The string landscape: On formulas for counting vacua

Abstract

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 G 2 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.

Counting conjugacy classes of elements of finite order in Lie groups

“…From Theorems 2.2 and 2.4, we deduce an intriguing symmetry between p and q. This symmetry has implications involving dualities of gauge theories; see [9].…”

“…Rep(π 1 (M), G) is the moduli space of isomorphism classes of flat connections on principal G-bundles over M; in grand unified theories arising from string or M-theory, these flat connections (called Wilson lines) serve as a symmetry breaking mechanism. For more on the physical applications and implications of these numbers, see [9].…”

“…In [15,3], the case of certain prime power orders was computed; and in [11], N(SU(n), m) was obtained. This topic was revived in [6,7], where N(G, m) was obtained for unitary, orthogonal, and symplectic Lie groups using simple combinatorial methods that apply to groups that are not necessarily connected, simply connected, or adjoint. Also in [6,7], E(G, m, s) and N(G, m, s) were introduced for the first time, motivated by an explicitly enumerative problem in string theory.…”

“…This topic was revived in [6,7], where N(G, m) was obtained for unitary, orthogonal, and symplectic Lie groups using simple combinatorial methods that apply to groups that are not necessarily connected, simply connected, or adjoint. Also in [6,7], E(G, m, s) and N(G, m, s) were introduced for the first time, motivated by an explicitly enumerative problem in string theory. 1 Formulas for N(G, m, s) were obtained there for the unitary, orthogonal, and symplectic Lie groups.…”

“…In[6,7], the number s denotes the number of distinct pairs of eigenvalues 2. Other aspects of elements of finite order in Lie groups have been studied.…”