Communications in Mathematical Physics

Published by Springer Nature

Online ISSN: 1432-0916

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Print ISSN: 0010-3616

Articles


Notes on Certain (0,2) Correlation Functions
  • Article
  • Full-text available

January 2006

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46 Reads

Sheldon Katz

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In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.
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Global Aspects of (0,2) Moduli Space: Toric Varieties and Tangent Bundles

September 2014

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29 Reads

We study the moduli space of A/2 half-twisted gauged linear sigma models for NEF Fano toric varieties. Focusing on toric deformations of the tangent bundle, we describe the vacuum structure of many (0,2) theories, in particular identifying loci in parameter space with spontaneous supersymmetry breaking or divergent ground ring correlators. We find that the parameter space of such an A/2 theory and its ground ring is in general a moduli stack, and we show in examples that with suitable stability conditions it is possible to obtain a simple compactification of the moduli space of smooth A/2 theories.

Calabi–Yau Black Holes and (0,4) Sigma Models

May 1999

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20 Reads

When an M-theory fivebrane wraps a holomorphic surface $\CP$ in a Calabi-Yau 3-fold $X$ the low energy dynamics is that of a black string in 5 dimensional $\CN=1$ supergravity. The infrared dynamics on the string worldsheet is an $\CN = (0,4)$ 2D conformal field theory. Assuming the 2D CFT can be described as a nonlinear sigma model, we describe the target space geometry of this model in terms of the data of $X$ and $\CP$. Variations of weight two Hodge structures enter the construction of the model in an interesting way.

Imperial/TP/2007/JG/03 Geometries with Killing Spinors and Supersymmetric AdS Solutions

November 2007

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27 Reads

The seven and nine dimensional geometries associated with certain classes of supersymmetric AdS 3 and AdS 2 solutions of type IIB and D = 11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in 2n + 2 dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for n ≥ 3, we show that when the geometry in 2n + 2 dimensions is a cone we obtain a class of geometries in 2n + 1 dimensions, specified by a Riemannian metric, a scalar field and a closed two-form, which includes the seven and nine-dimensional geometries mentioned above when n = 3, 4, respectively. We also consider various ansätze for the geometries and construct infinite classes of explicit examples for all n.

Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1,1)

June 2013

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38 Reads

We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra A(G) whose representing multipliers form an approximate identity in C_0(G) that is bounded in the M_0A(G) norm; the bound being an upper estimate for the associated Cowling-Haagerup constant. As an application, we find the appropriate approximation properties of the extended quantum SU(1,1) group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White.

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

May 2001

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47 Reads

S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer \(\) of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing \(\), a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of \(\). After defining the underlying von Neumann algebra of \(\) we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C * -algebra of \(\). The proofs of all these results depend on various properties of q-hypergeometric 1ϕ1 functions.

Integrable Structures in Classical Off-Shell 10D Supersymmetric Yang-Mills Theory

December 1999

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9 Reads

The field equations of supersymmetric Yang–Mills theory in ten dimensions may be formulated as vanishing curvature conditions on light-like rays in superspace. In this article, we investigate the physical content of the modified SO(7) covariant superspace constraints put forward earlier [11]. To this end, group-algebraic methods are developed which allow to derive the set of physical fields and their equations of motion from the superfield expansion of the supercurl, systematically. A set of integrable superspace constraints is identified which drastically reduces the field content of the unconstrained superfield but leaves the spectrum including the original Yang–Mills vector field completely off-shell. A weaker set of constraints gives rise to additional fields obeying first order differential equations. Geometrically, the SO(7) covariant superspace constraints descend from a truncation of Witten's original linear system to particular one-parameter families of light-like rays.

Figure 3: The bonds connecting the vertices of a stick Σ form a one-dimensional subsystem.
Finite-Volume Excitations of the¶111 Interface in the Quantum XXZ Model

August 1999

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56 Reads

We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet with a 111 interface have excitations localized in a subvolume of linear size R with energies bounded by O(1/R 2). As part of the proof we show the equivalence of ensembles for the 111 interface states in the following sense: In the thermodynamic limit the states with fixed magnetization yield the same expectation values for gauge invariant local observables as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant means commuting with the total third component of the spin, which is a conserved quantity of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed magnetization).

