Self-dual Yang-Mills fields and deformations of algebraic curves

Recently it has been shown that the methods of algebraic geometry first used for finding periodic and almost periodic solutions of KdV, HSh, SG and other equations [11–13] may be successfully applied to study the solutions of nonlinear equations with a variable spectral parameter in associated zero-curvature representation. In this work following [20] this treatment is extended to the case of the self-duality equation. It seems to be the first example of a four-dimensional non-linear equation solvable by the method of finite-gap integration. Two broad classes of finite-gap solutions for each —SU(2) andSU(1,1) gauge groups are constructed in terms of multidimensional theta-functions. The dynamics of the solutions is given by the movement of the hyperelliptic curve with moving branch points and a divisor of the poles in the moduli space of algebraic curves. In the general case our solutions have no periodicity property. We show how one-instanton solution and 5N-parametric t'Hooft family of instantons may be obtained by the degeneration of general formulae.     

The Hamiltonian formulation of general relativity: myths and reality

A conventional wisdom often perpetuated in the literature states that: (i) a 3 + 1 decomposition of spacetime into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific field-dependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [21] and of ADM [22] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i–iii) have been shown to be incorrect in [45] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac’s references to space-like surfaces are inessential and that such surfaces do not enter his calculations. In addition, we show that his assumption g _0k = 0, used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption g _0k = 0. The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary (also known as the “Hamiltonian” and “diffeomorphism” constraints). The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints. Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points (i–iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric _{g μν} to lapse and shift functions and the three-metric g _km , which is not canonical. This proves that point (iv) is incorrect. Points (i–iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein’s theory itself.     

Introducing boranorcaradienes with more stability than their corresponding borepins: Reversal of tautomerization via substituents at theoretical levels

In contrast to typical borepins, which appear about 40.0 kcalmol^?1 more stable than their corresponding tautomeric boranorcaradienes, we have found 2 species, which have reversed this trend and pushed the equilibrium in favor of their corresponding boranorcaradienes. They are namely 1a,9b‐dihydro‐1H‐borireno[2,3‐h]pyridazino[4,3‐f]cinnoline and 1a,9b‐dihydro‐1H‐borireno[2,3‐h]pyrimido[5,4‐f]quinazoline which stand up among 14 isomeric systems probed, at B3LYP/AUG‐cc‐pVTZ, M06‐2X/AUG‐cc‐pVTZ, MP2/AUG‐cc‐pVTZ, and HF/AUG‐cc‐pVTZ. Energy barriers are calculated in gas‐phase, where the possibility of a rapid interconversion of the isomers is ruled out. Generally, dibenzoboranorcaradienes assume C_s symmetry with planar geometry and dihedral angle of zero degree. In contrast, their corresponding borepins show a high tendency for puckering with dihedral angle of ~66°. The preference of the latter for puckered non‐planar geometries is evidenced by natural bonding orbitals calculations and visually through their frontier molecular orbitals. Main interactions appear to be hyperconjugations of σ and π bonds across the rings. Position and number of nitrogen atoms on the fused rings seem to affect the energy gap, dipole moment, symmetry, dihedral angle, the chemical shift, NICS, bond lengths, and charge distribution.     

Spectral Theory of Herglotz Functions and Their Compositions

Recent developments in the theory of value distribution for boundary values of Herglotz functions [5], with applications to the spectral analysis of Herglotz measures and differential operators [2, 3] lead in a natural way to the investigation of measures which relate (through the Herglotz representation theorem) to the composition of a pair of Herglotz functions F,G. The present paper provides results on the boundary values of composed Herglotz functions and on the terms of their Herglotz representation which are dominant at large |z|.     

Explorative computational study of the singlet fission process

Different ab initio methods, namely multi-reference and nonorthogonal configuration interaction techniques, are explored for their applicability in studying the singlet fission problem. It has been shown for 2-methyl-1,5-hexadiene that the ¹TT state can be identified using multi-reference techniques. The geometrical and vibrational properties of the ¹TT state are such that they can be approximated with those of the ⁵TT state. A proof of principle is given for the calculation of the singlet fission pathway driven by nuclear motion: efficient singlet fission can take place if the ¹TT and S₁ states are close in energy with a large non-adiabatic coupling matrix element at the S₁ geometry, and the energy of the S₀ state is well below that of the ¹TT state at the ¹TT geometry.

The nonorthogonal configuration interaction method was used to treat a tetracene trimer. It has been shown that the first excited states can be interpreted as delocalised states; interaction with charge-transfer base states plays an important role. The ¹TT states are localised on one pair of molecules. The electronic coupling between the diabatic S[n] and ¹TT[m] states is in the meV range, confirming previous estimates. The charge-transfer base states enhance the coupling between the S[1]/S[2] and ¹TT[2] excited states.     

Noncomputability arising in dynamical triangulation model of four-dimensional Quantum Gravity

Computations in dynamical triangulation models of four-dimensional Quantum Gravity involve weighted averaging over sets of all distinct triangulations of compact four-dimensional manifolds. In order to be able to perform such computations one needs an algorithm which for any givenN and a given compact four-dimensional manifoldM constructs all possible triangulations ofM with N simplices. Our first result is that such algorithm does not exist. Then we discuss recursion-theoretic limitations of any algorithm designed to perform approximate calculations of sums over all possible triangulations of a compact four-dimensional manifold.Communicated by N.Yu. Reshetikhin     

The Spin Density Distribution in the Hydrocarbon Radical Anions

In this note we wish to report our calculations of the spin density distribution in the several hydrocarbon radical anions using the “half-electron” SCF MO [variable beta] method of Dewar^1,2. This method calculates the energy of a system in which the unpaired electron is replaced by two “half-electrons” of the opposite spin. In this way one has the pseudo closed-shell system at equilibrium geometry and the resulting energy [E] differs from that [E_o] given by Roothaan's³ procedure only for ¼ J_mm

E_o=E - ¼ J_mm

Term ¼ J_mm represents the correction due to the spurious repulsion of the two “half-electrons”.     

Topology of lattice gauge fields

Non-Abelian gauge fields on a four-dimensional hypercubic lattice with small action density [Tr{U( )} for SU(2) gauge fields] are shown to carry an integer topological chargeQ, which is invariant under continuous deformations of the field. A concrete expression forQ is given and it is verified thatQ reduces to the familiar Chern number in the classical continuum limit.Work supported in part by Schweizerischer Nationalfonds     

Moduli and (un)attractor black hole thermodynamics

We investigate four-dimensional spherically symmetric black hole solutions in gravity theories with massless, neutral scalars non-minimally coupled to gauge fields. In the non-extremal case, we explicitly show that, under the variation of the moduli, the scalar charges appear in the first law of black hole thermodynamics. In the extremal limit, the near horizon geometry is AdS ₂ × S ² and the entropy does not depend on the values of moduli at infinity. We discuss the attractor behaviour by using Sen’s entropy function formalism as well as the effective potential approach and their relation with the results previously obtained through special geometry method. We also argue that the attractor mechanism is at the basis of the matching between the microscopic and macroscopic entropies for the extremal non-BPS Kaluza–Klein black hole.     

AdS 5 Solutions of Type IIB Supergravity and Generalized Complex Geometry

We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to ${\mathcal{N}=1}We use the formalism of generalized geometry to study the generic supersymmetric AdS ₅ solutions of type IIB supergravity that are dual to N=1{\mathcal{N}=1} superconformal field theories (SCFTs) in d = 4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a generalized complex geometry, which is an extension of K?hler-Einstein geometry. Remarkably, provided the five-form flux is non-vanishing, the cone is symplectic. The symplectic structure can be used to obtain Duistermaat-Heckman type integrals for the central charge of the dual SCFT and the conformal dimensions of operators dual to BPS wrapped D3-branes. We illustrate these results using the Pilch-Warner solution.     

