For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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The Poisson $\sigma$-model is a 2-dimensional sigma-model quantum field theory whose target space is a Poisson Lie algebroid. It is a 2-dimensional Chern-Simons theory. This may be thought of as encoding the quantum mechanics of a string propagating on the phase space of a system in classical mechanics.
In his solution of the problem of deformation quantization Maxim Kontsevich showed that correlators for the 2-string interaction (the correlator on the worldsheet that is a disk with three marked points on its boundary) describe a product operation which is a deformation of the Poisson bracket on the target space. This solves the formal deformation quantization problem of the phase space in quantum mechanics by identifying the quantum algebra with the open string algebra of a string theory on that target.
The principal variant of the nonlinear Poisson sigma model is sometimes called Cattaneo-Felder model who have shown the graphical expansion used in Kontsevich’s approach to the deformation quantization is explained via a Feynman diagram expansion in this model.
If one considers branes in the target space of the Poisson sigma-model, then then algebra of open strings that used to be just the deformation of the Poisson algebra becomes an A-infinity algebroid. (See the references below).
Probably something close to a Calabi-Yau category, hence identifying the Poisson sigma-model as a TCFT. Does anyone know more in this direction?
The target space of a Poisson $\sigma$-model is any Poisson manifold $(X, \{\})$, or rather the Poisson Lie algebroid $\mathfrak{P}$ corresponding to that.
A field configuration on a 2-dimensional $\Sigma$ is a connection
In components this is
a smooth function $\phi : \Sigma \to X$;
a 1-form $\eta \in \Omega^1(\Sigma, \phi^* T X)$ with values in the pullback of the tangent bundle of $X$ along $\phi$.
The action functional on the configuration space of all such connections for compact $\Sigma$ is defined to be
where $\pi \in \wedge^2_{C^\infty(C)}\Gamma(T X)$ is the Poisson tensor of $(X, \{-,-\})$ and where $\langle -,-\rangle$ is the canonical invariant polynomial on the Poisson Lie algebroid.
In (Cattaneo-Felder) it was shown that the 3-point function in the path integral quantization of the Poisson $\sigma$-model of a Poisson Lie algebroid associated with a Poisson manifold computes the star product in the deformation quantization of the Poisson manifold as given by (Kontsevich).
A higher geometric quantization that also yields the strict deformation quantization is discussed at extended geometric quantization of 2d Chern-Simons theory.
One may think of this relation between the 2d Poisson sigma-model and quantum mechanics = 1d quantum field theory as an example of the Chern-Simons type holographic principle. For more along these lines see below at holographic dual.
The branes of the Poisson sigma model are related to coisotropic submanifolds of the underlying Poisson manifold. Notice that these are the Lagrangian dg-submanifolds of the Poisson Lie algebroid. (Cattaneo-Felder 03).
By the Chern-Simons form of the holographic principle one expects the Poisson sigma-model to be related to a 1-dimensional quantum field theory. This is quantum mechanics. The above relation to the deformation quantization of Poisson manifolds goes in this direction. More explicit realizations have been attempted, for instance (Vassilevich).
Poisson sigma-model
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
The Poisson sigma model was first considered in
and later independently by P. Schaller, T. Strobl, motivated from an attempt to unify several two-dimensional models of gravity and to cast them into a common form with Yang-Mills theories.
P. Schaller, T. Strobl, Poisson structure induced (topological) field theories, Modern Phys. Lett. A 9 (1994), no. 33, 3129–3136, doi; Introduction to Poisson $\sigma$-models, Low-dimensional models in statistical physics and quantum field theory (Schladming, 1995), 321–333, Lecture Notes in Phys. 469, Springer 1996.
Thomas Strobl, Gravity from Lie algebroid morphisms, Comm. Math. Phys. 246 (2004), no. 3, 475–502, Algebroid Yang-Mills theories, Phys. Rev. Lett. 93 (2004), no. 21, 211601, 4 pp. doi
M. Bojowald, A. Kotov, T. Strobl, Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries, J. Geom. Phys. 54 (2005), no. 4, 400–426, doi
Ctirad Klimčík, T. Strobl, WZW-Poisson manifolds, J. Geom. Phys. 43 (2002), no. 4, 341–344, doi
The detailed argument by Cattaneo and Felder on how Maxim Kontsevich’s formula for the deformation quantization star product is the 3-point function of the Poisson sigma-model is in
See also
Alberto Cattaneo, Giovanni Felder, Poisson sigma models and symplectic groupoids , (ed. Klaas Landsman, M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198, 61–93 (Birkhäuser, 2001) math.SG/0003023.
Alberto Cattaneo, Giovanni Felder, On the AKSZ formulation of the Poisson sigma model , Lett. Math. Phys. 56, 163–179 (2001) math.QA/0102108.
The interpretation in terms of infinity-Chern-Simons theory is discussed in
Discussion in terms of holography is in
The study of branes in the Poisson sigma-model has been started in
Damien Calaque, Giovanni Felder, Andrea Ferrario, Carlo A. Rossi, Bimodules and branes in deformation quantization (arXiv:0908.2299)
Damien Calaque, Giovanni Felder, Carlo A. Rossi, Deformation quantization with generators and relations (arXiv:0911.4377)
Alberto Cattaneo, Giovanni Felder, Coisotropic submanifolds in Poisson geometry and branes in the Poisson $\sigma$-model, Lett.Math.Phys. 69 (2004) 157-175 (arXiv:0309180)
A review is in