Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes

We investigate the generic 3D topological field theory within AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpreted the Rozansky-Witten model in this light.Comment: 52 page

Quantum differential forms

Formalism of differential forms is developed for a variety of Quantum and noncommutative situations

Differential operators on supercircle: conformally equivariant quantization and symbol calculus

We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the M\"obius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator $L_Q$ and are represented by $Q$-invariant tensors made up of the homological vector field and a symmetric connection on $M$ by means of tensor operations.Comment: 17 pages, references and comments adde

Lectures on conformal field theory and Kac-Moody algebras

This is an introduction to the basic ideas and to a few further selected topics in conformal quantum field theory and in the theory of Kac-Moody algebras.Comment: 59 pages, LaTeX2e, extended version of lectures given at the Graduate Course on Conformal Field Theory and Integrable Models (Budapest, August 1996), to appear in Springer Lecture Notes in Physic

Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators Cϕ : A (R) → A (R) for a real analytic self map ϕ : R → R as well as an isomorphic description of corresponding eigenspaces. We completely characterize those ϕ for which Abel’s equation f ◦ ϕ = f + 1 has a real analytic solution on the real line. We find cases when the operator Cϕ has roots using a constructed embedding of ϕ into the so-called real analytic iteration semigroups.(1) The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200 and MTM2013-43540-P and the work of Bonet also by GV Project Prometeo II/2013/013. The research of Domanski was supported by National Center of Science, Poland, Grant No. NN201 605340. (2) The authors are very indebted to K. Pawalowski (Poznan) for providing us with references [26,27,47] and also explaining some topological arguments of [10]. The authors are also thankful to M. Langenbruch (Oldenburg) for providing a copy of [29].Bonet Solves, JA.; Domanski, P. (2015). Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory. 81(4):455-482. https://doi.org/10.1007/s00020-014-2175-4S455482814Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)Baker I.N.: Zusammensetzung ganzer Funktionen. Math. Z. 69, 121–163 (1958)Baker I.N.: Permutable power series and regular iteration. J. Aust. Math. Soc. 2, 265–294 (1961)Baker I.N.: Permutable entire functions. Math. Z. 79, 243–249 (1962)Baker I.N.: Fractional iteration near a fixpoint of multiplier 1. J. Aust. Math. Soc. 4, 143–148 (1964)Baker I.N.: Non-embeddable functions with a fixpoint of multiplier 1. Math. Z. 99, 337–384 (1967)Baker I.N.: On a class of nonembeddable entire functions. J. Ramanujan Math. Soc. 3, 131–159 (1988)Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 61, 1–48 (2001)Belitskii G., Lyubich Y.: The Abel equation and total solvability of linear functional equations. Studia Math. 127, 81–97 (1998)Belitskii G., Lyubich Yu.: The real analytic solutions of the Abel functional equation. Studia Math. 134, 135–141 (1999)Belitskii G., Tkachenko V.: One-Dimensional Functional Equations. Springer, Basel (2003)Belitskii G., Tkachenko V.: Functional equations in real analytic functions. Studia Math. 143, 153–174 (2000)Bonet J., Domański P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. 62, 69–83 (2011)Bonet J., Domański P.: Hypercyclic composition operators on spaces of real analytic functions. Math. Proc. Camb. Philos. Soc. 153, 489–503 (2012)Bracci, F., Poggi-Corradini, P.: On Valiron’s theorem. In: Proceedings of Future Trends in Geometric Function Theory. RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dept. Math. Stat., vol. 92, pp. 39–55 (2003)Contreras, M.D.: Iteración de funciones analíticas en el disco unidad. Universidad de Sevilla (2009). (Preprint)Contreras M.D., Díaz-Madrigal S., Pommerenke Ch.: Some remarks on the Abel equation in the unit disk. J. Lond. Math. Soc. 75(2), 623–634 (2007)Cowen C.: Iteration and the solution of functional equations for functions analytic in the unit disc. Trans. Am. Math. Soc. 265, 69–95 (1981)Cowen