We study the O(2N) model at criticality in three dimensions in the double scaling limit of large N and large charge. We show that the large-charge expansion is an asymptotic series, and we use resurgence techniques to study the non-perturbative corrections and to extend the validity of the effective field theory to any value of the charge. We conjecture the general form of the non-perturbative behavior of the conformal dimensions for any value of N and find very good agreement with previous lattice data.
Resurgence analysis of 2d Yang-Mills theory on a torus
研究了环面上二维$U(N)$ Yang-Mills理论配分函数的大$N$ t Hooft展开式.我们利用Kaneko和Zagier发现的递归关系，对二维Yang-Mills的手征配分函数和全配分函数进行了少许修改，计算了它们的$1/N$亏格展开式.然后我们研究了亏格展开的高阶行为，并利用复活关系提取了非微扰修正。结果表明，亏格展开不是波莱尔可和的，1-瞬子修正系数，即所谓的斯托克斯参数，是纯虚的。我们发现，从复苏获得的非微扰修正再现从一个非相对论性费米子系统的大配分函数的一个特定的解析延拓的圆。我们的解析延拓不同于hep-th/0504221中所考虑的延拓。
在本文中，基于作者在2015年5月的研讨会{ \it复苏，物理学和数字}，在比萨的Scuola Normale Monitoring的Centro di Ricerca Matematica Ennio De Giorgio，我们解释了Mathieu方程谱问题中复苏的trans-series的起源，使用均匀WKB和所有阶（精确）WKB。精确的量子化条件自然出现，它们在弱耦合区的扩展产生了复活的反级数表达式，这些表达式表现出不同瞬子扇区之间的精确关系。事实上，微扰膨胀编码的多瞬子膨胀，明确实现的一般概念“复苏”的所有订单。我们还讨论了从弱耦合到强耦合的过渡，一个明确的实现“瞬子凝聚”。
In Polyakov model, a non-perturbative mass gap is formed at leading order semi-classics by instanton effects. By using the notions of critical points at infinity, cluster expansion and Lefschetz thimbles, we show that a third order effect in semi-classics gives an imaginary ambiguous contribution to mass gap, which is supposed to be real and unambiguous. This is troublesome for the original analysis, and it is difficult to resolve this issue directly in QFT. However, we find a new compactification of Polyakov model to quantum mechanics, by using a background 't Hooft flux (or coupling to TQFT)…
通过一个非线性常微分方程给出了经历助推不变展开的均匀共形等离子体的二阶流体动力学描述，研究了其复苏渐近性质，进一步发展了Heller和Spalinski [Phys. Rev. Lett. 115，072501（2015）]。复苏清楚地确定了非流体动力学模式，在后期的指数抑制，类似于引力语言中的准正常模式，组织这些模式的trans-series扩展。这些模式类似于半经典展开中的瞬子，其中阻尼率扮演瞬子作用的角色。我们发现，该系统显示的通用功能的复苏，这些非流体动力学模式的不同订单的波动之间的明确的定量关系。反式级数参数的虚部由斯托克斯常数确定，实部由与初始条件相关的自由度确定。
In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in $1/N$, making the $1/N$ expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the $1/N$ expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first $1/N$ correction to the free energy in the n…
The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
本文利用费米子周期边界条件下的圆紧化方法，研究了所有规范群伴随费米子的四维规范理论的微扰和非微扰动力学.有新的规范现象。我们表明，在微扰理论的所有订单，许多规范群Higgsed的规范完整性周围的圆的产品的阿贝尔和nonabel规范群因子。在非微扰情况下，有费米子零模的夸克瞬子和两种夸克反夸克分子，称为生物子。一种类型是“磁生物子”，它携带净磁荷，并引起规范波动的质量间隙。另一种类型是磁中性的“中性生物子”，对它们的理解需要将量子力学中的多瞬子技术--我们称之为博戈莫尼-津恩-贾斯汀（BZJ）处方--推广到紧致化场论。应用于生物-反生物拓扑分子的BZJ处方预测了Borel平面的正实轴上的奇点（即，在微扰理论中，大阶求和的发散），它比领先的4-d BPST瞬子-反瞬子奇点更接近原点N倍，其中N是规范群的秩。的bion-反bion奇异的位置，从而定性类似的4-D IR renormalon奇异，我们猜想，它们是连续相关的紧化半径的变化。利用跨级数和Ecalle的复活理论，我们证明了一类允许半经典展开的场论的非微扰连续统定义是可能的。
Correlation functions of most composite operators decay exponentially with time at non-zero temperature, even in free field theories. This insight was recently codified in an OTH (operator thermalisation hypothesis). We reconsider an early example, with large $N$ free fields subjected to a singlet constraint. This study in dimensions $d>2$ motivates technical modifications of the original OTH to allow for generalised free fields. Furthermore, Huygens' principle, valid for wave equations only in even dimensions, leads to differences in thermalisation. It works straightforwardly when Huygens' pr…
We study the perturbative expansion of the ground state energy in the presence of an external field coupled to a conserved charge in the integrable two-dimensional O$(4)$ nonlinear sigma model. By solving Volin's algebraic equations for the perturbative coefficients we study the large order asymptotic behaviour of the perturbative series analytically. We confirm the previously numerically found leading behaviour and study the nearest singularities of the Borel transformed series and the associated alien derivatives. We find a 'resurgence' behaviour: the leading alien derivatives can be express…
