Showing papers in "Communications in Mathematical Physics in 2001"

Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence

Abstract: We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors.

Rational surfaces associated with affine root systems and geometry of the Painlevé equations

Abstract: We present a geometric approach to the theory of Painleve equations based on rational surfaces Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type We classify all such surfaces X To each X, there corresponds a root subsystem of E (1) 8 inside the Picard lattice of X We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces We show that the translation part of the affine Weyl group gives rise to discrete Painleve equations, and that the above action constitutes their group of symmetries by Backlund transformations The six Painleve differential equations appear as degenerate cases of this construction In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure

Local Wick polynomials and time ordered products of quantum fields in curved spacetime

Abstract: In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and Kohler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dutsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation.

Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations

Abstract: We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝn. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

Abstract: We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ+(ɛ) of the Riemann–Hilbert decomposition γ−(ɛ)− 1γ+(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g 0=gZ 1 Z 3 −3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ−(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue.

Bootstrap Multiscale Analysis and Localization¶in Random Media

Abstract: We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e − L ζ, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions.

Correlation functions for M**N / S(N) orbifolds

Abstract: We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M N /S N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is “universal” in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N – this is a manifestation of the stringy exclusion principle.

Multi-interval subfactors and modularity of representations in conformal field theory

Abstract: We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo–Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of ?. As a consequence, the index of ?(E)⊂?(E′)′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of ? form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

Weak transition matrix elements from finite volume correlation functions

Abstract: The two-body decay rate of a weakly decaying particle (such as the kaon) is shown to be proportional to the square of a well-defined transition matrix element in finite volume. Contrary to the physical amplitude, the latter can be extracted from finite-volume correlation functions in euclidean space without analytic continuation. The K→ππ transitions and other non-leptonic decays thus become accessible to established numerical techniques in lattice QCD.

Discrete Riemann Surfaces and the Ising Model

Abstract: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

Clebsch–Gordan and Racah–Wigner Coefficients for a Continuous Series of Representations of ?q (??(2, ℝ))

Abstract: The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of U q (sl(2,ℝ). It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch–Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah–Wigner coefficients. This kernel is explicitly calculated.

Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices

Abstract: Consider an N×N hermitian random matrix with independent entries, not necessarily Gaussian, a so-called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices.

Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

Abstract: Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of \(\), where \(\) for \(\) and \(\) otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of \(\). Of particular interest are \(\) and FN(λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities \(\) and FN(λ;a) for the other classical matrix ensembles.

On the Characteristic Polynomial of a Random Unitary Matrix

Abstract: We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞ First we show that \(\), evaluated at a finite set of distinct points, is asymptotically a collection of iid complex normal random variables This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for \(\) For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy

Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras

Abstract: We study finite-dimensional representations of quantum affine algebras using q-characters. We prove the conjectures from [FR] and derive some of their corollaries. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized R-matrices has a pole.

Finite-Volume Fractional-Moment Criteria¶for Anderson Localization

Abstract: A technically convenient signature of localization, exhibited by discrete operators with random potentials, is exponential decay of the fractional moments of the Green function within the appropriate energy ranges. Known implications include: spectral localization, absence of level repulsion, strong form of dynamical localization, and a related condition which plays a significant role in the quantization of the Hall conductance in two-dimensional Fermi gases. We present a family of finite-volume criteria which, under some mild restrictions on the distribution of the potential, cover the regime where the fractional moment decay condition holds. The constructive criteria permit to establish this condition at spectral band edges, provided there are sufficient “Lifshitz tail estimates” on the density of states. They are also used here to conclude that the fractional moment condition, and thus the other manifestations of localization, are valid throughout the regime covered by the “multiscale analysis”. In the converse direction, the analysis rules out fast power-law decay of the Green functions at mobility edges.

Quiescent cosmological singularities

Abstract: The most detailed existing proposal for the structure of spacetime singularities originates in the work of Belinskii, Khalatnikov and Lifshitz. We show rigorously the correctness of this proposal in the case of analytic solutions of the Einstein equations coupled to a scalar field or stiff fluid. More specifically, we prove the existence of a family of spacetimes depending on the same number of free functions as the general solution which have the asymptotics suggested by the Belinskii–Khalatnikov–Lifshitz proposal near their singularities. In these spacetimes a neighbourhood of the singularity can be covered by a Gaussian coordinate system in which the singularity is simultaneous and the evolution at different spatial points decouples.

Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation

Abstract: We study stationary measures for the two-dimensional Navier–Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes.

On Spherically Symmetric Solutions¶of the Compressible Isentropic Navier–Stokes Equations

Abstract: We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations in ℝ n (n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends Lions' existence theorem [15] to the case 1< γ <γ n for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ n = 3/2 for n= 2 and γ n = 9/5 for n= 3. Dedicated to Professor Rolf Leis on the occasion of his 70th birthday

A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas

Abstract: We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρa2 is small, where ρ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρa2)|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate.

Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions

Abstract: We show that wave maps from Minkowski space ℝ1+n to a sphere Sm−1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space \(\), in all dimensions n≥ 5. This generalizes the results in the prequel [40] of this paper, which addressed the high-dimensional case n≥ 5. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.

Strange attractors with one direction of instability

Abstract: We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy.

Strongly Coupled Quantum Discrete Liouville Theory.¶I: Algebraic Approach and Duality

Abstract: The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is self-dual: there are two exponential fields related by Hermitian conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

Conformal and Quasiconformal Realizations of Exceptional Lie Groups

Abstract: We present a nonlinear realization of E8(8) on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined “light cone” in ℝ57. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E7(7) on ℝ27 which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned.

Well Posedness for Pressureless Flow

Abstract: We study the uniqueness problem for pressureless gases. Previous results on this topic are only known for the case when the initial data is assumed to be a bounded measurable function. This assumption is unnatural because the solution is in general a Radon measure. In this paper, the uniqueness of the weak solution is proved for the case when the initial data is a Radon measure. We show that, besides the Oleinik entropy condition, it is also important to require the energy to be weakly continuous initially. Our uniqueness result is obtained in the same functional space as the existence theorem.

Phase Transitions for Countable Markov Shifts

Abstract: We study the analyticity of the topological pressure for some one-parameter families of potentials on countable Markov shifts. We relate the non-analyticity of the pressure to changes in the recurrence properties of the system. We give sufficient conditions for such changes to exist and not to exist. We apply these results to the Manneville–Pomeau map, and use them to construct examples with different critical behavior.

Polynomial invariants for torus knots and topological strings

Abstract: We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems

Noncommutative Instantons and Twistor Transform

Abstract: Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of ℝ4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one-to-one correspondences between three classes of objects: framed bundles on a noncommutative ℙ2, certain complexes of sheaves on a noncommutative ℙ3, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative ℙ2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative ℙ2. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative ℝ4 than the one considered by Nekrasov and Schwarz (a q-deformed ℝ4).

A Coupling Approach to Randomly Forced Nonlinear PDE's. II

Abstract: We consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure. The result obtained applies, in particular, to the 2D Navier-Stokes system and multidimensional complex Ginzburg-Landau equation with random kick-force.

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

Abstract: We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this construction shows, in particular, that in the simplest case of the sl (2|1) level 1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras.