nLab Poisson sigma-model

Idea

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?

Definition

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

$$(\phi,\eta) : \mathfrak{T}^*\Sigma \to \mathfrak{P} \,.$$

In components this is

1. a smooth function $\phi : \Sigma \to X$;

2. a 1-form $\eta \in \Omega^1(\Sigma, \phi^* T^* X)$ with values in the pullback of the cotangent bundle of $X$ along $\phi$.

The action functional on the configuration space of all such connections for compact $\Sigma$ is defined to be

$$S : (\phi,\eta) \mapsto \int_\Sigma \left( \langle \eta \wedge d_{dR}\phi\rangle + \frac{1}{2} \phi^*\pi(\eta) \right) \,,$$

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.

Properties

Relation to deformation quantization of Poisson manifolds

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.

Branes

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).

Holographic dual

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).

Related concepts

• ∞-Chern-Simons theory

• sigma-model

  • AKSZ sigma-model

    • Poisson sigma-model

      • A-model, B-model

    • Courant sigma-model

      • Chern-Simons theory

• ∞-Chern-Simons theory

• higher dimensional Chern-Simons theory

  • 1d Chern-Simons theory

  • 2d Chern-Simons theory

  • 3d Chern-Simons theory

  • 4d Chern-Simons theory

  • 5d Chern-Simons theory

  • 6d Chern-Simons theory

  • 7d Chern-Simons theory

  • infinite-dimensional Chern-Simons theory

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

General

The Poisson sigma model was first considered in

• Noriaki Ikeda, Two-dimensional gravity and nonlinear gauge theory , Ann.Phys.235(1994) 435- 464, [hep-th/9312059].

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

• Martin Bojowald, Alexei Kotov, Thomas Strobl, Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries, J. Geom. Phys. 54:4 (2005) 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

• Alberto Cattaneo, Giovanni Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212, 591–611 (2000) doi, math.QA/9902090.

• Alberto Cattaneo, Giovanni Felder, Poisson sigma models and deformation quantization, Mod. Phys. Lett. A 16, 179–190 (2001) hep-th/0102208.

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory (2011)

Discussion in terms of holography is in

• D. V. Vassilevich, Holographic duals to Poisson sigma models (arXiv:1301.7029)

With branes

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)

• Andrea Ferrario, Poisson Sigma Model with branes and hyperelliptic Riemann surfaces (arXiv:0709.0635)

A review is in

• F. Falceto, Branes in Poisson sigma models (2009) (pdf)

Recent developments

• Francesco Bonechi, Alberto Cattaneo, Pavel Mnev, The Poisson sigma model on closed surfaces (arXiv:1110.4850)