Observables and critical behaviour in fermionic matrix models

We review the properties of adjoint fermion one-, two-, and generic D-dimensional matrix models at large N. We derive and solve the complete sets of loop equations for the correlators of these models and examine the ensuing critical behavior. The topological 1/N-expansions are also constructed, and we discuss the applications of these matrix models to string theory and induced gauge theories.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.Work supported in part by a University of British Columbia Graduate Fellowship.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 78–122, July, 1995.     

Noncommutative geometry and spacetime gauge symmetries of string theory

We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac—Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories.     

Wilson spaces and homological algebra for coalgebraic modules

In an earlier work, Wilson spaces were used to compute certain CTor Hopf algebras. In this Note we show how one can replace a resolution by infinite loop spaces associated to the Brown–Peterson spectrum with a resolution by Wilson spaces.     

Non-Abelian gauge theories,prepotentials, and Abelian differentials

We discuss particular solutions of integrable systems (starting from the well-known dispersionless KdV and Toda hierarchies) that most directly define the generating functions for the Gromov-Witten classes in terms of a rational complex curve. From the mirror theory standpoint, these generating functions can be identified with the simplest prepotentials of complex manifolds, and we present some new exactly calculable examples of such prepotentials. For higher-genus curves, which in this context correspond to non-Abelian gauge theories via the topological string/gauge duality, we construct similar solutions using an extended basis of Abelian differentials, generally with extra singularities at the branch points of the curve. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 2, pp. 220–242, May, 2009.     

The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question

In “Beyond Einstein” the leading string theoretician and notable science writer Michio Kaku referred to what he labelled the ‘strange’ link between the E₈ exceptional Lie group and the various dimensionalities of strings and super string theories and commented on that by saying “If we could understand why the numbers 8, 10 and 26 continually crop up in super string theory, perhaps we could understand why the universe is four dimensional”.In the present work we demonstrate the existence of a Fibonacci code-like connection between the various coupling constants, charges and dimensionalities of super strings and P-Brane theories. This code is based on the Fibonacci numbers and the golden mean and in the final analysis, may be attributed to the deterministically chaotic nature of the hyperbolic Cantorian sets fixing the geometry and topology of quantum spacetime.     

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213-222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.     

Progress in physical concepts of string and superstring theory—thirty years of string theory

In this paper we discuss the evolution of physical concepts which led to the generation and development of string theories. The paper is conceived with the intention of summarizing and extending with new aspects the specific characteristics of strings which refer to the physical intuition and experiment. We hope to present new insights into the physics of strings and make it understandable from the point of view of a non-string theorist. Even if there exist some opinions that the (super)string theory appertains to the twenty-first or twenty-second century or that there are no concrete new predictions of string theory at low energies, we believe that string theory presents a rich field of research and a source of physical intuition not only for mathematicians but also for theoretical and experimental physicists. We offer as an example an atomic electron cloud which can also be interpreted in terms of a fixed point in a string theory We propose also an experiment to verify the fundamental hypotheses. Finally we deduce that the number of dimensions of spacetime must be infinite by virtue of the axiom of universality of motion.     

Integrable many-body systems and gauge theories

We review the study of the relation between integrable many-body systems and gauge theories. We show that the degrees of freedom of integrable systems are related to the topological degrees of freedom of gauge theories. We also describe the relation between families of integrable systems and N=2 supersymmetric gauge theories. We show that the degrees of freedom of many-body systems can be identified with the collective coordinates of string theory solitons, theD-branes. This article was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 3–56, October, 2000.     

On the notion of space in geometry

The development of the notion of space in geometry is traced from the early axiomatization in Euclid’s Elements over the discovery of non-Euclidean geometries to geometry of manifolds in relativity theory and in gauge and string theories in contemporary physics. The notion of space is considered in a historic-philosophical perspective including a short discussion of the contributions of artists to visualization of spatial objects.     

Hylomorphic solitons and charged Q-balls: Existence and stability

In this paper we present a general theoretical framework from which several known results (a some new ones) on the existence and stability of solitons can be recovered.We give an abstract definition of solitary wave and soliton and we develope an abstract existence theory. This theory provides a powerful tool to study the existence of solitons for the Klein–Gordon equations as well as for gauge theories. Applying this theory, we prove the existence of a continuous family of stable charged Q-balls.     

