Showing papers in "Letters in Mathematical Physics in 2010"
TL;DR: In this paper, the authors conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of SCFTs recently defined by one of the authors.
Abstract: We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of \({\mathcal{N}=2}\) SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.
1,881 citations
TL;DR: An infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT are derived and it is proved that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics.
Abstract: Using the thermodynamic Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in Gromov et al. (Integrability for the Full Spectrum of Planar AdS/CFT. arXiv:0901.3753 [hep-th]) for the spectrum of all operators in planar N = 4 SYM theory follows from these equations. In particular, we present the integral TBA type equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analog of DHM formula for the dressing kernel in the mirror kinematics.
545 citations
TL;DR: In this article, Nekrasov's instanton partition function of four-dimensional SU(2) quiver gauge theories in the presence of surface operators is studied and a modified version of the conformal blocks of affine SL (2) algebra is shown to provide the eigenfunctions of the Hitchin Hamiltonians.
Abstract: We study Nekrasov’s instanton partition function of four-dimensional $${\mathcal{N}=2}$$
gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.
195 citations
TL;DR: In this article, the wall crossing behavior of the refined BPS invariants was studied under wall crossing, and a new statistical model that counts refined pyramid partitions was proposed, which provided a combinatorial realization of wall crossing and clarified the relation between refined pyramid partition and the refined topological vertex.
Abstract: It is well known that in string compactifications on toric Calabi–Yau manifolds one can introduce refined BPS invariants that carry information not only about the charge of the BPS state but also about the spin content. In this paper we study how these invariants behave under wall crossing. In particular, by applying a refined wall crossing formula, we obtain the refined BPS degeneracies for the conifold in different chambers. The result can be interpreted in terms of a new statistical model that counts “refined” pyramid partitions; the model provides a combinatorial realization of wall crossing and clarifies the relation between refined pyramid partitions and the refined topological vertex. We also compare the wall crossing behavior of the refined BPS invariants with that of the motivic Donaldson–Thomas invariants introduced by Kontsevich–Soibelman. In particular, we argue that, in the context of BPS state counting, the three adjectives in the title of this paper are essentially synonymous.
164 citations
TL;DR: In this paper, the authors considered a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory, the Ponzano-Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex.
Abstract: We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang–Mills theory, the Ponzano–Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called ‘bubble divergences’. A common expectation is that the degree of these divergences is given by the number of ‘bubbles’ of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian – in both cases, the divergence degree is given by the second Betti number of the 2-complex.
93 citations
TL;DR: In this article, generalizations of the q-Onsager algebra are introduced and studied, and all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained.
Abstract: Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.
78 citations
TL;DR: In this article, a class of local and de Sitter invariant covariant tachyonic quantum fields for discrete negative values of the squared mass parameter and which have no Minkowskian counterpart were constructed.
Abstract: We provide a construction of a class of local and de Sitter covariant tachyonic quantum fields which exist for discrete negative values of the squared mass parameter and which have no Minkowskian counterpart. These quantum fields satisfy an anomalous non-homogeneous Klein–Gordon equation. The anomaly is a covariant field which can be used to select the physical subspace (of finite co-dimension) where the homogeneous tachyonic field equation holds in the usual form. We show that the model is local and de Sitter invariant on the physical space. Our construction also sheds new light on the massless minimally coupled field, which is a special instance of it.
49 citations
TL;DR: In this paper, a compact proof of Collins and Matsumoto's formula for the Weingarten matrix of the orthogonal group using Jucys-Murphy elements was provided.
Abstract: We provide a compact proof of the recent formula of Collins and Matsumoto for the Weingarten matrix of the orthogonal group using Jucys–Murphy elements.
47 citations
TL;DR: In this paper, a simple and case-independent proof for flip invariance of integrable quad-equations is provided, and a relation for Lagrangians within one elementary quadrilateral is found.
