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Register a new userJacobi - type identities in algebras and superalgebras
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We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity, we derive a set of four identities (one of which is the Jacobi identity) represented in terms of double commutators and anticommutators. We establish that two of the four identities are independent and show that if the fundamental identity holds for an algebra, then the multiplication operation in that algebra is associative. We find a generalization of the obtained results to the super case and give a generalization of the fundamental identity in the case of arbitrary elements. For nondegenerate even symplectic (super)manifolds, we discuss analogues of the fundamental identity.
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We consider the variation of the surface spanned by closed strings in a spacetime manifold. Using the Nambu-Goto string action, we induce the geodesic surface equation, the geodesic surface deviation equation which yields a Jacobi field, and we define the index form of a geodesic surface as in the case of point particles to discuss conjugate strings on the geodesic surface.
Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton-Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern-Simons theory, where the HJ action turns out to be the gauged Wess-Zumino-Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin-Vilkovisky (BV) formalism in the bulk and of the Batalin-Fradkin-Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern-Simons theory and the toy model for 7D Chern-Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [arXiv:2012.13983]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.
In this paper we express some simple random tensor models in a Givental-like fashion i.e. as differential operators acting on a product of generic 1-Hermitian matrix models. Finally we derive Hirotas equations for these tensor models. Our decomposition is a first step towards integrability of such models.
We consider properties of the operators D(r,M)=a^r(a^dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^dag are boson annihilation and creation operators respectively, satisfying [a,a^dag]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different classes are constructed. Applications to quantum mechanical systems including supersymmetric quantum mechanics are discussed
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