Triangle Seminars
Tuesday, 11 Oct 2016
An operator for the zeros of the Riemann zeta function
Dorje Brody
(Brunel)
Abstract:
The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta function should be of the form 1/2 + i E_n, where the set of numbers {E_n} are real. The so-called Hilbert-Pólya conjecture assumes that {E_n} should correspond to the eigenvalues of an operator that is Hermitian. The discovery of such an operator, if it exists, thus amounts to providing a proof of the Riemann hypothesis. In 1999 Berry and Keating conjectured that such an operator should correspond to a quantisation of the classical Hamiltonian H = xp. Since then, the Berry-Keating conjecture has been investigated intensely in the literature, but its validity has remained elusive up to now. In this talk I will derive a “Hamiltonian†(a differential operator), whose classical counterpart is H = xp, having the property that with a suitable boundary condition on its eigenstates, the eigenvalues {E_n} correspond to the nontrivial zeros of the Riemann zeta function. This Hamiltonian is not Hermitian, but is symmetric under space-time reflection (PT symmetric) in a special way. A formal argument will be given for the construction of the metric operator to define an inner-product space for the eigenstates, and the formally “Hermitian" counterpart Hamiltonian. The talk is based on the work carried out in collaboration with Carl M. Bender (Washington University) and Markus P. Müller (University of Western Ontario).
The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta function should be of the form 1/2 + i E_n, where the set of numbers {E_n} are real. The so-called Hilbert-Pólya conjecture assumes that {E_n} should correspond to the eigenvalues of an operator that is Hermitian. The discovery of such an operator, if it exists, thus amounts to providing a proof of the Riemann hypothesis. In 1999 Berry and Keating conjectured that such an operator should correspond to a quantisation of the classical Hamiltonian H = xp. Since then, the Berry-Keating conjecture has been investigated intensely in the literature, but its validity has remained elusive up to now. In this talk I will derive a “Hamiltonian†(a differential operator), whose classical counterpart is H = xp, having the property that with a suitable boundary condition on its eigenstates, the eigenvalues {E_n} correspond to the nontrivial zeros of the Riemann zeta function. This Hamiltonian is not Hermitian, but is symmetric under space-time reflection (PT symmetric) in a special way. A formal argument will be given for the construction of the metric operator to define an inner-product space for the eigenstates, and the formally “Hermitian" counterpart Hamiltonian. The talk is based on the work carried out in collaboration with Carl M. Bender (Washington University) and Markus P. Müller (University of Western Ontario).
Posted by: KCL
Wednesday, 12 Oct 2016
Vortices and Vermas
📍 London
Matthew Bullimore
(Oxford University)
Abstract:
In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d supersymmetric gauge theories in the presence of an omega background, and explain how it leads to a finite version of the AGT correspondence.
In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d supersymmetric gauge theories in the presence of an omega background, and explain how it leads to a finite version of the AGT correspondence.
Posted by: KCL
N=2 super Yang-Mills and the Geometry of Magnetic Monopoles
Andy Royston
(Texas A-M)
Abstract:
In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.
In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.
Posted by: IC
Covariant quantum spaces and emergent gravity in the IKKT matrix model
Harold Steinacker
(Vienna U.)
Abstract:
We study perturbations of 4-dimensional fuzzy spheres as backgrounds in the IKKT or IIB matrix model. Gauge fields and metric fluctuations
are identified among the excitation modes with lowest spin, supplemented by a tower of higher-spin fields.
They arise from an internal structure which can be viewed as a twisted bundle over S^4, leading to a covariant noncommutative geometry.
The linearized 4-dimensional Einstein equations are obtained from the classical matrix model action under certain conditions,
modified by an IR cutoff. Some one-loop contributions to the effective action are computed using the formalism of string states.
We study perturbations of 4-dimensional fuzzy spheres as backgrounds in the IKKT or IIB matrix model. Gauge fields and metric fluctuations
are identified among the excitation modes with lowest spin, supplemented by a tower of higher-spin fields.
They arise from an internal structure which can be viewed as a twisted bundle over S^4, leading to a covariant noncommutative geometry.
The linearized 4-dimensional Einstein equations are obtained from the classical matrix model action under certain conditions,
modified by an IR cutoff. Some one-loop contributions to the effective action are computed using the formalism of string states.
Posted by: QMW
Thursday, 13 Oct 2016
N=2 super Yang-Mills and the Geometry of Magnetic Monopoles
Andy Royston
(Texas U.)
Abstract:
In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.
In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.
Posted by: QMW