Quantization

Speculations

Motto: Everything is quantization.

Automorphic forms and Baum-Connes assembly

• These are related via K-theoretic quantization. The bridge being roughly the field of equivariant topology/semiclassical analysis/Poisson geometry: Duistermaat-Heckman, reduction and localization, Guillemin-Sternberg…
• TQFT and deformation quantization: TQFT is relevant through e.g. the interpretation of formal deformation quantizations of Poisson manifolds as TQFTs (Cattaneo & Felder). Underlyting formalisms: BV-BRST, factorization algebra, more geometrically symplectic/Poisson reduction
• Automorphic forms. Modular groups generally arise in the following way: given a space (classical system) $X$, a discrete group $G$ and a $G$-action on $X$, if the (symplectic/Poisson/Courant…) reduction of $g\cdot X$ w.r.t a group $H$ is isomorphic/Morita equivalent (Morita equivalence is much more natural since the quotient $X//H$ encodes the information about the representation ring i.e. quantization of the classical symmetry $H$) to that of $X$, then $G$ can be seen as a modular group. Automorphic forms are sections of a line bundle (prequantum bundle) or even a gerbe $L$ on $X$.
  • There is a buisiness about lifting the $H$-symmetry to $L$ here, it is not clear whether the lifted action on $L$ or the original action on $X$ is more relevant, I suspect the former. Thus maybe regarding the total space of the bundle $L\to X$ as the classical system would be a better option, but that doesn’t really make sense in the physics perspective. This is only relevant for studying projective representations of the quantum symmetry group.
• Baum-Connes assembly: Up to some Fourier transform, this is a generalized statement for quantization commutes with reduction. The $H$-equivariant Dirac operators, whose classes define Kasparov KK-groups, should be seen as classical equations of motions. Reduction is the process of ‘integrating out constraints’ (symmetries, including gauge symmetries, should also be seen as constraints), so the modular group $G$ relates those constrained classical systems of which the reductions yield “the same” quantum theories.
  • What is the precise meaning of the “the same” here?
  • When quantized the classical system is linearized, does it lose information?

Dimensional reduction

[Some quick notes]

• Poisson manifold quantization and field theory, Kaluza-Klein, AKSZ gauge fixing and dimensional reduction; QM as dimensional reduction in AKSZ
• Representation of motion groups of links via dimensional reduction of TQFTs ( https://arxiv.org/abs/2002.07642 ); basically quantization through reduction
• Index; fiber integration; moduli space of $G$-principal bundles
• Orbifoldization of TFTs ( https://arxiv.org/abs/1705.05171 ) - crossed product - fiber integration
• Foliation - bundle: leaf as sections and the base as the dimensional recution
• Dimensional reduction ( https://arxiv.org/abs/2004.04689 )

Reduction v.s. quantization in Guillemin-Sternberg picture: the same?