Schreiber master thesis Bongers

A thesis that I once supervised:

---

• Stephan Bongers,

  Geometric quantization of symplectic and Poisson manifolds

  MSc thesis, Utrecht, January 2014

  (pdf, official thesis archive entry, talk slides)

on the modern formulations of geometric quantization and leading towards higher geometric quantization.

---

Abstract

The first part of this thesis provides an introduction to recent developments in geometric quantization of symplectic and Poisson manifolds, including modern refinements involving Lie groupoid theory and index theory/K-theory. We start by giving a detailed treatment of traditional geometric quantization of symplectic manifolds, where we cover both the quantization scheme via polarization and via push-forward in K-theory. A different approach is needed for more general Poisson manifolds, which we treat by the geometric quantization of Poisson manifolds via the geometric quantization of their associated symplectic groupoids, due to Weinstein, Xu, Hawkins, et al.

In the second part of the thesis we show that this geometric quantization via symplectic groupoids can naturally be understood as an instance of higher geometric quantization in higher geometry, namely as the boundary theory of the 2d Poisson sigma-model. This thesis closes with an outlook on the implications of this change of perspective.

The following provides some hyperlinked keywords as to the content of the thesis.

1. Introduction

• The mathematical formalization of non-perturbative quantization of mechanical systems given by symplectic manifolds is geometric quantization;

• this needs to be generalized to smooth collections (foliations) of mechanical systems, hence to Poisson manifolds, but the traditional approach does not work here;

• to address this, Alan Weinstein once observed that every Poisson manifold induces a smooth groupoid (a Lie groupoid in favorable circumstances), called its symplectic groupoid, which does carry a suitable kind of symplectic structure;

• and there is a proposal in the literature to understand the quantization of Poisson manifolds as the geometric quantization of their symplectic groupoids – but existing constructions are somewhat ad hoc and break Morita equivalence both of Lie groupoids and of C*-algebras for interpreting their answer, which is unnatural.

• After reviewing all the ingredients that go into this in their modern incarnation (including geometric quantization by push-forward and Lie groupoids as smooth stacks);

• this thesis discusses how the pre-quantization of symplectic groupoids is naturally understood in higher differential geometry as the “differential Lie integration” of the canonical Lie algebroid cocycle on the Poisson Lie algebroid to a prequantum 2-bundle (bundle gerbe) on the symplectic groupoid.

• The thesis closes with an outlook on how this new perspective allows to understand the geometric quantization of Poisson manifolds as the 1-dimensional boundary field theory of the 2d Poisson-Chern-Simons theory induced from them, a non-perturbative version of the famous interpretation by Cattaneo and Felder of the formal (perturbative) deformation quantization formula by Maxim Kontsevich; this topic is further studied in (Nuiten, MSc 2013).

2. Geometric quantization of Symplectic Manifolds

• geometric quantization

3. Geometric quantization of Poisson Manifolds

• A Poisson manifold $(X,\pi)$ is not characterized by a differential 2-form (symplectic form) and hence the process of geometric quantization of symplectic manifolds does not directly apply;

• but the Poisson algebra-structure is equivalently encoded in the induced Lie algebroid $\mathfrak{P}(X,\pi)$ called the Poisson Lie algebroid, and its Lie integration is a smooth groupoid (in favorable cases a Lie groupoid) which canonically carries a multiplicative symplectic form on its space of morphisms, whence it is called the symplectic groupoid $SymplGrpd(X,\pi)$;

• so that a prequantization of $SymplGrpd(X,\pi)$ is naturally taken to be a multiplicative prequantum bundle on the space of morphisms of $SymplGrp(X,\pi)$ prequantizing this 2-form, an observation and suggestion that goes back to Alan Weinstein.

• From here one can try to mimic ordinary geometric quantization as above in a way that takes the composition structure on the morphisms into account, and so one arrives at some concept of geometric quantization of symplectic groupoids which one may think of as a stand-in for the quantization of the original Poisson manifold.

• Poisson Lie algebroid, Poisson cohomology

• symplectic groupoid

• groupoid convolution algebra, C*-algebra

4. Higher geometric perspective

• This program (reviewed in section 3) for quantization of Poisson manifolds has two drawbacks:

  1. It is somewhat ad-hoc and technically baroque, lacking a good conceptual explanation beyond the empirical observation that the results seem sensible to some extent;

  2. and in fact the results of the construction in the literature are a bit dubious in that they break the principle of equivalence“: the symplectic groupoid of a symplectic manifold is a pair groupoid and its symplectic groupoid quantization yields a algebra of observables of compact operators, both of which are Morita equivalent to the trivial case – so that to regard this quantization of a symplectic manifold as non-trivial one has to distinguish Morita-equivalent structures.

