arXiv 1602.035302016-02hep-th

非紧动力学的Witten指数

Witten Index for Noncompact Dynamics

Seung-Joo Lee, Piljin Yi

备注：: 54 pages, JHEP version, references updated with accompanying footnotes, an additional example added in Eq.(5.22)
DOI：: 10.1007/JHEP06(2016)089

摘要：

在由弦论激发的规范动力学中，我们发现许多规范动力学具有无间隙的渐近方向。虽然基态的自然边界条件是$L^2$，但人们经常使用化学势或超对称质量项来调节红外问题，并计算扭曲的配分函数。我们指出，这个过程一般无法捕捉物理$L^2$威滕指数往往误导的结果。我们还探讨了如何，然而，威滕指数有时错综复杂地嵌入在这种扭曲的配分函数。对于$d=1$理论与无间隙连续扇区从规范多重态，如非原始箭和纯杨米尔斯，更微妙的存在，导致分数表达式。然而，出乎意料的是，积分$L^2$威滕指数可以直接和容易地从扭曲的配分函数的理论。这种现象与自然出现在跨壁公式中的有理不变量的概念有关，并提供了直接从扭曲的配分函数中读出威滕指数的一般机制。在此过程中，我们修正了$\mathcal N=4,8,16$纯Yang-Mills量子力学的早期数值结果，并计算了$SU(N)$以外的一般规范群的阈值束缚态。

英文摘要：

Among gauged dynamics motivated by string theory, we find many with gapless asymptotic directions. Although the natural boundary condition for ground states is $L^2$, one often turns on chemical potentials or supersymmetric mass terms to regulate the infrared issues, instead, and computes the twisted partition function. We point out how this procedure generically fails to capture physical $L^2$ Witten index with often misleading results. We also explore how, nevertheless, the Witten index is sometimes intricately embedded in such twisted partition functions. For $d=1$ theories with gapless continuum sector from gauge multiplets, such as non-primitive quivers and pure Yang-Mills, a further subtlety exists, leading to fractional expressions. Quite unexpectedly, however, the integral $L^2$ Witten index can be extracted directly and easily from the twisted partition function of such theories. This phenomenon is tied to the notion of the rational invariant that appears naturally in the wall-crossing formulae, and offers a general mechanism of reading off Witten index directly from the twisted partition function. Along the way, we correct early numerical results for some of $\mathcal N=4,8,16$ pure Yang-Mills quantum mechanics, and count threshold bound states for general gauge groups beyond $SU(N)$.

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