Holomorphic stress-energy tensor ($T(z)$)

The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.

In a two-dimensional conformal field theory,The fourth-order pole is the central extension, while the remaining poles express that $T$ has conformal weight two.

The Virasoro algebra is the central extension of the algebra of holomorphic vector fields on a circle. Its generators are the Laurent modes of the holomorphic stress-energy tensor, and its central term is controlled by the central charge $c$.

A Virasoro character records the graded dimensions of a positive-energy Virasoro algebra module. A state of weight $h+N$ contributes $q^{h+N-c/24}$, and its multiplicity is the dimension of that energy eigenspace. The shift accounts for the Casimir energy of a two-dimensional conformal field theory. For a Virasoro Verma module, the character is $q^{h-c/24}{\prod }_{n\ge 1}(1-q^n{)}^{-1}$, by counting the ordered descendant basis.

The Virasoro Verma module $M(c,h)$ is the universal highest-weight representation with central charge $c$ and ground weight $h$. The Poincaré-Birkhoff-Witt theorem gives a basis $L_{-n_1}\cdots L_{-n_s}|h⟩$ with $n_1\ge \cdots \ge n_s>0$. Its invariant Hermitian form is fixed by $L_n^{†}=L_{-n}$ and the norm of the ground vector. Quotienting by its maximal proper submodule produces the irreducible highest-weight module; in the unitary cases this removes the null vectors.

For a normalized highest-weight vector with $L_0|h⟩=h|h⟩$, $L_{n>0}|h⟩=0$ and $L_n^{†}=L_{-n}$, the level-two descendants $L_{-2}|h⟩,L_{-1}^2|h⟩$ have the displayed Gram matrix. It follows from the Virasoro algebra: $⟨L_2{L}_{-2}⟩=4h+c/2$, $⟨L_2{L}_{-1}^2⟩=6h$ and $⟨L_1^2{L}_{-1}^2⟩=4h(2h+1)$. At $h=-1,c=26$ its null direction is $(1,3/2)$, giving the level-two scalar Virasoro null state. Nullity of a descendant is a metric statement; satisfying all physical constraints must be checked separately.

A unitary highest-weight Virasoro module is generated by a normalized vector satisfying the displayed relations, with a positive-definite inner product and $L_n^{†}=L_{-n}$. Its Virasoro descendants have $L_0$ eigenvalues $h+N$ for nonnegative integral levels $N$. The commutator gives $\Vert L_{-1}|h⟩{\Vert }^2=2h$, so unitarity requires $h\ge 0$. Such a module is a quotient of a Virasoro Verma module; its Unitary Virasoro discrete series describes the cases with central charge below one.

For a restricted representation of an affine current algebra with $[J_m^a,J_n^b]=i{f}^{abc}{J}_{m+n}^c+km{\delta }^{ab}{\delta }_{m+n,0}$ and $f^{abc}{f}^{abd}=Q{\delta }^{cd}$, the displayed normal ordering construction is defined when $2k+Q\ne 0$. Moving the currents past each other gives $[L_m,J_n^a]=-n{J}_{m+n}^a$; the additional $Q$ in the denominator is the adjoint-Casimir reordering term. The resulting Virasoro algebra has central charge $c=k\dim \mathfrak{g}/(k+Q/2)$. On a compact unitary positive-energy module, $L_m^{†}=L_{-m}$.

For a current subalgebra $\mathfrak{h}\subset \mathfrak{g}$, its currents have conformal weight one under both Sugawara construction stress tensors. Their difference therefore commutes with all $\mathfrak{h}$ currents and with the $\mathfrak{h}$ stress tensor. It obeys the Virasoro algebra with central charge $c_{\mathfrak{g}}-c_{\mathfrak{h}}$. For a positive-level compact unitary vacuum representation, $\Vert L_{-2}^{\mathfrak{g}/\mathfrak{h}}|0⟩{\Vert }^2=(c_{\mathfrak{g}}-c_{\mathfrak{h}})/2\ge 0$. A Cartan subalgebra has $c_{\mathfrak{h}}=\mathrm{rank}\mathfrak{g}$, giving the lower bound on $c_{\mathfrak{g}}$ under these hypotheses.

