Research ArticleNo Access

Anomaly-free TQFTs from the super Lie algebra $𝔰 𝔩 (2 | 1)$

Hung Vuong University, Faculty of Natural Sciences, Nong Trang, Viet Tri, Phu Tho, Vietnam

https://doi.org/10.1142/S0218216522500298Cited by:4 (Source: Crossref)

Abstract

It is known that the category $𝒞^{H}$ of nilpotent weight modules over the quantum group associated with the super Lie algebra $𝔰 𝔩 (2 | 1)$ is a relative pre-modular $G$ -category. Its modified trace enables one to define an invariant of $3$ -manifolds. In this paper, we show that the category $𝒞^{H}$ is a relative modular $G$ -category which allows one to constructing a family of non-semi-simple extended topological quantum field theories, and the family of TQFTs is anomaly free. The quantum group associated with $𝔰 𝔩 (2 | 1)$ is considered at odd roots of unity.

Keywords:

AMSC: 57M27, 17B37

References

1. B. Abdesselam, D. Arnaudon and M. Bauer , Centre and representations of $𝒰_{q} 𝔰 𝔩 (2 | 1)$ at roots of unity, J. Phys. A: Math. Gen. 30(3) (1997) 867–880. Crossref, Google Scholar
2. C. Blanchet, F. Costantino, N. Geer and B. Patureau-Mirand , Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math. 301 (2016) 1–78. Crossref, Web of Science, Google Scholar
3. F. Costantino, N. Geer and B. Patureau-Mirand , Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories, J. Topol. 7(4) (2014) 1005–1053. Crossref, Web of Science, Google Scholar
4. M. De Renzi , Non-semisimple extended topological quantum field theories, Mem. Amer. Math. Soc. 277(1364) (2022) 172. Web of Science, Google Scholar
5. M. De Renzi, N. Geer and B. Patureau-Mirand , Nonsemisimple quantum invariants and TQFTs from small and unrolled quantum groups, Algebr. Geom. Topol. 20(7) (2020) 3377–3422. Crossref, Web of Science, Google Scholar
6. N. Geer and B. Patureau-Mirand , Multivariable link invariants arising from $𝔰 𝔩 (2 | 1)$ and the Alexander polynomial, J. Pure Appl. Algebra 210(1) (2007) 283–298. Crossref, Web of Science, Google Scholar
7. N. Geer and B. Patureau-Mirand , An invariant supertrace for the category of representations of Lie superalgebras of type I, Pacific J. Math. 238(2) (2008) 331–348. Crossref, Web of Science, Google Scholar
8. N. Geer, B. Patureau-Mirand and V. Turaev , Modified quantum dimensions and re-normalized links invariants, Compos. Math. 145(1) (2009) 196–212. Crossref, Web of Science, Google Scholar
9. N. P. Ha , Topological invariants from quantum group $𝒰_{ξ} 𝔰 𝔩 (2 | 1)$ at roots of unity, Abh. Math. Semin. Univ. Hambg. 88(1) (2018) 1–26. Crossref, Web of Science, Google Scholar
10. C. Kassel , Quantum Groups, Graduate Texts in Mathematics (Springer-Verlag, 1995). Crossref, Google Scholar
11. S. M. Khoroshkin and V. N. Tolstoy , Universal $ℛ -$ matrix for quantized (super)algebras, Commun. Math. 141(3) (1991) 599–617. Crossref, Google Scholar
12. V. G. Turaev , Quantum Invariants of Knots and 3-Manifolds, Studies in Mathematiques (De Gruyter, 1994). Crossref, Google Scholar