Triangle Seminars
Tuesday, 19 Oct 2010
Use of equivalence theorems for the study of ambiguity in non-Abelian gauge theories
Andrei Slavnov
(Steklov Institute, Moscow)
Discrete integrable dynamics and Somos sequences
Andrew Hone
(Kent)
Abstract:
Somos sequences are generated by a rational recurrence, which is specified by a quadratic relation
between adjacent iterates. Michael Somos noticed that, for some special choices of initial values, such recurrences
could unexpectedly produce sequences of integers. Examples of Somos sequences were known somewhat earlier in
number theory, from Morgan Ward's elliptic analogues of Fibonacci and Lucas sequences. In algebraic combinatorics,
Somos recurrences provide a basic example of the Laurent phenomenon, which is a cornerstone of Fomin and Zelevinsky's
theory of cluster algebras. This introductory talk reviews the history of Somos sequences and their connections
with these and other areas of mathematics and theoretical physics, including solvable statistical mechanics (the hard hexagon
model) and discrete integrable systems (QRT maps and the discrete Hirota equation).
Somos sequences are generated by a rational recurrence, which is specified by a quadratic relation
between adjacent iterates. Michael Somos noticed that, for some special choices of initial values, such recurrences
could unexpectedly produce sequences of integers. Examples of Somos sequences were known somewhat earlier in
number theory, from Morgan Ward's elliptic analogues of Fibonacci and Lucas sequences. In algebraic combinatorics,
Somos recurrences provide a basic example of the Laurent phenomenon, which is a cornerstone of Fomin and Zelevinsky's
theory of cluster algebras. This introductory talk reviews the history of Somos sequences and their connections
with these and other areas of mathematics and theoretical physics, including solvable statistical mechanics (the hard hexagon
model) and discrete integrable systems (QRT maps and the discrete Hirota equation).
Posted by: KCL
Wednesday, 20 Oct 2010
A supermatrix model for ABJM theory
📍 London
Nadav Drukker
(Imperial)
Abstract:
I will review the matrix model which calculates the partition function of ABJM theory on S3 as well as the expectation value of Wilson loop operators. I will then explain how this matrix model is solved and present the results for these quantities at all values of the couplings. At strong coupling these calculations reproduce the results of supergravity on Ads4 x CP3 and in particular the N to the 3/2 scaling of the free energy of the theory.
I will review the matrix model which calculates the partition function of ABJM theory on S3 as well as the expectation value of Wilson loop operators. I will then explain how this matrix model is solved and present the results for these quantities at all values of the couplings. At strong coupling these calculations reproduce the results of supergravity on Ads4 x CP3 and in particular the N to the 3/2 scaling of the free energy of the theory.
Posted by: KCL
Thursday, 21 Oct 2010
Gauge theory non-local operators from two dimensional conformal field theories
Filippo Passerini
(Berlin)
Abstract:
The partition function of a family of four dimensional N=2 gauge theories has been related by Alday, Gaiotto and Tachikawa (AGT) to correlation functions of two dimensional conformal Toda field theories. For SU(2) gauge theories, the associated two dimensional theory is A1 conformal Toda field theory, i.e. Liouville theory. In this seminar I
will discuss how the AGT proposal is modified when the gauge theories include non-local operators. I will focus in particular on Wilson loop operators and a certain type of surface operators in SU(N) N=2 gauge theories. For the case of the surface operator, I will show that the relevant two dimensional field theory is a conformal theory with SL(N) affine algebra. This result generalizes the recent result by Alday and Tachikawa that relates SU(2) gauge theories with surface operators to SL(2) affine algebras.
The partition function of a family of four dimensional N=2 gauge theories has been related by Alday, Gaiotto and Tachikawa (AGT) to correlation functions of two dimensional conformal Toda field theories. For SU(2) gauge theories, the associated two dimensional theory is A1 conformal Toda field theory, i.e. Liouville theory. In this seminar I
will discuss how the AGT proposal is modified when the gauge theories include non-local operators. I will focus in particular on Wilson loop operators and a certain type of surface operators in SU(N) N=2 gauge theories. For the case of the surface operator, I will show that the relevant two dimensional field theory is a conformal theory with SL(N) affine algebra. This result generalizes the recent result by Alday and Tachikawa that relates SU(2) gauge theories with surface operators to SL(2) affine algebras.
Posted by: QMW
Friday, 22 Oct 2010
Freezing transition and replica symmetry breaking in decaying Burgers turbulence via random matrix dualities
Yan Fyodorov
(Nottingham)
Abstract:
We reveal a phase transition with decreasing viscosity in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities reflecting a spontaneous one step replica symmetry breaking in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory.
(Based on the joint work with Pierre Le Doussal and Alberto Rosso.)
We reveal a phase transition with decreasing viscosity in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities reflecting a spontaneous one step replica symmetry breaking in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory.
(Based on the joint work with Pierre Le Doussal and Alberto Rosso.)
Posted by: brunel