# Central charge Many interacting gapless quantum systems (including CFTs) do not have simple particle-like excitations, so it is hard to quantify the effective number of degrees of freedom. Central charge is useful in this regard as it plays the role of estimating the effective number of degrees of freedom. The central charge is an integer for free bosonic theories and a half-integer for free fermionic theories. Each free boson contributes 1 and each (Majorana-Weyl) fermion contributes $\frac{1}{2}$. Interacting theories have non-(half-)integer central charges. In a 2d CFT, the central charge can be defined using the stress tensor [[0030 Operator product expansion|OPE]] which takes the form $T(z) T(w)=\frac{c / 2}{(z-w)^{4}}+\frac{2 T(w)}{(z-w)^{2}}+\frac{\partial T(w)}{z-w}+\cdots.$The number $c$ that appears in the residue of the 4th order pole is then defined to be the central charge. One can define $\bar{c}$ analogously using the $\bar{T}\bar{T}$ OPE. ## See also - [[0032 Virasoro algebra]] - [[0572 Effective central charge]]