ALBERT

Beschreibung:    We give an algebraic proof of the formula on the norm of logarithmic primary of Virasoro algebra, which was proposed by Al. Zamolodchikov. This formula appears in the recursion formula for the norm of Gaiotto state, which guarantees the AGT relation for the four-dimensional SU(2) pure gauge theory. Content Type Journal Article Pages 1-24 DOI 10.1007/s11005-011-0502-0 Authors Shintarou Yanagida, Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501 Japan Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We prove Lieb-Thirring inequalities for Schrödinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities. Content Type Journal Article Pages 1-15 DOI 10.1007/s11005-011-0499-4 Authors Rupert L. Frank, Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Rikard Olofsson, Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We give an alternate definition of the free Dirac field featuring an explicit construction of the Dirac sea. The treatment employs a semi-infinite wedge product of Hilbert spaces. We also show that the construction is equivalent to the standard Fock space construction. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0496-7 Authors Jonathan Dimock, Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The integrated density of states of a Schrödinger operator with random potential given by a homogeneous Gaussian field whose covariance function is continuous, compactly supported and has positive mean, is locally uniformly Lipschitz-continuous. This is proven using a Wegner estimate. Content Type Journal Article Pages 1-3 DOI 10.1007/s11005-011-0465-1 Authors Ivan Veselić, Fakultät für Mathematik, Technische. Universität Chemnitz, 09107 TU-Chemnitz, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this paper, the Hamiltonian for the system of semi-relativistic particles interacting with a scalar Bose field is investigated. A scaled total Hamiltonian of the system is defined and its scaling limit is considered. Then, a semi-relativistic Schrödinger operator with an effective potential is derived. Content Type Journal Article Pages 1-13 DOI 10.1007/s11005-011-0487-8 Authors Toshimitsu Takaesu, Faculty of Mathematics, Kyushu University, Fukuoka, 812-8581 Japan Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung: Appendix: Publication List of Vladimir B. Matveev Content Type Journal Article Pages 1-8 DOI 10.1007/s11005-011-0486-9 Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey–de Witt coefficients and make explicit calculations for the case of n -spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes (J Geom Phys 57:121, 2006 ) and study the case of SU q (2). Content Type Journal Article Pages 1-16 DOI 10.1007/s11005-011-0498-5 Authors Andrzej Sitarz, Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Artur Zając, Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    To every 3-manifold M one can associate a two-dimensional N =(2, 2) supersymmetric field theory by compactifying five-dimensional N =2 super-Yang–Mills theory on M . This system naturally appears in the study of half-BPS surface operators in four-dimensional N =2 gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional N =(2, 2) supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT’s. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting. Content Type Journal Article Pages 1-63 DOI 10.1007/s11005-011-0531-8 Authors Tudor Dimofte, California Institute of Technology, Pasadena, CA 91125, USA Sergei Gukov, California Institute of Technology, Pasadena, CA 91125, USA Lotte Hollands, California Institute of Technology, Pasadena, CA 91125, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the spectral curve for the classical string in AdS 5 × S 5 . Classical integrability of the AdS 5  × S 5 string implies the existence of a flat connection, whose monodromies generate an infinite set of conserved charges. The spectral curve is constructed out of the quasi-momenta, which are eigenvalues of the monodromy matrix, and each finite-gap classical solution can be characterized in terms of such a curve. This provides a concise and powerful description of the classical solution space. In addition, semi-classical quantization of the string can be performed in terms of the quasi-momenta. We review the general frame-work of the semi-classical quantization in this context and exemplify it with the circular string solution which is supported on \mathbb R × S 3 Ì AdS 5 × S 5 . Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0525-6 Authors Sakura Schäfer-Nameki, Kavli Institute for Theoretical Physics, University of California, Santa Barbara, 93106 CA, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Let X be a smooth projective complex variety. The Hochschild homology HH • ( X ) of X is an important invariant of X , which is isomorphic to the Hodge cohomology of X via the Hochschild–Kostant–Rosenberg isomorphism. On HH • ( X ), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH • ( X ). The latter pairing on HH • ( X ) is intimately linked to the study of Fourier–Mukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu’s Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the “Noncommutative Riemann–Roch” implies the classical Riemann–Roch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties. Content Type Journal Article Pages 1-17 DOI 10.1007/s11005-011-0541-6 Authors Ajay C. Ramadoss, Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zurich, Switzerland Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Mathematical investigations on quantum Zeno effect (QZE) are presented, including the following aspects: (i) QZE by frequent measurements made by an arbitrary partition of a time interval [0,  t ] ( t  〉 0); (ii) non-occurrence of QZE for vector states which are not in the domain of the Hamiltonian of the quantum system under consideration; and (iii) asymptotic behavior of the survival probability characterizing QZE in the number N of divisions of [0,  t ]; and (iv) QZE along a curve in the Hilbert space of state vectors. Content Type Journal Article Pages 1-16 DOI 10.1007/s11005-011-0539-0 Authors Asao Arai, Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan Toru Fuda, Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010. Content Type Journal Article Pages 1-30 DOI 10.1007/s11005-011-0529-2 Authors Niklas Beisert, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, Potsdam, 14476 Germany Changrim Ahn, Department of Physics and Institute for the Early Universe, Ewha Womans University, Seoul, 120-750 South Korea Luis F. Alday, Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford, OX1 3LB UK Zoltán Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, 1117 Pázmány s. 1/A, Budapest, Hungary James M. Drummond, PH-TH Division, CERN, Geneva, Switzerland Lisa Freyhult, Division for Theoretical Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 08 Uppsala, Sweden Nikolay Gromov, Department of Mathematics, King’s College, The Strand, London, WC2R 2LS UK Romuald A. Janik, Institute of Physics, Jagiellonian University, ul. Reymonta 4, Kraków, 30-059 Poland Vladimir