Asymmetric Diffusion and the Energy Gap Above¶the 111 Ground State of the Quantum XXZ Model

March 2002

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32 Reads

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let L be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as L→∞. In [3] it was proved that, by perturbing in a sub-cylinder with basis of linear size R≪L the interface ground state, it is possible to construct excited states whose energy gap shrinks as R -2. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from above and below by const.L -2. We prove the result by first mapping the problem into an asymmetric simple exclusion process on ℤ3 and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in [7]. Along the way we improve some bounds on the equivalence of ensembles already discussed in [3] and we establish an upper bound on the density of states close to the bottom of the spectrum.

Global validity of the Boltzmann equation for a two-dimensional and three-dimensional rare gas in vacuum: Erratum and improved result, in. Communications of Mathematics Physiological, 121, 143-146

March 1989

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7 Reads

We point out an error in our earlier papers [1] and [2] and present a more direct and natural proof which, although based on the same physical ideas of the previous ones, saves and actually improves the validity results for the Boltzmann equation given in [1] and [2].

Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1

January 1996

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17 Reads

Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method.

Homoclinic orbits on compact hypersufaces in 293-1293-1293-1, of restricted contact type

September 1995

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3 Reads

Consider a smooth Hamiltonian system in [(x)\dot] = JH¢(x)\dot x = JH'(x) , the energy surface ={x/H(x)=H(0)} being compact, and 0 being a hyperbolic equilibrium. We assume, moreover, that {0} is of restricted contact type. These conditions are symplectically invariant. By a variational method, we prove the existence of an orbit homoclinic, i.e. non-constant and doubly asymptotic, to 0.

Local Borel summability of Euclidean 311-1311-1311-1a simple proof via differential flow equations

March 1994

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10 Reads

It is shown how the differential flow equation (or, equivalently, the continous renormalization group) method can be employed to give an astonishingly easy proof of the local Borel summability of the renormalized perturbative Euclidean massive 4 4 .

Critical limit one-point correlations of monodromy fields on 321-1321-1321-1

November 1990

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4 Reads

Monodromy fields on ℤ2 are a family of lattice fields in two dimensions which are a natural generalization of the two dimensional Ising field occurring in theC *-algebra approach to Statistical Mechanics. A criterion for the critical limit one point correlation of the monodromy field σa(M) at a ∈ ℤ2,$$\mathop {\lim }\limits_{s \uparrow 1} \left\langle {\sigma _a (M)} \right\rangle ,$$ is deduced for matrices M ∈ GL(p,ℂ) having non-negative eigenvalues. Using this criterion non-identity 2×2 matrices are found with finite critical limit one point correlation. The general set ofp×p matrices with finite critical limit one point correlations is also considered and a conjecture for the critical limitn point correlations postulated.

Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

October 1991

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13 Reads

We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.

Chern-Simons Gauge Theory and projectively flat vector bundles on 421-1421-1421-1

March 1990

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6 Reads

We consider a vector bundle on Teichmller space which arises naturally from Witten's analysis of Chern-Simons Gauge Theory, and define a natural connection on it. In the case when the gauge group isU(1) we compute the curvature, showing, in particular, that the connection is projectively flat.

Fields, observables and gauge transformations II, Commun. Math. Phys. 15: 173-200

January 1969

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6 Reads

We wish to study the construction of charge-carrying fields given the representation of the observable algebra in the sector of states of zero charge. It is shown that the set of those covariant sectors which can be obtained from the vacuum sector by acting with “localized automorphisms” has the structure of a discrete Abelian group . An algebra of fields\(\mathfrak{F}\) can be defined on the Hilbert space of a representation π of the observable algebra\(\mathfrak{A}\) which contains each of the above sectors exactly once. The dual group of acts as a gauge group on\(\mathfrak{F}\) in such a way that\(\pi (\mathfrak{A})\) is the gauge invariant part of\(\mathfrak{F}.\mathfrak{F}\) is made up of Bose and Fermi fields and is determined uniquely by the commutation relations between spacelike separated fields.