Abstract: The new idea of flip invariance of action functionals in multidimensional lattices was recently highlighted as a key feature of discrete integrable systems. Flip invariance was proved for several particular cases of integrable quad-equations by Bazhanov, Mangazeev and Sergeev and by Lobb and Nijhoff. We provide a simple and case-independent proof for all integrable quad-equations. Moreover, we find a new relation for Lagrangians within one elementary quadrilateral which seems to be a fundamental building block of the various versions of flip invariance.
47 citations
TL;DR: In this article, an explicit algebra isomorphism between the quantum reflection algebra for the R -matrix and a new type of current algebra (q-Onsager) was established.
Abstract: We establish an explicit algebra isomorphism between the quantum reflection algebra for the $${U_q(\widehat{sl_2}) R}$$
-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).
41 citations
TL;DR: In this paper, a local integral estimation of the squared length of the gradient of a distinguished smooth function on a constant mean curvature spacelike surface, under a suitable curvature condition on the ambient spacetime, is given.
Abstract: Several uniqueness and non-existence results on complete constant mean curvature spacelike surfaces lying between two slices in certain three-dimensional generalized Robertson–Walker spacetimes are given. They are obtained from a local integral estimation of the squared length of the gradient of a distinguished smooth function on a constant mean curvature spacelike surface, under a suitable curvature condition on the ambient spacetime. As a consequence, all the entire bounded solutions to certain family of constant mean curvature spacelike surface differential equations are found.
TL;DR: In this article, it was shown that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively.
Abstract: We show that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively. We also reformulate the results as a relationship between the relative invariants of some local Calabi-Yau geometries and integrable hierarchies and present some more examples using the topological vertex.
TL;DR: In this paper, the authors extend Tamarkin's formality of the little disk operad to the framed little disk (FLD) operad and propose a formality for the chain operad of Framed Little Disks.
Abstract: We extend Tamarkin's formality of the little disk operad to the framed little disk operad. SEVERA, Pavol. Formality of the Chain Operad of Framed Little Disks. Letters in Mathematical Physics, 2010, vol. 93, no. 1, p. 29 35 DOI : 10.1007/s11005-010-0399-z
TL;DR: In this paper, the authors conjecture that correlations in a quantum bipartite state can be broadcast if and only if they are classical-quantum, or equivalently, the quantum discord, as defined by Ollivier and Zurek (Phys Rev Lett 88:017901, 2002), vanishes.
Abstract: We address the issue of one-side local broadcasting for correlations in a quantum bipartite state, and conjecture that the correlations can be broadcast if and only if they are classical–quantum, or equivalently, the quantum discord, as defined by Ollivier and Zurek (Phys Rev Lett 88:017901, 2002), vanishes. We prove this conjecture when the reduced state is maximally mixed and further provide various plausible arguments supporting this conjecture. Moreover, we demonstrate that the conjecture implies the following two elegant and fundamental no-broadcasting theorems: (1) The original no-broadcasting theorem by Barnum et al. (Phys Rev Lett 76:2818, 1996), which states that a family of quantum states can be broadcast if and only if the quantum states commute. (2) The no-local-broadcasting theorem for quantum correlations by Piani et al. (Phys Rev Lett 100:090502, 2008), which states that the correlations in a single bipartite state can be locally broadcast if and only if they are classical. The results provide an informational interpretation for classical–quantum states from an operational perspective and shed new lights on the intrinsic relation between non-commutativity and quantumness.
TL;DR: In this paper, a new proof of the regularity away from the nuclei of the electronic density of a molecule obtained by Fournais et al. was given based on regularity properties of the Coulomb interactions.
Abstract: We give a new proof of the regularity away from the nuclei of the electronic density of a molecule obtained by Fournais et al (Commun Math Phys 228(3):401–415, 2002; Ark Math 42(1):87–106, 2004) The new argument is based on the regularity properties of the Coulomb interactions underlined by Hunziker (Ann Inst Henri Poincare, section A, tome 45, no 4, pp 339–358, 1986) and by Klein et al (Commun Math Phys 143(3):607–639, 1992) Well-known pseudodifferential techniques for elliptic operators are used and the method works in a larger framework
TL;DR: A truly elementary proof of the convexity of metric-adjusted skew information is given following an idea of Effros and it is shown that a recently introduced extension to parameter values 1 < p ≤ 2 of the WYD-information is a special case of (unbounded) metric- adjusted skew information.