Kazakov, LPT, Ecole Normale Superiéure, 24 rue Lhomond, 75231 Paris Cedex 05, France Thomas Klose, Division for Theoretical Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 08 Uppsala, Sweden Gregory P. Korchemsky, Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Charlotte Kristjansen, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark Marc Magro, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, Potsdam, 14476 Germany Tristan McLoughlin, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, Potsdam, 14476 Germany Joseph A. Minahan, Division for Theoretical Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 08 Uppsala, Sweden Rafael I. Nepomechie, Physics Department, University of Miami, P. O. Box 248046, Coral Gables, FL 33124, USA Adam Rej, Blackett Laboratory, Imperial College London, London, SW7 2AZ UK Radu Roiban, Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Sakura Schäfer-Nameki, Department of Mathematics, King’s College, The Strand, London, WC2R 2LS UK Christoph Sieg, Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Johann von Neumann-Haus, Rudower Chaussee 25, 12489 Berlin, Germany Matthias Staudacher, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, Potsdam, 14476 Germany Alessandro Torrielli, Department of Mathematics, University of York, Heslington, York, YO10 5DD UK Arkady A. Tseytlin, Blackett Laboratory, Imperial College London, London, SW7 2AZ UK Pedro Vieira, Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Dmytro Volin, Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Konstantinos Zoubos, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We consider the class of quantum spin chains with arbitrary U q (\mathfrak sl 2 ) -invariant nearest-neighbor interactions, sometimes called SU q (2) for the quantum deformation of SU (2), for q 〉 0. We derive sufficient conditions for the Hamiltonian to satisfy the property we call ferromagnetic ordering of energy levels . This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron–Frobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of “cascade operators,” a special new basis of U q (\mathfrak sl 2 ) intertwiners we define. Content Type Journal Article Pages 1-30 DOI 10.1007/s11005-011-0538-1 Authors Bruno Nachtergaele, Department of Mathematics, University of California, Davis, Davis, CA 95616, USA Stephen Ng, Department of Mathematics, University of California, Davis, Davis, CA 95616, USA Shannon Starr, Department of Mathematics, University of Rochester, Rochester, NY 14627, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Superconformal indices (SCIs) of 4 d N = 4 SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For F 4 , E 6 , E 7 , E 8 gauge groups this yields first examples of integrals of such type. S -duality transformation for G 2 and F 4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP (2 N ) and SO (2 N  + 1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3 d N =2 SYM theories with one matter field in the adjoint representation is investigated, corresponding 3 d dual partners are found, and some new related hyperbolic beta integrals are conjectured. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0537-2 Authors Vyacheslav P. Spiridonov, Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Moscow Region 141980, Russia Grigory S. Vartanov, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Let V be a braided vector space, i.e., a vector space together with a solution ^ R   Î End ( V Ä V ) of the Yang–Baxter equation. Denote T ( V ):= Å k   V Ä k . We associate to ^ R   a one-parameter family of solutions T ( ^ R   ) Î End ( T ( V ) Ä T ( V )) of the Yang–Baxter equation on the tensor space T ( V ). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association ^ R   \rightsquigarrow T ( ^ R   ) is functorial with respect to V . Content Type Journal Article Pages 1-12 DOI 10.1007/s11005-011-0533-6 Authors Thomas Grapperon, Center of Theoretical Physics, Luminy, 13288 Marseille, France Oleg Ogievetsky, Center of Theoretical Physics, Luminy, 13288 Marseille, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We consider the low-density limit of a Fermi gas in the BCS approximation. We show that if the interaction potential allows for a two-particle bound state, the system at zero temperature is well approximated by the Gross–Pitaevskii functional, describing a Bose–Einstein condensate of fermion pairs. Content Type Journal Article Pages 1-20 DOI 10.1007/s11005-011-0535-4 Authors Christian Hainzl, Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle, 10 72076 Tübingen, Germany Robert Seiringer, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Canada Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    A \mathbb Z N -curve is one of the form y N =( x - l 1 ) m 1 ¼ ( x - l s ) m s . When N  = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ 1 , . . . , λ s . In his work on Fermionic fields on \mathbb Z N -curves with arbitrary N , Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m i equal 1. This work was continued by several authors and new Thomae’s type formulae for \mathbb Z N -curves with other rotation numbers m i were found. In this article we prove that for some choices of the rotation numbers the corresponding \mathbb Z N -curves do not admit such generalized Thomae’s formulae. Content Type Journal Article Pages 1-13 DOI 10.1007/s11005-011-0497-6 Authors Gabino González-Diez, Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain David Torres-Teigell, Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We propose an expansion of the definition of almost-commutative spectral triple that accommodates non-trivial fibrations and is stable under inner fluctuation of the metric and then prove a reconstruction theorem for almost-commutative spectral triples under this definition as a simple consequence of Connes’s reconstruction theorem for commutative spectral triples. Along the way, we weaken the orientability hypothesis in the reconstruction theorem for commutative spectral triples and, following Chakraborty and Mathai, prove a number of results concerning the stability of properties of spectral triples under suitable perturbation of the Dirac operator. