# In 2d CFT, if the partition function is $Z=\text{Tr}q^{L_0-c/24}$, where is the analogue of temperature?

In 2d CFT, restricting to the holomorphic part of the theory, the torus partition function is given by $$Z(q)=\text{Tr}q^{L_0-c/24}$$ where $$L_0$$ is the element of the Virasoro algebra, i.e. mode of energy-momentum tensor, $$c$$ is the central charge, and the trace is over the Hilbert space in radial quantization.

This is analogous to the partition function in the statistical mechanics $$Z(q)=\text{Tr}e^{-\beta H}$$ where $$\beta=1/T$$ and $$H$$ is the Hamiltonian.

I can see that the $$L_0$$ is can be associated with the Hamiltonian in the statistical mechanics. However, what does the temperature (i.e. $$\beta$$) in statistical mechanics correspond to in the 2d CFT?

Should I think of it as a partition function with external thermal bath that sets temperature fixed?

• The $q=e^{2\pi i \tau}$ insert the analogue of $\beta=\frac{1}{T}$ as well as a "twisting" parameter since $\tau$ is a complex number. Dec 9, 2021 at 20:36
• Exactly, the temperature is part of $\tau$. Also, keep in mind that most theories don't holomorphically factorize. Dec 9, 2021 at 20:43
• @Nogueira Thank you for your answer. I understand that $\tau$ is $\beta$ here. Do you mean twisting by 'multiplied by $i$'? Or do you mean 'complex temperature'?
– Nugi
Dec 9, 2021 at 21:24
• @ConnorBehan Thank you. So the inverse temperature is the complex structure, i.e. magnitude of momentum (or distance in momentum space) is temperature? Also, I am thinking about four-dimensional theories. (More specifically, 4d $\mathcal{N}=2 chiral algebras constructed by Beem, Rastelli, et.al.) Except the even dimensions, what other information can we immediately use to determine criteria of holomorphic factorization? – Nugi Dec 9, 2021 at 21:28 • @Nugi$\tau$is the modular parameter of the torus. The torus basically compute the partition function of the CFT in a$S^{1}$, where the "size" of the torus is the inverse of the temperature. The modular parameter is complex is because we can also twist the$S^{1}$along the torus, i.e. rotate the$S^{1}$by acting with the generator$L_0-\overline L_0\$. Dec 9, 2021 at 21:50