• The claim of this section 4 (and section 5) is that both of these aspects are fixed by properly regarding the situation in higher differential geometry. In particular a multiplicative prequantum bundle on a Lie groupoid is more intrinsically really a bundle gerbe, hence a prequantum 2-bundle over the Lie groupoid, a higher categorical analog of a bundle. The claim that this perspective is more natural for the problem

4.1 Motivating examples

• local prequantum field theory

• extended Chern-Simons theory

(Fiorenza-Sati-Schreiber 12, Fiorenza-Sati-Schreiber 13)

4.2 Higher prequantum geometry

• General background:

  • higher differential geometry,

  • Lie groupoids, stacks, smooth stacks

  • bundle gerbes, circle n-bundle with connection

• prequantum n-bundle

(Fiorenza-Rogers-Schreiber 13)

4.3 Higher symplectic geometry

• higher symplectic geometry

4.3.1 Simplicial de Rham cohomology

• simplicial de Rham complex

4.3.2 Lie $\infty$-Algebroids

• infinity-Lie algebroid

4.3.3 Cocycles, invariant polynomials and Chern-Simons elements

• L-infinity cocycle

• invariant polynomial

• Chern-Simons element

4.3.4 Poisson $\sigma$-model

4.3.5 Lie $\infty$-integration

• Lie integration

4.3.6 2d Poisson-Chern-Simons theory

• The key observation in section 4.3 is that the Poisson Lie algebroid $\mathfrak{P}(X,\pi)$ of a Poisson manifold is, as a symplectic Lie n-algebroid, an L-infinity cocycle

  $$\pi \;\colon\; \mathfrak{P}(X,\pi) \longrightarrow b^2 \mathbb{R}$$

  which has a Chern-Simons element $cs$ and as such a “differential Lie integration” to a morphism of the form

  $$\exp(cs) \;\colon\; \tau_1 \exp(\mathfrak{P}(X,\pi)) \longrightarrow \mathbf{B}^2 U(1)_{conn} \,.$$

  This uses (Fiorenza-Schreiber-Stasheff 10, Fiorenza-Rogers-Schreiber 11).

• The key theorem (prop. 3.5.9, p. 87) in the thesis shows that under 1-truncation this is equivalent to a morphism of smooth stacks of the form

  $$\exp(cs) \;\colon\; SymplGrpd(X,\pi) \longrightarrow \mathbf{B}(\mathbf{B}U(1)_{conn})$$

  and that this is the map that modulates the prequantum 2-bundle on the symplectic groupoid that in the traditional literature on geometric quantization of symplectic groupoids is defined in components.

  This uses in addition(Bonechi-Cattaneo-Zabzine 05).

• Viewed via this fact now the process has a good conceptualization (section 4.3.4):

  1. $SymplGrpd(X,\pi)$ may be understood as the moduli stack of fields of a non-perturbative TQFT 2d Poisson-Chern-Simons theory;

  2. $\exp(cs)$ is the prequantum 2-bundle of this theory regarded as a local prequantum field theory;

  3. the actual prequantum bundle on a symplectic manifold is exhibited as that of a 1d boundary field theory to this 2d theory.

4.3.7 Boundary field theory

This perspective suggests a more encompassing perspective in which geometric quantization of symplectic groupoids is a realization of the holographic principle in local quantum field theory and a non-perturbative refinement of how deformation quantization of Poisson manifolds is the perturbative boundary field theory of the perturbative Poisson sigma-model.

Expanding on this is the topic of the Outlook section.

5. Outlook

• motivic quantization

• master thesis Nuiten

References

• F. Bonechi, Alberto Cattaneo, Maxim Zabzine, Geometric quantization and non-perturbative Poisson sigma-model, Adv. Theor. Math. Phys. 10, 683-712 (2006), (arXiv:0507223).

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

  • cohomology

    • Principal ∞-bundles – models and general theory

    • Fivebrane structures

    • Cosimplicial C-infinity rings and the de Rham complex of Euclidean space

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via Cohesive homotopy types

    • Local prequantum field theory

    • L-∞ algebras of local observables from higher prequantum bundles

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory