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Let $A$ be a graded vector space, and suppose that two differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.

We now construct $C(A)$, the "chiral ring" of $A$, defined as follows: Let $A^\prime$ be the intersection of $\textrm{Ker}(d_1)$ and $\textrm{Ker}(d_2)$ inside $A$. That is, it is the space of vectors that are annihilated by both differentials. $C(A)$ is then a particular quotient of $A^\prime$. Then, we take its quotient by $\textrm{Im}(d_1)$ and $\textrm{Im}(d_2)$. Because space may not be a subspace of $A^\prime$, the quotient is taken by the intersection of ($\textrm{Im}(d_1)\cup \textrm{Im}(d_1)$) with $A^\prime$. The definition does not involve the commutation relations between $d_1$ and $d_2$. Also, the chiral ring $C(A)$ is NOT the cohomology of $A$, with respect to any differential.

If $A$ is endowed with a ring structure (commutative differential graded algebra), $A^\prime$ is obviously a subalgebra. So the product of two chiral operators is chiral. It remains to check whether or not the quotient is by an ideal in A'. Since the kernel of the quotient $A^\prime \to C(A)$ is an ideal, $C(A)$ inherits a multiplication from $A$. The chiral ring is a ring.

Question: Is this construction well-known in the mathematical literature, and does it have a name?

In physics, the chiral ring is defined in this way (see the paper for a nice review). In supersymmetric gauge theories, chiral operators are particularly interesting because a correlation function of chiral operators is independent of positions of the operators. I wonder if mathematicians study properties of a chiral ring.

Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.

We now construct $C(A)$, the "chiral ring" of $A$, defined as follows: Let $A^\prime$ be the intersection of $\textrm{Ker}(d_1)$ and $\textrm{Ker}(d_2)$ inside $A$. That is, it is the space of vectors that are annihilated by both differentials. $C(A)$ is then a particular quotient of $A^\prime$. Then, we take its quotient by $\textrm{Im}(d_1)$ and $\textrm{Im}(d_2)$. Because space may not be a subspace of $A^\prime$, the quotient is taken by the intersection of ($\textrm{Im}(d_1)\cup \textrm{Im}(d_1)$) with $A^\prime$. In fact, the chiral ring $C(A)$ is NOT the cohomology of $A$, with respect to any differential.

If $A$ is endowed with a ring structure (commutative differential graded algebra), $A^\prime$ is obviously a subalgebra. So the product of two chiral operators is chiral. It remains to check whether or not the quotient is by an ideal in $A^\prime$. Since the kernel of the quotient $A^\prime \to C(A)$ is an ideal, $C(A)$ inherits a multiplication from $A$. The chiral ring is a ring.

Question: Is this construction well-known in the mathematical literature, and does it have a name?

In physics, the chiral ring is defined in this way (see the paper for a nice review). In supersymmetric gauge theories, chiral operators are particularly interesting because a correlation function of chiral operators is independent of positions of the operators. I wonder if mathematicians study properties of a chiral ring.

Let $A$ be a graded vector space, and suppose that two differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.

We now construct $C(A)$, the "chiral ring" of $A$, defined as follows: Let $A^\prime$ be the intersection of $\textrm{Ker}(d_1)$ and $\textrm{Ker}(d_2)$ inside $A$. That is, it is the space of vectors that are annihilated by both differentials. $C(A)$ is then a particular quotient of $A^\prime$. Then, we take its quotient by $\textrm{Im}(d_1)$ and $\textrm{Im}(d_2)$. Because space may not be a subspace of $A^\prime$, the quotient is taken by the intersection of ($\textrm{Im}(d_1)\cup \textrm{Im}(d_1)$) with $A^\prime$. The definition does not involve the commutation relations between $d_1$ and $d_2$. Also, the chiral ring $C(A)$ is NOT the cohomology of $A$, with respect to any differential.

If $A$ is endowed with a ring structure (commutative differential graded algebra), $A^\prime$ is obviously a subalgebra. So the product of two chiral operators is chiral. It remains to check whether or not the quotient is by an ideal in A'. Since the kernel of the quotient $A^\prime \to C(A)$ is an ideal, $C(A)$ therefore inherits a multiplication from $A$. The chiral ring is a ring.

Question: Is this construction well-known in the mathematical literature, and does it have a name?

In physics, the chiral ring is defined in this way (see the paper for a nice review). In supersymmetric gauge theories, chiral operators are particularly interesting because a correlation function of chiral operators is independent of positions of the operators. I wonder if mathematicians study properties of a chiral ring.

Let $A$ be a graded vector space, and suppose that two differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex.

We now construct $C(A)$, the "chiral ring" of $A$, defined as follows: Let $A^\prime$ be the intersection of $\textrm{Ker}(d_1)$ and $\textrm{Ker}(d_2)$ inside $A$. That is, it is the space of vectors that are annihilated by both differentials. $C(A)$ is then a particular quotient of $A^\prime$. Then, we take its quotient by $\textrm{Im}(d_1)$ and $\textrm{Im}(d_2)$. Because space may not be a subspace of $A^\prime$, the quotient is taken by the intersection of ($\textrm{Im}(d_1)\cup \textrm{Im}(d_1)$) with $A^\prime$. The definition does not involve the commutation relations between $d_1$ and $d_2$. Also, the chiral ring $C(A)$ is NOT the cohomology of $A$, with respect to any differential.

If $A$ is endowed with a ring structure (commutative differential graded algebra), $A^\prime$ is obviously a subalgebra. So the product of two chiral operators is chiral. It remains to check whether or not the quotient is by an ideal in A'. Since the kernel of the quotient $A^\prime \to C(A)$ is an ideal, $C(A)$ inherits a multiplication from $A$. The chiral ring is a ring.

Question: Is this construction well-known in the mathematical literature, and does it have a name?

In physics, the chiral ring is defined in this way (see the paper for a nice review). In supersymmetric gauge theories, chiral operators are particularly interesting because a correlation function of chiral operators is independent of positions of the operators. I wonder if mathematicians study properties of a chiral ring.