String theory, knot theory, thermal QCD and string cosmology

我的研究计划可以分为四大类：弦宇宙学、弦理论、热QCD和纽结理论。让我从弦宇宙学开始。最近的挑战是从弦理论构建德西特真空，即具有正宇宙学常数的真空。由于某些禁行条件（也称为沼泽地标准），这变得非常具有挑战性。我的目标之一是找到一种具体的方法来克服不可行的条件，并展示德西特真空是如何产生的，或者证明弦理论中不可能有德西特解。我的第二个研究计划是弦理论。弦理论中重要的问题之一是稳定真空的选择。模量的稳定需要切换背景通量以及非微扰效应，从而产生非卡拉比-丘流形，或者更一般地，非卡勒流形。尽管无处不在，但这些流形的具体构造比较困难，因为它们需要复杂的微分几何技术。因此，我的建议之一是对这些真空进行通用构造。这很重要，否则弦理论将无法重现标准模型。我的第三个建议与热 QCD 有关。我们已经能够构建对偶引力描述，可以解释所有能量尺度上类似 QCD 的动力学理论。然而，还有很多事情有待展示。例如，目前尚不清楚这种理论中的体积粘度如何在中间耦合（即弱和强't Hooft耦合之间的耦合）下发挥作用。了解这一点将是文献中的巨大进步，因为目前用于研究中间耦合动力学的技术几乎没有结论。另一个相关的事情是颜色超导性，可以使用相同的双引力模型来研究它。有趣的是，我们构建的引力对偶不仅仅是为了研究QCD，也是为了研究其他全息问题，例如纠缠熵等。由于我们的模型产生了非共形理论的引力对偶，因此回答与纠缠熵（EE）等相关的问题将在非共形性开启时阐明这些问题。我的最后一个建议是关于纽结理论的。这是我最近开始的一个新方向，基于我们的发现，拓扑场论的许多结果可能源自M理论中的某些引力背景。这种令人惊讶的构造不仅解释了拓扑扭曲的概念，而且直接再现了边界陈-西蒙斯理论以及结不变量。然而令人费解的是，给定的结多项式的系数与某个微分方程的解的个数是一一对应的。我的建议之一就是证明这个猜想。彩色琼斯和考夫曼多项式的构造等相关主题需要进一步研究。

My research proposal may be divided into four broad categories: string cosmology, string theory, thermal QCD and knot theory. Let me start by string cosmology. A recent challenge is to construct de Sitter vacua, i.e a vacua with positive cosmological constant, from string theory. This has turned out to be very challenging due to certain no-go conditions, also called the swampland criteria. One of my aim is to find a concrete way to overcome the no-go conditions and show how de Sitter vacua may come about or justify that there could be no de Sitter solution possible in string theory. My second research proposal is in string theory. One of the issue that is important in string theory is the choice of the stable vacuum. Stabilization of moduli require switching on background fluxes, and also non-perturbative effects, leading to non-Calabi-Yau manifolds, or more generically, non-Kahler manifolds. Despite ubiquitous, concrete constructions of these manifolds are harder because they require sophisticated techniques in differential geometry. Thus one of my proposal here is to have generic constructions of these vacua. This is important otherwise string theory will be unable to reproduce the standard model. My third proposal is related to thermal QCD. We have been able to construct the dual gravitational description that can explain the dynamics of QCD like theories at all energy scales. However many things remain to be shown. For example it is as yet unknown how bulk viscosity in such a theory works at intermediate couplings, i.e couplings between weak and strong 't Hooft couplings. Knowing this will be an immense progress in the literature because the present techniques used to study dynamics at intermediate couplings are pretty much inconclusive. Another related thing is color superconductivity which may be studied using the same dual gravitational model. Interestingly, the gravity dual constructed by us is not just to study QCD, but also to study other holographic questions like entanglement entropies etc. Since our model gives rise to the gravity dual of a theory that is non-conformal, answering questions related to entanglement entropies (EE) etc. will shed light on these issues when non-conformality is switched on. My last proposal is on knot theory. This is a new direction that I have started recently, and is based on our findings that many of the results of topological field theory my be derived from certain gravitational background in M-theory. This surprising construction not only explains the concept of topological twisting, but also directly reproduces the boundary Chern-Simons theory as well as the knot invariants. However the puzzling thing is that the coefficients of a given knot polynomial have one-to-one correspondence to the number of solutions of a certain differential equation. One of my proposal is to prove this conjecture. Related topics like the construction of the colored Jones and Kaufmann polynomials require further research.