Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 7

Two-dimensional perturbative scalar QFT and Atiyah–Segal gluing

Pages: 1847 – 1952



Santosh Kandel (Mathematics Institute, University of Freiburg, Germany)

Pavel Mnev (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.; and St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia)

Konstantin Wernli (Centre for Quantum Mathematics, University of Southern Denmark, Odense, Denmark)


We study the perturbative quantization of $2$-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorial quantum field theory framework of Atiyah–Segal. In particular, we prove that the perturbative partition function defined in terms of integrals over configuration spaces of points on the surface satisfies an Atiyah–Segal type gluing formula. Tadpoles (short loops) behave nontrivially under gluing and play a crucial role in the result.

S.K. and K.W. would like to thank the University of Zurich where a part of this work was written, and acknowledge partial support of NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology), and the SNF grant No. 200020 172498/1 during their affilitation with the University of Zurich. K. W. acknowledges further support from a BMS Dirichlet postdoctoral fellowship and the SNF Postdoc Mobility grant P2ZHP2 184083, and would like to thank the Humboldt-Universität Berlin, in particular the group of Dirk Kreimer, and the University of Notre Dame for their hospitality.

Published 11 July 2022