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BPZ equation from a Virasoro level-two null vector

Statement

Let $F (z)$ be a fixed-coordinate sphere Virasoro block family with primary insertions $v_{j}$ of weights $h_{j}$ . Suppose that in the chosen module the $i$ th primary obeys the actual relation

(L_{- 2} - β L_{- 1}^{2}) v_{i} = 0,

not merely that this vector is singular in a Verma module. Then

[β \partial_{z_{i}}^{2} - \sum_{j \neq i} (\frac{h_{j}}{(z_{i} - z_{j})^{2}} + \frac{1}{z_{i} - z_{j}} \partial_{z_{j}})] F (z) = 0.

Facts & Assumptions

Given: The Virasoro conventions of Virasoro algebra in the $L_{m}$ convention, a block family as in Fixed-coordinate sphere Virasoro block family, and an actual level-two null relation in the $i$ th module.

[F1]

Stress-tensor Ward identity on the sphere identifies the $L_{- 2}$ insertion with the displayed sum over $j \neq i$ .

[F2]

Fixed-coordinate sphere Virasoro block family makes translations act by insertion of $L_{- 1}$ , for all module states.

[A1]

The relation $(L_{- 2} - β L_{- 1}^{2}) v_{i} = 0$ holds as equality to zero in the chosen module.

Proof

1.1F1given

By [F1], evaluating the block on $L_{- 2}^{(i)} v_{i}$ gives

F (L_{- 2}^{(i)} v_{i}) = \sum_{j \neq i} (\frac{h_{j}}{(z_{i} - z_{j})^{2}} + \frac{1}{z_{i} - z_{j}} \partial_{z_{j}}) F .

This is a coefficient identity in the fixed local coordinate $t_{i} = w - z_{i}$ .

2.1A1F2step 1.1algebra∎

The actual module relation [A1] gives $F (L_{- 2}^{(i)} v_{i}) = β F ((L_{- 1}^{(i)})^{2} v_{i})$ . Applying the translation property [F2] twice gives $F ((L_{- 1}^{(i)})^{2} v_{i}) = \partial_{z_{i}}^{2} F$ ; the modes are fixed operators and the family is holomorphic, so the second derivative is defined. Substitution in step 1.1 and moving the sum to the left proves the BPZ equation.

Remarks

The generic integral- $L_{0}$ carrier Coinvariants and conformal blocks on a pointed curve is separate formulation context; it supplies neither the actual null relation [A1] nor coordinate descent for fractional weights. Singularity in a Verma module alone is not used in the deduction.

BPZ equation from a Virasoro level-two null vector

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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