The equation curl 73-173-173-1(x)=0 in quantum field theory

March 1972

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6 Reads

In one time and arbitrarily many space dimensions we obtain necessary and sufficient conditions for the existence of a local operator solution of the equation v =W v . Here the given local fieldsW v satisfy W v - v W =0 and the spectrum of the two point function (,W (x)W v (y)) is assumed to have a mass gap.

1976: The Existence of Maximal Slicings in Asymptotically Flat Spacetimes

July 1976

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78 Reads

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We consider Cauchy data on IR^3 that are asymptotically Euclidean and that satisfy the vacuum constraint equations of general relativity. Only those Cauchy data are treated that can be joined by a curve of sufficiently bounded initial data to the trivial data (\delta,0). It is shown that in the Cauchy developments of such data, the maximal slicing condition tr \pi =0 can always be satisfied. The proof uses the recently introduced weighted Sobolev spaces of Nirenberg, Walker, and Cantor.

Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

March 2004

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41 Reads

In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any positive U to the limit of large interaction is also shown. This continuity property, when combined with the proof that the wavefunction obtained with the generalized Bethe Ansatz is normalizable, is relevant to the question of whether or not the solution gives us the ground state of the finite system, as suggested by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the well known solution of the 1D Hubbard model. Comment: 18 pages

On the Second Mixed Moment of the Characteristic Polynomials of 1D Band Matrices

September 2012

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41 Reads

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices, i.e. of the hermitian matrices $H_n$ with independent Gaussian entries such that $< H_{ij}H_{lk}>=\delta_{ik}\delta_{jl}J_{ij}$, where $J=(-W^2\triangle+1)^{-1}$. Assuming that $W^2=n^{1+\theta}$, $0<\theta<1$, we show that this asymptotic behavior (as $n\to\infty$) in the bulk of the spectrum coincides with those for the Gaussian Unitary Ensemble.

Instability Zones of a Periodic 1D Dirac Operator and Smoothness of its Potential

October 2005

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15 Reads

Let L be the differential operator where P(x),Q(x) are 1-periodic functions such that The operator L, considered on [0,1] with periodic (y(0)=y(1)), or antiperiodic (y(0)=−y(1)) boundary conditions, is self-adjoint, and moreover, for large |n| it has, close to n π, a pair of periodic (if n is even), or antiperiodic (if n is odd) eigenvalues λ+n , λ-n . We study the relationship between the decay rate of the instability zone sequence γn = λn + - λn -, n → ± ∞, and the smoothness of the potential function P(x).

Nonuniformly Expanding 1D Maps

May 2006

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37 Reads

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ``Misiurewicz points'' have invariant densities for positive measure sets of parameters.

Figure 1. A sketch of the initial data (ρ 0 , ω 0 ). 
Figure 2. A sketch of the choice of {x n }. 
Finite Time Blow Up for a 1D Model of 2D Boussinesq System

December 2013

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138 Reads

The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.

FIG. 1: A possible function ϕ[F] before reflection.  
Periodic Minimizers in 1D Local Mean Field Theory

December 2007

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17 Reads

Using reflection positivity techniques we prove the existence of minimizers for a class of mesoscopic free-energies representing 1D systems with competing interactions. All minimizers are either periodic, with zero average, or of constant sign. If the local term in the free energy satisfies a convexity condition, then all minimizers are either periodic or constant. Examples of both phenomena are given. This extends our previous work where such results were proved for the ground states of lattice systems with ferromagnetic nearest neighbor interactions and dipolar type antiferromagnetic long range interactions.

