Nigel Hitchin
In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry. A Hamiltonian flow arising from this corresponds to a version of the Nahm equations using the Courant bracket, and we investigate the six-dimensional geometrical structure this describes.
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The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of a functional. Both of these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither this Laplacian nor the anti-bracket is well-defined for sections of an arbitrary vector bundle. We show that if the functional…
Invariant Reduction for Partial Differential Equations. III: Poisson Brackets
本文研究了偏微分方程有限维系统在不变性缩减下的局部哈密顿算子继承机制，展示了其在点、接触及更高对称性下的统一应用，并通过实例证明继承的泊松括号与守恒量之间的关系。
How to give a natural geometric definition of a covariant Poisson bracket in classical field theory has for a long time been an open problem - as testified by the extensive literature on "multisymplectic Poisson brackets", together with the fact that all these proposals suffer from serious defects. On the other hand, the functional approach does provide a good candidate which has come to be known as the Peierls - De Witt bracket and whose construction in a geometrical setting is now well understood. Here, we show how the basic "multisymplectic Poisson bracket" already proposed in the 1970s can…
We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.
On Gauge Poisson Brackets with Prescribed Symmetry
该研究在平均法框架下，提出从紧李群在纤维相空间的典型作用重构依赖不变联络的泊松结构的方法，并据此推导出Wong型方程的若干对称性质。
It is shown that the new formula for the field theory Poisson brackets arise naturally in the extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect to divergences. The bilinear operations, such as the action of vector fields on functionals, the commutator of vector fields, the interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator …
Rankin-Cohen brackets for orthogonal Lie algebras and bilinear conformally invariant differential operators
本文利用代数傅里叶变换的李理论方法，分类了正交李代数情形下标量广义Verma模对角分支规则中的奇异向量，并应用于分类共形球面上的双线性共形不变微分算子。
Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $ω_1|_{X^r_\mathfrak{h}}$ and $ω_2|_{X^r_\mathfrak{h}}$ generates an $…
The Chern-Simons Functional Integral, Kauffman's Bracket Polynomial, and other link invariants
本文在轴向规范下研究Chern-Simons理论，通过显式公式计算Wilson环期望值并推导 skein 关系，证明其与Kauffman括号及Jones多项式紧密相关，且耦合常数无需量子化，并推广到高秩群和非紧致群，同时展示二维情形可导出Goldman括号。
It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect to divergences. Bilinear operations, such as action of vector fields onto functionals, commutator of vector fields, interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition o…
We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer delta-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems.
A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antib…
In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector bundle. We consider those automorphisms of the vector bundle which induce mappings on the space of functionals preserving the Poisson bracket and refer to such automorphisms as canonical automorphisms. We determine how such automorphisms relate to the corresponding sh-Lie structu…
We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant. We define a corresponding notion of a Nambu-Lie algebra. We give several examples of Nambu-Lie groups and algebras.
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