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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 $$\langle \mathcal{O}(\tau_1) \dots \mathcal{O}(\tau_s) \rangle = \int \mathcal{D}z \mathcal{D}{\psi} \mathcal{D}{\overline{\psi}} \vert_P e^{-S(z, \psi, \overline{\psi})} \mathcal{O}(\tau_1) \dots \mathcal{O}(\tau_s).\tag{10.190}$$

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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.
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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.

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

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