Authors:Michele Cirafici Pages: 1 - 70 Abstract: Abstract A large class of \({\mathcal {N}}=2\) quantum field theories admits a BPS quiver description, and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modeled by framed quivers. The associated spectral problem characterizes the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution to this problem. We detail our formalism with several explicit examples. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0611-0 Issue No:Vol. 19, No. 1 (2018)

Authors:Sylvain Lacroix; Benoît Vicedo Pages: 71 - 139 Abstract: Abstract Let \(\mathfrak {g}\) be a semisimple Lie algebra over \(\mathbb {C}\) . Let \(\nu \in \hbox {Aut}\, \mathfrak {g}\) be a diagram automorphism whose order divides \(T \in \mathbb {Z}_{\ge 1}\) . We define cyclotomic \(\mathfrak {g}\) -opers over the Riemann sphere \(\mathbb {P}^1\) as gauge equivalence classes of \(\mathfrak {g}\) -valued connections of a certain form, equivariant under actions of the cyclic group \(\mathbb {Z}/ T\mathbb {Z}\) on \(\mathfrak {g}\) and \(\mathbb {P}^1\) . It reduces to the usual notion of \(\mathfrak {g}\) -opers when \(T = 1\) . We also extend the notion of Miura \(\mathfrak {g}\) -opers to the cyclotomic setting. To any cyclotomic Miura \(\mathfrak {g}\) -oper \(\nabla \) , we associate a corresponding cyclotomic \(\mathfrak {g}\) -oper. Let \(\nabla \) have residue at the origin given by a \(\nu \) -invariant rational dominant coweight \(\check{\lambda }_0\) and be monodromy-free on a cover of \(\mathbb {P}^1\) . We prove that the subset of all cyclotomic Miura \(\mathfrak {g}\) -opers associated with the same cyclotomic \(\mathfrak {g}\) -oper as \(\nabla \) is isomorphic to the \(\vartheta \) -invariant subset of the full flag variety of the adjoint group G of \(\mathfrak {g}\) , where the automorphism \(\vartheta \) depends on \(\nu \) , T and \(\check{\lambda }_0\) . The big cell of the latter is isomorphic to \(N^\vartheta \) , the \(\vartheta \) -invariant subgroup ... PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0616-8 Issue No:Vol. 19, No. 1 (2018)

Authors:Jochen Zahn Pages: 163 - 187 Abstract: Abstract We discuss a free scalar field subject to generalized Wentzell boundary conditions. On the classical level, we prove well posedness of the Cauchy problem and in particular causality. Upon quantization, we obtain a field that may naturally be restricted to the boundary. We discuss the holographic relation between this boundary field and the bulk field. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0629-3 Issue No:Vol. 19, No. 1 (2018)

Authors:Michele Correggi; Marco Falconi Pages: 189 - 235 Abstract: Abstract We study the quasi-classical limit of a quantum system composed of finitely many nonrelativistic particles coupled to a quantized field in Nelson-type models. We prove that, as the field becomes classical and the corresponding degrees of freedom are traced out, the effective Hamiltonian of the particles converges in resolvent sense to a self-adjoint Schrödinger operator with an additional potential, depending on the state of the field. Moreover, we explicitly derive the expression of such a potential for a large class of field states and show that, for certain special sequences of states, the effective potential is trapping. In addition, we prove convergence of the ground-state energy of the full system to a suitable effective variational problem involving the classical state of the field. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0612-z Issue No:Vol. 19, No. 1 (2018)

Authors:Jake Fillman; May Mei Pages: 237 - 247 Abstract: Abstract We study continuum Schrödinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters. We show that the Hausdorff dimension of the spectrum tends to one in the small coupling and high-energy regimes, regardless of the shape of the potential pieces. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0624-8 Issue No:Vol. 19, No. 1 (2018)

Authors:Rui Han; Chris A. Marx Pages: 249 - 265 Abstract: Abstract We quantify the coupling asymptotics for the Lyapunov exponent of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. The result refines the well-known lower bound of the Lyapunov exponent by Sorets and Spencer. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0626-6 Issue No:Vol. 19, No. 1 (2018)

Authors:Jan P. Boroński; Jiří Kupka; Piotr Oprocha Pages: 267 - 281 Abstract: Abstract Motivated by a recent result of Ciesielski and Jasiński we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0623-9 Issue No:Vol. 19, No. 1 (2018)

Authors:Gianfausto Dell’Antonio; Alessandro Michelangeli; Raffaele Scandone; Kenji Yajima Pages: 283 - 322 Abstract: Abstract We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schrödinger operators with multi-centre local point interactions are bounded in \(L^p({\mathbb {R}}^3)\) for \(1<p<3\) and unbounded otherwise. PubDate: 2018-01-01 DOI: 10.1007/s00023-017-0628-4 Issue No:Vol. 19, No. 1 (2018)

