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On non-semisimple quantum invariants

Mori, Akihito 東北大学

2023.03.24

概要

Inspired by Witten [Wit89] who had given a three-dimensional interpretation of Jones
polynomials, Reshetikhin–Turaev [RT91] constructed topological invariants of three-manifolds.
They defined a modular Hopf algebra and proved that it gives rise to a topological
quantum field theory (TQFT for short) in dimension three, in particular invariants of
three-manifolds. The invariants of three-manifolds are called Witten–Reshetikhin–Turaev
(WRT for short) invariants.
Reshetikhin and Turaev reduced three-manifolds to framed links in S 3 and used representations of small quantum groups at roots of unity U˜q sl2 which are examples of modular
Hopf algebras. That is, the WRT invariant is a weighted sum of the Reshetikhin–Turaev
link invariants F for links coloured by representations of the small quantum group. An
important property of U˜q sl2 is that it has only a finite stock of irreducible representations.
Turaev [Tur92] generalised a modular Hopf algebra to a modular tensor category which is
a semisimple category. A typical example of such a category is the representation category
of U˜q sl2 . The WRT invariant is called a semisimple quantum invariant named after the
semisimplicity of U˜q sl2 .
There are two quantum invariants closely related to WRT invariants. One of these
is a non-semisimple quantum invariant. Costantino–Geer–Patureau-Mirand [CGPM14]
defined a relative G-modular category for a commutative group which is a non-semisimple
ribbon category. An example of such a category is a representation category of the unrolled
quantum group U¯qH sl2 . This quantum group is a version of the small quantum group and
has an infinite stock of irreducible representations. This is one of the differences between
U˜q sl2 and U¯qH sl2 .
A non-semisimple quantum invariant is constructed as a weighted sum of the renormalised Reshetikhin–Turaev link invariants F0 for links coloured by representations
of the unrolled quantum group. Costantino, Geer, and Patureau-Mirand controlled the
infiniteness by degree-1 cohomology classes and constructed invariants of three-manifolds
with cohomology classes. We call the invariant Costantino–Geer–Patureau-Mirand (CGP
for short) invariant. The CGP invariant is more fruitful than WRT invariant. For example, the CGP invariant distinguishes homotopically equivalent three-manifolds that WRT
invariant can not. Moreover the CGP invariant allows a version of the volume conjecture.
This version was proved for an infinite class of links [CGPM14]. It is worthwhile to study
the CGP invariant and the WRT invariant in relation to each other.
In this thesis, we study the following problem: find relations between semisimple and
non-semisimple quantum invariants. This problem leads to revealing differences between
semisimple and non-semisimple theory and positioning the semisimple theory as a part of
the non-semisimple theory. Specifically, we consider the following problems.
(i) Find relations between the Reshetikhin–Turaev link invariant and the re-normalised
link invariant. ...

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参考文献

[ADO92]

Yasuhiro Akutsu, Tetsuo Deguchi, and Tomotada Ohtsuki. Invariants of

colored links. J. Knot Theory Ramifications, 1(2):161–184, 1992.

[AM22]

Jørgen Ellegaard Andersen and William Elbæk Mistegård. Resurgence analysis of quantum invariants of Seifert fibered homology spheres. J. Lond.

Math. Soc. (2), 105(2):709–764, 2022.

[And13]

Jørgen Ellegaard Andersen. The Witten-Reshetikhin-Turaev invariants of

finite order mapping tori I. J. Reine Angew. Math., 681:1–38, 2013.

[Ati88]

Michael Atiyah. Topological quantum field theories. Inst. Hautes Études

Sci. Publ. Math., (68):175–186 (1989), 1988.

[Ati90]

Michael Atiyah. The geometry and physics of knots. Cambridge University

Press, Cambridge, 1990.

[BCGPM16] Christian Blanchet, Francesco Costantino, Nathan Geer, and Bertrand

Patureau-Mirand. Non-semi-simple TQFTs, Reidemeister torsion and

Kashaev’s invariants. Adv. Math., 301:1–78, 2016.

[BH21]

Anna Beliakova and Kazuhiro Hikami. Non-semisimple invariants and

Habiro’s series. Topology and geometry, 33:161–174, 2021.

