407 research outputs found
Extremal discs and the holomorphic extension from convex hypersurfaces
Let D be a convex domain with smooth boundary in complex space and let f be a
continuous function on the boundary of D. Suppose that f holomorphically
extends to the extremal discs tangent to a convex subdomain of D. We prove that
f holomorphically extends to D. The result partially answers a conjecture by
Globevnik and Stout of 1991
Low-energy spectrum of N = 4 super-Yang-Mills on T^3: flat connections, bound states at threshold, and S-duality
We study (3+1)-dimensional N=4 supersymmetric Yang-Mills theory on a spatial
three-torus. The low energy spectrum consists of a number of continua of states
of arbitrarily low energies. Although the theory has no mass-gap, it appears
that the dimensions and discrete abelian magnetic and electric 't Hooft fluxes
of the continua are computable in a semi-classical approximation. The
wave-functions of the low-energy states are supported on submanifolds of the
moduli space of flat connections, at which various subgroups of the gauge group
are left unbroken. The field theory degrees of freedom transverse to such a
submanifold are approximated by supersymmetric matrix quantum mechanics with 16
supercharges, based on the semi-simple part of this unbroken group. Conjectures
about the number of normalizable bound states at threshold in the latter theory
play a crucial role in our analysis. In this way, we compute the low-energy
spectra in the cases where the simply connected cover of the gauge group is
given by SU(n), Spin(2n+1) or Sp(2n). We then show that the constraints of
S-duality are obeyed for unique values of the number of bound states in the
matrix quantum mechanics. In the cases based on Spin(2n+1) and Sp(2n), the
proof involves surprisingly subtle combinatorial identities, which hint at a
rich underlying structure.Comment: 28 pages. v2:reference adde
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