We introduce three universality classes of chiral random matrix ensembles with a nonzero chemical potential and real, complex or quaternion real matrix elements. In the thermodynamic limit we find that the distribution of the eigenvalues in the complex plane does not depend on the Dyson index, and is given by the solution proposed by Stephanov. For a finite number of degrees of freedom, N , we ...

Now choose a Borel subgroup B in G and a maximal torus T in B so that, if B is the opposite Borel subgroup, then B ∩ H and B ∩ H are a Borel subgroups in H and T ∩H is a maximal torus in H . We follow the convention that the roots of B are positive. If λ ∈ X(T ), then indGB(−λ) is the dual Weyl module ∇G(λ ) with highest weight λ = −w0λ and lowest weight −λ. Its dual is the Weyl module ∆G(λ). I...

ABSTRACT We organize the eight variables of the four-dimensional bosonic string (Ẋ, X ) into a 2×2×2 hypermatrix aAA′A′′ and show that in signature (2, 2) the Nambu-Goto Lagrangian is given by √ Det a where Det is Cayley’s hyperdeterminant. This is invariant not only under [SL(2, R)] but also under interchange of the indices A, A and A. This triality reveals hitherto hidden discrete symmetries ...

We give the full set of S matrices for extensions of D(n)1 permutation orbifolds, extending our previous work to the yet unknown case of integer spin spinor currents. The main tool is triality of SO(8). We also provide fixed point resolution matrices for spinor currents of D(n)1 permutation orbifolds with n even and not multiple of four, where the spinor currents have half-integer spin.

The second homology group H2(st(m,n,R)) of the Steinberg Lie superalgebra st(m,n,R)(m + n ≥ 5), which is trivial, has been studied by A.V.Mikhalev and I.A.Pinchuk in [MP]. In this paper, we will work out H2(st(m,n,R)) explicitly for m + n = 3, 4 which is also trivial when m + n = 3, but not necessarily trivial when m + n = 4. Introduction Steinberg Lie algebras stn(R) and/or their universal cov...