research35 min studyLesson 738 in Physics

Two-dimensional Virasoro symmetry

Distinguish local holomorphic symmetry, central extension and Virasoro descendants from the global subalgebra.

Before you begin

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The question for today

What you’ll understand

Define holomorphic weights and stress-tensor modes.
Compute descendant norms including the central term.
Separate global primaries, Virasoro primaries and quasiprimaries.

Two Euclidean dimensions permit locally holomorphic coordinate transformations, producing a much larger symmetry algebra than the finite global conformal group. Quantum stress-tensor modes realize its centrally extended Virasoro algebra. This extra structure can determine correlators and representation constraints unavailable in generic higher dimensions. It still requires care about global coordinate maps, central charge, left and right sectors and null states. A two-dimensional global block does not automatically sum the additional Virasoro descendants.

Use complex coordinates and two weights

Introduce a complex coordinate built from the two real Euclidean coordinates and its conjugate. An orientation-preserving local conformal transformation is holomorphic where its derivative is nonzero. A field carries holomorphic and antiholomorphic weights, whose sum is its scaling dimension and whose difference is its Euclidean spin. The barred sector is independent algebraically, while reflection and reality conditions relate appropriate physical observables. Chiral fields and full local fields should not be identified without specifying their locality and spin structure.

The global holomorphic transformations are generated by three modes, conventionally labeled minus one, zero and one. Together with the antiholomorphic copies they form the finite global algebra. Infinitely many additional modes arise from local transformations on patches; they are not all globally nonsingular maps of the sphere. Statements about the infinite algebra are therefore statements about local symmetry and operator relations, rather than a claim that every mode exponentiates to a smooth global sphere transformation.

z=x^1+ix^2,\qquad \Delta=h+\bar h,\qquad s=h-\bar h,\qquad T(z)=\sum_{n\in\mathbb Z}L_nz^{-n-2}

Here x¹, x² and z are dimensionless coordinates obtained by dividing physical lengths by a fixed reference length; T is correspondingly rescaled by the square of that reference length. The radial modes Ln and weights h, h̄, Δ and s are dimensionless. The expansion defines the rescaled weight-two holomorphic stress tensor.↔ Scroll wide equations horizontally.

Derive the central term from the stress-tensor OPE

The stress-tensor operator product contains a fourth-order pole proportional to the central charge. Contour integration converts its singular terms into commutators of modes. The central term vanishes on the three global modes but survives for more general integers, so two theories can share a global conformal algebra while having different Virasoro representation data. With a unit-normalized vacuum, the norm of the state made by the minus-two stress-tensor mode is one half the central charge.

Reflection positivity gives the radial adjoint of a mode as the oppositely numbered mode. For a highest-weight state annihilated by positive modes, commute a positive mode through its negative partner to compute a descendant norm. The result contains both the weight and the central charge. Nonnegative individual mode norms are necessary but not sufficient: Gram matrices of all descendant combinations at a given level can impose stronger constraints and generate null relations.

[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0},\qquad L_n^\dagger=L_{-n},\qquad \|L_{-n}|h\rangle\|^2=2nh+\frac{c}{12}n(n^2-1)

The norm formula assumes n > 0, a unit-normalized highest-weight state and Ln|h⟩ = 0 for positive n. The antiholomorphic algebra has its own central charge c̄ and commutes with this one.↔ Scroll wide equations horizontally.

Track what the word primary means in two dimensions

A Virasoro primary is annihilated by every positive mode, while a global primary need only satisfy the positive global-generator condition. A quasiprimary transforms covariantly under the global subgroup but need not be primary under the full local algebra. The stress tensor is a standard warning: it has an anomalous transformation proportional to central charge and is a global quasiprimary rather than an ordinary Virasoro primary when the central term is nonzero.

Virasoro descendants arise from all negative modes, so their number exceeds that generated by repeated ordinary derivatives alone. Null states can remove combinations and yield differential equations for correlators. A central charge and a list of highest weights must still be supplemented by consistent left-right pairing and operator products to form a full local theory. The spectral lab only compares finite harmonic content; it cannot generate a Virasoro module, central extension, null state or conformal anomaly.

T(z)T(0)\sim\frac{c/2}{z^4}+\frac{2T(0)}{z^2}+\frac{\partial T(0)}{z},\qquad L_n|h\rangle=0\ (n>0),\quad L_0|h\rangle=h|h\rangle

The stress-tensor OPE and highest-weight conditions use the same holomorphic normalization. A full local primary also has an antiholomorphic highest-weight condition; the singular OPE does not include all regular terms.↔ Scroll wide equations horizontally.

Work it through

Two norms in the Ising chiral representation

Take a unit-normalized Virasoro highest-weight state with $c=1/2$ and $h=1/16$ . Compute the norms of $L_{-1}|h\rangle$ and $L_{-2}|h\rangle$ , including the central contribution.

For n = 1 the central term vanishes because $n(n^2-1)=0$ . The norm is $2h=1/8$ .
For n = 2 the weight contribution is $4h=1/4$ . The central contribution is $(c/12)2(4-1)=c/2=1/4$ .
Adding them gives norm $1/2$ for the minus-two descendant. Both particular states have positive norm, but their positivity alone does not test every linear combination with other level-two descendants or prove the absence of a null vector.

The result

The norms are 1/8 and 1/2, with half of the second value arising from the central term.

The living notebook

Explore the model

Interactive lab

Interactive lab · spectral

More modes do not mean a Virasoro module

Reset and set BOTH Mode count N = 2 and Coefficient decay p = 1. Reset and set N = 8 and p = 1 for a fresh equal-time comparison; predict finer spatial structure with the additional harmonics. These sine modes are not stress-tensor modes, representation states, central charge or Virasoro conformal blocks.

Finite sumHighest included mode

Sum at x = π/2: 0.867
Smallest wavelength: 2π / 5
Highest-mode coefficient: 0.2

Mode count N5Coefficient decay p1

t = 0 s

A finite sum of sin(nx − t)/nᵖ. Playback changes t; increasing the cutoff includes finer spatial scales. This is not a proof about a continuum limit or renormalization. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forAn infinite algebra specifies commutators and representations, beyond merely having many Fourier modes.

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Read the checkpoint explanation

The central term produces the anomalous local transformation. Global covariance is weaker than being primary under every positive Virasoro mode.

Your turn

Try it yourself

For the unit-normalized vacuum with h = 0 and c = 1, compute the norms of $L_{-1}|0\rangle$ and $L_{-2}|0\rangle$ . Explain why a vanishing first norm does not make the stress-tensor state vanish.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

The central contribution is zero for n = 1 but nonzero for n = 2.

Compare with the solution

The first norm is zero, consistent with translation invariance and the null vacuum derivative. The second norm is $c/2=1/2$ , which is positive. Thus the stress-tensor state can be nonzero even though translating the vacuum gives zero. Distinguishing these modes is precisely why a vacuum Virasoro module contains more structure than repeated ordinary derivatives of the identity, and why the global subalgebra alone cannot reconstruct all its descendant states.

Keep the idea

Take it with you

Two-dimensional local conformal symmetry enlarges the global algebra.
Central charge contributes to descendant norms and stress-tensor transformations.
Virasoro primaries and global quasiprimaries satisfy different conditions.

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