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 9e6b
X-WR-CALNAME:QM Research Seminar: The TQFT Factorization Axiom and Bohr-So
 mmerfeld Quantization
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DTSTAMP:20260414T021918Z
DTSTART:20261028T030000
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DESCRIPTION:[b]Speaker: William Elbæk Mistegård[/b] (University of Souther
 n Denmark) [nl]\n[b]Abstract:[/b][nl]\nIn his seminal work on quantum Che
 rn-Simons field theory and the Jones polynomial\, Witten envisioned a 2+1
  dimensional TQFT. Further\, he presented two procedures for constructing
  the underlying modular functor\, namely by conformal field theory (CFT) 
 and by geometric Kähler quantization of the moduli spaces of flat princip
 al bundles on two-manifolds. Subsequently\, Reshetikhin and Turaev constr
 ucted a TQFT using surgery of three-manifolds and representation theory o
 f quantum groups\, and this is now known as the WRT-TQFT. The CFT approac
 h was initially developed by Tsuchiya\, Ueno and Yamada\, and then furthe
 r refined to a full modular functor by Andersen and Ueno\, who also estab
 lished that this is equivalent to the WRT-TQFT.  The geometric Kähler qua
 ntization approach was initialized by Axelrod\, Witten and Pietra\, and H
 itchin. The quantization approach results in a projective representation 
 of the mapping class group of a two-manifold\, which is known due to Lasz
 lo to be projectively equivalent to the representation of the WRT-TQFT. H
 owever\, the full TQFT is not yet described from the point of view of qua
 ntization. In particular\, the so-called TQFT factorization axiom has not
  been proved using only geometric quantization with respect to Kähler pol
 arizations. However\, Andersen have understood factorization from the poi
 nt of view of geometric quantization with respect to reducible non-negati
 ve polarizations.\n \nJeffrey and Weitsman introduced a different quantiz
 ation procedure. Given a trinion decomposition of a closed two-manifold S
 \, one can obtain a system of functions on the moduli space of flat bundl
 es on the two-manifold S. This is given by taking the trace of the holono
 my representations of the boundary circles of the trinions. This results 
 in a fibration of the moduli space\, the fibres of this map being Lagrang
 ian tori. The vertical distribution of this fibration is a polarization i
 n the sense of geometric quantization. Jeffrey and Weitsman considered th
 e so-called Bohr-Sommerfeld quantization of this system (i.e. this polari
 zation)\, and proved that it results in a finite dimensional Hilbert spac
 e\, which we denote by V(S). Further\, they showed that V(S) has dimensio
 n equal to the dimension of the Hilbert space of the WRT-TQFT (this dimen
 sion is the famous Verlinde formula).\n \nIn this talk\, we present aspec
 ts of an ongoing project on quantization and TQFT\, which is complementar
 y to the factorization construction of Andersen. We consider the two-mani
 fold S' obtained by cutting along a circle of the trinion decomposition. 
 This two-manifold have two boundaries and an induced trinion decompositio
 n. Following Jeffrey and Weitsman\, we consider for each element j in the
  gauge group\, the associated Lagrangian torus fibration of the moduli sp
 ace of flat bundles on S' with boundary holonomy conjugate to j. We consi
 der the Bohr-Sommerfeld quantization of this system\, which we denote by 
 V(S'\,j). We show that in accordance with the TQFT factorization axiom\, 
 the Hilbert space of the closed surface\, denoted above by V(S)\, splits 
 as a direct sum indexed by elements of the WRT-TQFT label set\, the summa
 nd indexed by j being equal to V(S'\,j). Further\, we show that this dire
 ct sum decomposition is conjugate to factorization isomorphism of the WRT
 -TQFT. Thus these results gives a geometric illumination of the factoriza
 tion axiom of the WRT-TQFT by means of the Bohr-Sommerfeld quantization o
 f Chern-Simons theory.[nl]
DTEND:20230328T140000Z
DTSTAMP:20260414T021918Z
DTSTART:20230328T130000Z
LOCATION:Syddansk Universitet\, Campusvej 55\, 5230\, Odense M
SEQUENCE:0
SUMMARY:QM Research Seminar: The TQFT Factorization Axiom and Bohr-Sommerf
 eld Quantization
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