Automorphism Groups Of Lattice Vertex Operator Superalgebras And Their Zhu Algebra

Lattice vertex operator superalgebra is the main research object,which belongs to the category of vertex operator algebra.In this paper,we give the series of notions of vertex operator superalgebra,including the automorphism group,various kinds of modules,the rationality and regularity and Zhu’s algebra.Then we construct the vertex operator superalgebra associated with an arbitrary positive definite integral lattice in detail and determine the automorphism group of the lattice vertex operator superalgebra by using the structure of the lattice vertex operator superalgebra,the conjugacy theorem of Lie algebra theory and central extension of the positive definite integral lattice,which is the product of the normal subgroup we induced by the inner derivations and the subgroup induced by the isometries of the positive definite integral lattice.Next,we introduce a class of certain associative algebras,and prove the Zhu’s algebras of the lattice vertex operator superalgebra is isomorphic to the certain associative algebra.Finally,we prove the regularity of lattice vertex operator superalgebras which implies its rationality.