of locally symmetric spaces of rank one and Chern-Simons invariants

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