Abstract: We give a truly elementary proof of the convexity of metric-adjusted skew information following an idea of Effros. We extend earlier results of weak forms of superadditivity to general metric-adjusted skew information. Recently, Luo and Zhang introduced the notion of semi-quantum states on a bipartite system and proved superadditivity of the Wigner–Yanase–Dyson skew informations for such states. We extend this result to the general metric-adjusted skew information. We finally show that a recently introduced extension to parameter values 1 < p ≤ 2 of the WYD-information is a special case of (unbounded) metric-adjusted skew information.
TL;DR: In this article, the authors derived bounds on the integrated density of states for a class of Schrodinger operators with a random potential, where the potential depends on a sequence of random variables, not necessarily in a linear way.
Abstract: We derive bounds on the integrated density of states for a class of Schrodinger operators with a random potential. The potential depends on a sequence of random variables, not necessarily in a linear way. An example of such a random Schrodinger operator is the breather model, as introduced by Combes, Hislop and Mourre. For these models, we show that the integrated density of states near the bottom of the spectrum behaves according to the so called Lifshitz asymptotics. This result can be used to prove Anderson localization in certain energy/disorder regimes.
TL;DR: In this paper, the authors proved the completeness of the Bethe ansatz eigenfunctions of the Laplacian on a Weyl alcove with repulsive boundary condition at the walls.
Abstract: We prove the completeness of the Bethe ansatz eigenfunctions of the Laplacian on a Weyl alcove with repulsive boundary condition at the walls. For the root system of type A this amounts to the result of Dorlas of the completeness of the Bethe ansatz eigenfunctions of the quantum Bose gas on the circle with repulsive delta-function interaction.
TL;DR: In this article, the authors describe a canonical reduction of AKSZ-BV theories to the cohomology of the source manifold, and present the possible generalizations of theAKSZ theory.
Abstract: We describe a canonical reduction of AKSZ–BV theories to the cohomology of the source manifold. We get a finite-dimensional BV theory that describes the contribution of the zero modes to the full QFT. Integration can be defined and correlators can be computed. As an illustration of the general construction, we consider two-dimensional Poisson sigma model and three-dimensional Courant sigma model. When the source manifold is compact, the reduced theory is a generalization of the AKSZ construction where we take as source the cohomology ring. We present the possible generalizations of the AKSZ theory.
TL;DR: In this paper, the authors describe a way to deform a spectral triple with a 2-torus action parametrized by a real deformation parameter, motivated by the Connes-Landi deformation of manifolds.
Abstract: We describe a way to deform a spectral triple with a 2-torus action parametrized by a real deformation parameter, motivated by the Connes–Landi deformation of manifolds. Such deformations are shown to have naturally isomorphic K-theoretic invariants independent of the deformation parameter.
TL;DR: In this paper, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz) was introduced to analyze the monodromy problem of the cubic oscillator, which is a system of five coupled nonlinear integral equations, which in particular case reduces to the Zamolodchikov TBA equation for the three-state Potts model.
Abstract: We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed TBA is a system of five coupled nonlinear integral equations, which in a particular case reduces to the Zamolodchikov TBA equation for the three-state Potts model. Our method generalizes the Dorey–Tateo analysis of the (monomial) cubic oscillator. We introduce a Y-system corresponding to the Deformed TBA and give it an elegant geometric interpretation.
TL;DR: In this paper, the Batalin-Vilkovisky Quantum Master Equation (QME) was used for open-closed string theory and showed that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains.
Abstract: We set up a Batalin–Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L
∞ and A
∞ algebraic structures associated to it.
TL;DR: In this article, the authors apply standard techniques from Toeplitz operator theory to analyze the asymptotics of the Hilbert-Schmidt norms of the TQFT operators coming from isotopy classes of one dimensional oriented labeled submanifolds on a closed oriented surface.
