Volkov pentagon for the modular quantum dilogarithm

The new form of pentagon equations suggested by Volkov (Int. Math. Res. Notices (2011); ) for the q-exponential on the basis of formal series is derived within the Hilbert space framework for the modular version of the quantum dilogarithm.     

An invariant of triangulated links from the quantum dilogarithm

An invariant of triangulated links in triangulated three-dimensional manifolds is constructed by means of the cyclic quantum dilogarithm. Apparently, it is an isotopic invariant of nonoriented links in an oriented closed three-dimensional manifold. Bibliography: 13 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 208–214. Translated by B. Bekker.     

Conjectures on the dilogarithm

The paper expresses Bloch's dilogarithm D(z) in terms of Clausen's function Cl₂(t). Using Lichtenbaum's conjecture on #K ₂ O _F for quadratic imaginary fields F, we get more than 50 conjectural linear relations between the values of Cl₂(t). We cannot prove most of them.     

On matrix generalizations of the dilogarithm

On the basis of generalized comultiplication depending on a continuous parameter, solutions of the matrix equation generalizing the five-term Rogers identity for the dilogarithm are found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 398–404, March, 1999.     

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH²(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.     

Sums of series of Rogers dilogarithm functions

Some sums of series of Rogers dilogarithm functions are established by Abel’s functional equation.      

Geometry in Grassmannians and a generalization of the dilogarithm

In this paper we introduce some polyhedra in Grassman manifolds which we call Grassmannian simplices. We study two aspects of these polyhedra: their combinatorial structure (Section 2) and their relation to harmonic differential forms on the Grassmannian (Section 3). Using this we obtain results about some new differential forms, one of which is the classical dilogarithm (Section 1). The results here unite two threads of mathematics that were much studied in the 19th century. The analytic one, concerning the dilogarithm, goes back to Leibnitz (1696) and Euler (1779) and the geometric one, concerning Grassmannian simplices, can be traced to Binet (1811). In Section 4, we give some of this history along with some recent related results and open problems. In Section 0, we give as an introduction an account in geometric terms of the simplest cases.     

The inner structure of the dilogarithm in algebraic fields

A certain combination of dilogarithms of powers of an algebraic base is constructed and shown to have significant properties when the powers are divisors of the highest power (index) in the relation. The combination is called a ladder and under some circumstances is zero when all the coefficients involved are rational. When this happens it is found empirically that the base satisfies an equation of cyclotomic form whose structure is obtainable by inspection from the ladder. A proof of this equation is given for the case where the ladder is obtainable by a finite number of steps from Abel's functional equation. A number of conjectures are made and used to discover many new relations, all of which are confirmed numerically, but which do not appear to be capable of analytic proof with presently available methods. The paper concludes with some conjectures on the cyclotomic equations which occur in this context.     

A bilinear form of dilogarithm and motivic regulator map

The Bloch-Wigner function D₂ is a single-valued version of a dilogarithm function and is used by Bloch to describe the Borel regulator map from K₃(C) into R explicitly (c.f. [Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, American Mathematical Society, Providence, RI, 2000]). We introduce a new way to formulate a single-valued dilogarithm function and use it to explicitly define a motivic regulator map for , defined in terms of the motivic complex of Goodwillie and Lichtenbaum. We also detect certain explicit nonzero elements in the motivic cohomology group. Throughout this paper, a path will be a C¹-function from the unit interval [0,1] into C-{0}.     

Identities for the Rogers dilogarithm function connected with simple Lie algebras

New identities are proved for the dilogarithm function connected with the Lie algebras of the series A_n and with classical Lie algebras of rank 4.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 121–133 1987.     

Additive blending of local approximations into a globally-valid approximation with application to the dilogarithm

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.     

The Borel regulator map on pictures, I: A dilogarithm formula

We give a dilogarithm formula for the value of the Borel regulator map on an element ofK ₃ of the complex numbers given by a picture.This research is supported by NSF Grant No. MCS-90-02512.     

Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H_n)_n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then is the quantum Gromov-Hausdorff limit of any sequence for the natural quantum metric structures and when the lifts of σ_n to converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C^*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.     

The quantum euler class and the quantum cohomology of the Grassmannians

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.