Thomas H. Otway
A lemma from elliptic theory is used to improve a recent result by Li concerning the removability of an isolated point singularity from solutions of the coupled Yang-Mills-Dirac equations.
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Quantum Calogero-Moser spin system is a superintegable system with the spectrum of commuting Hamiltonians that can be described entirely in terms of representation theory of corresponding simple Lie group. In this paper the underlying Lie group G is a compact connected, simply connected simple Lie group. It has a natural generalization known as quantum Calogero-Moser spin chain. In the first part of the paper we show that quantum Calogero-Moser spin chain is a quantum superintegrable systems. Then we show that the Euclidean multi-time propagator for this model can be written as a partition fun…
We consider the Schramm-Loewner evolution (SLE$_κ$) with $κ=4$, the critical value of $κ> 0$ at or below which SLE$_κ$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($γ=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\m…
We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of General Relativity. Compactness is the essential tool of mathematical analysis for establishing validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with $L^p$ curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative ov…
In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated \emph{Yang-Mills heat flow}. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings. Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy…
A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the causality and stability principles. This paper presents a fully revised, simplified, and corrected version of the corresponding material in the previous papers DYNIN[11] and [12]. The principal result is established anew: due to the quartic self-interaction term in the Yang-Mills Lagrangian along with the semisimplicity of the gauge group, the quantum Yang-Mil…
Yangian symmetry of amplitudes in $\mathcal{N}=4$ super Yang-Mills theory is formulated in terms of eigenvalue relations for monodromy matrix operators. The Quantum Inverse Scattering Method provides the appropriate tools to treat the extended symmetry and to recover as its consequences many known features like cyclic and inversion symmetry, BCFW recursion, Inverse Soft Limit construction, Grassmannian integral representation, $\mathrm{R}$-invariants and on-shell diagram approach.
Deconfinement For $\mathrm{SO}(3)$ Lattice Yang-Mills at Strong Coupling
研究强耦合下SO(3)晶格杨-米尔斯理论是否满足威利森准则，证明其不满足，揭示了非平凡中心群对束缚态的影响。
Four-dimensional $\mathcal{N}=4$ super Yang-Mills theory admits interfaces that preserve integrability of the theory. It was shown that addition of fundamental hypermultiplets living on a co-dimension one defect is an example of such an interface. We consider NS5 interface as a different example of half-BPS defect in this theory and prove that integrability is generically broken already at one loop. We show that one-loop dilatation operator acting on boundary local operators does not lead to an integrable spin chain. The same is true also for D5-like and NS5-like boundaries.
Considering the $B$-branes over a complex manifold $Y$ as objects of the bounded derived category $D^b(Y)$, we define holomorphic gauge fields on $B$-branes and the Yang-Mills functional for these fields.These definitions are a generalization to $B$-branes of concepts that are well known in the context of vector bundles. Given ${\mathscr F}^{\bullet}\in D^b(Y)$, we show that the Atiyah class $a({\mathscr F}^{\bullet})\in{\rm Ext}^1({\mathscr F}^{\bullet},\,Ω^1({\mathscr F}^{\bullet}))$ is the obstruction to the existence of gauge fields on ${\mathscr F}^{\bullet}$. We determine the $B$-branes …
Yang-Mills theory with a symmetry algebra that is the semidirect product $\mathfrak{h}\ltimes\mathfrak{h}^*$ defined by the coadjoint action of a Lie algebra $\mathfrak{h}$ on its dual $\mathfrak{h}^*$ is studied. The gauge group is the semidirect product ${\rm G}_{\mathfrak{h}}\ltimes{\mathfrak{h}^*}$, a noncompact group given by the coadjoint action on $\mathfrak{h}^*$ of the Lie group ${\rm G}_{\mathfrak{h}}$ of $\mathfrak{h}$. For $\mathfrak{h}$ simple, a method to construct the self-antiself dual instantons of the theory and their gauge non\-equivalent deformations is presented. Every ${\…
A quadratic Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ],κ)$ gives rise to a canonical Yang-Mills type functional $S$ over every space-time manifold. The gauge fields consist of 1-forms $A$ taking values in $\mathbb{V}$ and 2-forms $B$ with values in the subspace $\mathbb{W} \subset \mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\equiv 0$, and $S$ becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz alg…
I describe how my involvement with monopoles related to the multimonopole existence proof of Taubes, and how my later work on quaternionic quantum mechanics led to the classification theorem for generalized projective group representations of Tao and Millard.
The resurgence of errors in the localization of $\mathcal{N} = 2$ superconformal Yang-Mills
本文研究了$\mathcal{N}=2$超共形杨-米尔斯理论在四维球面上的局部化误差问题，通过二维构造将不稳定瞬子与复奇点关联，基于chiral代数子部分并与Higgs分支局部化相一致。
We consider 5d Yang-Mills-Higgs theory with a compact ADE-type gauge group $G$ and one adjoint scalar field on $\mathbb{R}^{3,1}\times\mathbb{R}_+$, where $\mathbb{R}_+=[0,\infty)$ is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d ${\cal N}{=}\,(2,0)$ superconformal field theory on $\mathbb{R}^{3,1}\times\mathbb{R}_+\times S^1$ along the circle $S^1$. We show that in the low-energy limit, when momenta along $\mathbb{R}^{3,1}$ are much smaller than along $\mathbb{R}_+$, the 5d Yang-Mills-Higgs theory reduces to a n…
'Just another field theory' approach to $\mathcal{N}$=1 Super Yang-Mills and the origin of intrinsic SuperGeometry
本文提出 $\mathcal{N}$=1 Super Yang-Mills 的非几何推导，通过超空间一阶描述将理论表为含单一三次项的三阶多项式，避免无穷级数与特殊规范，几何解释自然涌现。
The instanton partition functions of $\mathcal{N}=1$ 5d super Yang-Mills are built using elements of the representation theory of quantum $\mathcal{W}_{1+\infty}$ algebra: Gaiotto state, intertwiner, vertex operator. This algebra is also known under the names of Ding-Iohara-Miki and quantum toroidal $\widehat{\mathfrak{gl}}(1)$ algebra. Exploiting the explicit action of the algebra on the partition function, we prove the regularity of the 5d qq-characters. These characters provide a solution to the Schwinger-Dyson equations, and they can also be interpreted as a quantum version of the Seiberg-…
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