arXiv subjects

Antun Milas

Publications and source records attributed to Antun Milas.

65 records · Page 4Linked to original sources

arXiv2003

Virasoro Algebra, Dedekind $η$-function and Specialized Macdonald's Identities

We motivate and prove a series of identities which form a generalization of the Euler's pentagonal number theorem, and are closely related to specialized Macdonald's identities for powers of the Dedekind $η$--function. More precisely, we show that what we call ``denominator formula'' for the Virasoro algebra has ``higher analogue'' for all $c_{s,t}$-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as {\em fusion} and {\em modular invariance} that require all the irreducible modules of the series. In particular, in the case of $c_{2,2k+1}$--minimal models we give a new proof of a family of specialized Macdonald's identities associated with twisted affine Lie algebras of type $A^{(2)}_{2k}, k \geq 2$ (i.e., $BC_k$-affine root system) which involve $(2k^2-k)$-th powers of the Dedekind $η$-function. Our paper is in many ways a continuation of math.QA/0309201.

math.QA↗

arXiv2003

Twisted modules for vertex operator algebras and Bernoulli polynomials

Using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call ``modified weak associativity.'' This paper is an announcement. The detailed proofs will appear elsewhere.

math.QA↗

arXiv2003

Formal differential operators, vertex operator algebras and zeta--values, I

We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $ζ$--functions at the negative integers. By using formal calculus techniques we discuss the appearance of values of $ζ$--functions at the negative integers underlying the construction. In addition we provide a conceptual explanation of this phenomena through several different notions of normal ordering via vertex operator algebra theory. We also derive a general Jacobi--type identity generalizing our previous construction. At the end we discuss related constructions associated to Dirichlet $L$--functions.

math.QA↗

arXiv2003

Formal differential operators, vertex operator algebras and zeta--values , II

We introduce certain correlation functions (graded $q$--traces) associated to vertex operator algebras and superalgebras which we refer to as $n$--point functions. These naturally arise in the studies of representations of Lie algebras of differential operators on the circle \cite{Le1}--\cite{Le2}, \cite{M}. We investigate their properties and consider the corresponding graded $q$--traces in parallel with the passage from genus 0 to genus 1 conformal field theory. By using the vertex operator algebra theory we analyze in detail correlation functions in some particular cases. We obtain elliptic transformation properties for $q$--traces and the corresponding $q$--difference equations. In particular, our construction leads to correlation functions and $q$--difference equations investigated by S. Bloch and A. Okounkov \cite{BO}.

math.QA↗

arXiv2002

Weak modules and logarithmic intertwining operators for vertex operator algebras

We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible logarithmic intertwiners at the level $c=0$.

math.QA↗

arXiv2002

Fusion rings for degenerate minimal models

We study fusion rings for degenerate minimal models ($p=q$ case) for N=0 and N=1 (super)conformal algebras. We consider a distinguished family of modules at the level $c=1$ and $c=3/2$ and show that the corresponding fusion rings are isomorphic to the representation rings for ${\mathfrak sl}(2, {\bf C})$ and ${\mathfrak osp}(1|2)$ respectively.

math.QA↗

arXiv2001

Vertex operator superalgebra structure for degenerate minimal models: Neveu-Schwarz algebra

The $\mathbb{Z}/2\mathbb{Z}$--graded intertwining operators are introduced. We study these operators in the case of ``degenerate'' N=1 minimal models, with the central charge $c=3/2$. The corresponding fusion ring is isomorphic to the Grothendieck ring for the Lie superalgebra ${\mathfrak osp}(1|2)$. We also discus multiplicity--2 fusion rules and logarithmic intertwiners.

math.QA↗

arXiv2000

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.

math.QA↗

arXiv1999

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, I

This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining operator superalgebras and vertex tensor categories in the N=1 case.

math.QA↗

arXiv1996

Tensor product of Vertex operator algebras

Let $V_1 \otimes V_2$ be a tensor product of VOAs. Using Zhu theory we discuss the theory of representations of V (associative algebra, modules and fusion rules). We prove that this theory is more or less the same as representation theory of tensor product of the associative algebras.

q-alg↗

arXiv1995

Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.

q-alg↗