Symplectic geometry and Hamiltonian flow of the renormalisation group equation

It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate \lq\lq momenta", which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace of the energy-momentum operator. For theories with massive couplings the identity operator plays a central role and its associated coupling gives rise to a potential in the flow equations. The evolution of any quantity , such as $N$-point Green functions, under renormalisation group flow can be obtained from its Poisson bracket with the Hamiltonian. Ward identities can be represented as constants of the motion which act as symmetry generators on the phase space via the Poisson bracket structure.Comment: 30 page

Bose condensation and branes

When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary $SU(N)$ gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of $AdS_5\times S^5$, for ${\cal N}=4$ SUSY Yang-Mills at large $N$, and the variable thermodynamically conjugate to $N$, a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.Comment: 9 pages, 1 figur

Potential Flow Of The Renormalisation Group In A Simple Two Component Model

The renormalisation group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with $\phi^4$ self interaction coupled, via Yukawa coupling, to a fermion in flat four dimensional space. The RG flow on the two dimensional space of couplings, ${\cal G}$, is shown to be derivable from a potential to sixth order in the couplings, which requires two loop calculations of the $\beta$-functions. The identification of a potential requires the introduction of a metric on ${\cal G}$ and it is shown that the metric defined by Zamalodchikov, in terms of two point correlation functions of composite operators, gives potential flow to this order.Comment: 7 pages Typset in PlainTeX, C Version 3.14

The intrinsic curvature of thermodynamic potentials for black holes with critical points

The geometry of thermodynamic state space is studied for asymptotically anti-de Sitter black holes in D-dimensional space times. Convexity of thermodynamic potentials and the analytic structure of the response functions is analysed. The thermodynamic potentials can be used to define a metric on the space of thermodynamic variables and two commonly used such metrics are the Weinhold metric, derived from the internal energy, and the Ruppeiner metric, derived from the entropy. The intrinsic curvature of these metrics is calculated for charged and for rotating black holes and it is shown that the curvature diverges when heat capacities diverge but, contrary to general expectations, the singularities in the Ricci scalars do not reflect the critical behaviour. When a cosmological constant is included as a state space variable it can be interpreted as a pressure and the thermodynamically conjugate variable as a thermodynamic volume. The geometry of the resulting extended thermodynamic state space is also studied, in the context of rotating black holes, and there are curvature singularities when the heat capacity at constant angular velocity diverges and when the black hole is incompressible. Again the critical behaviour is not visible in the singularities of the thermodynamic Ricci scalar.Comment: 35 pages, 3 figure

Thermodynamic stability of asymptotically anti-de Sitter rotating black holes in higher dimensions

Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D-dimensions are determined. Local thermodynamic stability requires not only positivity conditions on the specific heat and the moment of inertia tensor but it is also necessary that the adiabatic compressibility be positive. It is shown that, in the absence of a cosmological constant, neither rotation nor charge is sufficient to ensure full local thermodynamic stability of a black hole. Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes are investigated for both singly spinning and multi-spinning black holes. Simple expressions are obtained for the specific heat and moment of inertia tensor in any dimension. An analytic expression is obtained for the boundary of the region of parameter space in which such space-times are thermodynamically stable.Comment: 30 pages, 3 figures. References added, minor typos corrected in v

The Modular and Renormalisation Groups in the Quantum Hall Effect

An analytic form for the crossover of the conductivity tensor between two Hall plateaux, as a function of the external magnetic field, is proposed. The form of the crossover is obtained from the action of a symmetry group, a particular subgroup of the modular group, on the upper-half complex conductivity plane, by assuming that the beta-function describing the crossover is a holomorphic function of the conductivity. The group action also leads to a selection rule, |p_1q_2-p_2q_1|=1, for allowed transitions between Hall plateaux with filling factors p_1/q_1 and p_2/q_2, where q_1 and q_2 are odd.Comment: Talk presented at the Workshop on the Exact Renormalisation Group, Faro, Portugal, 10-12th September, 1998. Typeset in Latex, 9 pages, 3 figure

Renormalisation Flow and Geodesics on the Moduli Space of Four Dimensional N=2 Supersymmetric Yang-Mills Theory

It is shown that the beta functions for four dimensional N=2 supersymmetric Yang-Mills theory without matter give integral curves on the moduli space some of which are geodesics of the natural metric on the moduli space. In particular the flow lines which cross-over from from the weak coupling limit (asymptotically free theory) to the singular points, representing the strong coupling limit, are geodesics. A possible connection with irreversibility is discussed.Comment: 9 pages, 1 figure, typeset in PlainTe

Geodesic Renormalisation Group Flow

It is shown that the renormalisation group flow in coupling constant space can be interpreted in terms of a dynamical equation for the couplings analogous to viscous fluid flow under the action of a potential. For free scalar field theory the flow is geodesic in two dimensions, while for $D \neq 2$ it is only geodesic in certain limits, e.g. for vanishing external source. For the 1-D Ising model the renormalisation flow is geodesic when the external magnetic field vanishes.Comment: PlainTeX file, 11 pages, 5 figure

The Weyl-Lanczos Equations and the Lanczos Wave Equation in 4 Dimensions as Systems in Involution

Using the work by Bampi and Caviglia, we write the Weyl-Lanczos equations as an exterior differential system. Using Janet-Riquier theory, we compute the Cartan characters for all spacetimes with a diagonal metric and for the plane wave spacetime since all spacetimes have a plane wave limit. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet-Riquier theory, we find that it forms a system in involution. This result can be derived from the scalar wave equation itself. We compute its Cartan characters and compare them with those of the Weyl-Lanczos equations.Comment: 18 pages, latex, no figures, references correcte