122884461
On the composition laws of Beltrami-Liouville theories and 2D quantum gravity☆
Abstract We present the composition laws or cocycle conditions of the Liouville theory and Beltrami-Liouville theory as well as the quantum gravity in two dimensions. We show that for the 2D gravity, their special cases give rise to the composition laws of Polyakov's light-cone action and that of the geometric action of the Virasoro group. By means of the composition laws, we give the gravitational Ward identities and a relation between the actions of the Liouville theory and the topological gravity and find that the principal action of the 2D topological gravity is a coboundary which should obey a cocylce condition in the Liouville complex.
122786546
Field equations, conservation laws in 2D quantum gravity and the Beltrami-KdV-Liouville system
Abstract We present the equations of motion and the conservation laws under various gauges of 2D induced gravity as a kind of the Beltrami-Liouville theory. We generalize the schwarzian derivative and derive its composition law. We also find the relation between the Beltrami-Liouville theory and the KdV system as well as the reccursion relations due to Gel'fand and Dikii based on the properties of the quasiconformal transformations.
10586906
Intrinsic Vertex Regularization and Renormalization in $\phi^4$ Field Theory
Based upon the intrinsic relation between the divergent lower point functions and the convergent higher point ones in the renormalizable quantum field theories, we propose a new method for regularization and renormalization in QFT. As an example, we renormalize the $\phi^{4}$ theory at the one loop order by means of this method.
121280592
The exchange algebra for Liouville field on Riemann surfaces with h>1 genus
We consider the classical Liouville field theory on the Riemann surface with h>1 genus. In terms of the uniformization theorem of the Riemann surface, we show explicitly the classical exchange algebra (CEA) for the chiral components of the Liouville field. We find that this classical exchange algebra is just the same as that on a Riemann sphere with 2h punctures.
22580845
Dynamical mass generation of composite Dirac fermions and fractional quantum Hall effects near charge neutrality in graphene.
We develop a composite Dirac fermion theory for the fractional quantum Hall effects (QHE) near charge neutrality in graphene. We show that the interactions between the composite Dirac fermions lead to a dynamical mass generation through exciton condensation. The four-fold spin-valley degeneracy is fully lifted due to the mass generation and exchange effects such that the odd-denominator fractional QHE observed in the vicinity of charge neutrality can be understood in terms of the integer QHE of composite Dirac fermions. At a filling factor ν = 1/2, we show that the massive composite Dirac fermion liquid is unstable against chiral p-wave pairing for weak Coulomb interactions and the ground state is a paired non-Abelian quantum Hall state described by the Moore-Read Pfaffian in the long wavelength limit.
119126315
Corrected Loop Vertex Expansion for Phi42 Theory
This paper is an extended erratum to J. Math. Phys.53, 042302 (2012) and arXiv:1104.3443, in which the classic construction and Borel summability of the phi^4_2 Euclidean quantum field theory was revisited combining a multi-scale analysis with the constructive method called Loop Vertex Expansion (LVE). Unfortunately we discovered an important error in the method of J. Math. Phys.53, 042302 (2012). We explain the mistake, and provide a new, correct construction of the phi^4_2 theory according to the LVE.
17172847
Topological graph polynomials and quantum field theory Part I: heat kernel theories
We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first paper we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are particular versions of the Bollob\'as-Riordan polynomials.
19017470
Topological properties of Abelian and non-Abelian quantum Hall states classified using patterns of zeros
It has been shown that different Abelian and non-Abelian fractional quantum Hall states can be characterized by patterns of zeros described by sequences of integers ${{S}_{a}}$. In this paper, we will show how to use the data ${{S}_{a}}$ to calculate various topological properties of the corresponding fraction quantum Hall state, such as the number of possible quasiparticle types and their quantum numbers, as well as the actions of the quasiparticle tunneling and modular transformations on the degenerate ground states on torus.
54712303
Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model
We study a quartic matrix model with partition function $$Z=\int d\ M\exp \mathrm{Tr}\ (-\Delta M^2-\frac{\lambda }{4}M^4)$$Z=∫dMexpTr(-ΔM2-λ4M4). The integral is over the space of Hermitian $$(\varLambda +1)\times (\varLambda +1)$$(Λ+1)×(Λ+1) matrices, the matrix $$\Delta $$Δ, which is not a multiple of the identity matrix, encodes the dynamics and $$\lambda >0$$λ>0 is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series and hence is a well-defined analytic function of the coupling constant in certain analytic domain of $$\lambda $$λ, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual $$\phi ^4$$ϕ4 theory on the 2-dimensional Moyal space also called the 2-dimensional Grosse–Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.