On the $Δ_a$ invariants in non-perturbative complex Chern-Simons theory
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| Format: | Preprint |
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2023
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| _version_ | 1866909948426846208 |
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| author | Harichurn, Shimal |
| author_facet | Harichurn, Shimal |
| contents | Recently a set of $q$-series invariants, labelled by $\operatorname{Spin}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the $\widehat{Z}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the $\widehat{Z}_a$ invariants are invariants themselves denoted by $Δ_a$. In this paper we further analyze the structure of these $Δ_a$ invariants. We review some of the foundations of the $Δ_a$ invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the $Δ_0$ invariants for Brieskorn spheres. Along the way we show that the $Δ_a$ invariants are not homology cobordism invariants, thereby answering an open question in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_11298 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $Δ_a$ invariants in non-perturbative complex Chern-Simons theory Harichurn, Shimal Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra 57M27 Recently a set of $q$-series invariants, labelled by $\operatorname{Spin}^c$ structures, for weakly negative definite plumbed $3$-manifolds called the $\widehat{Z}_a$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the $\widehat{Z}_a$ invariants are invariants themselves denoted by $Δ_a$. In this paper we further analyze the structure of these $Δ_a$ invariants. We review some of the foundations of the $Δ_a$ invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the $Δ_0$ invariants for Brieskorn spheres. Along the way we show that the $Δ_a$ invariants are not homology cobordism invariants, thereby answering an open question in the literature. |
| title | On the $Δ_a$ invariants in non-perturbative complex Chern-Simons theory |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics Quantum Algebra 57M27 |
| url | https://arxiv.org/abs/2306.11298 |