Finite-size corrections of the entanglement entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement alpha-Renyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry such as the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of a, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the alpha-Renyi entropies. We conjecture that the exponent of the leading finite-size correction of the alpha-Renyi entropies is p(alpha) = 2X(epsilon)/alpha for alpha > 1 and p(1) = nu, where X-epsilon denotes the dimensions of the energy operator of the model and nu = 2 for all the models.

Aspects of Integrability in Gauge/String Correspondence

In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences. We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality. In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic (1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences. In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side. One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size. At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood by the rapidly evolving developments of this extremely exciting research field.

A dictionary of modular threefolds

The thesis deals with the modularity conjecture for three-dimensional Calabi-Yau varieties. This is a generalization of the work of A. Wiles and others on modularity of elliptic curves. Modularity connects the number of points on varieties with coefficients of certain modular forms. In chapter 1 we collect the basics on arithmetic on Calabi-Yau manifolds, including general modularity results and strategies for modularity proofs. In chapters 2, 3, 4 and 5 we investigate examples of modular Calabi-Yau threefolds, including all examples occurring in the literature and many new ones. Double octics, i.e. Double coverings of projective 3-space branched along an octic surface, are studied in detail. In chapter 6 we deal with examples connected with the same modular forms. According to the Tate conjecture there should be correspondences between them. Many correspondences are constructed explicitly. We finish by formulating conjectures on the occurring newforms, especially their levels. In the appendices we compile tables of coefficients of weight 2 and weight 4 newforms and many examples of double octics.

Investigations on asymptotic safety of metric, tetrad and Einstein-Cartan gravity

Among the different approaches for a construction of a fundamental quantum theory of gravity the Asymptotic Safety scenario conjectures that quantum gravity can be defined within the framework of conventional quantum field theory, but only non-perturbatively. In this case its high energy behavior is controlled by a non-Gaussian fixed point of the renormalization group flow, such that its infinite cutoff limit can be taken in a well defined way. A theory of this kind is referred to as non-perturbatively renormalizable. In the last decade a considerable amount of evidence has been collected that in four dimensional metric gravity such a fixed point, suitable for the Asymptotic Safety construction, indeed exists. This thesis extends the Asymptotic Safety program of quantum gravity by three independent studies that differ in the fundamental field variables the investigated quantum theory is based on, but all exhibit a gauge group of equivalent semi-direct product structure. It allows for the first time for a direct comparison of three asymptotically safe theories of gravity constructed from different field variables. The first study investigates metric gravity coupled to SU(N) Yang-Mills theory. In particular the gravitational effects to the running of the gauge coupling are analyzed and its implications for QED and the Standard Model are discussed. The second analysis amounts to the first investigation on an asymptotically safe theory of gravity in a pure tetrad formulation. Its renormalization group flow is compared to the corresponding approximation of the metric theory and the influence of its enlarged gauge group on the UV behavior of the theory is analyzed. The third study explores Asymptotic Safety of gravity in the Einstein-Cartan setting. Here, besides the tetrad, the spin connection is considered a second fundamental field. The larger number of independent field components and the enlarged gauge group render any RG analysis of this system much more difficult than the analog metric analysis. In order to reduce the complexity of this task a novel functional renormalization group equation is proposed, that allows for an evaluation of the flow in a purely algebraic manner. As a first example of its suitability it is applied to a three dimensional truncation of the form of the Holst action, with the Newton constant, the cosmological constant and the Immirzi parameter as its running couplings. A detailed comparison of the resulting renormalization group flow to a previous study of the same system demonstrates the reliability of the new equation and suggests its use for future studies of extended truncations in this framework.

