The WDVV Equations in Pure Seiberg–Witten Theory

We review the relationship between pure four-dimensional Seiberg–Witten theory and the periodic Toda chain. We discuss the definition of the prepotential and give two proofs that it satisfies the generalized Witten–Dijkgraaf–Verlinde–Verlinde equations. A number of steps in the definitions and proofs that is missing in the literature is supplied.Mathematics Subject Classifications (2000) 14H10, 14H20, 14H40, 14H70, 14D45.     

Tachyon Effective Actions in String Theory

We discuss the derivation of string effective actions for tachyons and massless modes within the sigma-model approach. We mostly consider the open string case, demonstrating that the renormalized partition function of the boundary sigma model gives the effective action for the massless vector and tachyon in the derivative expansion. We give a manifestly gauge-invariant definition of Z(T,A) in the non-Abelian Neveu–Schwarz–Ramond (NSR) open string theory and verify that its derivative reproduces the tachyon beta function in a particular scheme. We also comment on the derivation of similar actions for tachyons and massless modes in the closed bosonic and the NSR string theories.     

Infinite-Dimensional Analysis and Analytic Number Theory

We consider arithmetical aspects of analysis on Fock spaces (Boson Fock space, Fermion Fock space, and Boson–Fermion Fock space) with applications to analytic number theory.     

Exponential Sums and Coding Theory: A Review

This article is a survey of several recent applications of methods from analytic number theory to research in coding theory, including results on Kloosterman codes, binary Goppa codes, and prime phase shift sequences. The mathematical methods focus on exponential sums, in particular Kloosterman sums. The interrelationships with the Weil–Carlitz–Uchiyama bound, results on Hecke operators, theorems of Bombieri and Deligne and the Eichler–Selberg trace formula are reviewed.     

Probabilistic Number Theory in Additive Arithmetic Semigroups, III

We extend the investigation of quantitative mean-value theorems of completely multiplicative functions on additive arithmetic semigroups given in our previous paper. Then the new and old quantitative mean-value theorems are applied to the investigation of local distribution of values of a special additive function *(a). The result is unexpected from the point of view of classical number theory. This reveals the fact that the essential divergence of the theory of additive arithmetic semigroups from classical number theory is not related to the existence of a zero of the zeta function Z(y) at y = –q ^–1.     

Relativistically Covariant Quantum Field Theory of the Maslov Complex Germ

We consider an explicitly covariant formulation of the quantum field theory of the Maslov complex germ (semiclassical field theory) in the example of a scalar field. The main object in the theory is the “semiclassical bundle” whose base is the set of classical states and whose fibers are the spaces of states of the quantum theory in an external field. The respective semiclassical states occurring in the Maslov complex germ theory at a point and in the theory of Lagrangian manifolds with a complex germ are represented by points and surfaces in the semiclassical bundle space. We formulate semiclassical analogues of quantum field theory axioms and establish a relation between the covariant semiclassical theory and both the Hamiltonian formulation previously constructed and the axiomatic field theory constructions Schwinger sources, the Bogoliubov S-matrix, and the Lehmann-Symanzik-Zimmermann R-functions. We propose a new covariant formulation of classical field theory and a scheme of semiclassical quantization of fields that does not involve a postulated replacement of Poisson brackets with commutators.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 492–512, September, 2005.     

Yang–Mills Theory over Compact Symplectic Manifolds

In this paper, Yang–Mills theory over a compactKähler manifold is naturally extended to a compactsymplectic manifold. The relation betweenthe Yang–Mills equation and symplecticstructure is explicitly clarified, and the moduli spaceof Yang–Mills connections over a compactsymplectic manifold is constructed. Furthermore, theabsolute minima of the Yang–Mills functional arecharacterized, and finite dimensionality ofthe moduli space of the minimizers of the Yang–Millsfunctional is shown.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

Field Theory Analysis of Critical Behavior of a Symmetric Binary Fluid

A method is elaborated for constructing an effective field theory Hamiltonian of the Landau–Ginzburg–Wilson type for off-lattice models of binary fluids. We show that all coefficients of the effective Hamiltonian for a symmetric binary fluid can be expressed in terms of some known characteristics of the model hard-sphere fluid, namely, compressibility and its derivatives with respect to density. Application of the effective Hamiltonian is demonstrated by an example of determining the curve of critical layering points in the mean-field approximation. This curve agrees well with numerical experiment results for symmetric binary fluids.     

