Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. We study the Generalized Gibbs Ensemble with chemical potentials for these charges at high temperature. In a large central charge limit, the partition function can be computed in a saddle-point approximation. We compare the ensemble values of the KdV charges to the values in a microstate, and find that they match irrespective of the values of the chemical potentials. We study the partition function at finite central charge perturbatively in the chemical potentials, and find that this degeneracy is broken. We also study the statistics of the KdV charges at high level within a Virasoro representation, and find that they are sharply peaked.

Article funded by SCOAP3

1$ CFTs, where the Virasoro representations are (except for the vacuum representation) Verma modules. The Verma module character is \begin{equation}\label{chiV} \chi_{h}=\frac{q^{h-k}}{\prod_{n=1}^{\infty}(1-q^n)}, \end{equation} We will denote the thermal expectation value of a KdV charge within this representation as \be \langle I_{2m_1-1}\dots I_{2m_n-1} \rangle_h \equiv {\rm Tr}_h \left[ q^{L_0} I_{2m_1-1}\dots I_{2m_n-1}\right] \ee These can be computed as differential operators applied to the character~\eqref{chiV}. For example, \begin{equation} \label{exppa} \langle I_3 \rangle_h = \left[ \partial^2 -{1\over 6} E_2 \partial +{k \over 60} E_4\right] \chi_h, \end{equation} where $\partial = q \partial_q$ and $E_{2n}$ is an Eisenstein series. We refer to appendix C of~\cite{paper1} for a more complete list of differential operators. It is easy to evaluate the derivative of the character~\eqref{chiV}: \begin{equation} \partial \frac{1}{\prod_{n=1}^{\infty}(1-q^n)} = \frac{1}{\prod_{n=1}^{\infty}(1-q^n)} \sum_{n=1}^\infty \frac{ n q^n}{1-q^n} = \frac{1}{\prod_{n=1}^{\infty}(1-q^n)} \frac{1}{24} (1-E_2), \end{equation} so that \begin{equation} \label{expp} \partial \chi_{h}= \left[ h-k + \frac{1}{24} - \frac{E_2}{24} \right] \chi_h = \left[\tilde h - \frac{E_2}{24} \right] \chi_h, \end{equation} Here for future convenience we have introduced a shifted level $\tilde h \equiv h-k+\frac{1}{24}$ to simplify our formulae. The result is \begin{equation} \label{I3exp} \langle I_3 \rangle_h = \left[ \tilde h^2 - \frac{1}{4} \tilde h E_2 + \frac{E_2^2}{192} + \left( \frac{k}{60} + \frac{1}{288} \right) E_4 \right] \chi_h. \end{equation} Under modular transformations, we have $E_4(-1/\tau) = \tau^4 E_4(\tau)$, $E_6(-1/\tau) = \tau^6 E_6(\tau)$, $E_2(-1/\tau) = \tau^2 E_2(\tau) + \frac{6 \tau}{\pi i}$, which give the high temperature behaviour \begin{equation} \label{eiss} E_2 \approx - \left( \frac{2\pi}{\beta} \right)^2 + \frac{12}{\beta}, \quad E_4 \approx \left( \frac{2\pi}{\beta} \right)^4, \quad E_6 \approx - \left( \frac{2\pi}{\beta} \right)^6, \end{equation} up to exponentially suppressed corrections. Using~\eqref{eiss} in~\eqref{I3exp}, we find \begin{equation} \label{I3h} \frac{\langle I_3 \rangle_h}{\chi} = \tilde h^2 + \frac{\pi^2}{\beta^2} \tilde h - \frac{3 \tilde h}{\beta} + \frac{(25+48k) \pi^4}{180 \beta^4} - \frac{\pi^2}{2\beta^3} + \frac{3}{4 \beta^2} + \dots \end{equation} where the $\dots$ denote terms which are exponentially suppressed at high temperature. We note that if we take $\beta\to 0$ while holding $\tilde h$ fixed this goes like $\beta^4$, just like the result in a full CFT~\eqref{I3high}. But the coefficient is different, and is suppressed by one order of the central charge. We can nevertheless reproduce the correct result for the full partition function by summing this over all representations in the theory: \begin{equation} \langle I_3 \rangle= \int dh \left(\frac{1}{\sqrt{2(h-k+\frac{1}{24})}} e^{2\pi \sqrt{\frac{(24k-1)}{6}(h-k+\frac{1}{24})} } \right)\langle I_3 \rangle_h, \end{equation} where the expression in parenthesis is the Cardy formula for the density of states of primary operators of dimension $h$ in a CFT with $c>1$. This density of states can be derived by looking at the modular transformation properties of the partition function which counts primary states (note that we have been careful to keep the power-law correction to the usual exponential factor). At high temperature the character is \begin{equation} \label{chihighT} \chi(q) \approx e^{\frac{\pi^2}{6 \beta} - \beta \tilde h} \sqrt{ \frac{\beta}{2\pi}}, \end{equation} up to exponentially suppressed corrections. Evaluating the integral at the saddle point (which is at $h_\star=(k-1/24)\frac{(2\pi)^2}{\beta^2}$) gives \begin{equation} \langle I_3 \rangle/Z \approx \frac{(2\pi)^4}{\beta^4} k \left( k + \frac{11}{60} \right) + \dots. \end{equation} exactly reproducing our previous result. We note that it is necessary to carefully keep track of the subleading terms in the saddle point analysis in order to see that all of the potential power law corrections cancel and that the $\dots$ in this formula are indeed exponentially suppressed in $\beta$. It is important to note that the order $k^2$ part of this result, which dominates at large central charge, comes just from the ${\cal O}(h^2)$ contribution in $\langle I_3\rangle_h$. This is consistent with the expectation that the contribution from the conformal dimension of the primary determines the high temperature behaviour at large central charge. For the higher correlation functions, we find cancellations in the connected correlation functions, just as in the previous section. Using the differential