The Virasoro Algebra and the Stress Tensor

The holomorphic stress tensor generates local conformal transformations, but its operator product contains a central term that has no classical Witt-algebra counterpart. Contour integration turns that OPE into the Virasoro algebra; the same central charge produces the Schwarzian transformation and the universal cylinder vacuum-energy shift. The derivation below fixes contour, radial-ordering, Hermiticity, and anomaly signs in one convention.

Required background. Complex coordinates and local conformal symmetry distinguish local holomorphic maps from global Möbius transformations. Currents and the stress tensor provide the general Ward-identity construction.

Helpful background. The cylinder map and Hamiltonian explain why radial dilatations become cylinder time translations.

Stress-tensor OPE and Laurent modes

On the Euclidean plane, use counterclockwise contours and radial ordering. The holomorphic normalization is

$$T(z)T(w) \sim \frac{c/2}{(z-w)^4} +\frac{2T(w)}{(z-w)^2} +\frac{\partial T(w)}{z-w}.$$

Define

$$L_n=\frac{1}{2\pi i}\oint_0 dz\,z^{n+1}T(z), \qquad T(z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}.$$

For a local field $\mathcal O(w)$, the infinitesimal transformation generated by a holomorphic test function $\epsilon$ is the contour operation

$$\delta_\epsilon\mathcal O(w) =-\frac{1}{2\pi i}\oint_w dz\, \epsilon(z)T(z)\mathcal O(w).$$

The minus sign matches the active-field convention on the preceding page. For a primary of weight $h$,

$$T(z)\phi_h(w) \sim\frac{h\phi_h(w)}{(z-w)^2} +\frac{\partial\phi_h(w)}{z-w},$$

so $\delta_\epsilon\phi_h=-(\epsilon\partial+h\partial\epsilon)\phi_h$. With several separated insertions, contour deformation gives the holomorphic Ward identity

$$\left\langle T(z)\prod_i\phi_i(z_i,\bar z_i)\right\rangle =\sum_i\left[ \frac{h_i}{(z-z_i)^2} +\frac{1}{z-z_i}\partial_{z_i} \right] \left\langle\prod_i\phi_i\right\rangle.$$

This identity is an equality away from coincident points. Distributional contact terms can appear when one takes $\bar\partial$ or varies background sources; they must not be silently promoted to separated-point OPE coefficients. The OPE and contour conventions follow Di Francesco, Mathieu, and Sénéchal 1997, §§5.3–6.2.

Deriving the Virasoro commutator

Radial ordering implies that a commutator is the difference between nested contours. Shrinking the outer contour onto the inner insertion gives

$$[L_m,L_n] =\frac{1}{(2\pi i)^2} \oint_0dw\,w^{n+1} \oint_wdz\,z^{m+1}T(z)T(w).$$

The three singular terms contribute separately. Cauchy’s formula gives

$$\frac{1}{2\pi i}\oint_wdz\, \frac{z^{m+1}}{(z-w)^4} =\frac{m(m^2-1)}{6}w^{m-2},$$

while the $T/(z-w)^2$ and $\partial T/(z-w)$ terms combine, after one integration by parts in $w$, to $(m-n)L_{m+n}$. Hence

$$\boxed{ [L_m,L_n]=(m-n)L_{m+n} +\frac{c}{12}m(m^2-1)\delta_{m+n,0}.}$$

The central term vanishes for $m=-1,0,1$, so the globally regular generators retain the $\mathfrak{sl}(2)$ algebra. In a reflection-positive theory, radial conjugation $z\mapsto1/\bar z$ gives

$$L_n^\dagger=L_{-n}, \qquad c\in\mathbb R,$$

and positivity of $\lVert L_{-2}|0\rangle\rVert^2=c/2$ requires $c\ge0$. The algebra itself does not require this Hermiticity condition; nonunitary representations may have indefinite norms.

