Renormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor
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Springer
Received: September 5, 2020 Accepted: September 11, 2020 Published: October 15, 2020
ˇ J´ an Pulmann, Pavol Severa and Donald R. Youmans Section of Mathematics, University of Geneva, rue du Li`evre 2-4, Geneva, Switzerland
E-mail: [email protected], [email protected], [email protected] Abstract: We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim σ-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant σ-model or possibly by a more general AKSZ model. Keywords: Boundary Quantum Field Theory, Chern-Simons Theories, Renormalization Group, String Duality ArXiv ePrint: 2009.00509
1
Supported in part by the NCCR SwissMAP of the Swiss National Science Foundation.
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)096
JHEP10(2020)096
Renormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor1
Contents 1
2 Chern-Simons propagator with the chiral boundary condition 2.1 Chiral boundary condition 2.2 Propagator 2.3 The case of a non-trivial g 2.4 Generalized metrics
2 2 3 4 4
3 2-dim σ-models on Chern-Simons boundary and Poisson-Lie T-duality 3.1 A tale of two boundary conditions 3.1.1 A (classically) topological boundary condition 3.1.2 Generalized metrics, again 3.2 2-dim σ-models on Chern-Simons boundary 3.3 Poisson-Lie T-duality
4 4 5 5 5 6
4 Diagramatics, RG flow, and the generalized Ricci tensor 4.1 Regularization 4.2 Diagramatics 4.3 Generalized Ricci tensor and conformal invariance 4.4 Effective action and Weyl anomaly 4.5 RG flow of V+ 4.5.1 Renormalization 4.5.2 Field redefinition 4.5.3 The RG flow
7 7 8 10 10 11 11 12 12
5 Further developments 5.1 Courant σ-model and generalized Ricci tensor 5.1.1 Courant σ-model 5.1.2 Boundary condition 5.1.3 Propagators 5.1.4 Diagramatics 5.1.5 1-loop RG flow and generalized Ricci tensor 5.2 AKSZ models, higher dualities, and RG flow
13 13 13 13 14 14 14 15
A 1-loop divergent diagrams A.1 Chern-Simons A.2 Courant model A.3 Diagrams with at least 3 vertices converge
16 16 16 17
–i–
JHEP10(2020)096
1 Introduction
1
Introduction
Ω0,1 (Σ, V+ ) ⊕ Ω1,0 (Σ, V− ). To get a 2-dim σ-model we then need to consider the Chern-Simons theory on Σ×[0, 1], with the V+ -boundary condition on Σ × {0}, and with a topological boundary condition on Σ × {1} given by a Lagrangian Lie subalgebra h ⊂ g. h-b.c.
Σ × [0, 1]
V+ -b.c. Σ
This “Chern-Simons sandwich” is equivalent to a 2-dim σ-model with the worldsheet Σ and the target G/H. Different choices of h give different targets, all of them linked by Poisson-Lie T-duality (the usual T-duality corresponds to G being a torus). It is thus natural to study the V+ -boundary condition on its own, since that is where the dynamical degrees of freedom of the
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