Abstract: Applying standard techniques from Toeplitz operator theory, we analyze the asymptotics of the Hilbert–Schmidt norms of the TQFT operators coming from isotopy classes of one dimensional oriented labeled submanifolds on a closed oriented surface. We thereby obtain a Toeplitz operator interpretation and generalization of the asymptotic formula obtained recently by Marche and Narimannejad (Duke Math J 141(3):573–587, 2008).
TL;DR: In this article, the singularities of the Borel transform for both positive and negative values of the parameter are compared for the four-dimensional supersymmetric Wess-Zumino model and the complex scalar field.
Abstract: Efficient computation methods are devised for the perturbative solution of Schwinger--Dyson equations for propagators. We show how a simple computation allows to obtain the dominant contribution in the sum of many parts of previous computations. This allows for an easy study of the asymptotic behavior of the perturbative series. In the cases of the four-dimensional supersymmetric Wess--Zumino model and the $\phi_6^3$ complex scalar field, the singularities of the Borel transform for both positive and negative values of the parameter are obtained and compared.
TL;DR: In this paper, the authors consider the case of weighted graphs and give an optimal condition to ensure that every self-adjoint realization of the adjacency matrix is also unbounded from below.
Abstract: Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note, we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.
TL;DR: In this article, a complex vector bundle is endowed with a connection over an eight-dimensional manifold, where G/H = SU(3)/U(1) × U(1), provided with a never-integrable almost-complex structure.
Abstract: We consider a complex vector bundle \({\mathcal{E}}\) endowed with a connection \({\mathcal{A}}\) over the eight-dimensional manifold \({\mathbb{R}^2\times G/H}\), where G/H = SU(3)/U(1) × U(1) is a homogeneous space provided with a never-integrable almost-complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions \({\mathcal{A}}\) of the Spin(7)-instanton equations on \({\mathbb{R}^2\times G/H}\) and general solutions of non-Abelian coupled vortex equations on \({\mathbb{R}^2}\). These vortices are BPS solitons in a d = 4 gauge theory obtained from \({\mathcal{N} =1}\) supersymmetric Yang–Mills theory in ten dimensions compactified on the coset space G/H with an SU(3)-structure. The novelty of the obtained vortex equations lies in the fact that Higgs fields, defining morphisms of vector bundles over \({\mathbb{R}^2}\), are not holomorphic in the generic case. Finally, we introduce BPS vortex equations in \({\mathcal{N} =4}\) super Yang–Mills theory and show that they have the same feature.
TL;DR: Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases as discussed by the authors, and the definition of kth partial sum under the prescribed order of terms is given.
Abstract: Extensions of Fannes’ inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann’s partial fidelities.
TL;DR: In this paper, kernel functions related to quantum many-body systems of Calogero-Sutherland type are discussed, in particular for the elliptic case, and the main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions.
Abstract: Kernel functions related to quantum many-body systems of Calogero–Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh–Feigin–Veselov–Sergeev-type deformations of the elliptic Calogero–Sutherland model for special parameter values.
TL;DR: In this article, several Euler-Lagrange equations for variational functions defined on a set of Holder curves are obtained for the cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives.
Abstract: We obtain several Euler–Lagrange equations for variational functionals defined on a set of Holder curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.
TL;DR: In this article, a compact formulation of the orthogonal Weingarten formula is presented, with the traditional quantity I(i_1,\ldots,i_{2k}) replaced by a more advanced quantity, I(a), i.i.
Abstract: We present a compact formulation of the orthogonal Weingarten formula, with the traditional quantity \({I(i_1,\ldots,i_{2k}:j_1,\ldots,j_{2k})=\int_{O_n}u_{i_1j_1}\cdots u_{i_{2k}j_{2k}}\,{\rm d}u}\) replaced by the more advanced quantity \({I(a)=\int_{O_n}\Pi u_{ij}^{a_{ij}}\,{\rm d}u}\), depending on a matrix of exponents \({a\in M_n(\mathbb{N})}\). Among consequences, we establish a number of basic facts regarding the integrals I(a): vanishing conditions, possible poles, asymptotic behavior.