Genus zero BPS invariants for local \(\mathbb{P}^{1}\) (Q2878691)

Let \(X\) be a Calabi-Yau 3-fold. In the celebrated work [``M-theory and topological strings. II'', \url{arXiv:hep-th/9812127}], \textit{R. Gopakumar} and \textit{C. Vafa} introduced \textit{the BPS invariants} \(n_{\beta}^{g}(X)\), which are related to the Gromov-Witten invariants \(N_{\beta}^{g}(X):=\int_{[\overline{M}_{g,0}(X,\beta)]^{vir}}1 \in \mathbb{Q}\) by the identify NEWLINE\[NEWLINE\sum_{\beta \neq0}\sum_{g\geq0}N_{\beta}^{g}(X)\lambda^{2g-2}q^{\beta} =\sum_{\beta \neq0}\sum_{g\geq0}n_{\beta}^{g}(X)\sum_{k>0}\frac{1}{k}(2\sin(\frac{k\lambda}{2}))^{2g-2}q^{k\beta}.NEWLINE\]NEWLINE Matching the coefficients of the two series yields equations determining \(n_{\beta}^{g}(X)\) recursively in terms \(N_{\beta}^{g}(X)\). Moreover, it is conjectured that the BPS invariants \(n_{\beta}^{g}(X)\) are all integers, which heuristically count the genus \(g\) curves \(C \subset X\) with \([C]=\beta\).NEWLINENEWLINE For example, the genus zero part reads NEWLINE\[NEWLINEN_{\beta}^{0}(X)=\sum_{m | \beta}\frac{n_{\beta/m}^{0}(X)}{m^3}. \tag{1}NEWLINE\]NEWLINE However, a mathematical framework to define the BPS invariants \(n_{\beta}^{g}(X)\) has been missing. In [J. Differ. Geom. 79, No. 2, 185--195 (2008; Zbl 1142.32011)], \textit{S. Katz} proposed a rigorous definition of the BPS invariants \(n_{\beta}^{0}(X)\) using the Donaldson-Thomas type invariants [\textit{K. Behrend}, Ann. Math. (2) 170, No. 3, 1307--1338 (2009; Zbl 1191.14050)]. Namely they are the weighted Euler number of the moduli space of pure \(1\)-dimensional sheaves \(E\) on \(X\) of \(\chi(E)=1\) and class \(\mathrm{ch}_2(E)=\beta\).NEWLINENEWLINE A natural question is whether or not these invariants \(n_{\beta}^{0}(X)\) satisfy the above genus zero formula (1).NEWLINENEWLINEIn the article under review, the author verified the local version of the formula (1) in certain cases. Let \(X\) be the total space of the split vector bundle \(\mathcal{O}(k)\oplus \mathcal{O}(-k-2) \rightarrow \mathbb{P}^1\), which is a toric Calabi-Yau 3-fold. Since \(X\) is not compact, the virtual cycle technique does not work.NEWLINENEWLINE The author uses a natural \((\mathbb{C}^\times)^2\)-action, which preserves the Calabi-Yau structure, and defined the equivariant BPS invariants via the equivariant residue integrals of the virtual cycles at the fixed loci of the moduli space.NEWLINENEWLINE A similar technique has been used by Bryan and Pandharipande in [\textit{J. Bryan} and \textit{R. Pandharipande}, J. Am. Math. Soc. 21, No. 1, 101--136 (2008; Zbl 1126.14062)] to define the equivariant Gromov-Witten invariants of the total space of a rank \(2\) split vector bundle over a curve.NEWLINENEWLINE In the article under review, the author explicitly computes the above equivariant BPS invariants by classifying the stable equivariant sheaves on \(\mathbb{P}^1\).NEWLINENEWLINE The main result asserts that the genus zero formula (1) holds for these equivariant invariants \(n_{\beta}^{0}(X)\) and \(N_{\beta}^{0}(X)\) for \(1 \leq d \leq 3\) for arbitrary \(k\), and for \(d=4\) for \(k\leq 100\), where \(\beta=d[\mathbb{P}^1]\) is the multiple of the class of the 0-section. This result provides another important supporting evidence for the proposal of Katz.NEWLINENEWLINEThe proof is based on the virtual localization [\textit{T. Graber} and \textit{R. Pandharipande}, Invent. Math. 135, No. 2, 487--518 (1999; Zbl 0953.14035)] and the classification of equivariant vector bundles [\textit{T. Kaneyama}, Nagoya Math. J. 57, 65--86 (1975; Zbl 0283.14008)], together with a careful analysis of the stable equivariant sheaves on \(\mathbb{P}^1\). Although the main techniques of calculation are by now standard, it is a tour-de-force to bring these to a good end. In this, the author succeeds very well and has written a nice article that reads rather well.