The Bi-Hamiltonian Theory of the Harry Dym Equation

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg–de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev–Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD–KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD–KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev–Petviashivili hierarchy.     

Duality Principles in Frame Theory

Duality principles in Gabor theory such as the Ron–Shen duality principle and the Wexler–Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. In this article we present a general approach to derive duality principles in abstract frame theory. For each sequence in a separable Hilbert space we define a corresponding sequence dependent only on two orthonormal bases. Then we characterize exactly properties of the first sequence in terms of the associated one, which yields duality relations for the abstract frame setting. In the last part we apply our results to Gabor systems.     

Existence of a Global Classical Solution of One Problem Arising in Combustion Theory

We consider a multidimensional free-boundary problem for a parabolic equation that arises in combustion theory. We prove the existence of a global classical solution. The idea of the method is as follows: first, we perform the differential–difference approximation of the problem and establish its solvability; then we prove uniform estimates and perform a limit transition.     

Standard Monomial Theory for Bott–Samelson Varieties

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts [22, 23]). Standard Monomial Theory, developed by Seshadri and the first author [15, 16], and recently completed by the second author [20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of [20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in [12] (see also [13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar [10] and Mathieu [25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.     

Massless Elementary Particles in a Quantum Theory over a Galois Field

We consider massless elementary particles in a quantum theory based on a Galois field (GFQT). We previously showed that the theory has a new symmetry between particles and antiparticles, which has no analogue in the standard approach. We now prove that the symmetry is compatible with all operators describing massless particles. Consequently, massless elementary particles can have only half-integer spin (in conventional units), and the existence of massless neutral elementary particles is incompatible with the spin–statistics theorem. In particular, this implies that the photon and the graviton in the GFQT can only be composite particles.     

Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete L ₂-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.     

Quantum Mechanics and Representation Theory: The New Synthesis

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C ^*-algebras. In particular, the study of the K-theory of the reduced C ^*-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.     

Theory of self-adjoint extensions of symmetric operators. entire operators and boundary-value problems

This is a brief survey of M. G. Krein's contribution to the theory of self-adjoint extensions of Hermitian operators and to the theory of boundary-value problems for differential equations. The further development of these results is also considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, Nos. 1–2, pp. 55–62, January–February, 1994.     

Spectral Theory of Pauli–Fierz Operators

We study spectral properties of Pauli–Fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field. We give precise estimates of the location and multiplicity of the singular spectrum of such operators. Applications of these estimates, which will be discussed elsewhere, concern spectral and ergodic theory of non-relativistic QED. Our proof has two ingredients: the Feshbach method, which is developed in an abstract framework, and Mourre theory applied to the operator restricted to the sector orthogonal to the vacuum.     

Pathological Birth-and-Death Processes and the Spectral Theory of Strings

In a 1957 paper, Karlin and McGregor discovered the existence of birth-and-death processes for which the Chapman-Kolmogorov equation does not hold. Such processes are said to be pathological. We disclose the probability nature of pathological birth-and-death processes with the help of M. Krein’s spectral theory of strings.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 74–78, 2005Original Russian Text Copyright © by I. S. Kac     

A Differential Analog of the Main Lemma of the Theory of Markov Branching Processes and Its Applications

We obtain a differential analog of the main lemma of the theory of Markov branching processes μ(t), t ≥ 0, with continuous time. We show that the results obtained can be used in the proof of limit theorems of the theory of branching processes by the known Stein-Tikhomirov method. Moreover, in contrast to the classical condition of nondegeneracy of the branching process {μ(t) > 0}, we consider the condition of its nondegeneracy in the distant future {μ(∞) > 0} and justify it in terms of generating functions. Under this condition, we study the asymptotic behavior of the trajectory of the process considered.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 258–264, February, 2005.