Monodromy calculations for some differential equations

In vielen Teilgebieten der Mathematik ist es w"{u}nschenswert, die Monodromiegruppe einer homogenen linearen Differenzialgleichung zu verstehen. Es sind nur wenige analytische Methoden zur Berechnung dieser Gruppe bekannt, daher entwickeln wir im ersten Teil dieser Arbeit eine numerische Methode zur Approximation ihrer Erzeuger.rnIm zweiten Abschnitt fassen wir die Grundlagen der Theorie der Uniformisierung Riemannscher Fl"achen und die der arithmetischen Fuchsschen Gruppen zusammen. Auss erdem erkl"aren wir, wie unsere numerische Methode bei der Bestimmung von uniformisierenden Differenzialgleichungen dienlich sein kann. F"ur arithmetische Fuchssche Gruppen mit zwei Erzeugern erhalten wir lokale Daten und freie Parameter von Lam'{e} Gleichungen, welche die zugeh"origen Riemannschen Fl"achen uniformisieren. rnIm dritten Teil geben wir einen kurzen Abriss zur homologischen Spiegelsymmetrie und f"uhren die $widehat{Gamma}$-Klasse ein. Wir erkl"aren wie diese genutzt werden kann, um eine Hodge-theoretische Version der Spiegelsymmetrie f"ur torische Varit"aten zu beweisen. Daraus gewinnen wir Vermutungen "uber die Monodromiegruppe $M$ von Picard-Fuchs Gleichungen von gewissen Familien $f:mathcal{X}rightarrow bbp^1$ von $n$-dimensionalen Calabi-Yau Variet"aten. Diese besagen erstens, dass bez"uglich einer nat"urlichen Basis die Monodromiematrizen in $M$ Eintr"age aus dem K"orper $bbq(zeta(2j+1)/(2 pi i)^{2j+1},j=1,ldots,lfloor (n-1)/2 rfloor)$ haben. Und zweitens, dass sich topologische Invarianten des Spiegelpartners einer generischen Faser von $f:mathcal{X}rightarrow bbp^1$ aus einem speziellen Element von $M$ rekonstruieren lassen. Schliess lich benutzen wir die im ersten Teil entwickelten Methoden zur Verifizierung dieser Vermutungen, vornehmlich in Hinblick auf Dimension drei. Dar"uber hinaus erstellen wir eine Liste von Kandidaten topologischer Invarianten von vermutlich existierenden dreidimensionalen Calabi-Yau Variet"aten mit $h^{1,1}=1$.

Computations of Vector-Valued Siegel Modular Forms

We carry out some computations of vector-valued Siegel modular forms of degree two, weight (k, 2) and level one, and highlight three experimental results: (1) we identify a rational eigenform in a three-dimensional space of cusp forms; (2) we observe that non-cuspidal eigenforms of level one are not always rational; (3) we verify a number of cases of conjectures about congruences between classical modular forms and Siegel modular forms. Our approach is based on Satoh's description of the module of vector-valued Siegel modular forms of weight (k, 2) and an explicit description of the Hecke action on Fourier expansions. (C) 2013 Elsevier Inc. All rights reserved.

Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes $A$ and $B$ must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than Schur equality: that sA and sB have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.

Centralization versus Delegation in an Experimental Capital Budgeting Setting

In an experiment, we model two stylized facts about capital budgeting practice, budgetary slack creation and delegation of decision-making authority. In our setting, under centralization, headquarters announces a budget, the division manager gives a cost report, and headquarters decides on the project. Under delegation, headquarters allocates a budget to the manager, and the manager is authorized to make the investment decision. We argue that the ability of headquarters to commit to a budget moderates the effect of delegation, and we find evidence in favor of our argument as there is an interaction effect of delegation and commitment to budgets. The effects of delegation are particularly strong when budgets are non-binding as delegation serves as a substitute for commitment in this case. This leads to smaller expenditures and to a higher headquarters’ payoff under delegation than under centralization. In contrast, when headquarters can commit to the budget, the descriptive data are consistent with our conjectures about the effects of honesty preferences, but the effects are too small to be significant.

Newton’s Metaphysics of Space as God’s Emanative Effect

In several of his writings, Isaac Newton proposed that physical space is God’s “emanative effect” or “sensorium,” revealing something interesting about the metaphysics underlying his mathematical physics. Newton’s conjectures depart from Plato and Aristotle’s metaphysics of space and from classical and Cambridge Neoplatonism. Present-day philosophical concepts of supervenience clarify Newton’s ideas about space and offer a portrait of Newton not only as a mathematical physicist but an independent-minded rationalist philosopher.

Brief aus Italien

Mose Beilinson

Von den russisch-jüdischen Beziehungen

Mose Beilinson

Jabotinsky als Schriftsteller

Mose Beilinson

Der Untergang der Assimilation

Mose Beilinson

Kongressnotizen

Mose Beilinson

Zionistische Krise und Jüdischer Bund

Mose Beilinson