Pure gauge configurations and tachyon solutions for cubic fermionic string field theory equations of motion

Special perturbative pure gauge solutions parameterized by a pair of wedge states are parts of the nontrivial (not purely gauge) tachyon solutions of the cubic fermionic string field theory describing the non-BPS brane true vacuum. We demonstrate explicitly that for the large parameter of the perturbation expansion, these pure gauge configurations are no longer solutions of the equations of motion. We show that this problem is solved by adding an extra term that is just the term needed for the first Sen conjecture to hold.     

Temperature Independent Renormalization of Finite Temperature Field Theory

We analyze 4-dimensional massive .4 theory at .nite temperature T in the imaginary-time formalism. We present a rigorous proof that this quantum .eld theory is renormalizable, to all orders of the loop expansion. Our main point is to show that the counterterms can be chosen temperature independent, so that the temperature .ow of the relevant parameters as a function of T can be followed. Our result con.rms the experience from explicit calculations to the leading orders. The proof is based on .ow equations, i.e. on the (perturbative) Wilson renormalization group. In fact we will show that the di.erence between the theories at T > 0 and at T = 0 contains no relevant terms. Contrary to BPHZ type formalisms our approach permits to lay hand on renormalization conditions and counterterms at the same time, since both appear as boundary terms of the renormalization group .ow. This is crucial for the proof.     

Constraints, gauge symmetries, and noncommutative gravity in two dimensions

After a short introduction, we show how the canonical formalism is constructed in space-time noncommutative theories such that it allows defining the notion of first-class constraints and analyzing gauge symmetries. We use this formalism to noncommutatively deform the two-dimensional string gravity (also known as the Witten black hole). Dedicated to Yu. V. Novozhilov on the occasion of his 80th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 64–79, July, 2006.     

Geometry as seen by string theory

This is an introductory review of the topological string theory from physicist’s perspective. I start with the definition of the theory and describe its relation to the Gromov–Witten invariants. The BCOV holomorphic anomaly equations, which generalize the Quillen anomaly formula, can be used to compute higher genus partition functions of the theory. The open/closed string duality relates the closed topological string theory to the Chern–Simons gauge theory and the random matrix model. As an application of the topological string theory, I discuss the counting of bound states of D-branes.     

Instanton partition functions and M-theory

We discuss instanton partition functions in various spacetime dimensions. These partition functions capture some information about the spectrum of the supersymmetric gauge theories and their low-energy dynamics. Some of these theories can be defined microscopically only through string theory. Remarkably, they even know about the M-theory. Our conjectures include the identities between the generalization of the MacMahon formula and the character of M-theory, compactified down to 0 + 1 dimension. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

Quantum fields on the light front, formulation in coordinates close to the light front, lattice approximation

We review the fundamental ideas of quantizing a theory on a light front including the Hamiltonian approach to the problem of bound states on the light front and the limiting transition from formulating a theory in Lorentzian coordinates (where the quantization occurs on space-like hyperplanes) to the theory on the light front, which demonstrates the equivalence of these variants of the theory. We describe attempts to find such a form of the limiting transition for gauge theories on the Wilson lattice. This paper is based on a lecture delivered at the V. A. Fock International School of Physics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 89–101, July, 2006.     

Symmetry algebra of the AdS4×ℂℙ3 superstring

By direct calculation in the classical theory, we derive the central extension of the off-shell symmetry algebra for a string propagating in AdS ₄ ×?? ³ . It turns out to be the same as in the case of the AdS ₅ ×S ⁵ string. We consider the choice of the κ-symmetry gauge in detail and also explain how this gauge can be chosen without breaking the bosonic symmetries.     

New approach to vacuum string field theory

We discuss the possibility of applying a purely ghost kinetic operator in open string field theory from the perspective of a modern analytic method based on the KBc subalgebra. A purely ghost kinetic operator is obtained as a result of gauge fixing string field theory around the identity-based tachyon vacuum solution. We show that the obtained kinetic operator is not equivalent to the midpoint insertion of a conformal ghost, to which an extensive literature is devoted. We also show that the equation of motion does not admit nontrivial solutions.