Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwo-dimensional conformal field theory, current-current deformation and mass formula
96
0
0.0
(
0
)
Authors
Yuto Moriwaki
Ask ChatGPT about the research
No Arabic abstract
The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of a full vertex algebra which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformation is expressed as a double coset of an orthogonal group, a quotient of an orthogonal Grassmannian. As an application, we consider a deformation of chiral conformal field theories, vertex operator algebras. A current-current deformation of a vertex operator algebra may produce new vertex operator algebras. We give a formula for counting the number of the isomorphic classes of vertex operator algebras obtained in this way. We demonstrate it for some holomorphic vertex operator algebra of central charge $24$.
rate research
Read More
A two-dimensional chiral conformal field theory can be viewed mathematically as the representation theory of its chiral algebra, a vertex operator algebra. Vertex operator algebras are especially well suited for studying logarithmic conformal field theory (in which correlation functions have logarithmic singularities arising from non-semisimple modules for the chiral algebra) because of the logarithmic tensor category theory of Huang, Lepowsky, and Zhang. In this paper, we study not-necessarily-semisimple or rigid braided tensor categories $mathcal{C}$ of modules for the fixed-point vertex operator subalgebra $V^G$ of a vertex operator (super)algebra $V$ with finite automorphism group $G$. The main results are that every $V^G$-module in $mathcal{C}$ with a unital and associative $V$-action is a direct sum of $g$-twisted $V$-modules for possibly several $gin G$, that the category of all such twisted $V$-modules is a braided $G$-crossed (super)category, and that the $G$-equivariantization of this braided $G$-crossed (super)category is braided tensor equivalent to the original category $mathcal{C}$ of $V^G$-modules. This generalizes results of Kirillov and M{u}ger proved using rigidity and semisimplicity. We also apply the main results to the orbifold rationality problem: whether $V^G$ is strongly rational if $V$ is strongly rational. We show that $V^G$ is indeed strongly rational if $V$ is strongly rational, $G$ is any finite automorphism group, and $V^G$ is $C_2$-cofinite.
We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.
We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduce two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov--Witten invariants) and quantum (with examples found in the works of Buryak--Rossi on integrable systems). We introduce a new version of Kontsevichs graph complex, enriched with tautological classes on the moduli spaces of stable curves. We use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs, by methods due to Merkulov--Willwacher. This group is shown to contain both the prounipotent Grothendieck--Teichmuller group and the Givental group.
Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.
For $mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $text{Aut}, mathcal O$-equivariant identification between $text{Fun},text{Op}_{mathfrak g}(D)$, the algebra of functions on the space of ${mathfrak g}$-opers on the disc, and $Wsubset pi_0$, the intersection of kernels of screenings inside a vacuum Fock module $pi_0$. This kernel $W$ is generated by two states: a conformal vector, and a state $delta_{-1}left|0right>$. We show that the latter endows $pi_0$ with a canonical notion of translation $T^{text{(aff)}}$, and use it to define the densities in $pi_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $text{Aut},mathcal O$-action defines a bundle $Pi$ over $mathbb P^1$ with fibre $pi_0$. We show that the product bundles $Pi otimes Omega^j$, where $Omega^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $ abla^{text{(aff)}} - alpha T^{text{(aff)}}$, $alphain mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[mathbf v_j dt^{j+1} ] in H^1(mathbb P^1, Piotimes Omega^j, abla^{text{(aff)}})$ of the de Rham cohomology of $ abla^{mathrm{(aff)}}$. Any choice of ${mathfrak g}$-Miura oper $chi$ gives a connection $ abla^{mathrm{(aff)}}_chi$ on $Omega^j$. Using coinvariants, we define a map $mathsf F_chi$ from sections of $Pi otimes Omega^j$ to sections of $Omega^j$. We show that $mathsf F_chi abla^{text{(aff)}} = abla^{text{(aff)}}_chi mathsf F_chi$, so that $mathsf F_chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(mathbb P^1, Omega^j, abla^{text{(aff)}}_chi)$ defined by the ${mathfrak g}$-oper underlying $chi$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
