There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. Some people told me that chiral algebras are $2$-dimensional analogue of associative algebras, which are considered to be $1$-dimensional. However, I don't understand its precise meaning since the definition of a vertex operator algebra is so complicated. Does the term chiral has something to do with this $2$-dimensionality?

For a vertex operator algebra $V$, Yongchang Zhu constructed an associative algebra $A(V)$ out of $V$, such that there is a bijection between the set of isomorphism classes of irreducible positive energy representations of $V$ and that of simple $A(V)$-modules. For an associative algebra $A$, Tomoyuki Arakawa calls $V$ to be the *chiralization* of $A$ if $A\simeq A(V)$ as associative algebras. What's the meaning of chiralization here?

There are some other explanations for the term chiral that I have ever heard. For example, in electromagnetism, chirality means the handedness of electromagnetic waves associated with their polarization. Some others also told me that in the $2$-dimensional setting, chiral means holomorphic.

I want to know the geometry/physics behind the term chiral. A philosophical answer is welcome, but a mathematical/physical answer is better.

kheri(χέρι) means hand, and chirality, referring to this in physics and chemistry, was coined by Lord Kelvin. $\endgroup$1more comment