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0534-5 Authors Branimir Ćaćić, Department of Mathematics, California Institute of Technology, MC 253-37, 1200 E., California Blvd., Pasadena, 91125 CA, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence, and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence. Content Type Journal Article Pages 1-34 DOI 10.1007/s11005-011-0536-3 Authors Stefan Jansen, Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann-Herder-Straße 3, 79104 Freiburg, Germany Nikolai Neumaier, Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann-Herder-Straße 3, 79104 Freiburg, Germany Gregor Schaumann, Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany Stefan Waldmann, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B bus 2400, 3001 Heverlee, Belgium Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the derivation of the S -matrix for planar \fancyscript N =4 supersymmetric Yang–Mills theory and type IIB superstring theory on an AdS 5  × S 5 background. After deriving the S -matrix for the su (2) and su (3) sectors at the one-loop level based on coordinate Bethe ansatz, we show how su (2|2) symmetry leads to the exact asymptotic S -matrix up to an overall scalar function. We then briefly review the spectrum of bound states by relating these states to simple poles of the S -matrix. Finally, we review the derivation of the asymptotic Bethe equations, which can be used to determine the asymptotic multiparticle spectrum. Content Type Journal Article Pages 1-21 DOI 10.1007/s11005-011-0478-9 Authors Changrim Ahn, Department of Physics and Institute for the Early Universe, Ewha Womans University, Seoul, 120-750 South Korea Rafael I. Nepomechie, Physics Department, University of Miami, P. O. Box 248046, Coral Gables, FL 33124, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this survey article, we explain a few ideas behind the fermionic projector approach and summarize recent results which clarify the connection to quantum field theory. The fermionic projector is introduced, which describes the physical system by a collection of Dirac states, including the states of the Dirac sea. Formulating the interaction by an action principle for the fermionic projector, we obtain a consistent description of interacting quantum fields which reproduces the results of perturbative quantum field theory. We find a new mechanism for the generation of boson masses and obtain small corrections to the field equations which violate causality. Content Type Journal Article Pages 1-19 DOI 10.1007/s11005-011-0473-1 Authors Felix Finster, Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    This paper deals with the dynamics of linear Hamiltonian systems which have almost periodic Hamiltonians and symplectic phase transitions over almost periodic lattices. By introducing some discrete skew-product dynamical systems based on certain joint hulls of Hamiltonians and lattices, it will be proved that such a system admits a well-defined rotation number, which gives a global, topological understanding on the motion of these systems in symplectic groups and manifolds of Lagrangian planes. Content Type Journal Article Pages 1-25 DOI 10.1007/s11005-011-0475-z Authors Long Li, Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 People’s Republic of China Meirong Zhang, Department of Mathematical Sciences, Tsinghua University, Beijing, 100084 People’s Republic of China Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We study properties of transfer matrices in the sl ( N ) spin chain models. The transfer matrices with an infinite-dimensional auxiliary space are factorized into the product of N commuting Baxter Q -operators. We consider the transfer matrices with auxiliary spaces of a special type (including the finite-dimensional ones). It is shown that they can be represented as the alternating sum over the transfer matrices with infinite- dimensional auxiliary spaces. We show that certain combinations of the Baxter Q -operators can be identified with the Q -functions, which appear in the Nested Bethe Ansatz. Content Type Journal Article Pages 1-18 DOI 10.1007/s11005-011-0472-2 Authors Sergey E. Derkachov, St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia Alexander N. Manashov, Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We construct a 2-colored operad Ger ∞ which, on the one hand, extends the operad Ger ∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras. We show that Tamarkin’s Ger ∞ -structure on the Hochschild cochain complex C • ( A , A ) of an A ∞ -algebra A extends naturally to a Ger + ¥ -structure on the pair ( C • ( A , A ), A ). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this Ger + ¥ -structure to the cohomology of the pair ( C • ( A , A ), A ). We show that Ger + ¥ is a sub DG operad of the first sheet E 1 (SC) of the homology spectral sequence for the Fulton–MacPherson version SC of Voronov’s Swiss Cheese operad. Finally, we prove that the DG operads Ger + ¥ and E 1 (SC) are non-formal. Content Type Journal Article Pages 1-41 DOI 10.1007/s11005-011-0476-y Authors Vasily Dolgushev, Department of Mathematics, Temple University, 1805 N. Broad St., Philadelphia, PA 19122, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra \mathfrak pgl ( p +1 | q ) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of \mathfrak pgl ( n | n )\not @ \mathfrak sl ( n | n ) ), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields. Content Type Journal Article Pages 1-21 DOI 10.1007/s11005-011-0474-0 Authors Pierre Mathonet, Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, 1359 Luxembourg, Luxembourg Fabian Radoux, Institute of Mathematics, University of Liège, Grande Traverse, 12-B37, 4000 Liège, Belgium Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    A relation between circular 1/2 BPS ’t Hooft operators in 4d N =4 SYM and instantonic solutions in 2D Yang-Mills theory (YM 2 ) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM 2 , belonging to representations of the dual group L G . This conjecture has been tested in the case G  =  U ( N ) =  L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G  =  SU ( N ) and L G  =  SU ( N )/ Z N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out. Content Type Journal Article Pages 1-14 DOI 10.1007/s11005-011-0480-2 Authors Antonio Bassetto, Dipartimento di Fisica “G. Galilei” and INFN, Sezione di Padova, Via Marzolo 8, 35131 Padua, Italy Shiyamala Thambyahpillai, Dipartimento di Fisica “G. Galilei” and INFN, Sezione di Padova, Via Marzolo 8, 35131 Padua, Italy Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In the mean-field regime we prove convergence, with explicit bounds, of N -particle density matrices satisfying the time-dependent von Neumann equation with factorized initial data to a product of one particle density matrices satisfying the Hartree–von Neumann equation. To prove explicit bounds we generalize techniques developed by Pickl (in A simple derivation of mean field limits for quantum systems. ArXiv:0907.4464, 2009 ) and Knowles–Pickl (in Commun. Math. Phys. 298(1):101–138, 2010 ). Content Type Journal Article Pages 1-31 DOI 10.1007/s11005-011-0477-x Authors Ioannis Anapolitanos, Department of Mathematics, University of Toronto, Toronto, Canada Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We show that the Yang–Mills quantum field theory with momentum and spacetime cutoffs in four Euclidean dimensions is equivalent, term by term in an appropriately resummed perturbation theory, to a Fermionic theory with nonlocal interaction terms. When a further momentum cutoff is imposed, this Fermionic theory has a convergent perturbation expansion. To zeroth order in this perturbation expansion, the correlation function E ( x , y ) of generic components of pairs of connections is given by an explicit, finite-dimensional integral formula, which we conjecture will behave as E ( x , y ) ~ | x - y | - 2 - 2 d G , for | x - y | 〉〉 0 , where d G is a positive integer depending on the gauge group G . In the case where G  =  SU ( N ), we conjecture that d G = dim    SU ( N ) - dim    S ( U ( N - 1) × U (1)), so that the rate of decay of correlations increases as N → ∞. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0460-6 Authors Jonathan Weitsman, Department of Mathematics, Northeastern University, Boston, MA 02115, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Given a semi-Hamiltonian system, we construct an F -manifold with a connection satisfying a suitable compatibility condition with the product. We exemplify this procedure in the case of the so-called e -system. The corresponding connection turns out to be flat, and the flat coordinates give rise to additional chains of commuting flows. Content Type Journal Article Pages 1-24 DOI 10.1007/s11005-011-0469-x Authors Paolo Lorenzoni, Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Roberto Cozzi 53, 20125 Milan, Italy Marco Pedroni, Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of \mathbb Z generated by the class of a point. This clarifies the extent to which the graph hypersurfaces ‘generate the Grothendieck ring of varieties’: while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of \mathbb Z [\mathbb L ] in which \mathbb L becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive \mathbb L to zero. In particular, this shows that the graph hypersurfaces do not generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne–Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0461-5 Authors Paolo Aluffi, Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review basic examples of classical string solutions in AdS 5  × S 5 . We concentrate on simplest rigid closed string solutions of circular or folded type described by integrable 1-d Neumann system but mention also various generalizations and related open-string solutions. Content Type Journal Article Pages 1-23 DOI 10.1007/s11005-011-0466-0 Authors Arkady A. Tseytlin, Blackett Laboratory, Imperial College London, London, SW7 2AZ UK Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We investigate ( d  + 1)-dimensional quasilinear systems which are integrable by the method of hydrodynamic reductions. In the case d  ≥ 3 we formulate a conjecture that any such system with an irreducible dispersion relation must be linearly degenerate. We prove this conjecture in the 2-component case, providing a complete classification of multi- dimensional integrable systems in question. In particular, our results imply the non- existence of 2-component integrable systems of hydrodynamic type for d  ≥ 6. In the second half of the paper we discuss a numerical and analytical evidence for the impossibility of the breakdown of smooth initial data for linearly degenerate systems in 2 + 1 dimensions. Content Type Journal Article Pages 1-31 DOI 10.1007/s11005-011-0462-4 Authors Evgeny V. Ferapontov, Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU UK Karima R. Khusnutdinova, Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU UK Christian Klein, Institut de Mathématiques de Bourgogne, 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this paper, we identify q -deformed \mathfrak gl l +1 -Whittaker functions with a specialization of the Macdonald polynomials. This provides a representation of q -deformed \mathfrak gl l +1 -Whittaker functions in terms of the Demazure characters of affine Lie algebra ^ \mathfrak gl   l +1   . We also define a system of dual Hamiltonians for q -deformed \mathfrak gl l +1 -Toda chains and give a new integral representation for the q -deformed \mathfrak gl l +1 -Whittaker functions. Finally, we represent the q -deformed \mathfrak gl l +1 -Whittaker function as a matrix element of a quantum torus algebra. Content Type Journal Article Pages 1-24 DOI 10.1007/s11005-011-0468-y Authors Anton Gerasimov, Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia Dimitri Lebedev, Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia Sergey Oblezin, Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We study the general model of self-financing trading strategies in illiquid markets introduced by Schönbucher and Wilmott (SIAM J Appl Math 61(1):232–272, 2000 ). A hedging strategy in the framework of this model satisfies a nonlinear partial differential equation (PDE) which contains some function g ( α ). This function is deeply connected to a marginal utility function. We describe the Lie symmetry algebra of this PDE and provide a complete set of reductions of the PDE to ordinary differential equations (ODEs). In addition, we show the way how to describe all types of functions g ( α ) for which the PDE admits an extended Lie group. Two of these special type functions correspond to the models introduced before by different authors, whereas one is new. We clarify the connection between these three special models and the general model for trading strategies in the illiquid markets. We also apply the Lie group analysis to the new special case of the PDE describing the self-financing strategies. For the general model, as well as for the new special model, we provide the optimal systems of subalgebras and study the complete set of reductions of the PDEs to ODEs. We provide explicit solutions to the new special model in all reduced cases. Moreover, in one of the cases the solutions describe power derivative products. Content Type Journal Article Pages 1-17 DOI 10.1007/s11005-011-0463-3 Authors Ljudmila A. Bordag, IDE, MPE Lab, Halmstad University, Box 823, 301 18 Halmstad, Sweden Anna Mikaelyan, IDE, MPE Lab, Halmstad University, Box 823, 301 18 Halmstad, Sweden Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We perform separation of variables for the symplectic Weyl character using Sklyanin’s scheme. Viewing the characters as eigenfunctions of a quantum integrable system, we explicitly construct the separating operator using the Q -operator method. We also construct the inverse of the separating operator, as well as the factorised Hamiltonian. Content Type Journal Article Pages 1-23 DOI 10.1007/s11005-011-0467-z Authors Jan de Gier, Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia Anita Ponsaing, Department of Mathematics and Statistics, The University of Melbourne, Melbourne, VIC 3010, Australia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi,  μ CH and  μ DP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle. Content Type Journal Article Pages 1-16 DOI 10.1007/s11005-011-0464-2 Authors Feride Tığlay, Section de Mathématiques, EPFL, 1015 Lausanne, Switzerland Cornelia Vizman, Department of Mathematics, West University of Timişoara, Bd. V. Pârvan 4, 300223 Timişoara, Romania Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We study the question of magnetic confinement of quantum particles on the unit disk \mathbb D in \mathbb R 2 , i.e. we wish to achieve confinement solely by means of the growth of the magnetic field B ( ® x   ) near the boundary of the disk. In the spinless case, we show that B ( ® x   ) ³ \frac Ö 32·\frac1(1 - r ) 2 - \frac1 Ö 3\frac1(1 - r ) 2 ln\frac11 - r , for | ® x   | close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants \frac Ö 32 and - \frac1 