International Journal of Mathematics

This is the third part in a series of papers developing a twisted tensor product theory for nonlocal vertex algebras and its modules. In this paper we introduce and study twisted tensor product modules over Möbius twisted tensor product nonlocal vertex algebras. Among the main results, we find the isomorphic relation between the opposite Möbius twisted tensor product nonlocal vertex algebra and twisted tensor product of opposite Möbius nonlocal vertex algebras. And we also establish the isomorphism between two twisted tensor product contragredient modules. Furthermore, we study iterated twisted tensor product modules over iterated twisted tensor product nonlocal vertex algebras and find conditions for constructing an iterated twisted tensor product module of three factors.

By defining the Rund Laplacian, we obtain the first and the second holomorphic variation formulas for the strongly pseudoconvex complex Finsler metric. Using the holomorphic variation formulas, we get an estimate for the Levi forms of distance function on complex Finsler manifolds. Further, an estimate for the Rund Laplacians of distance function on strongly pseudoconvex complex Finsler manifolds is obtained. As its applications, we get the Bonnet theorem and maximum principle on complex Finsler manifolds.

We prove that any Kähler–Einstein metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.

We consider generalizations of commuting squares, called twisted commuting squares, obtained by having the commuting square orthogonality condition hold with respect to the inner product given by a faithful state on a finite-dimensional *-algebra. We present various examples of twisted commuting squares, most of which are computationally easy to work with, and we prove an isolation result. We also show how parametric families of (not necessarily) twisted commuting squares yield associative deformations of the matrix multiplication.

In this paper we study an integral equation of the first kind concerning an indirect boundary integral method for the Dirichlet problem in the theory of Cosserat continuum. Our method hinges on the theory of reducible operators and on the theory of differential forms.

Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson–Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count framed twisted quiver sheaves with the moment map relation and directly generalize the ADHM invariants of Diaconescu.

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with ℤ coefficients, of certain sutured manifolds of the form (Σ × S¹, F × S¹) where Σ is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with ℤ coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot's theorem that the contact invariant vanishes for a contact structure on (Σ × S¹, F × S¹) described by an isolating dividing set.

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra for the transition matrix A^G of the directed edges of G. The generalized Catalan numbers enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

Generalizing local Gromov–Witten theory, in this paper we define a local version of symplectic field theory. When the symplectic manifold with cylindrical ends is four-dimensional and the underlying simple curve is regular by automatic transversality, we establish a transversality result for all its multiple covers and discuss the resulting algebraic structures.