Schwarzian transformation and the cylinder shift

The central term forces $T$ to transform anomalously. If $z=f(w)$, then

$$T_w(w)=\bigl(f'(w)\bigr)^2T_z(f(w)) +\frac{c}{12}\{f,w\},$$

where

$$\{f,w\}=\frac{f'''(w)}{f'(w)} -\frac32\left(\frac{f''(w)}{f'(w)}\right)^2.$$

For a Möbius map the Schwarzian vanishes, consistent with $T$ being quasiprimary. For the map $z=e^w$ from a cylinder $w=\tau+i\sigma$, $\sigma\sim\sigma+2\pi$, one finds

$$\{e^w,w\}=1-\frac32=-\frac12, \qquad T_{\rm cyl}(w)=z^2T_{\rm plane}(z)-\frac{c}{24}.$$

The vacuum has $\langle T_{\rm plane}\rangle=0$, hence $\langle T_{\rm cyl}\rangle=-c/24$. For circumference $L$, rescaling $w\mapsto(2\pi/L)w$ gives the chiral energy operator $(2\pi/L)(L_0-c/24)$. The full Hamiltonian includes the barred copy:

$$H=\frac{2\pi}{L} \left(L_0+\bar L_0-\frac{c+\bar c}{24}\right).$$

The finite-size shift and its relation to the conformal anomaly are reviewed in Ginsparg 1990, §§3.1–3.4. With $W=-\log Z$ and the curvature sign used in this volume, the two-dimensional trace response is $\langle T^\mu{}_{\mu}\rangle=+cR/(24\pi)$. This positive trace coefficient does not conflict with the negative cylinder energy: the first is a local Weyl variation, whereas the second follows from the negative Schwarzian $\{e^w,w\}$.

The figure summarizes the path from this shift to modular data. At this stage, inspect only the first three boxes: the plane–cylinder transformation fixes the vacuum offset, and quotienting a Virasoro module fixes the character that enters later traces.

The schematic chain uses $z=e^w$, $T_{\rm cyl}=z^2T_{\rm plane}-c/24$, and $\chi_i=\operatorname{Tr}_{\mathcal H_i}q^{L_0-c/24}$. A single chiral character or block is not yet a full modular-invariant CFT.

The same chain, with its required checks, is:

Step	Formula or object	Required check
Coordinate map	$z=e^w$, $w\sim w+2\pi i$	Image is the punctured plane and the contour orientation is preserved
Anomalous shift	$T_{\rm cyl}=z^2T_{\rm plane}-c/24$	$\{e^w,w\}=-1/2$ in the declared transformation law
Module trace	$\chi_i=\operatorname{Tr}_{\mathcal H_i}q^{L_0-c/24}$	Null states have been quotiented and state multiplicities are nonnegative
Full torus object	$Z=\sum_{i,j}M_{ij}\chi_i\bar\chi_j$	Left–right sectors are complete and $M$ is physically admissible
Modular action	$S:\tau\mapsto-1/\tau$, $T:\tau\mapsto\tau+1$	The chosen character basis closes under both transformations

A normalization check from the vacuum

The Virasoro algebra gives

$$\langle0|L_2L_{-2}|0\rangle =\langle0|[L_2,L_{-2}]|0\rangle =\frac{c}{2},$$

because $L_0|0\rangle=0$. This equals the coefficient of $z^{-4}$ in $\langle T(z)T(0)\rangle$. Thus the OPE normalization, the central term in the mode algebra, and the level-two vacuum norm agree. A factor-of-two mismatch in any one of these formulas diagnoses inconsistent conventions.

Common pitfalls

Reversing the Schwarzian. The displayed formula transforms the $z$-plane tensor into the $w$-coordinate tensor. Solving it for $T_z$ reverses the sign and Jacobian placement; state which direction is used before substituting $z=e^w$.

Ignoring the contour orientation. Clockwise contours change the sign of every mode integral. Fix counterclockwise orientation before deriving the commutator.

Calling $T$ a primary. The Schwarzian vanishes only for Möbius maps or when $c=0$. At nonzero central charge, $T$ is quasiprimary under the global subgroup, not primary under arbitrary local maps.

Exercises

Use the Virasoro algebra to compute the norm of $L_{-n}|0\rangle$ for $n\ge2$.

Solution

Since $L_n|0\rangle=0$ for $n\ge-1$ and $L_0|0\rangle=0$,

$$\lVert L_{-n}|0\rangle\rVert^2 =\langle0|[L_n,L_{-n}]|0\rangle =\frac{c}{12}n(n^2-1).$$

It is nonnegative for all $n\ge2$ when $c\ge0$ and is negative for every such $n$ when $c<0$, immediately excluding reflection positivity in that vacuum module.

References

• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Ginsparg, Paul. “Applied Conformal Field Theory.” In Fields, Strings and Critical Phenomena, Les Houches Session XLIX, edited by Édouard Brézin and Jean Zinn-Justin, 1–168. Amsterdam: North-Holland, 1990. arXiv.