Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds

Suppose $(X,\Omega,g)$ is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin(7), or a Calabi-Yau 4-fold. Let $G$ be U$(m)$ or SU$(m)$, and $P\to X$ be a principal $G$-bundle. We show that the infinite-dimensional moduli space ${\mathcal B}_P$ of all connections on $P$ modulo gauge is orientable, in a certain sense. We deduce that the moduli space ${\mathcal M}_P^{Spin(7)}\subset{\mathcal B}_P$ of irreducible Spin(7)-instanton connections on $P$ modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung arXiv:1502.01141 and Mu\~noz and Shahbazi arXiv:1707.02998. If $X$ is a Calabi-Yau 4-fold, the derived moduli stack $\boldsymbol{\mathscr M}$ of (complexes of) coherent sheaves on $X$ is a $-2$-shifted symplectic derived stack $(\boldsymbol{\mathcal M},\omega)$ by Pantev-To\"en-Vaqui\'e-Vezzosi arXiv:1111.3209, and so has a notion of orientation by Borisov-Joyce arXiv:1504.00690. We prove that $(\boldsymbol{\mathscr M},\omega)$ is orientable, by relating algebro-geometric orientations on $(\boldsymbol{\mathscr M},\omega)$ to differential-geometric orientations on ${\mathcal B}_P$ for U$(m)$-bundles $P\to X$, and using orientability of ${\mathcal B}_P$. This has applications to the programme of defining Donaldson-Thomas type invariants counting moduli spaces of (semi)stable coherent sheaves on a Calabi-Yau 4-fold, as in Donaldson and Thomas 1998, Cao and Leung arXiv:1407.7659, and Borisov and Joyce arXiv:1504.00690. This is the third in a series arXiv:1811.01096, arXiv:1811.02405 on orientations of gauge-theoretic moduli spaces.Comment: 57 pages. (v2) Major rewrite: new title, added author, new material on Calabi-Yau manifold

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Motivation to Lead: College Students\u27 Perceptions of their Roles as Leaders on Campus

Student leadership in college has long been studied from the perspective of the university, and more specifically, how the university supports its student leaders in hopes of enhancing institutional reputation. With this in mind, there is little attention given to the micro-level experience of college student leaders. As such, researchers are called to shift the rhetoric on student leadership away from an institutionally-driven conceptualization to a more student-focused discourse. The central aim of this study was to gather student leader perceptions of their roles as campus leaders and construct meaning behind those perceptions. This qualitative research study examined the phenomenology behind college student leadership and the subsequent motivation to lead. Using the Motivation to Lead framework (MTL), this study added to the already dense discussion on leadership and produced a clearer understanding as to how college students make sense of their leadership experience. Findings suggested college student leaders experience the leadership phenomenon from several perspectives; acting with intention in their respective roles, how the administration impacts their role, how prior experiences shape leadership motivation, and finally, how being of service creates increased enjoyment and motivation to continue leading. As a result of this qualitative, phenomenological study, more focused attention was placed on the student leadership experience at the micro level. The findings may support future leadership development programs, program evaluation projects, and provide a better understanding of the student-administration relationship in higher education

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