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six-dimensional minimal conformal matter of types SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are \(\mathcal{N}=1\) quiver gauge theories with SU(3) and SU(4) gauge groups, respectively. The defects are engineered by giving space-time-dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter, the defects carry SU(2) flavor symmetry which is not a symmetry of the four-dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors, and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four-dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six- dimensional compactifications and four-dimensional models.

     

Midpoints in gyrogroups

The obscured Thomas precessionof the special theory of relativity (STR) has been soared into prominence by exposing the mathematical structure, called a gyrogroup,to which it gives rise [A. A. Ungar, Amer. J. Phys.59,824 (1991)], and the role that it plays in the study of Lorentz groups [A. A. Ungar, Amer. J. Phys.60,815 (1992); A. A. Ungar, J. Math. Phys.35,1408 (1994); A. A. Ungar, J. Math. Phys.35,1881 (1994)]. Thomas gyrationresults from the abstraction of Thomas precession.As such, its study sheds light on relativistic velocity spaces and their symmetries which are concealed in Thomas precession. In order to uncover new properties of relativistic gyrogroups, we employ in this article the group theoretic concepts of divisible groupsand two-torsion free groupsto construct midpointsin gyrogroups. Systems of successive midpoints then describe straight gyrolinesand suggest a new, natural distance function that involves a Thomas gyration. These, in turn, reveal a new, interesting geometry that underlies relativistic velocity spaces. In this resulting gyrogeometrythe straight gyrolines form geodesics under a distance function which turns out to be the Poincaré hyperbolic distance function relaxed by a Thomas gyration. These geodesics do obey the parallel axiom of Euclidean geometry. Ironically, (i) attempts to understand the parallel postulate of Euclidean geometry gave rise to hyperbolic geometry in which the parallel postulate disappears;(ii) hyperbolic geometry gave rise to Einstein's STR; (iii) Einstein's STR established the bizarre and counterintuitive relativistic effect called Thomas precession, the abstraction of which is called Thomas gyration; and (iv) Thomas gyration now repairs in this article the Poincaré distance function of hyperbolic geometry to the point where the parallel postulate reappears.     

A Method for q-Calculus

Abstract

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts.

The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward–AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition.

In contrast to the the Ward–AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product.

We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz’ [26] operator expression for Laguerre polynomials. The notation for Cigler’s [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.     

Study of disorder-order transitions in FexAl1−x binary alloys using the augmented space recursion based orbital peeling technique

In this communication we propose a method for the study of disorder-order transitions in Fe_xAl_1−x binary alloys. We turn to our earlier development [1] of a combination of the recursion method introduced by Haydock et al. [2] and our augmented space approach [3] with the orbital peeling technique proposed by Burke [4] to determine the small energy differences required in obtaining the pair energies which go as input to the generalized perturbation technique [5] of studying disorder-order transitions.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

HYPERKÄHLER STRUCTURE OF THE TAUB-NUT METRIC

The Taub-NUT four-dimensional space-time can be obtained from Euclidean eight-dimensional one through a momentum map construction; the HKLR theorem [9] guarantees the hyperkähler structure of R ⁸ descends to a hyperkähler structure in the Taub-NUT space. Here we present a detailed and fully explicit construction of the hyperkähler structure of a space-time with a Taub-NUT metric.     

The Low Energy Limit of String Theory and its Compactifications with Background Fluxes

String theory is consistently defined in ten dimensions. In order to extract any information about four-dimensional physics, we need to understand the properties of the six-dimensional compact manifold orthogonal to our four-dimensional universe. A possibility that is being very much explored lately is to look at manifolds on which background fluxes are turned on. In this article, we present an introduction to string theory, focusing on its massless sector. We then review traditional compactifications to four–dimensions, and finally motivate and describe the so-called flux compactifications. We interpret the allowed six-dimensional manifolds from the point of view of generalized complex geometry.