Ö 3 are optimal. This answers, in this context, an open question from Colin de Verdière and Truc (Ann Inst Fourier 2011 , Preprint, arXiv:0903.0803v3). We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles. Content Type Journal Article Pages 1-17 DOI 10.1007/s11005-011-0506-9 Authors Gheorghe Nenciu, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Griviţei, Sector 1, 010702 Bucharest, Romania Irina Nenciu, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21, Calea Griviţei, Sector 1, 010702 Bucharest, Romania Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The purpose of this paper is to describe an analogue of a construction of Costello in the context of finite-dimensional differential graded Frobenius algebras which produces closed forms on the decorated moduli space of Riemann surfaces. We show that this construction extends to a certain natural compactification of the moduli space which is associated with the modular closure of the associative operad, due to the absence of ultra-violet divergences in the finite-dimensional case. We demonstrate that this construction is equivalent to the “dual construction” of Kontsevich. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0507-8 Authors Alastair Hamilton, Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2 h . We prove that if the coefficients of the discrete Wronskian are real and the imaginary parts of its roots are bounded by | h |, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0521-x Authors Evgeny Mukhin, Department of Mathematical Sciences, Indiana University—Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA Vitaly Tarasov, Department of Mathematical Sciences, Indiana University—Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA Alexander Varchenko, Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review recent applications of the integrable discrete Hirota dynamics ( Y -system) in the context of calculation of the planar AdS/CFT spectrum. We start from the description of solution of Hirota equations by the Bäcklund method where the requirement of analyticity results in the nested Bethe ansatz equations. Then, we discuss applications of the Hirota dynamics for the analysis of the asymptotic limit of long operators in the AdS/CFT Y -system. Content Type Journal Article Pages 1-27 DOI 10.1007/s11005-011-0513-x Authors Nikolay Gromov, Department of Mathematics, King’s College London, London, WC2R 2LS UK Vladimir Kazakov, Ecole Normale Superieure, Laboratoire de Physique Théorique, 75231 Paris Cedex 5, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The aim of the chapter is to introduce in a pedagogical manner the concept of Thermodynamic Bethe Ansatz designed to calculate the energy levels of finite volume integrable systems and to review how it is applied in the planar AdS/CFT setting. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0512-y Authors Zoltán Bajnok, Theoretical Physics Research Group of the Hungarian Academy of Sciences, H-1117 Pázmány s. 1/A, Budapest, Hungary Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the role of integrability in the planar spectral problem of four-dimensional superconformal gauge theories besides \fancyscript N =4 SYM. The cases considered include the Leigh–Strassler marginal deformations of \fancyscript N =4 SYM, quiver theories which arise as orbifolds of AdS 5  × S 5 on the dual gravity side, as well as various theories involving open spin chains. Content Type Journal Article Pages 1-26 DOI 10.1007/s11005-011-0515-8 Authors Konstantinos Zoubos, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In integrable quantum field theories, the large volume spectrum is given by the Bethe Ansatz. The leading corrections, due to virtual particles circulating around the cylinder, are encoded in so-called Lüscher corrections. In order to apply these techniques to the AdS/CFT correspondence, one has to generalize these corrections to the case of generic dispersion relations and to multiparticle states. We review these various generalizations and the applications of Lüscher’s corrections to the study of the worldsheet QFT of the superstring in AdS 5 ×  S 5 and, consequently, to anomalous dimensions of operators in N =4 SYM theory. Content Type Journal Article Pages 1-21 DOI 10.1007/s11005-011-0511-z Authors Romuald A. Janik, Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the duality and integrability of \fancyscript N  = 6 superconformal Chern–Simons theory in three dimensions and IIA superstring theory on the background Ads 4  × CP 3 . We introduce both of these models and describe how their degrees of freedom are mapped to excitations of a long-range integrable spin-chain. Finally, we discuss the properties of the Bethe equations, the S -matrix and the algebraic curve that are special to this correspondence and differ from the case of \fancyscript N  = 4 SYM theory and strings on Ads 5 × S 5 . Content Type Journal Article Pages 1-23 DOI 10.1007/s11005-011-0520-y Authors Thomas Klose, Department of Physics and Astronomy, Uppsala University, 751 08 Uppsala, Sweden Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the constructions and tests of the dilatation operator and of the spectrum of composite operators in the flavour SU (2) subsector of N =4 SYM in the planar limit by explicit Feynman graph calculations with emphasis on analyses beyond one loop. From four loops on, the dilatation operator determines the spectrum only in the asymptotic regime, i.e. to a loop order which is strictly smaller than the number of elementary fields of the composite operators. We review also the calculations which take a first step beyond this limitation by including the leading wrapping corrections. Content Type Journal Article Pages 1-26 DOI 10.1007/s11005-011-0508-7 Authors Christoph Sieg, Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Johann von Neumann Haus, Rudower Chaussee 25, 12489 Berlin, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010 ) proposed a relation between N =2 four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra A =\sf Vir Ä H which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of A . The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called Z \sf bif appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. ( 2010 ). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy. Content Type Journal Article Pages 1-32 DOI 10.1007/s11005-011-0503-z Authors Vasyl A. Alba, Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Vladimir A. Fateev, Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Alexey V. Litvinov, Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Grigory M. Tarnopolskiy, Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this paper, we present a concrete counterexample to the conjecture of Kontsevich about the polynomial countability of graph hypersurfaces. In contrast to this, we show that the “wheel with spokes” graphs WS n are polynomially countable. Content Type Journal Article Pages 1-13 DOI 10.1007/s11005-011-0501-1 Authors Dzmitry Doryn, FB6, Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this note, we prove the following inequality: 2 || D h j \frac s 2 x j || 2 ³ j (1)+ h (1) - | j - h | (1) , where j and η are positive normal linear functionals over a von Neumann algebra. This is a generalization of the famous Powers–Størmer inequality (Powers and Størmer proved the inequality for L ( H ) in Commun Math Phys 16:1–33, 1970 ; Takesaki in Theory of Operator Algebras II, 2001 ). For matrices, this inequality was proven by Audenaert et al. (Phys Rev Lett 98:160501, 2007 ). We extend their result to general von Neumann algebras. Content Type Journal Article Pages 1-8 DOI 10.1007/s11005-011-0504-y Authors Yoshiko Ogata, Graduate School of Mathematics, University of Tokyo, Tokyo, Japan Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Given a formal symplectic groupoid G over a Poisson manifold ( M , π 0 ), we define a new object, an infinitesimal deformation of G , which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation p 0 + e p 1 of the Poisson bivector field π 0 . To any pair of natural star products ( * , ~ *   ) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G . We call it the deformation groupoid of the pair ( * , ~ *   ) . To each star product with separation of variables * on a Kähler–Poisson manifold M we relate another star product with separation of variables ^ *   on M . We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables * . This algorithm is based upon the deformation groupoid of the pair ( * , ^ *   ) . Content Type Journal Article Pages 1-23 DOI 10.1007/s11005-011-0495-8 Authors Alexander Karabegov, Department of Mathematics, Abilene Christian University, ACU Box 28012, Abilene, TX 79699-8012, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the construction of the AdS/CFT dressing factor, its analytic properties, and several checks of its validity. Content Type Journal Article Pages 1-23 DOI 10.1007/s11005-011-0482-0 Authors Pedro Vieira, Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Dmytro Volin, Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    This review is devoted to the classical integrability of the AdS 5 × S 5 superstring theory. It starts with a reminder of the corresponding action as a coset model. The symmetries of this action are then reviewed. The classical integrability is then considered from the lagrangian and hamiltonian points of view. The second part of this review deals with the gauge fixing of this theory. Finally, some aspects of the pure spinor formulation are also briefly reviewed. Content Type Journal Article Pages 1-19 DOI 10.1007/s11005-011-0481-1 Authors Marc Magro, Laboratoire de Physique, ENS Lyon et CNRS UMR 5672, Université de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Aspects of the D  = 4, N =4 superconformal symmetry relevant to the AdS/CFT duality and integrability are reviewed. These include the Lie superalgebra \mathfrak psu (2,2 | 4) , its representations, conformal transformations and correlation functions in N =4 super Yang–Mills theory as well as an illustration of the AdS 5  × S 5 superspace on which the dual string theory is formulated. Content Type Journal Article Pages 1-17 DOI 10.1007/s11005-011-0479-8 Authors Niklas Beisert, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the study of Hopf algebras, classical and quantum R -matrices, infinite-dimensional Yangian symmetries and their representations in the context of integrability for the N = 4 versus AdS 5 × S 5 correspondence. Content Type Journal Article Pages 1-19 DOI 10.1007/s11005-011-0491-z Authors Alessandro Torrielli, Institute for Theoretical Physics and Spinoza Institute, Utrecht University, 3508 TD Utrecht, The Netherlands Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    For bounded linear operators A , B on a Hilbert space H we show that å l   dist ( l , Num ( A )) p is bounded from above by the Schatten- p -norm of B − A . Here, the sum is taken over all discrete eigenvalues of B and Num( A ) denotes the numerical range of A . We apply this estimate to recover and improve some Lieb–Thirring type inequalities for non-selfadjoint Jacobi and Schrödinger operators. Content Type Journal Article Pages 1-17 DOI 10.1007/s11005-011-0494-9 Authors Marcel Hansmann, Faculty of Mathematics, Chemnitz University of Technology, Chemnitz, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this paper, we quantize universal gauge groups such as SU (∞), as well as their homogeneous spaces, in the σ - C *-algebra setting. More precisely, we propose concise definitions of σ - C *-quantum groups and σ - C *-quantum homogeneous spaces and explain these concepts here. At the same time, we put these definitions in the mathematical context of countably compactly generated spaces as well as C *-compact quantum groups and homogeneous spaces. We also study the representable K-theory of these spaces and compute these groups for the quantum homogeneous spaces associated to the quantum version of the universal gauge group SU (∞). Content Type Journal Article Pages 1-15 DOI 10.1007/s11005-011-0492-y Authors Snigdhayan Mahanta, Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia Varghese Mathai, Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We consider a magnetic Laplacian −Δ A = (i d +  A ) * (i d + A ) on a non-compact hyperbolic surface M with finite area. A is a real one-form and the magnetic field d A is constant in each cusp. When the harmonic component of A satisfies some quantified condition, the spectrum of −Δ A is discrete. In this case, we prove that the counting function of the eigenvalues of −Δ A satisfies the classical Weyl formula, even when d A =0. Content Type Journal Article Pages 1-9 DOI 10.1007/s11005-011-0489-6 Authors Abderemane Morame, Dpt. Mathématiques, Faculté des Sciences, UMR 6629 du CNRS, Université de Nantes, B.P. 99208, 44322 Nantes Cedex 3, France Françoise Truc, Institut Fourier, UMR 5582 CNRS-UJF, Université de Grenoble I, B.P. 74, 38402 St Martin d’Hères Cedex, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on \mathbb P 2 and its blow-up ~ \mathbb P   2   . Wall-crossing is used to obtain the Betti numbers for ~ \mathbb P   2   . These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for \mathbb P 2 follow from those for ~ \mathbb P   2   by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions. Content Type Journal Article Pages 1-14 DOI 10.1007/s11005-011-0490-0 Authors Jan Manschot, Institut de Physique Théorique, CEA Saclay, CNRS-URA 2306, 91191 Gif sur Yvette, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Belov-Kanel and Kontsevich (Lett Math Phys 74:181–199, 2005 ) conjectured that the group of automorphisms of the n th Weyl algebra and the group of polynomial symplectomorphisms of \mathbb C 2 n are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0471-3 Authors Erik Backelin, Departamento de Matemáticas, Universidad de los Andes, Carrera 1 N. 18A-10, Bogotá, Colombia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We extend Milnor’s  μ -invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for  μ -invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday’s diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads. Content