[BHMV95]

C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel. Topological quantum

field theories derived from the Kauffman bracket. Topology, 34(4):883–927,

1995.

[BK01]

Bojko Bakalov and Alexander Kirillov, Jr. Lectures on tensor categories and

modular functors. American Mathematical Society, Providence, RI, 2001.

[BKM19]

Kathrin Bringmann, Jonas Kaszian, and Antun Milas. Higher depth quantum modular forms, multiple Eichler integrals, and sl3 false theta functions.

Res. Math. Sci., 6(2):Paper No. 20, 41, 2019.

[BM15]

Kathrin Bringmann and Antun Milas. W-algebras, false theta functions and

quantum modular forms, I. Int. Math. Res. Not. IMRN, (21):11351–11387,

2015.

74

[BMM20]

Kathrin Bringmann, Karl Mahlburg, and Antun Milas. Higher depth

quantum modular forms and plumbed 3-manifolds. Lett. Math. Phys.,

110(10):2675–2702, 2020.

[CCF+ 19]

Miranda C.N. Cheng, Sungbong Chun, Francesca Ferrari, Sergei Gukov, and

Sarah M. Harrison. 3d modularity. J. High Energy Phys., (10):010, 93, 2019.

[CGPM14]

Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. Quantum invariants of 3-manifolds via link surgery presentations and non-semisimple categories. J. Topol., 7(4):1005–1053, 2014.

[CGPM15a] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. Relations between Witten-Reshetikhin-Turaev and nonsemisimple sl(2) 3manifold invariants. Algebr. Geom. Topol., 15(3):1363–1386, 2015.

[CGPM15b] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand. Some

remarks on the unrolled quantum group of sl(2). J. Pure Appl. Algebra,

219(8):3238–3262, 2015.

[Chu17]

Sungbong Chun. A resurgence analysis of the SU (2) Chern-Simons partition

functions on a Brieskorn homology sphere ⌃(2, 5, 7). arXiv:1701.03528v1,

2017.

[CP94]

Vyjayanthi Chari and Andrew Pressley. A guide to quantum groups. Cambridge University Press, Cambridge, 1994.

[DR22]

Marco De Renzi. Non-semisimple extended topological quantum field theories. Mem. Amer. Math. Soc., 277(1364):v+161, 2022.

[DT07]

Florian Deloup and Vladimir Turaev. On reciprocity. J. Pure Appl. Algebra,

208(1):153–158, 2007.

[FG91]

Daniel S. Freed and Robert E. Gompf. Computer calculation of Witten’s

3-manifold invariant. Comm. Math. Phys., 141(1):79–117, 1991.

[FIMT21]

Hiroyuki Fuji, Kohei Iwaki, Hitoshi Murakami, and Yuji Terashima. WittenReshetikhin-Turaev function for a knot in Seifert manifolds. Comm. Math.

Phys., 386(1):225–251, 2021.

[FYH+ 85]

P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomial invariant of knots and links. Bull. Amer. Math.

Soc. (N.S.), 12(2):239–246, 1985.

[GMnP16]

Sergei Gukov, Marcos Mariño, and Pavel Putrov. Resurgence in complex

Chern-Simons theory. arXiv:1605.07615v2, 2016.

75

[GPMT09]

Nathan Geer, Bertrand Patureau-Mirand, and Vladimir Turaev. Modified

quantum dimensions and re-normalized link invariants. Compos. Math.,

145(1):196–212, 2009.

[GPPV20]

Sergei Gukov, Du Pei, Pavel Putrov, and Cumrun Vafa. BPS spectra and

3-manifold invariants. J. Knot Theory Ramifications, 29(2):2040003, 85,

2020.

[Han05]

Søren Kold Hansen. Analytic asymptotic expansions of the Reshetikhin–

Turaev invariants of Seifert 3-manifolds for SU(2). arXiv:math/0510549v1,

2005.

[Hik05a]

Kazuhiro Hikami. On the quantum invariant for the Brieskorn homology

spheres. Internat. J. Math., 16(6):661–685, 2005.

[Hik05b]

Kazuhiro Hikami. Quantum invariant, modular form, and lattice points.

Int. Math. Res. Not., (3):121–154, 2005.

