Non-hermitian holography

Quantum theory can be formulated with certain non-Hermitian Hamiltonians. An anti-linear involution, denoted by PT, is a symmetry of such Hamiltonians. In the PT-symmetric regime the non-Hermitian Hamiltonian is related to a Hermitian one by a Hermitian similarity transformation. We extend the concept of non-Hermitian quantum theory to gauge-gravity duality. Non-Hermiticity is introduced via boundary conditions in asymptotically AdS spacetimes. At zero temperature the PT phase transition is identified as the point at which the solutions cease to be real. Surprisingly at finite temperature real black hole solutions can be found well outside the quasi-Hermitian regime. These backgrounds are however unstable to fluctuations which establishes the persistence of the holographic dual of the PT phase transition at finite temperature.

Some generalizations of the Borsuk–Ulam theorem

In a recent paper, Fenn (3) has established the following theorem: if ø is a piecewise linear involution on Sn without fixed points, if f: Sn → Sn and g: are continuous and the degree of f is odd, then there are points x and y on Sn such that f(x) = ø(f(y)) and g(x) = g(y). The Borsuk–Ulam theorem is the special case of this in which f is the identity and ø corresponds to reflexion in the origin. Since an infinite-dimensional version of the Borsuk–Ulam theorem is known, involving compact maps (see, for example, page 72 of (2)), it is natural to ask whether Fenn's result can also be extended to general Banach spaces, and in this paper we give such an extension when ø is reflexion in the origin. More precisely, we prove that if B is the closed unit ball in a Banach space X and f, g: B → X are compact, with deg (I − f, B, 0) odd (this is the Leray–Schauder degree and I is the identity map) and (I − g) (B) contained in a proper, closed subspace of X, then there exist x, y on the boundary ∂B of B and apositive numberα such that α(I − f) (x) = − (I − f) (y) and (I − g) (x) = (I – g) (y).