We compute non-perturbative contributions to the Adler function, the derivative of the vacuum polarization function in gauge theory, using resurgence methods and Borel-summed gauge field propagators. At 2-loop, to order $1/N_f$, we construct the full 2-parameter transseries and perform the sum over the non-perturbative sectors. We then introduce a convolution-based method to derive the transseries structure of product series, which can also be used to study higher orders in the expansion in $1/N_f$. We compute 3-loop planar diagrams, at order $1/N_f^2$, and for each diagram study the asymptoti…
Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model
这项工作是一个非微扰连续定义的量子场论（QFT）的一步，开始与渐近自由的二维非线性σ模型，使用最近的想法从数学和QFT。来自数学的想法是复苏理论，trans-series框架和Borel-Ecalle重申。QFT的思想利用了R^1 x S^1_L上的连续性，也就是说，当N \to无穷大时没有任何相变，或者有限N的快速交叉，以及小L弱耦合极限，使得半经典扇区是明确定义的和可计算的。我们分类的半经典配置与行动1/N（扭结瞬子），2/N（bions和bi-kinks），在单位的2d瞬子行动归一化为1。微扰理论具有IR-重正马龙歧义，这是由于高阶微扰级数（Gevrey-1型）的非Borel求和性而产生的，其中微观抵消机制是未知的。这种发散必然存在，因为相应的展开是在复杂的耦合常数平面上的奇异Stokes射线上，并且总和表现出穿过射线的Stokes现象。我们发现，也有一个非微扰的模糊性固有的某些中性拓扑分子（中性生物和生物反生物）的半经典扩展。我们找到一组“汇合方程”，编码的两种不同类型的模糊度的精确取消。我们表明，一个新的概念“分级复苏三角形”是必要的，以捕捉复苏的路径积分方法，分级复苏的基础是一个潜在的严格定义的一般QFT。用第一性原理计算了真空能的质量间隙和θ角依赖关系，与大N和晶格结果一致。
We show that the (3+1)-dimensional gauged non-linear sigma model minimally coupled to a U(1) gauge field possesses analytic solutions representing gauged solitons at finite Baryon density whose electromagnetic field is a Force Free Plasma. These gauged solitons manifest a crystalline structure and generate in a very natural way persistent currents able to support Force Free Plasma electromagnetic fields. The trajectories of charged test particles moving within these configurations can be characterized. Quite surprisingly, despite the non-integrable nature of the theory, some of the perturbatio…
As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625) of strong-coupling expansion of non-planar corrections in $\mathcal N=2$ 4d superconformal models we consider two special theories with gauge groups $SU(N)$ and $Sp(2N)$. They contain $N$-independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding $\mathcal N=4$ SYM theories. These $\mathcal N=2$ theories can be realised on a system of $N$ D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particula…
We investigate the analytic properties of the fixed charge expansion for a number of conformal field theories in different space-time dimensions. The models investigated here are $O(N)$ and $QED_3$. We show that in $d=3-ε$ dimensions the contribution to the $O(N)$ fixed charge $Q$ conformal dimensions obtained in the double scaling limit of large charge and vanishing $ε$ is non-Borel summable, doubly factorial divergent, and with order $\sqrt{Q}$ optimal truncation order. By using resurgence techniques we show that the singularities in the Borel plane are related to worldline instantons that w…
Perturbative expansions in quantum field theory diverge for at least two reasons: the number of Feynman diagrams increases dramatically with the loop number and the process of renormalization may make the contribution of some diagrams large. We give an example of the second problem, from an ultra-violent renormalon of $ϕ^3$ theory in 6 dimensions, where we can compute to very high loop-order. Taming this renormalon involves recent work on resurgence. This challenge is much more demanding than the corresponding problem for Yukawa theory in 4 dimensions.
