Gauge theory deformations and novel Yang-Mills Chern-Simons field theories with torsion
Stephen C. Anco
备注：
Published version (misprints corrected and additional references included) in the special issue of IJGMMP, "Advanced Geometric Techniques in Gauge Theory"
A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories.
The first part of this paper summarizes and significantly elaborates a field- theoretic deformation method developed in earlier work. As a key contribution presented here, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang-Mills theory, Freedman-Townsend theory, Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years.
The second part of the paper presents a new geometrical type of Yang-Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang-Mills Chern-Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual 2+2 dimensional Yang-Mills theory under reduction to 2+1 dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.
We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.
Geometric quantization of 2+1 dimensional pure Yang-Mills theory is studied with focusing on finite large scales. It is previously shown that (Yildirim, 2015, Int. J. Mod. Phys A, 30(7), 1550034), topologically massive Yang-Mills theory exhibits a Chern-Simons splitting behavior at large scales, similar to the topologically massive AdS gravity model in its near Chern-Simons limit. This splitting occurs as two Chern-Simons parts with levels $+k/2$ each, where $k$ is the original Chern-Simons level in the Lagrangian. In the pure Chern-Simons limit, split parts combine to give the original Chern-…
The Frenet equation governs the extrinsic geometry of a string in three-dimensional ambient space in terms of the curvature and torsion, which are both scalar functions under string reparameterisations. The description engages a local SO(2) gauge symmetry, which emerges from the invariance of the extrinsic string geometry under local frame rotations around the tangent vector. Here we inquire how to construct the most general SO(2) gauge invariant Hamiltonian of strings, in terms of the curvature and torsion. The construction instructs us to introduce a long-range (self-) interaction between st…
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions generalize that of the pure Yang-Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of dimensions two, three and four and due to Rade (1992) for dimensions two and thre…
Topologically massive Yang-Mills theory is studied in the framework of geometric quantization. Since this theory has a mass gap proportional to the topological mass m, Yang-Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern-Simons theory with level number k. In this paper, the near Chern-Simons limit is studied where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of two copies of Chern-Simons with level number k/2, v…
Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
利用二维Yang-Mills与布朗运动之间的关系，证明了在一类透镜空间中，柱体上的二维Yang-Mills与Chern-Simons理论之间的关系.或者，这可以被看作是计算共形场论中的某些标量的2dYM。我们发现，对于特定的YM耦合值但有限的k和N，2dYM的配分函数约化为Chern-Simons理论中的U=ST^pS型算子. U是一个算子，从中可以得到S^3/Z_p上的陈-西蒙斯配分函数，以及威尔逊圈的期望值。对应关系涉及杨-米尔斯耦合的虚部是一个有理数，可以看作是陈-西蒙斯/WZW理论和拓扑2dYM的Witten，和Blau ant Thompson之间的关系的推广。本重新制定的一些性质的2dYM的圆柱明确。特别是，我们表明，模变换性质的配分函数是密切相关的仿射字符。
In this thesis, topologically massive Yang-Mills theory is studied in the framework of geometric quantization. This theory has a mass gap that is proportional to the topological mass $m$. Thus, Yang-Mills contribution decays exponentially at very large distances compared to $1/m$, leaving a pure Chern-Simons theory with level number $k$. The focus of this research is the $near$ Chern-Simons limit of the theory, where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of …
We determine the dimension of the moduli space of non-Abelian vortices in Yang-Mills-Chern-Simons-Higgs theory in 2+1 dimensions for gauge groups $G=({\rm U}(1)\times G')/\mathbb{Z}_{n_0}$ with $G'$ being an arbitrary semi-simple group and $n_0$ the greatest common divisor of the Abelian charges of the $G'$ invariants. The calculation is carried out using a Callias-type index theorem, the moduli matrix approach and a D-brane setup in Type IIB string theory. We prove that the index theorem gives the number of zeromodes or moduli of the non-Abelian vortices, extend the moduli matrix approach to …
The Yang-Mills (YM) equation in three spacetime dimensions (3D) can be modified to include a novel parity-preserving interaction term, with inverse mass parameter, in addition to a possible topological mass term. The novelty is that the modified YM equation is not the Euler-Lagrange equation of any gauge-invariant local action for the YM gauge potential alone. Instead, consistency is achieved in the "third way" exploited by 3D "minimal massive gravity". We relate our results to the "novel Higgs mechanism" for Chern-Simons gauge theories.
Geometric quantization of topologically massive and pure Yang- Mills theories is studied in 2+1 dimensions. Analogous to the topologically massive AdS gravity model, both topologically massive Yang-Mills and pure Yang-Mills theories are shown to exhibit a Chern-Simons splitting behavior at large distances. It is also shown that, this large distance topological behavior can be further used to incorporate link invariants.
In this article we establish the notion of classical Yangian symmetry for planar N=4 supersymmetric Yang-Mills theory and for related planar gauge theories. After revisiting Yangian invariance for the equations of motion, we describe how the bi-local generators act on the action of the model such that the latter becomes exactly invariant. In particular, we elaborate on the relevance of the planar limit and how to act non-linearly with bi-local generators on the cyclic action.
For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Lojasiewicz-Simon gradient inequality for abstract Morse-Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis in arXiv:1510.03817. We apply this result to prove the optimal Lojasiewicz-Si…
We provide an elegant homological construction of the extended phase space for linear Yang-Mills theory on an oriented and time-oriented Lorentzian manifold $M$ with a time-like boundary $\partial M$ that was proposed by Donnelly and Freidel [JHEP 1609, 102 (2016)]. This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern-Simons theory, in which case we obtain the extended phase space introduced by Geiller [Nucl. Phys. B 924, 312 (2017)].
