# Vanishing correlation function

Mirror Symmetry p. 206, Eq. 10.192.

I have an operator $$\mathcal{O}$$ that commutes with my supercharge $$\overline{Q}_+$$, $$\left[\overline{Q}_+, \mathcal{O} \right]=0$$. Why does the correlation function vanish? $$\left< \{\overline{Q}_+, \psi^i\} \mathcal{O} \right> = 0\tag{10.192}$$ Where $$\psi^i$$ is a fermionic variable and $$\langle \mathcal{O}(\tau_1) \dots \mathcal{O}(\tau_s) \rangle = \int \mathcal{D}z \mathcal{D}{\psi} \mathcal{D}{\overline{\psi}} \vert_P e^{-S(z, \psi, \overline{\psi})} \mathcal{O}(\tau_1) \dots \mathcal{O}(\tau_s).\tag{10.190}$$

## 2 Answers

Eq. (10.192) follows because the supercharge $$\overline{Q}_+$$

• supercommutes with the operator $${\cal O}$$;
• annihilates the (implicitly written) vacuum ket and bra in the operator formalism.

Looking at equation 10.191, 10.192 was just the operator multiplication and taking the expectation value, and equation 10.191 was given and derived from the previous cases.

• Thanks for your answer, but i do not see how this answers my question. Why is the expectation value equal to zero? May 24, 2019 at 12:08