String theory, knot theory, thermal QCD and string cosmology

弦理论、纽结理论、热 QCD 和弦宇宙学

• 批准号: SAPIN-2019-00039
• 负责人: Dasgupta, Keshav
• 金额: $ 3.64万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Subatomic Physics Envelope - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739419
• 关键词: String; theory; knot; thermal; QCD

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.

我的研究建议可以分为四大类：弦宇宙论、弦理论、热QCD和纽结理论。让我从弦宇宙学开始。最近的一个挑战是用弦理论构造de Sitter真空，即具有正宇宙常数的真空。事实证明，由于某些禁止条件，也被称为沼泽标准，这是非常具有挑战性的。我的目标之一是找到一种具体的方法来克服不能进行的条件，并展示德西特真空是如何产生的，或者证明在弦理论中可能没有德西特解。我的第二个研究建议是弦理论。弦理论中的一个重要问题是稳定真空的选择。模的稳定化需要打开背景磁通，以及非微扰效应，从而导致非Calabi-Yau流形，或者更一般的非Kahler流形。尽管无处不在，但这些流形的具体构造要困难得多，因为它们需要复杂的微分几何技术。因此，我在这里的建议之一是建立这些真空的一般结构。这一点很重要，否则弦理论将无法复制标准模型。我的第三个建议与热QCD有关。我们已经能够构造出能够解释QCD类理论在所有能量尺度下的动力学的对偶引力描述。然而，仍有许多东西有待展示。例如，这种理论中的体粘性如何在中间耦合，即弱耦合和强t Hooft耦合之间起作用，目前还是个未知数。知道这一点在文献中将是一个巨大的进步，因为目前用于研究中间耦合动力学的技术几乎没有定论。另一件相关的事情是颜色超导，它可以用同样的双引力模型来研究。有趣的是，我们构造的引力对偶不仅是为了研究QCD，也是为了研究其他全息问题，如纠缠熵等。由于我们的模型引入了非共形理论的引力对偶，当非共形开启时，回答与纠缠熵(EE)相关的问题将有助于揭示这些问题。我的最后一个建议是关于纽结理论。这是我最近开始的一个新方向，是基于我们的发现，即拓扑场论的许多结果可以从M理论中的某些引力背景中推导出来。这种令人惊讶的结构不仅解释了拓扑扭曲的概念，而且直接复制了边界Chern-Simons理论以及纽结不变量。然而，令人费解的是，给定的结点多项式的系数与某一微分方程解的个数一一对应。我的建议之一就是证明这个猜想。有色琼斯多项式和考夫曼多项式的构造等相关课题还需要进一步研究。