An irreducible unitary highest weight module of the Virasoro algebra with $c<1$ has integer $m\ge 2$, $1\le r\le m-1$, $1\le s\le m$, and the displayed weights. Labels $(r,s)$ and $(m-r,m+1-s)$ represent the same module. It is obtained from the Verma module by quotienting its zero-norm submodules; the two generating singular-vector levels are $rs$ and $(m-r)(m+1-s)$. For $m=2$ this is the trivial $c=h=0$ module. For $m=3$, the three weights are $0$, $1/2$, $1/16$.

Virasoro generators are the contour modes of the holomorphic stress tensor in a two-dimensional conformal field theory. Their commutators are $[L_m,L_n]=(m-n)L_{m+n}+c(m^3-m){\delta }_{m+n,0}/12$. A contour around an operator insertion extracts the corresponding coefficient of its stress-tensor operator product expansion. $L_{-1}$ generates translations, $L_0$ measures holomorphic weight, and all positive generators annihilate a primary state. An independent antiholomorphic copy acts on the other weight.

Shifting the zero mode changes the algebra's central representative:The linear term follows from replacing $L_{m+n}$ by ${\mathcal{L}}_{m+n}+a{\delta }_{m+n,0}$. It changes the zero-mode convention, not the cubic coefficient or central charge. This distinction separates the intercept from the Virasoro central extension.

The Virasoro central extension is the term $c(m^3-m){\delta }_{m+n,0}/12$ in $[L_m,L_n]$. It records the conformal anomaly and commutes with every Virasoro generator.

For $d$ free target coordinates, normal ordering of $L_m={\sum }_n:{\alpha }_{m-n}\cdot {\alpha }_n:/2$ gives a matter central charge $d$. On the formal zero-momentum vacuum, the double contractions yield $⟨L_m{L}_{-m}⟩=(d/2){\sum }_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. Two target-metric contractions give ${\eta }_{ab}{\eta }^{ab}=d$, so timelike target coordinates do not subtract from this central charge. Transverse-only and covariant generators must not be confused.

The central charge is the coefficient of the fourth-order pole in the stress-tensor operator-product expansion, $T(z)T(w)\sim c/[2(z-w{)}^4]+\cdots$. A free real boson contributes one and a free real chiral fermion contributes one half.

In a two-dimensional supersymmetric conformal field theory, the superconformal current has weight $3/2$ and its self-OPE closes on the stress tensor and the central charge.

For RNS string matter normalized by $[{\dot{X}}^{\mu }(\sigma ),X^{\nu }({\sigma }^{\prime })]=-i\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$ and $\{{\psi }_1^{\mu }(\sigma ),{\psi }_1^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. Then $[A^{\mu }(\sigma ),A^{\nu }({\sigma }^{\prime })]=(i\pi /2){\eta }^{\mu \nu }{\partial }_{\sigma }\delta (\sigma -{\sigma }^{\prime })$. The noncentral part of $\{J(\sigma ),J({\sigma }^{\prime })\}$ is $\pi \delta A^2+(i\pi /2){\partial }_{\sigma }\delta {\psi }_1(\sigma )\cdot {\psi }_1({\sigma }^{\prime })$. The distributional derivative identity $f(\sigma ,{\sigma }^{\prime }){\delta }^{\prime }=f(\sigma ,\sigma ){\delta }^{\prime }+({\partial }_{{\sigma }^{\prime }}f)(\sigma ,\sigma )\delta$, with a vanishing coincident Grassmann bilinear, converts this to $\pi \delta {\theta }_{++}$. Use the chiral equation of motion to identify ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. Quantum normal ordering adds a central contact term proportional to ${\delta }^{\prime \prime }$, the local form of the N=1 super-Virasoro algebra central extension.

The holomorphic N=1 super-Virasoro algebra extends the Virasoro algebra by fermionic modes $G_r$ satisfying $\{G_r,G_s\}=2L_{r+s}+(c/12)(4r^2-1){\delta }_{r,-s}$.

With the standard string oscillator algebra and $G_0={\sum }_m{\alpha }_m\cdot {\psi }_{-m}$, pairing positive and negative modes gives $G_0^2={\alpha }_0^2/2+{\sum }_{m>0}({\alpha }_{-m}\cdot {\alpha }_m+m{\psi }_{-m}\cdot {\psi }_m)$. This is the vacuum-normal-ordered Ramond sector generator $L_0^R$, so $\{G_0,G_0\}=2L_0^R$. Rescaling the superconformal current rescales its square quadratically; its normalization cannot be ignored when comparing it with a fixed Virasoro generator.