Type Journal Article Pages 1-20 DOI 10.1007/s11005-010-0459-4 Authors Olga Kravchenko, Université de Lyon, Université Lyon 1, ICJ, UMR 5208 CNRS, 43 blvd 11 novembre 1918, 69622 Villeurbanne Cedex, France Michael Polyak, Department of Mathematics, Technion, Haifa, 32000 Israel Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    It is shown that the periodic initial value problem (i.v.p.) for a novel integrable generalization of the nonlinear Schrödinger equation (igNLS) is well-posed in Sobolev spaces with exponent greater than 3/2. The proof is based on a Galerkin-type approximation method. When 1/ ν is not an integer, then a mollified version of the i.v.p. is solved first by applying the fundamental ODE theorem in Banach spaces. Then, deriving appropriate energy estimates, it is shown that the family of the approximate solutions thus obtained has a convergent subsequence, which at the limit gives a solution to the igNLS equation. Finally, again using energy estimates, it is shown that this solution is unique and that the data-to-solution map is continuous. When 1/ ν is an integer then well-posedness is proved for the nonlocal version of this equation in the corresponding homogeneous Sobolev spaces. Content Type Journal Article Pages 1-21 DOI 10.1007/s11005-011-0488-7 Authors Athanasios S. Fokas, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA UK A. Alexandrou. Himonas, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    A manifold is multisymplectic, or more specifically n -plectic, if it is equipped with a closed nondegenerate differential form of degree n  + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n -algebras are examples of L ∞ -algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n -plectic manifold, we explicitly construct a corresponding Lie n -algebra on a complex consisting of differential forms whose multi-brackets are specified by the n -plectic structure. We also show that any n -plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0493-x Authors Christopher L. Rogers, Department of Mathematics, University of California, Riverside, CA 92521, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean-field description of a many particle quantum system directly into a mathematical algorithm. It is effective because, with less effort, the strategy yields better results than previously achieved. As an instructional example we treat a simple model for the time-dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean-field scalings leading, e.g. to the Gross-Pitaevskii equation can also be treated (Pickl in Derivation of the time dependent Gross Pitaevskii equation with external fields, preprint; Pickl in Derivation of the time dependent Gross Pitaevskii equation without positivity condition on the interaction, preprint). Content Type Journal Article Pages 1-14 DOI 10.1007/s11005-011-0470-4 Authors Peter Pickl, Institute of Mathematics, University of Munich, Theresienstr. 39, 80333 Munich, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the computation of the anomalous dimension of twist operators in the planar limit of N =4 SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations; this aspect of the story is emphasized. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0483-z Authors Lisa Freyhult, Division for Theoretical Physics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Following Bettelheim et al. (Phys Rev Lett 95:251601, 2005 ), we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm–Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik–Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G  =  SU (2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G  =  SU ( n ) and the level k  = 1, there are several situations when boundary one-point correlators are SLE κ -martingales. If the boundary field is labelled by the defining n -dimensional representation of SU ( n ), we obtain \varkappa = 2 . For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get \varkappa = 8/( n + 2) . We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the SLE \varkappa evolution is replaced by SLE \varkappa, r with r = \varkappa - 6 . Content Type Journal Article Pages 1-19 DOI 10.1007/s11005-011-0500-2 Authors Anton Alekseev, Section of Mathematics, University of Geneva, 2-4 rue de Lièvre, C.P. 64, 1211 Geneva 4, Switzerland Andrei Bytsko, Steklov Mathematics Institute, Fontanka 27, St. Petersburg, 191023 Russia Konstantin Izyurov, Section of Mathematics, University of Geneva, 2-4 rue de Lièvre, C.P. 64, 1211 Geneva 4, Switzerland Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The Gauge/Bethe correspondence of Nekrasov and Shatashvili relates the spectrum of integrable spin chains to the ground states of supersymmetric gauge theories. Up to now, this correspondence has been an observation; the underlying reason for its existence remaining elusive. We argue here that geometrical representation theory is a mathematical foundation of the Gauge/Bethe correspondence, and it provides a framework to study families of gauge theories in a unified way. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0526-5 Authors Domenico Orlando, Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa-no-Ha 5-1-5, Kashiwa, Chiba 277-8568, Japan Susanne Reffert, Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa-no-Ha 5-1-5, Kashiwa, Chiba 277-8568, Japan Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the semiclassical analysis of strings in AdS 5 × S 5 with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in AdS 3 ⊂ AdS 5 as our main example. Furthermore, we review the perturbative light-cone quantisation of the string theory and the calculation of the worldsheet S-matrix. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0510-0 Authors Tristan McLoughlin, Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We give a one-dimensional interpretation of the four-dimensional twisted N  = 1 superYang–Mills theory on a Kähler manifold by performing an appropriate dimensional reduction. We prove the existence of a 6-generator superalgebra, which does not possess any invariant Lagrangian but contains two different subalgebras that determine the twisted and untwisted formulations of the N  = 1 superYang–Mills theory. Content Type Journal Article Pages 1-11 DOI 10.1007/s11005-011-0505-x Authors Laurent Baulieu, Theoretical Division, CERN, 1211 Geneva 23, Switzerland Francesco Toppan, CBPF, Rua Dr. Xavier Sigaud 150, CEP 22290-180 Rio de Janeiro, RJ, Brazil Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    Scattering amplitudes in planar N =4 super Yang–Mills theory reveal a remarkable symmetry structure. In addition to the superconformal symmetry of the Lagrangian of the theory, the planar amplitudes exhibit a dual superconformal symmetry. The presence of this additional symmetry imposes strong restrictions on the amplitudes and is connected to a duality relating scattering amplitudes to Wilson loops defined on polygonal light-like contours. The combination of the superconformal and dual superconformal symmetries gives rise to a Yangian, an