<p>Recently, a set of <i>q</i>-series invariants, labeled by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\operatorname {Spin}^c$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mo>Spin</mo> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation> structures, for weakly negative definite plumbed 3-manifolds called the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\widehat{Z}_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\widehat{Z}_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>Z</mi> <mo stretchy="false">^</mo> </mover> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> invariants are invariants themselves denoted by <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Delta _a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>. In this paper, we further analyze the structure of these <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Delta _a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> invariants. We review some of the foundations of the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Delta _a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Delta _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> invariants for Brieskorn spheres. Along the way we show that the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Delta _a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> invariants are not homology cobordism invariants, thereby answering an open question in the literature.</p>

On the $\Delta _{a}$ invariants in non-perturbative complex Chern–Simons theory

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发表日期：

2025年11月18日

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Shimal Harichurn

摘要

Recently, a set of q-series invariants, labeled by $\operatorname {Spin}^c$ ${Spin}^{c}$ structures, for weakly negative definite plumbed 3-manifolds called the $\widehat{Z}_a$ ${\hat{Z}}_{a}$ invariants were discovered by Gukov, Pei, Putrov and Vafa. The leading rational power of the $\widehat{Z}_a$ ${\hat{Z}}_{a}$ invariants are invariants themselves denoted by $\Delta _a$ $Δ_{a}$ . In this paper, we further analyze the structure of these $\Delta _a$ $Δ_{a}$ invariants. We review some of the foundations of the $\Delta _a$ $Δ_{a}$ invariants and analyze their structure for a subclass of integer homology spheres. In particular, we provide a complete description of the $\Delta _0$ $Δ_{0}$ invariants for Brieskorn spheres. Along the way we show that the $\Delta _a$ $Δ_{a}$ invariants are not homology cobordism invariants, thereby answering an open question in the literature.

On the \(\Delta _{a}\) invariants in non-perturbative complex Chern–Simons theory

摘要