# Showing papers in "Communications in Mathematical Physics in 2005"

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Harvard University

^{1}, Humboldt University of Berlin^{2}, CERN^{3}, California Institute of Technology^{4}TL;DR: In this paper, a cubic field theory was constructed for all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefold.

Abstract: We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

881 citations

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TL;DR: In this article, the authors prove that the real four-dimensional Euclidean noncommutative ϕ4 model is renormalisable to all orders in perturbation theory.

Abstract: We prove that the real four-dimensional Euclidean noncommutative ϕ4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative ℝ4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

523 citations

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TL;DR: In this article, the smoothness of time functions and slicings by Cauchy hypersurfaces has been studied in the context of Lorentzian geometries, and it has been shown that any globally hyperbolic spacetime admits a smooth time function.

Abstract: The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function Open image in new window whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting Open image in new window if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function Open image in new window with timelike gradient Open image in new window on all M.

450 citations

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TL;DR: In this paper, an expression for the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel was derived, generalizing both the classical and the quantum capacities of the channel.

Abstract: An expression is derived characterizing the set of admissible rate pairs for simultaneous transmission of classical and quantum information over a given quantum channel, generalizing both the classical and quantum capacities of the channel. Although our formula involves regularization, i.e. taking a limit over many copies of the channel, it reduces to a single-letter expression in the case of generalized dephasing channels. Analogous formulas are conjectured for the simultaneous public-private capacity of a quantum channel and for the simultaneously 1-way distillable common randomness and entanglement of a bipartite quantum state.

413 citations

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TL;DR: In this paper, the existence and nonexistence of ground state solutions of N coupled nonlinear Schrodinger equations is established. But the sign of the coupling constants is not crucial for the existence of ground-state solutions.

Abstract: We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants β
ij
’s is crucial for the existence of ground state solutions. When all β
ij
’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all β
ij
’s are negative, or one of β
ij
’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.

386 citations

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TL;DR: In this article, the universal perturbative invariants of rational homology spheres up to order five were derived from the Chern-Simons partition function with arbitrary simply-laced group for these spaces.

Abstract: The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented.

363 citations

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TL;DR: In this paper, the authors studied three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ) and showed that the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot.

Abstract: We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,ℂ) partition function and the colored Jones polynomial.

291 citations

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TL;DR: In this article, Christodoulou and Klainerman this article proved global stability of Minkowski space for the Einstein vacuum equations in harmonic coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity.

Abstract: We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are “unstable in the large” and provides an alternative approach to the stability problem originally solved ( for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a weak null condition and showed that the Einstein equations in harmonic coordinates satisfy this condition.The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all “small” asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

250 citations

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TL;DR: In this paper, the authors studied the long time behavior of the solutions to the two dimensional dissipative quasi-geostrophic equations and obtained a new positivity lemma which improved a previous version of A. Cordoba and D. Duygulu.

Abstract: 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 Lp for anyp ∈ [2,+∞) when f is zero. As a second application of the new positivity lemma, for the sub-critical dissipative case with Open image in new window the existence of the global attractor for the solutions in the space Hs 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 Hs for any s≥0 and the estimate of the Hausdorff and fractal dimensions of the global attractor is also available.

229 citations

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TL;DR: In this paper, the authors prove an almost sure invariance principle for general classes of nonuniformly expanding and non-uneiformly hyperbolic dynamical systems, and apply it to planar periodic Lorentz flows with finite horizon.

Abstract: We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.

225 citations

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TL;DR: In this paper, the authors consider finite time blow up solutions to the critical nonlinear Schrodinger equation, and prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part correspond to the regular part and has a strong L2 limit at blow up time.

Abstract: We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.

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TL;DR: In this article, it was shown that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called Oseen's vortex.

Abstract: Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called “Oseen’s vortex”. This implies that the Oseen vortices are dynamically stable for all values of the circulation Reynolds number, and our approach also shows that these vortices are the only solutions of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. Finally, under slightly stronger assumptions on the vorticity distribution, we give precise estimates on the rate of convergence toward the vortex.

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TL;DR: In this paper, a supersymmetric relative of the Poisson sigma model was constructed for deformation quantization in generalized complex geometry, a notion introduced by Hitchin which interpolates between complex and symplectic manifolds.

Abstract: We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two–dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in the context of deformation quantization.

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TL;DR: In this paper, the exact renormalisation group equation is solved perturbatively, and a power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs.

Abstract: Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative ℝD in matrix formulation.

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TL;DR: In this article, it was shown that every principal T2-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize in this paper) it is a non-classical and is a bundle of non-commutative tori.

Abstract: It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious “missing T-duals.” Here we show that this problem is resolved using noncommutative topology. It turns out that every principal T2-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

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TL;DR: In this paper, the authors classify disordered fermion systems with quadratic Hamiltonians by their unitary and anti-unitary symmetries, based on Dyson's fundamental 1962 article known in random-matrix theory as the threefold way.

Abstract: Building upon Dyson’s fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.