Authors:Felix Finster; Alexander Strohmaier Abstract: Abstract In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion. PubDate: 2017-12-28 DOI: 10.1007/s00023-017-0632-8

Authors:Wojciech Dybalski; Alessandro Pizzo Abstract: Abstract Let \(H_{P,\sigma }\) be the single-electron fiber Hamiltonians of the massless Nelson model at total momentum P and infrared cut-off \(\sigma >0\) . We establish detailed regularity properties of the corresponding n-particle ground state wave functions \(f^n_{P,\sigma }\) as functions of P and \(\sigma \) . In particular, we show that $$\begin{aligned} \ \ \partial _{P^j}f^{n}_{P,\sigma }(k_1,\ldots , k_n) , \ \ \partial _{P^j} \partial _{P^{j'}} f^{n}_{P,\sigma }(k_1,\ldots , k_n) \!\le \! \frac{1}{\sqrt{n!}} \frac{(c\lambda _0)^n}{\sigma ^{\delta _{\lambda _0}}} \prod _{i=1}^n\frac{ \chi _{[\sigma ,\kappa )}(k_i)}{ k_i ^{3/2}}, \end{aligned}$$ where c is a numerical constant, \(\lambda _0\mapsto \delta _{\lambda _0}\) is a positive function of the maximal admissible coupling constant which satisfies \(\lim _{\lambda _0\rightarrow 0}\delta _{\lambda _0}=0\) and \(\chi _{[\sigma ,\kappa )}\) is the (approximate) characteristic function of the energy region between the infrared cut-off \(\sigma \) and the ultraviolet cut-off \(\kappa \) . While the analysis of the first derivative is relatively straightforward, the second derivative requires a new strategy. By solving a non-commutative recurrence relation, we derive a novel formula for \(f^n_{P,\sigma }\) with improved infrared properties. In this representation \(\partial _{P^{j'}}\partial _{P^{j}}f^n_{P,\sigma }\) is amenable to sharp estimates obtained by iterative analytic perturbation theory in part II of this series of papers. The bounds stated above are instrumental for scattering theory of two electrons in the Nelson model, as explained in part I of this series. PubDate: 2017-12-28 DOI: 10.1007/s00023-017-0642-6

Authors:Xinliang An; Xuefeng Zhang Abstract: Abstract The vacuum Einstein equations in \(5+1\) dimensions are shown to admit solutions describing naked singularity formation in gravitational collapse from nonsingular asymptotically locally flat initial data that contain no trapped surface. We present a class of specific examples with topology \(\mathbb {R}^{3+1} \times S^2\) . Thanks to the Kaluza–Klein dimensional reduction, these examples are constructed by lifting continuously self-similar solutions of the 4-dimensional Einstein-scalar field system with a negative exponential potential. The latter solutions are obtained by solving a 3-dimensional autonomous system of first-order ordinary differential equations with a combined analytic and numerical approach. Their existence provides a new test-bed for weak cosmic censorship in higher-dimensional gravity. In addition, we point out that a similar attempt of lifting Christodoulou’s naked singularity solutions of massless scalar fields fails to capture formation of naked singularities in \(4+1\) dimensions, due to a diverging Kretschmann scalar in the initial data. PubDate: 2017-12-20 DOI: 10.1007/s00023-017-0631-9

Authors:Gerhard Bräunlich; David Hasler; Markus Lange Abstract: Abstract We consider expansions of eigenvalues and eigenvectors of models of quantum field theory. For a class of models known as generalized spin–boson model, we prove the existence of asymptotic expansions of the ground state and the ground state energy to arbitrary order. We need a mild but very natural infrared assumption, which is weaker than the assumption usually needed for other methods such as operator theoretic renormalization to be applicable. The result complements previously shown analyticity properties. PubDate: 2017-12-19 DOI: 10.1007/s00023-017-0625-7

Authors:Vincenzo Morinelli Abstract: Abstract We discuss the Bisognano–Wichmann property for localPoincaré covariant nets of standard subspaces. We provide a sufficient algebraic condition on the covariant representation ensuring the Bisognano–Wichmann and the duality properties without further assumptions on the net. We call it modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano–Wichmann property. Furthermore, we give an outlook on the relation between the Bisognano–Wichmann property and the split property in the standard subspace setting. PubDate: 2017-12-19 DOI: 10.1007/s00023-017-0636-4

Authors:Peter Hintz Abstract: Abstract We show that a stationary solution of the Einstein–Maxwell equations which is close to a non-degenerate Reissner–Nordström–de Sitter solution is in fact equal to a slowly rotating Kerr–Newman–de Sitter solution. The proof uses the nonlinear stability of the Kerr–Newman–de Sitter family of black holes with small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions. PubDate: 2017-12-15 DOI: 10.1007/s00023-017-0633-7