algebraic structure which is known to be related to the appearance of integrability in other regimes of the theory. We discuss two dual formulations of the symmetry and address the classification of its invariants. Content Type Journal Article Pages 1-25 DOI 10.1007/s11005-011-0519-4 Authors James M. Drummond, PH-TH Division, CERN, 1211 Geneva 23, Switzerland Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi–Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi–Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost. Content Type Journal Article Pages 1-20 DOI 10.1007/s11005-011-0528-3 Authors Jimmy Dillies, Department of Mathematics, University of Utah, Salt Lake City, UT, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We investigate convergence properties of the spectral shift functions associated with a pair of Schrödinger operators with Dirichlet boundary conditions at the end points of a finite interval as the length of interval approaches infinity. Content Type Journal Article Pages 1-15 DOI 10.1007/s11005-011-0524-7 Authors Vita Borovyk, Department of Mathematics, University of Cincinnati, Cincinnati, OH 45221, USA Konstantin A. Makarov, Department of Mathematics, University of Missouri, Columbia, MO 63211, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the role of integrability in certain aspects of \fancyscript N =4 SYM which go beyond the planar spectrum. In particular, we discuss integrability in relation to non-planar anomalous dimensions, multi-point functions and Maldacena–Wilson loops. Content Type Journal Article Pages 1-26 DOI 10.1007/s11005-011-0514-9 Authors Charlotte Kristjansen, The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review the computation of scattering amplitudes of planar maximally super-symmetric Yang–Mills at strong coupling. By using the AdS / CFT duality the problem boils down to the computation of the area of certain minimal surfaces on AdS . The integrability of the model can then be efficiently used in order to give an answer for the problem in terms of a set of integral equations. Content Type Journal Article Pages 1-22 DOI 10.1007/s11005-011-0518-5 Authors Luis F. Alday, Mathematical Institute, University of Oxford, 24–29 St. Giles, Oxford, OX1 3LB UK Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this chapter of Review of AdS/CFT Integrability we introduce N =4 Super Yang-Mills. We discuss the global superalagebra PSU (2, 2|4) and its action on gauge invariant operators. We then discuss the computation of the correlators of certain gauge invariant operators, the so-called single trace operators in the large N limit. We show that interactions in the gauge theory lead to mixing of the operators. We compute this mixing at the one-loop level and show that the problem maps to a one-dimensional spin chain with nearest neighbor interactions. For operators in the SU (2) sector we show that the spin chain is the ferromagnetic Heisenberg spin chain whose eigenvalues are determined by the Bethe equations. Content Type Journal Article Pages 1-26 DOI 10.1007/s11005-011-0522-9 Authors Joseph A. Minahan, Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In this contribution, we review long-range integrable spin chains that originate from the recently discovered integrability in the planar AdS/CFT correspondence. We also briefly summarise the theory of generic integrable perturbatively long-range spin chains. Content Type Journal Article Pages 1-18 DOI 10.1007/s11005-011-0509-6 Authors Adam Rej, Blackett Laboratory, Imperial College, London, SW7 2AZ UK Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We review current efficient techniques for the construction of multi-leg and multi-loop on-shell scattering amplitudes in supersymmetric gauge theories. Examples in the maximally supersymmetric Yang-Mills theory in four dimensions are included. Content Type Journal Article Pages 1-25 DOI 10.1007/s11005-011-0517-6 Authors Radu Roiban, Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    There is a growing amount of evidence that QCD (and four-dimensional gauge theories in general) possesses a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. In this review we consider the scale dependence of local gauge invariant operators and high-energy (Regge) behavior of scattering amplitudes to explain that the effective QCD dynamics in both cases is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations. Content Type Journal Article Pages 1-29 DOI 10.1007/s11005-011-0516-7 Authors Gregory P. Korchemsky, Institut de Physique Théorique, Unité de Recherche Associée au CNRS URA 2306, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We provide a new unified construction of the two-parameter Podleś two-spheres as characterised by a projector e with trace q ( e ) = 1 + λ. In our formulation the limit in which q → 1 with λ fixed is the fuzzy sphere, while the limit λ → 0 with q fixed is the standard q -deformed sphere. We show further that the non-standard Podleś spheres arise geometrically as ‘constant time slices’ of the unit hyperboloid in q -Minkowski space viewed as the braided group B q [ SU 2 ]. Their localisations are then isomorphic to quotients of U q ( su 2 ) at fixed values of the q -Casimir precisely q -deforming the fuzzy case. We also use transmutation and twisting theory to introduce a C q [ G \mathbb C ] -covariant differential calculus on general B q [ G ] and U q ( g ), with Ω( B q [ SU 2 ]) and Ω( U q ( su 2 ) given in detail. To complete the picture, we show how the covariant calculus on the 3D bicrossproduct spacetime arises from Ω( C q [ SU 2 ]) prior to twisting. Content Type Journal Article Pages 1-25 DOI 10.1007/s11005-011-0523-8 Authors Shahn Majid, School of Mathematics, Queen Mary University of London, Mile End Rd., London, E1 4NS UK Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    In the operator theoretic renormalization analysis introduced by Bach, Fröhlich, and Sigal, we prove uniqueness of the ground state. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0532-7 Authors David Hasler, Department of Mathematics, College of William and Mary, Williamsburg, VA, USA Ira Herbst, Department of Mathematics, University of Virginia, Charlottesville, VA, USA Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally, we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe’s ansatz. Content Type Journal Article Pages 1-18 DOI 10.1007/s11005-011-0530-9 Authors Matthias Staudacher, Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Johann von Neumann-Haus, Rudower Chaussee 25, 12489 Berlin, Germany Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017

Beschreibung:    We consider systems of ODEs with the right-hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich (private communication). We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries. Content Type Journal Article Pages 1-10 DOI 10.1007/s11005-011-0527-4 Authors Thomas Wolf, Brock University, St. Catharines, Ontario, Canada Olga Efimovskaya, Moscow State University, Moscow, Russia Journal Letters in Mathematical Physics Online ISSN 1573-0530 Print ISSN 0377-9017