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TL;DR: In this paper, it was shown that the principal eigenvalue of linear elliptic equations with high first-order coefficients is bounded as the amplitude of the coefficients of the first order derivatives goes to infinity if and only if the associated dynamical system has a first integral.

Abstract: This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.

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TL;DR: In this article, a general class of T3-fibered geometries admitting SU(3) structure was considered, and an exchange of pure spinors (eiJ and Ω) was found in dual geometry under fiberwise T-duality, and the transformations of the NS flux and the components of intrinsic torsion.

Abstract: When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T3-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (eiJ and Ω) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux.

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TL;DR: In this paper, the authors present nonabelian bundle gerbes as a higher version of principal bundles and study connection, curving, curvature and gauge transformations both in a global coordinate independent formalism and in local coordinates.

Abstract: Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of Yang-Mills theories with 2-form gauge potential.

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TL;DR: In this article, a 3+summable spectral triple Open Image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action was constructed.

Abstract: We construct a 3+-summable spectral triple Open image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Open image in new window The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

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TL;DR: In this article, the authors study a class of geometric Lorenz flows, introduced independently by Afraimovic, Bykov & Sil'nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing.

Abstract: We study a class of geometric Lorenz flows, introduced independently by Afraimovic, Bykov & Sil'nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.

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TL;DR: In this paper, for free-field theories associated with BRST first-quantized gauge systems, the authors identify generalized auxiliary fields and pure gauge variables already at the firstquantized level as the fields associated with algebraically contractible pairs for the BRST operator.

Abstract: For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space–time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.

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TL;DR: In this paper, the authors proved the existence of the universal attractor for strongly damped semilinear wave equation, in the presence of a quite general nonlinearity of critical growth.

Abstract: We prove the existence of the universal attractor for the strongly damped semilinear wave equation, in the presence of a quite general nonlinearity of critical growth. When the nonlinearity is subcritical, we prove the existence of an exponential attractor of optimal regularity, having a basin of attraction coinciding with the whole phase-space. As a byproduct, the universal attractor is regular and of finite fractal dimension. Moreover, we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient.

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TL;DR: In this article, the authors studied the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm.

Abstract: In this paper, we study the well-posedness of the Cauchy problem and the scattering problem for a new nonlinear dispersive shallow water wave equation (the so-called DGH equation) which was derived by Dullin, Gottwald and Holm. The issue of passing to the limit as the dispersive parameter tends to zero for the solution of the DGH equation is investigated, and the convergence of solutions to the DGH equation as α2→0 is studied, and the scattering data of the scattering problem for the equation can be explicitly expressed; the new exact peaked solitary wave solutions are obtained in the DGH equation. After giving the condition of existing peakon in the DGH equation, it turns out to be nonlinearly stable for the peakon in the DGH equation.

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TL;DR: The theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group was developed in this paper.

Abstract: We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H4(BG, ℤ) for a compact semi-simple Lie group G. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H4(BG, ℤ) to H3(G, ℤ). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

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TL;DR: In this article, the motion of a perfect liquid body in vacuum is modeled as a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order.

Abstract: We study the motion of a compressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a ``physical condition'', related to the fact that the pressure of a fluid has to be positive.

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TL;DR: In this paper, the authors derived effective mass theorems in solid state physics for a Schrodinger equation with a large periodic potential, denoting by ∈ the period, the potential is scaled as ∈−2.

Abstract: We study the homogenization of a Schrodinger equation with a large periodic potential: denoting by ∈ the period, the potential is scaled as ∈−2. We obtain a rigorous derivation of so-called effective mass theorems in solid state physics. More precisely, for well-prepared initial data concentrating on a Bloch eigenfunction we prove that the solution is approximately the product of a fast oscillating Bloch eigenfunction and of a slowly varying solution of an homogenized Schrodinger equation. The homogenized coefficients depend on the chosen Bloch eigenvalue, and the homogenized solution may experience a large drift. The homogenized limit may be a system of equations having dimension equal to the multiplicity of the Bloch eigenvalue. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.

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TL;DR: In this article, the integrable structure of the Dirichlet boundary problem in two dimensions was studied and a quasiclassical tau-function was proposed for the multiply-connected case.

Abstract: We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.

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TL;DR: The existence of self-similar solutions to Smoluchowski's coagulation equation has been conjectured for several years by physicists, and numerical simulations have confirmed the validity of this conjecture as mentioned in this paper.

Abstract: The existence of self-similar solutions to Smoluchowski’s coagulation equation has been conjectured for several years by physicists, and numerical simulations have confirmed the validity of this conjecture. Still, there was no existence result up to now, except for the constant and additive kernels for which explicit formulae are available. In this paper, the existence of self-similar solutions decaying rapidly at infinity is established for a wide class of homogeneous coagulation kernels.

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TL;DR: In this article, the authors considered the Bogoliubov-Dirac-Fock model and showed the existence of a unique minimizer of the BDF energy in the presence of an external electrostatic field, by means of a fixed-point approach.

Abstract: According to Dirac’s ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D0. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B 22, 3791–3814 (1989)), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is the solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.