Quenched Limit Theorems for Nearest Neighbour Random Walks in 1D Random Environment

December 2010

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49 Reads

It is well known that random walks in one dimensional random environment can exhibit subdiffusive behavior due to presence of traps. In this paper we show that the passage times of different traps are asymptotically independent exponential random variables with parameters forming, asymptotically, a Poisson process. This allows us to prove weak quenched limit theorems in the subdiffusive regime where the contribution of traps plays the dominating role.

1D-quasiperiodic operators. Latent symmetries

January 1991

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11 Reads

A large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofSL(2,Z) action with equal densities of states. Moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum. Some applications of these results are presented.

Phase Transition in the 1d Random Field Ising Model with Long Range Interaction

April 2008

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267 Reads

We study the one dimensional Ising model with ferromagnetic, long range interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an external random filed. we assume that the random field is given by a collection of independent identically distributed random variables, subgaussian with mean zero. We show that for temperature and strength of the randomness (variance) small enough with P=1 with respect to the distribution of the random fields there are at least two distinct extremal Gibbs measures.

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

July 2003

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25 Reads

 The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in Σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations.

Typical Gibbs Configurations for the 1d Random Field Ising Model with Long Range Interaction

November 2010

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54 Reads

We study a one--dimensional Ising spin systems with ferromagnetic, long--range interaction decaying as $n^{-2+\a}$, $\a \in [0,\frac 12]$, in the presence of external random fields. We assume that the random fields are given by a collection of symmetric, independent, identically distributed real random variables, gaussian or subgaussian with variance $\theta$. We show that for temperature and variance of the randomness small enough, with an overwhelming probability with respect to the random fields, the typical configurations, within volumes centered at the origin whose size grow faster than any power of $\th^{-1}$, % {\bf around the origin} are intervals of $+$ spins followed by intervals of $-$ spins whose typical length is $ \simeq \th^{-\frac{2}{(1-2\a)}}$ for $0\le \a<1/2$ and $\simeq e^{\frac 1 {\th^{2}}}$ for $\a=1/2$.

KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

April 2000

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123 Reads

In this paper, one-dimensional (1D) nonlinear wave equations with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u≡0. It is proved that for “most” potentials V(x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem which allows for multiple normal frequencies.

Figure 1. Non-colliding Brownian bridges: a typical path configuration in W β N (a, b), contributing to the partition function Z N (β).
Figure 2. A path configuration in the set Γ xjky with j = 3 and k = 6
Figure 3. An "open loop" from x and y with winding number 4. The endpoints of dotted lines are identified.
Wigner Crystallization in the Quantum 1D Jellium at All Densities

June 2013

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71 Reads

The jellium is a model, introduced by Wigner (1934), for a gas of electrons moving in a uniform neutralizing background of positive charge. Wigner suggested that the repulsion between electrons might lead to a broken translational symmetry. For classical one-dimensional systems this fact was proven by Kunz (1974), while in the quantum setting, Brascamp and Lieb (1975) proved translation symmetry breaking at low densities. Here, we prove translation symmetry breaking for the quantum one-dimensional jellium at all densities.

ClassicalN=1W-superalgebras from Hamiltonian reduction

March 1992

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23 Reads

A combinatorial proof is presented of the fact that the space of supersymmetric Lax operators admits a Poisson structure analogous to the second Gel'fand-Dickey bracket of the generalized KdV hierarchies. This allows us to prove that the space of Lax operators of odd order has a symplectic submanifold-defined by (anty)symmetric operators-which inherits a Poisson structure defining classicalW-superalgebras extending theN=1 supervirasoro algebra. This construction thus yields an infinite series of extended superconformal algebras.

Instantons on Noncommutative ℝ4, and (2,0) Superconformal Six Dimensional Theory

January 1998

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33 Reads

We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative ℝ4. This moduli space appears to be the Higgs branch of the theory of k D0-branes bound to N D4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2,0) superconformal theories in six dimensions.

All unitary ray representations of the conformal group SU(2,2) with positive energy

February 1977

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41 Reads

We find all those unitary irreducible representations of the [(G)\tilde]\tilde G of the conformal group SU(2,2)/4 which have positive energyP 00. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j 1,j 2) of the Lorentz group SL(2). They all decompose into a finite number of unitary irreducible representations of the Poincar subgroup with dilations.

