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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.

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    $\begingroup$ Surely chirality always refers to handedness, and mirror symmetry? The Greek kheri (χέρι) means hand, and chirality, referring to this in physics and chemistry, was coined by Lord Kelvin. $\endgroup$ Feb 28, 2023 at 9:20
  • $\begingroup$ @Estwald: I refer you to the following Wikipedia pages: https://en.wikipedia.org/wiki/Chirality#Mathematics and https://en.wikipedia.org/wiki/Chirality#Physics. $\endgroup$ Feb 28, 2023 at 9:21
  • $\begingroup$ @Carl-FredrikNybergBrodda Yes, that's the origin of the word chiral. But I want to know the mathematical meaning. $\endgroup$
    – Estwald
    Feb 28, 2023 at 9:23
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    $\begingroup$ @JoseArnaldoBebitaDris I've already read the wikipedia pages, but I think it's explaining in a naive way. More precisely, I want to know the meaning of chiral in the context of vertex operator algebras. In particular, I want to know why the Zhu's algebra is called chiralization. $\endgroup$
    – Estwald
    Feb 28, 2023 at 9:25
  • $\begingroup$ Since you seem to ask about the meaning of "chiral" in the very specific context of operator algebras, and to avoid too general answers and comments, I suggest to include this context in the title. E.g. "What is the meaning of 'chiral' in the context of operator algebras?". $\endgroup$
    – M. Winter
    Feb 28, 2023 at 11:25

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A vertex operator algebra describes the algebra of local operators in the chiral part of a 2d CFT. Typically one sees a VOA depending on a complex coordinate $z$. To describe a full 2d CFT, you would need to also include an "anti-chiral" VOA depending on a conjugate coordinate $\bar{z}$. So by considering only a single vertex algebra depending on one complex variable, you are only considering a "chiral half" of the CFT.

In physics more generally, people will often refer to "chiral algebras" of local operators in other types of theories of different dimensions as well, so in that field the terminology is quite broad, but relates to the chirality of a theory in a traditional sense.

Mathematicians have since extended and generalized a number of things relating to vertex algebras. For example Beilinson-Drinfeld chiral algebras generalise vertex algebras, the Chiral de Rham complex and chiral differential operators are VOA versions of differential forms and differential operators etc... In the past, constructions such as these were the only examples of mathematically well-defined and well-studied chiral algebras (in the physics sense of algebras of local operators). Hence in the mathematical community, I suspect it became practice to name constructions relating to vertex algebras and BD chiral algebras "chiral" since this was the only real example of a chiral algebra of local operators they were looking at in those communities.

In Arakawa's work, "chiralization" essentially means going from an associative algebra type object to a VOA type object. More generally, I believe chiralization/adding the word "chiral" to something in these communities will describe working with something like an affine or loopy version of a previously known construction.

For example, taking the Weyl algebra of differential operators, one can consider its VOA counter-part which is the $\beta\gamma$ VOA, also known as "chiral differential operators". The modes of this VOA satisfy similar relations to the Weyl algebra and the Weyl algebra can be recovered from it in various constructions.

I guess the point to emphasise though is that the word "chiral" itself is often not-literal in the mathematics community, but is rather describing something like "VOA-ization" (which in BD language is chiralization).

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Q: What is the geometry/physics behind the term chiral?

In the physics context, a Hamiltonian $H$ is said to possess chiral symmetry if it anticommutes with a unitary involution $C$. Eigenstates $\Psi_\pm$ of $C$ are said to be of left-handed or right-handed chirality, depending on the sign of the eigenvalue $\pm 1$. Chiral symmetry ensures that the dynamics generated by the Hamiltonian conserves the chirality.

In specific contexts the chirality is related to a spin degree of freedom, such that the particle moves parallel or antiparallel to the direction of the spin, depending on its chirality.

Q: What is the meaning of chiral in the context of vertex algebras?

A conformal field theory is called chiral if it contains only particles of one single chirality. It is my understanding that vertex algebras were introduced as an algebraic description of a chiral CFT. The connection with the physics concept of chirality seems to have been lost.

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