C.C., MacCluer B.D.: Composition operators on spaces of analytic functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)Domański, P.: Notes on real analytic functions and classical operators. In: Topics in Complex Analysis and Operator Theory (Winter School in Complex Analysis and Operator Theory, Valencia, February 2010). Contemporary Math., vol. 561, pp. 3–47. Am. Math. Soc., Providence (2012)Domański P., Goliński M., Langenbruch M.: A note on composition operators on spaces of real analytic functions. Ann. Polon. Math. 103, 209–216 (2012)P. Domański M. Langenbruch 2003 Language="En"Composition operators on spaces of real analytic functions Math. Nachr. 254–255, 68–86 (2003)Domański P., Langenbruch M.: Coherent analytic sets and composition of real analytic functions. J. Reine Angew. Math. 582, 41–59 (2005)Domański P., Langenbruch M.: Composition operators with closed image on spaces of real analytic functions. Bull. Lond. Math. Soc. 38, 636–646 (2006)Domański P., Vogt D.: The space of real analytic functions has no basis. Studia Math. 142, 187–200 (2000)Fuks D.B., Rokhlin V.A.: Beginner’s Course in Topology. Springer, Berlin (1984)Greenberg M.J.: Lectures on Algebraic Topology. W. A. Benjamin Inc., Reading (1967)Hammond, C.: On the norm of a composition operator, PhD. dissertation, Graduate Faculty of the University of Virginia (2003). http://oak.conncoll.edu/cnham/Thesis.pdfHandt T., Kneser H.: Beispiele zur Iteration analytischer Funktionen. Mitt. Naturwiss. Ver. für Neuvorpommernund Rügen, Greifswald 57, 18–25 (1930)Heinrich T., Meise R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Karlin S., McGregor J.: Embedding iterates of analytic functions with two fixed points into continuous group. Trans.Am. Math. Soc. 132, 137–145 (1968)Kneser H.: Reelle analytische Lösungen der Gleichung $${\varphi(\varphi(x))=e^x}$$ φ ( φ ( x ) ) = e x und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187, 56–67 (1949)Königs, G.: Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. (3) 1, Supplément, 3–41 (1884)Kuczma M.: Functional Equations in a Single Variable. PWN-Polish Scientific Publishers, Warszawa (1968)Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Milnor, J.: Dynamics in One Complex Variable. Vieweg, Braunschweig (2006)Schröder E.: über iterierte Funktionen. Math. Ann. 3, 296–322 (1871)Shapiro J.H.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer, New York (1993)Shapiro, J.H.: Notes on the dynamics of linear operators. Lecture Notes. http://www.mth.msu.edu/~hapiro/Pubvit/Downloads/LinDynamics/LynDynamics.htmlShapiro, J.H.: Composition operators and Schröder functional equation. In: Studies on Composition Operators (Laramie, WY, 1996), Contemp. Math., vol. 213, pp. 213–228. Am. Math. Soc., Providence (1998)Szekeres G.: Regular iteration of real and complex functions. Acta Math. 100, 203–258 (1958)Szekeres G.: Fractional iteration of exponentially growing functions. J. Aust. Math. Soc. 2, 301–320 (1961)Szekeres G.: Fractional iteration of entire and rational functions. J. Aust. Math. Soc. 4, 129–142 (1964)Szekeres G.: Abel’s equations and regular growth: variations on a theme by Abel. Exp. Math. 7, 85–100 (1998)Trappmann H., Kouznetsov D.: Uniqueness of holomorphic Abel function at a complex fixed point pair. Aequ. Math. 81, 65–76 (2011)Viro, O.: 1-manifolds. Bull. Manifold Atlas. http://www.boma.mpim-bonn.mpg.de/articles/48 (a prolonged version also http://www.map.mpim-bonn.mpg.de/1-manifolds#Differential_structures )Walker P.L.: A class of functional equations which have entire solutions. Bull. Aust. Math. Soc. 39, 351–356 (1988)Walker P.L.: The exponential of iteration of e x −1. Proc. Am. Math. Soc. 110, 611–620 (1990)Walker P.L.: On the solution of an Abelian functional equation. J. Math. Anal. Appl. 155, 93–110 (1991)Walker P.L.: Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57, 723–733 (1991

Geometric quantities associated to differential operators

Denote by F_lambda the space of fields of tensor densities of weight -lambda over a manifold M. The space D^p_{lambda,mu} of differential operators of order at most p that map F_lambda onto F_mu are modules over the Lie algebra of vector fields Vect(M). We compute all the Vect(M)-invariant mappings from D^p_{lambda,mu} onto F_nu