Authors:Matthias Christandl; M. Burak Şahinoğlu; Michael Walter Abstract: Abstract We prove that the asymptotic behavior of the recoupling coefficients of the symmetric group \(S_k\) is characterized by a quantum marginal problem: they decay polynomially in k if there exists a quantum state of three particles with given eigenvalues for their reduced density operators and exponentially otherwise. As an application, we deduce solely from symmetry considerations of the coefficients the strong subadditivity property of the von Neumann entropy, first proved by Lieb and Ruskai (J Math Phys 14:1938–1941, 1973). Our work may be seen as a non-commutative generalization of the representation-theoretic aspect of the recently found connection between the quantum marginal problem and the Kronecker coefficient of the symmetric group, which has applications in quantum information theory and algebraic complexity theory. This connection is known to generalize the correspondence between Weyl’s problem on the addition of Hermitian matrices and the Littlewood–Richardson coefficients of SU(d). In this sense, our work may also be regarded as a generalization of Wigner’s famous observation of the semiclassical behavior of the recoupling coefficients (here also known as 6j or Racah coefficients), which decay polynomially whenever a tetrahedron with given edge lengths exists. More precisely, we show that our main theorem contains a characterization of the possible eigenvalues of partial sums of Hermitian matrices thus presenting a representation-theoretic characterization of a generalization of Weyl’s problem. The appropriate geometric objects to SU(d) recoupling coefficients are thus tuples of Hermitian matrices and to \(S_k\) recoupling coefficients they are three-particle quantum states. PubDate: 2017-12-15 DOI: 10.1007/s00023-017-0639-1

Authors:José M. Gracia-Bondía; Jens Mund; Joseph C. Várilly Abstract: Abstract We show how chirality of the weak interactions stems from string independence in the string-local formalism of quantum field theory. PubDate: 2017-12-14 DOI: 10.1007/s00023-017-0637-3

Authors:Giulio Bonelli; Alba Grassi; Alessandro Tanzini Abstract: Abstract We describe the magnetic phase of SU(N) \({\mathcal {N}}=2\) super-Yang–Mills theories in the self-dual \(\Omega \) -background in terms of a new class of multi-cut matrix models. These arise from a non-perturbative completion of topological strings in the dual four-dimensional limit which engineers the gauge theory in the strongly coupled magnetic frame. The corresponding spectral determinants provide natural candidates for the \(\tau \) -functions of isomonodromy problems for flat spectral connections associated with the Seiberg–Witten geometry. PubDate: 2017-12-14 DOI: 10.1007/s00023-017-0643-5

Authors:David Klein; Jake Reschke Abstract: Abstract Robertson–Walker spacetimes within a large class are geometrically extended to larger cosmologies that include spacetime points with zero and negative cosmological times. In the extended cosmologies, the big bang is lightlike, and though singular, it inherits some geometric structure from the original spacetime. Spacelike geodesics are continuous across the cosmological time zero submanifold which is parameterized by the radius of Fermi space slices, i.e., by the proper distances along spacelike geodesics from a comoving observer to the big bang. The continuous extension of the metric, and the continuously differentiable extension of the leading Fermi metric coefficient \(g_{\tau \tau }\) of the observer, restrict the geometry of spacetime points with pre-big bang cosmological time coordinates. In our extensions the big bang is two dimensional in a certain sense, consistent with some findings in quantum gravity. PubDate: 2017-12-14 DOI: 10.1007/s00023-017-0634-6

Authors:Pietro Longhi Abstract: Abstract A new construction of BPS monodromies for 4d \({\mathcal {N}}=2\) theories of class \({\mathcal {S}}\) is introduced. A novel feature of this construction is its manifest invariance under Kontsevich–Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve C of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For \(A_1\) -type theories, the graphs encoding the monodromy are “dessins d’enfants” on C, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg–Witten curve. PubDate: 2017-12-11 DOI: 10.1007/s00023-017-0635-5

Authors:Raphaël Belliard; Bertrand Eynard; Olivier Marchal Abstract: Abstract To any flat section equation of the form \(\nabla _0\Psi =\Phi \Psi \) in a principal bundle over a Riemann surface ( \(\nabla _0\) is a reference connection), we associate an infinite sequence of “correlators”, symmetric n-differentials on \(\Sigma \) that we denote \(\{W-n\}_{n \in \mathcal {N}}\) . The goal of this article is to prove that these correlators are solutions to “loop equations,” the same ones satisfied by correlation functions in random matrix models, or equivalently Ward identities of Virasoro or \({\mathcal {W}}\) -symmetric CFT. PubDate: 2017-12-02 DOI: 10.1007/s00023-017-0622-x