operator for $\langle I_3^2\rangle_h$ from~\cite{paper1} we have \begin{equation} \begin{aligned} \label{I3sqexp} \langle I_3^2 \rangle_h &= \bigg[ \tilde h^4 - \frac{1}{2} \tilde h^3 E_2 - \frac{5}{96} \tilde h^2 E_2^2 + \frac{ 24k+95}{720} \tilde h^2 E_4 + \frac{25}{1152} \tilde h E_2^3 - \frac{(7+24k)}{360} \tilde h E_6\\&\qquad + \frac{168k-19}{2880} \tilde h E_2 E_4 -\frac{5}{12288}E_2^4-\frac{19+120k}{46080}E_2^2E_4\\&\qquad +\frac{(19536 k (12 k+5)+12425)}{14515200}E_4^2+\frac{(7-24 k (120 k+29))}{181440}E_2 E_6 \bigg] \chi_h. \end{aligned} \end{equation} Evaluating this expression in the high temperature limit using~\eqref{eiss}, there will be a contribution which goes like $\beta^{-8}$ from the terms in the second line. When we consider the connected correlation function, there are cancellations. If we consider first finite temperature, the expression is most cleanly given in terms of derivatives of the Eisenstein series, \begin{eqnarray} \frac{\langle I_3^2 \rangle_h}{\chi} - \frac{\langle I_3 \rangle^2_h}{\chi^2} &=& - \frac{3}{2} \partial E_2 \tilde h^2 + \left( \frac{7}{4} \partial^2 E_2 + \frac{48k+49}{240} \partial E_4 \right) \tilde h\\\nonumber&& - \frac{1}{8} \partial^3 E_2 - \frac{(211+192k)}{9600} \partial^2 E_4 - \frac{(1883+6816k+11520 k^2)}{362880} \partial E_6. \end{eqnarray} Evaluating this expression in the high temperature limit, the most divergent term comes from the last term, $\partial E_6 \sim \beta^{-7}$, so \begin{equation} \frac{\langle I_3^2 \rangle_h}{\chi} - \frac{\langle I_3 \rangle^2_h}{\chi^2} = \frac{(1883 + 6816 k + 11520 k^2) \pi^6}{945 \beta^7} + \dots, \end{equation} where the subleading terms include subleading powers of $\beta$, which we can determine as in~\eqref{I3h}, but have not written explicitly for simplicity. We see that the connected correlation function is suppressed by one power of $\beta$ relative to the full correlation function, just as in the previous section when we computed correlation functions in the full CFT. Similarly, $\langle I_3^3 \rangle_h$ involves contributions going like $\beta^{-12}$, but in the connected correlation functions the first two orders cancel, giving \begin{equation} \frac{\langle I_3^3 \rangle_h}{\chi} - 3 \frac{\langle I_3 \rangle_h \langle I_3^2 \rangle_h}{\chi^2} + 2 \frac{\langle I_3 \rangle_h^3}{\chi^3} = \frac{(25925 + 135504 k + 407808 k^2 + 552960 k^3) \pi^8}{225 \beta^{10}} + \dots \end{equation} where again the subleading terms include subleading powers of $\beta$. Our conclusion is that the distribution of eigenvalues of the KdV charges is sharply peaked, with the variance and higher cumulants being suppressed by powers of the temperature relative to the mean. This is a novel result: at low temperatures the correlation functions factorize and the primary state dominates because the KdV charges are simply powers of $L_0$. But at high temperatures the character is dominated by descendants with large level (much larger than $h$). In this limit the $L_0^m$ term scales like $1/\beta^{2m}$ with a numerical ($k$-independent) coefficient. The $k$-dependent part of the expectation value comes from the other terms in the KdV charge, making the cancellations in the connected correlators (and hence the fact that the distribution of KdV charges is sharply peaked) non-trivial. Finally, we note that we are calculating here a thermal average over all the states in the Verma module, but at high temperature the calculation is dominated by a narrow range of levels with $n \approx \frac{\pi^2}{6 \beta^2}$. This is the usual equivalence between canonical and microcanonical ensembles at high temperature. In fact, the statistics of KdV charges at a particular level can be computed exactly (see section 7 of~\cite{paper1}); the results agree with the canonical ensemble computation described here. \paragraph{Conclusions.} We have studied the structure of the Generalised Gibbs ensemble with chemical potentials for the KdV charges in the high temperature limit, and the comparison of the expectation values of the KdV charges in this ensemble to their values in a particular microstate. We found that in the large central charge limit, the ensemble partition function could be obtained from a saddle-point approximation, and the expectation values of the KdV charges match those of a primary state. The conformal dimension of this primary state depends on the temperature and chemical potentials, but for given conformal dimension of the primary there is a temperature for which the expectation values agree for any values of the chemical potentials. We made some steps towards calculating the partition function at finite central charge, perturbatively in the chemical potentials. At finite central charge matching the expectation values of the KdV charges in a particular microstate is expected to fix the chemical potentials. To determine the appropriate values, we would need to understand the structure of the coefficients in the expansion of the partition function to all orders in the chemical potentials; we were only able to calculate the first few terms in the expansion. If one could find values of the chemical potentials which matched the KdV charges in a particular microstate, one could then further explore ETH for two-dimensional CFTs by comparing the expectation values of other simple operators, in the microstate and in the ensemble with these values of the chemical potentials. ]]>