[Hik06a]

Kazuhiro Hikami. On the quantum invariants for the spherical Seifert manifolds. Comm. Math. Phys., 268(2):285–319, 2006.

[Hik06b]

Kazuhiro Hikami. Quantum invariants, modular forms, and lattice points.

II. J. Math. Phys., 47(10):102301, 32, 2006.

[Jef92]

Lisa C. Jeffrey. Chern-Simons-Witten invariants of lens spaces and torus

bundles, and the semiclassical approximation. Comm. Math. Phys.,

147(3):563–604, 1992.

[Kas95]

Christian Kassel. Quantum groups. Springer-Verlag, New York, 1995.

[KL01]

Thomas Kerler and Volodymyr V. Lyubashenko. Non-semisimple topological

quantum field theories for 3-manifolds with corners. Springer-Verlag, Berlin,

2001.

[Lic62]

W. B. R. Lickorish. A representation of orientable combinatorial 3manifolds. Ann. of Math. (2), 76:531–540, 1962.

[LR99]

Ruth Lawrence and Lev Rozansky. Witten-Reshetikhin-Turaev invariants

of Seifert manifolds. Comm. Math. Phys., 205(2):287–314, 1999.

[LZ99]

Ruth Lawrence and Don Zagier. Modular forms and quantum invariants of

3-manifolds. Asian J. Math., 3(1):93–107, 1999.

[MM22]

Akihito Mori and Yuya Murakami. Witten-Reshetikhin-Turaev invariants,

homological blocks, and quantum modular forms for unimodular plumbing

H-graphs. SIGMA Symmetry Integrability Geom. Methods Appl., 18:Paper

No. 034, 20, 2022.

76

[MN08]

Jun Murakami and Kiyokazu Nagatomo. Logarithmic knot invariants arising

from restricted quantum groups. Internat. J. Math., 19(10):1203–1213, 2008.

[Mor22]

Akihito Mori. Relations between Reshetikhin–Turaev and re-normalized link

invariants. arXiv:2209.03608v1, 2022.

[Mur08]

Jun Murakami. Colored Alexander invariants and cone-manifolds. Osaka J.

Math., 45(2):541–564, 2008.

[Mur18]

Jun Murakami. From colored Jones invariants to logarithmic invariants.

Tokyo J. Math., 41(2):453–475, 2018.

[O+ 02]

Tomotada Ohtsuki et al. Problems on invariants of knots and 3-manifolds.

Geom. Topol. Monogr, 4:377–572, 2002.

[Oht02]

Tomotada Ohtsuki. Quantum invariants. A study of knots, 3–manifolds,

and their sets. World Scientific Publishing Co., Inc., River Edge, NJ, 2002.

[Rei27]

Kurt Reidemeister. Elementare Begründung der Knotentheorie. Abh. Math.

Sem. Univ. Hamburg, 5(1):24–32, 1927.

[Ren17]

Marco De Renzi. Quantum Invariants of 3-Manifolds Arising from NonSemisimple Categories. arXiv:1703.07319v1, 2017.

[Rol76]

Dale Rolfsen. Knots and links. Publish or Perish, Inc., Berkeley, Calif., 1976.

[RT91]

N. Reshetikhin and V. G. Turaev. Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991.

[Tur92]

Vladimir G. Turaev. Modular categories and 3-manifold invariants. Internat.

J. Modern Phys. B, 6(11-12):1807–1824, 1992.

[Tur16]

Vladimir G. Turaev. Quantum invariants of knots and 3-manifolds. De

Gruyter, Berlin, 2016.

[Wal60]

Andrew H. Wallace. Modifications and cobounding manifolds. Canadian J.

Math., 12:503–528, 1960.

[Wit89]

Edward Witten. Quantum field theory and the Jones polynomial. Comm.

Math. Phys., 121(3):351–399, 1989.

[Wu21]

David H. Wu. Resurgent analysis of SU(2) Chern-Simons partition function

on Brieskorn spheres ⌃(2, 3, 6n + 5). J. High Energy Phys., (2):Paper No.

008, 18, 2021.

[Zag10]

Don Zagier. Quantum modular forms. Quanta of maths, 11:659–675, 2010.

77

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