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arXiv 1703.00511hep-thmath-phmath.MP
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再论同调超对称量子力学

More on Homological Supersymmetric Quantum Mechanics

Alireza Behtash

备注：
29 pages, 13 figures. v3 Improved version, more references added, conclusions added, typos fixed, accepted for publication in PRD
发表信息：
Phys. Rev. D 97, 065002 (2018)
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摘要：

在这项工作中，我们首先解决复杂的Morse流方程的最简单的情况下，玻色子谐振子，讨论本地化的皮卡德-莱夫谢茨理论。本文简要地讨论了玻色化超对称量子力学在代数几何基础上的精确非BPS解，并指出它们的复相位可以通过对非零亏格进行必要的上同调修正后的WKB 1-形式的奇异谱曲线的上同调得到.受Picard-Lefschetz理论的启发，我们给出了具有规范场背景$R$对称性的$\mathcal{N} = 4$量子力学指数的一般公式.我们推测，某些对称性的细化Witten指数和奇异性的模空间可以用来确定正确的交叉系数。有几个例子，这个猜想成立，显示在线性和封闭的箭图与秩1的规范群。沿“Morsified”相对同调环（也称为“Lefschetz顶针”）的$R$ -异常去除导致斯托克斯线的出现。我们证明了Fayet-Iliopoulos（FI）参数出现在$2d$ $\mathcal{N}=(2,2)$规范理论在圆上降维所产生的量子力学的相对同调的交叉系数中，并明确地计算了$\mathcal{N}=4$ $\mathbb{CP}^{k-1}$模型中沿Lefschetz环的积分.简要讨论了系数的Stokes跳跃及其与跨壁现象的关系。我们还发现，“在墙上”指数的概念是有关的不变莱夫谢茨顶针下斯托克斯现象。指出了莱夫谢茨顶针在构造量子量子力学中的一个意义。

英文摘要：

In this work, we first solve complex Morse flow equations for the simplest case of a bosonic harmonic oscillator to discuss localization in the context of Picard-Lefschetz theory. We briefly touch on the exact non-BPS solutions of the bosonized supersymmetric quantum mechanics on algebraic geometric grounds and report that their complex phases can be accessed through the cohomology of WKB 1-form of the underlying singular spectral curve subject to necessary cohomological corrections for non-zero genus. Motivated by Picard-Lefschetz theory, we write down a general formula for the index of $\mathcal{N} = 4$ quantum mechanics with background $R$-symmetry gauge fields. We conjecture that certain symmetries of the refined Witten index and singularities of the moduli space may be used to determine the correct intersection coefficients. A few examples, where this conjecture holds, are shown in both linear and closed quivers with rank-one quiver gauge groups. The $R$-anomaly removal along the "Morsified" relative homology cycles also called "Lefschetz thimbles" is shown to lead to the appearance of Stokes lines. We show that the Fayet-Iliopoulos (FI) parameters appear in the intersection coefficients for the relative homology of the quiver quantum mechanics resulting from dimensional reduction of $2d$ $\mathcal{N}=(2,2)$ gauge theory on a circle and explicitly calculate integrals along the Lefschetz thimbles in $\mathcal{N}=4$ $\mathbb{CP}^{k-1}$ model. The Stokes jumping of coefficients and its relation to wall crossing phenomena is briefly discussed. We also find that the notion of "on-the-wall" index is related to the invariant Lefschetz thimbles under Stokes phenomena. An implication of the Lefschetz thimbles in constructing knots from quiver quantum mechanics is indicated.

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