Unitary irreducible representations ofSU(2,2)

June 1966

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8 Reads

Using a Lie algebra method based on works byHarish-Chandra, several series of unitary, irreducible representations of the groupSU(2,2) are obtained.


Comment on “Random Quantum Circuits are Approximate 2-designs” by A.W. Harrow and R.A. Low (Commun. Math. Phys. 291, 257–302 (2009))

May 2011

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81 Reads

In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291(1):257–302 (2009)], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper’s main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence.

Phase transitions in anisotropic lattice spin systems. Commun. Math. Phys. 60(3): 233-267

October 1978

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132 Reads

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.

On the Constructions of Holomorphic Vertex Operator Algebras of Central Charge 24

July 2011

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24 Reads

In this article, we construct explicitly several holomorphic vertex operator algebras of central charge 24 using Virasoro frames. The Lie algebras associated to their weight one subspaces are of the types A1,2 A3,44, A1,2D5,8, A1,13A7,4{A_{1,2} {A_{3,4}}^4, A_{1,2}D_{5,8}, {A_{1,1}}^3A_{7,4}} , A1,12 C3,2 D5,4, A2,12 A5,22 C2,1, A3,1 A7,2 C3,12, A3,1C7,2{{A_{1,1}}^2 C_{3,2} D_{5,4}, {A_{2,1}}^2 {A_{5,2}}^2 C_{2,1}, A_{3,1} A_{7,2} {C_{3,1}}^2, A_{3,1}C_{7,2}} , A4,1 A9,2B3,1, B4,1 C6,12{A_{4,1} A_{9,2}B_{3,1}, B_{4,1} {C_{6,1}}^2} and B 6,1 C 10,1. These vertex operator algebras correspond to number 7, 10, 18, 19, 26, 33, 35, 40, 48 and 56 in Schellekens’ list Schellekens (Commun Math Phys 153:159–185, 1993).

Elliptic Genera of 2d N = 2 Gauge Theories

August 2013

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56 Reads

We compute the elliptic genera of general two-dimensional N=(2,2) and N=(0,2) gauge theories. We find that the elliptic genus is given by the sum of Jeffrey-Kirwan residues of a meromorphic form, representing the one-loop determinant of fields, on the moduli space of flat connections on T^2. We give several examples illustrating our formula, with both Abelian and non-Abelian gauge groups, and discuss some dualities for U(k) and SU(k) theories. This paper is a sequel to the authors' previous paper arXiv:1305.0533.

The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations

April 2005

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113 Reads

The long time behavior of the solutions to the two dimensional dissipative quasi-geostrophic equations is studied. We obtain a new positivity lemma which improves a previous version of A. Cordoba and D. Cordoba [10] and [11]. As an application of the new positivity lemma, we obtain the new maximum principle, i.e. the decay of the solution in L p for any p ∈ [2,+∞) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with the existence of the global attractor for the solutions in the space H s for any s>2(1−α) is proved for the case when the time independent f is non-zero. Therefore, the global attractor is infinitely smooth if f is. This significantly improves the previous result of Berselli [2] which proves the existence of an attractor in some weak sense. For the case α=1, the global attractor exists in H s for any s≥0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.

N Dependence of Upper Bounds of Critical Temperatures of 2D O(N) Spin Models

April 1999

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40 Reads

We investigate critical temperature of the classical O(N) spin model in two dimensions. We show that if N is large and there is a phase transition in the system, the critical inverse temperature βc obeys the bound βc (N)> const. N log N.

