Abstract
First, we give a functorial construction of a group associated to a symmetric operad. Applied to the endomorphism operad it gives the group of formal diffeomorphisms. Second, we associate a symmetric operad to any family of decorated graphs stable by contraction. In the case of Quantum Field Theory models it gives the renormalization group. As an example we get an operadic interpretation of the group of “diffeographisms” attached to the Connes–Kreimer Hopf algebra.
Professor Jean-Louis Loday passed away on 6 June 2012.
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Notes
- 1.
According to the meaning of a formal power series (cf. [5, Chap. 2]) the notation \(\overrightarrow{\mathrm{y} } =\overrightarrow{\mathrm{f} }(\overrightarrow{\mathrm{x} })\) is just an abbreviation of the formal sum in the right hand side of (1), which in turn, is nothing but just the list of coefficients \((f_{\nu ;\mu _{1},\ldots ,\mu _{n}})_{\nu ,\mu _{1},\ldots ,\mu _{n}\,=\,1}^{N}\). One defines the formal derivative series \(\frac{{\partial }^{m}{ \overrightarrow{f}}} {\partial x_{\nu _{1}}\cdots \partial x_{\nu _{m}}}{({\overrightarrow{X}})}\) : = \(\sum \limits _{ n\,=\,1}^{\infty }\frac{1} {n!}\) \(\sum \limits _{ \mu _{1},\ldots ,\mu _{n}\,=\,1}^{N}\overrightarrow{\mathrm{f} }_{\nu _{1},\ldots ,\nu _{m},\mu _{1},\ldots ,\mu _{n}}\) \(x_{\mu _{1}}\cdots x_{\mu _{n}}\) and then the coefficient \(\overrightarrow{\mathrm{f} }_{\nu _{1},\ldots ,\nu _{m}}\) coincides with the leading term of the derivative series \(\frac{{\partial }^{m}\overrightarrow{\mathrm{f} }} {\partial x_{\nu _{1}}\cdots \partial x_{\nu _{m}}}{\bigl (\overrightarrow{\mathrm{x} }\bigr )}\). In particular, the coefficients \(\overrightarrow{\mathrm{f} }_{\nu _{1},\ldots ,\nu _{m}}\) must be symmetric in the indices, which is equivalent to the symmetry of the derivatives.
- 2.
Equation (3) follows from the formula for the nth formal derivative of the composition series \(\overrightarrow{g}{\rm o}\overrightarrow{f}(\overrightarrow{X})\)
$$\displaystyle\begin{array}{rcl} \frac{{\partial }^{n}h_{\mu }} {\partial x_{\mu _{1}}\cdots \partial x_{\mu _{n}}}{\bigl (\overrightarrow{\mathrm{x} }\bigr )}\,& =& \displaystyle\sum \limits _{\mathfrak{P}\, \in \,\mathrm{Part}\{1,\ldots ,n\}}\ \displaystyle\sum \limits _{\rho _{1},\ldots ,\rho _{k}\,=\,1}^{N}\ {\Bigl ( \frac{{\partial }^{k}g_{\nu }} {\partial x_{\rho _{1}}\cdots \partial x_{\rho _{k}}} \circ \overrightarrow{\mathrm{f} }\Bigr )}{\bigl (\overrightarrow{\mathrm{x} }\bigr )}\, \\ & & \times \, \frac{{\partial }^{j_{1}}f_{\rho _{ 1}}} {\partial \mu _{i_{1,1}}\cdots \partial \mu _{i_{1,j_{ 1}}}} {\bigl (\overrightarrow{\mathrm{x} }\bigr )}\cdots \frac{{\partial }^{j_{k}}f_{\rho _{ k}}} {\partial \mu _{i_{k,1}}\cdots \partial \mu _{i_{k,j_{ k}}}}{\bigl (\overrightarrow{\mathrm{x} }\bigr )}, \\ \end{array}$$which in turn is derived by induction in n.
- 3.
Thus, \(f_{n}(x_{1;1},\ldots ,x_{1;N}; \ldots ; x_{n;1},\ldots ,x_{n;N})_{\nu }\) : = \(\sum \limits _{ \mu _{1},\ldots ,\mu _{n}\,=\,1}^{N}f_{\nu ;\mu _{1},\ldots ,\mu _{n}}\) \(x_{1;\mu _{1}}\cdots x_{n;\mu _{n}}\).
- 4.
∘i is the ith operadic partial composition.
- 5.
In terms of formal power series; note that the series K → (κ → ; ε) starts from n = 1 but for U(κ → ; ε) we do not have such a restriction.
- 6.
However, we remark that the algebra structure induced by the monoid structure of \(\mathfrak{M}(n)\) is quite different from the algebra structure on the space of diagrams that is usually used in the Connes–Kreimer approach.
References
Brunetti, R., Duetsch, M., Fredenhagen, K.: Adv. Theor. Math. Phys. 13, 1–56 (2009)
Chapoton, F.: Rooted trees and an exponential-like series [arXiv:math/0209104]
Connes, A., Kreimer, D.: Comm. Math. Phys. 199, 203–242 (1998)
Frabetti, A.: J. Algebra 319(1), 377–413 (2008)
Lang, S.: Complex analysis, 4th edn. In: Graduate Texts in Mathematics, vol. 103. Springer, New York, (1999)
Loday, J.-L., Vallette, B.: Algebraic operads. In: Grundlehren Math. Wiss. vol. 346. Springer, Heidelberg (2012)
Loday, J.-L., Nikolov, N.M.: Renormalization from the operadic point of view (in preparation)
Nikolov, N.M.: Anomalies in quantum field theory and cohomologies of configuration spaces [arXiv:0903.0187]
Stora, R.: Causalité et Groupes de Renormalisation Perturbatifs. In: Boudjedaa, T., Makhlouf, A. (eds.) Théorie Quantique des Champs Méthode et Applications, p. 67 (2007)
Acknowledgements
We thank Dorothea Bahns, Kurusch Ebrahimi-Fard, Alessandra Frabetti, Klaus Fredenhagen and Raymond Stora for fruitful discussions. The work was partially supported by the French-Bulgarian Project Rila under the contract Egide-Rila N112 and by grant DO 02-257 of the Bulgarian National Science Foundation. N.N. thanks the Courant Research Center “Higher order structures in mathematics” (Göttingen) and the II. Institute for Theoretical Physics at the University of Hamburg for support and hospitality.”
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Loday, JL., Nikolov, N.M. (2013). Operadic Construction of the Renormalization Group. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_13
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