# ConformalBlocks -- for vector bundles of conformal blocks on the moduli space of curves

## Description

Vector bundles of conformal blocks are vector bundles on the moduli stack of Deligne-Mumford stable n-pointed genus g curves Mg,n that arise in conformal field theory. Each triple (g,l,(λ1,...,λn)) with g a simple Lie algebra, l a nonnegative integer called the level, and 1,...,λn) an n-tuple of dominant integral weights of g specifies a conformal block bundle V=V(g,l,(λ1,...,λn)). This package computes ranks and first Chern classes of conformal block bundles on M0,n using formulas from Fakhruddin’s paper [Fakh].

Most of the functions are in this package are for Sn symmetric divisors and/or symmetrizations of divisors, but a few functions are included for non-symmetric divisors as well.

Some of the documentation nodes refer to books, papers, and preprints. Here is a link to the Bibliography.

Between versions 1.x and 2.0, the package was rewritten in a more object-oriented way, and the basic Lie algebra functions were moved into a separate package called LieTypes.

## Author

• Dave Swinarski

## Certification

Version 0.5 of this package was accepted for publication in volume 8 of the journal The Journal of Software for Algebra and Geometry on 2 August 2018, in the article Software for computing conformal block divisors on bar M_0,n. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/LieTypes.m2, commit number 923fbcc7c77b23f510bb0d740e00fc1722a2f397.

## Version

This documentation describes version 2.4 of ConformalBlocks.

## Source code

The source code from which this documentation is derived is in the file ConformalBlocks.m2.

## Exports

• Types
• ConformalBlockVectorBundle -- the class of conformal block vector bundles on the moduli space of n-pointed genus g curves
• SymmetricDivisorM0nbar -- the class of S_n symmetric divisors on the moduli space of stable n-pointed genus 0 curves
• Functions and commands
• Methods
• - SymmetricDivisorM0nbar -- negate a symmetric divisor
• coefficientList(SymmetricDivisorM0nbar), see coefficientList -- the coefficients of a symmetric divisor D in the standard basis
• conformalBlockDegreeM04bar(ConformalBlockVectorBundle), see conformalBlockDegreeM04bar -- computes the degree of a conformal block bundle on $\bar{M}_{0,4}$
• conformalBlockRank(ConformalBlockVectorBundle), see conformalBlockRank -- computes the rank of the conformal block vector bundle
• expression(ConformalBlockVectorBundle) (missing documentation)
• expression(SymmetricDivisorM0nbar) (missing documentation)
• FCurveDotConformalBlockDivisor(List,ConformalBlockVectorBundle), see FCurveDotConformalBlockDivisor -- intersection of an F-curve with a conformal block divisor
• isExtremalSymmetricFDivisor(SymmetricDivisorM0nbar), see isExtremalSymmetricFDivisor -- tests whether an S_n symmetric divisor spans an extremal ray of the cone of symmetric F-divisors
• isSymmetricFDivisor(SymmetricDivisorM0nbar), see isSymmetricFDivisor -- checks whether a symmetric divisor intersects all the F-curves nonnegatively
• killsCurves(SymmetricDivisorM0nbar), see killsCurves -- given an S_n symmetric divisor D, produces a list of symmetric F-curves C such that C dot D = 0
• net(ConformalBlockVectorBundle) (missing documentation)
• net(SymmetricDivisorM0nbar) (missing documentation)
• Number * SymmetricDivisorM0nbar -- multiply a symmetric divisor by a number
• scale(SymmetricDivisorM0nbar), see scale -- reduces a list or divisor by the gcd of its coefficients
• symmetricCurveDotDivisorM0nbar(List,SymmetricDivisorM0nbar), see symmetricCurveDotDivisorM0nbar -- the intersection number of a symmetric F-curve C with the symmetric divisor D
• SymmetricDivisorM0nbar + SymmetricDivisorM0nbar -- add two $S_n$ symmetric divisors
• SymmetricDivisorM0nbar == SymmetricDivisorM0nbar -- test equality of two symmetric divisor classes on $\bar{M}_{0,n}$
• symmetrizedConformalBlockDivisor(ConformalBlockVectorBundle), see symmetrizedConformalBlockDivisor -- computes the symmetrization of the first Chern class of a conformal block vector bundle