Bosons in Disc-Shaped Traps: From 3D to 2D

November 2005

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50 Reads

We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, \(N\gg 1\), the two-dimensional extension, \(\bar L\), of the gas cloud in the trap, the thickness, \(h\ll \bar L\) of the trap, and the scattering length a of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premises that the confining energy, ~ 1/h 2, is much larger than the internal energy per particle, and a/h→ 0, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length a 2D = hexp(−(const.)h/a). In the parameter region where \(a/h\ll |\ln(\bar\rho h^2)|^{-1}\), with \(\bar\rho\sim N/\bar L^2\) the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter ~ Na/h. If \(|\ln(\bar\rho h^2)|^{-1}\lesssim a/h\) the coupling parameter is \(\sim N |\ln(\bar\rho h^2)|^{-1}\) and thus independent of a. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.

Rotational Invariance and Discrete Analyticity in the 2d Dimer Model

March 2004

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19 Reads

We exploit the discrete analytic structure present in the 2d dimer model to rigorously compute the exponents of a class of two point functions in all directions. This is a dimer analogue of the critical 2d Ising spin spin correlation function.

Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

September 2007

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23 Reads

Given g and f = gǵ, we consider solutions to the following non linear wave equation : {(u, ut)|t=0 = (u0, u1). utt - urr - 1/r ur = -f(u)/r2, Under suitable assumptions on g, this equation admits non-constant stationary solutions : we denote Q one with least energy. We characterize completely the behavior as time goes to ±∞ of solutions (u, u t ) corresponding to data with energy less than or equal to the energy of Q : either it is (Q, 0) up to scaling, or it scatters in the energy space. Our results include the cases of the 2 dimensional corotational wave map system, with target double script S sign2 , in the critical energy space, as well as the 4 dimensional, radially symmetric Yang-Mills fields on Minkowski space, in the critical energy space.

A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation

May 2007

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244 Reads

We show a new Bernstein’s inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.

The quantum group structure of 2D gravity and minimal models I

January 1991

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4 Reads

On the unit circle, an infinite family of chiral operators is constructed, whose exchange algebra is given by the universalR-matrix of the quantum groupSL(2) q . This establishes the precise connection between the chiral algebra of two dimensional gravity or minimal models and this quantum group. The method is to relate the monodromy properties of the operator differential equations satisfied by the generalized vertex operators with the exchange algebra ofSL(2) q . The formulae so derived, which generalize an earlier particular case worked out by Babelon, are remarkably compact and may be entirely written in terms of q-deformed factorials and binomial coefficients.

Duality Between Spin Networks and the 2D Ising Model
The goal of this paper is to exhibit a deep relation between the partition function of the Ising model on a planar trivalent graph and the generating series of the spin network evaluations on the same graph. We provide respectively a fermionic and a bosonic Gaussian integral formulation for each of these functions and we show that they are the inverse of each other (up to some explicit constants) by exhibiting a supersymmetry relating the two formulations. We investigate three aspects and applications of this duality. First, we propose higher order supersymmetric theories which couple the geometry of the spin networks to the Ising model and for which supersymmetric localization still holds. Secondly, after interpreting the generating function of spin network evaluations as the projection of a coherent state of loop quantum gravity onto the flat connection state, we find the probability distribution induced by that coherent state on the edge spins and study its stationary phase approximation. It is found that the stationary points correspond to the critical values of the couplings of the 2D Ising model, at least for isoradial graphs. Third, we analyze the mapping of the correlations of the Ising model to spin network observables, and describe the phase transition on those observables on the hexagonal lattice. This opens the door to many new possibilities, especially for the study of the coarse-graining and continuum limit of spin networks in the context of quantum gravity.

Fig.1: The matrices associated with the links < 12 >, < 13 >, < 14 > and the corners (214 >, (312 >, (413 > associated with the point 1.
Fig 2. Phase diagram in the t, A T plane.
Complex Matrix Models and Statistics of Branched Coverings of 2D Surfaces

March 1997

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We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model's partition function in terms of a sum over representations of the group U(N). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum limit allowing the branch points to freely float over the surface. The simplest such limit reproduces two-dimensional chiral U(N) Yang-Mills theory and its string description due to Gross and Taylor. Comment: 21 pages, 2 figures, TeX, harvmac.tex, epsf.tex, TeX "big"

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