4,518 research outputs found
The String Landscape: On Formulas for Counting Vacua
We derive formulas for counting certain classes of vacua in the string/M theory landscape. We do so in the context of the moduli space of M-theory compactifications on singular manifolds with G2 holonomy. Particularly, we count the numbers of gauge theories with different gauge groups but equal numbers of U (1) factors which are dual to each other. The vacua correspond to various symmetry breaking patterns of grand unified theories. Counting these dual vacua is equivalent to counting the number of conjugacy classes of elements of finite order inside Lie groups. We also point out certain cases where the conventional expectation is that symmetry breaking patterns by Wilson lines and Higgs fields are the same, but we show they are in fact different
Brewing moonshine for Mathieu
We propose a moonshine for the sporadic Mathieu group M_12 that relates its
conjugacy classes to various modular forms and Borcherds Kac-Moody Lie
superalgebras.Comment: 21 pages; LaTeX; no figure
Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven
Let G(q) be a finite Chevalley group, where q is a power of a good prime p,
and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a
conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in
U(q) is given by a polynomial in q with integer coefficients. In an earlier
paper, the first and the third authors developed an algorithm to calculate the
values of k(U(q)). By implementing it into a computer program using GAP, they
were able to calculate k(U(q)) for G of rank at most 5, thereby proving that
for these cases k(U(q)) is given by a polynomial in q. In this paper we present
some refinements and improvements of the algorithm that allow us to calculate
the values of k(U(q)) for finite Chevalley groups of rank six and seven, except
E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman
conjecture holds for these groups. Moreover, if we write k(U(q)) as a
polynomial in q-1, then the coefficients are non-negative.
Under the assumption that k(U(q)) is a polynomial in q-1, we also give an
explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.Comment: 16 page
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