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KZ equations for WZW conformal blocks on the sphere

Statement

Let $g, B, k, λ_{i}$ be as in def-cft-wzw-insertion-data, and let $V = ⨂_{i = 1}^{n} V_{λ_{i}}$ be the ordered tensor of finite horizontal tops. Let $F (z) \in V^{*}$ be the top restriction of a differentiable local family of block functionals from def-cft-wzw-sphere-current-coinvariants. Assume the moving-coordinate trivialization $t_{i} = w - z_{i}$ is translation-covariant, so differentiating the $i$ th insertion replaces its top vector by $L_{- 1}^{(i)}$ of that vector. For each $i$ , define $Ω_{i j} = \sum_{a} ρ_{i} (x_{a}) ρ_{j} (x^{a}), ρ^{\lor} (x) ℓ = - ℓ \circ ρ (x), Ω_{i j}^{\lor} = \sum_{a} ρ_{i}^{\lor} (x_{a}) ρ_{j}^{\lor} (x^{a}),$ where $(x_{a}), (x^{a})$ are $B$ -dual bases. Then $(k + h^{\lor}) \partial_{z_{i}} F (z) = \sum_{j \neq i} \frac{Ω_{i j}^{\lor} F (z)}{z_{i} - z_{j}} .$ In row-functional notation, $Ω_{i j}^{\lor} F = F \circ Ω_{i j}$ . The equation is a chiral, fixed-coordinate deduction. If “block” is instead taken to mean the generic integral- $L_{0}$ carrier def-cft-coinvariants-and-conformal-blocks, its hypotheses must also be imposed; the present argument does not identify that carrier with fractional- $L_{0}$ WZW modules or assert local freeness.

Statement references

Coinvariants and conformal blocks on a pointed curve

Integrable WZW insertions on a punctured sphere

Fixed-coordinate WZW sphere current coinvariants

Facts & Assumptions

[A1]

Full AC is assumed in Integrable WZW insertions on a punctured sphere for the existence of the modules and tops used in this equation; no additional choice is used in the local calculation.

Given: The fixed normalized form, $k \in Z_{\geq 0}$ , admissible labels, distinct insertion points, and a differentiable local current-block family with the translation convention in the Statement.

[F1]

Integrable WZW insertions on a punctured sphere: each module has level $k$ , restricted affine modes, and top $V_{λ_{i}}$ .

[F2]

Current Ward identity for WZW sphere coinvariants gives the rational-current Ward identity on top vectors.

[F3]

Sugawara modes and their normal ordering: the normal-ordered Sugawara mode is the dual-basis quadratic current sum with denominator $2 (k + h^{\lor})$ ; it is defined away from the critical level.

[F4]

Dual Coxeter number for the normalized invariant form defines $h^{\lor}$ using the same long-root-square- $2$ normalization; $h^{\lor} > 0$ .

[F5]

Sugawara modes satisfy Virasoro at every noncritical complex level: these modes form the Virasoro action on every restricted affine-current module when $k + h^{\lor} \neq 0$ .

[F6]

Coinvariants and conformal blocks on a pointed curve defines generic algebraic blocks using integrally graded conformal modules and explicitly makes no finite-rank or local-freeness assertion.

[A2]

The family is differentiable in the moving coordinates and satisfies $\partial_{z_{i}} F (v) = F (L_{- 1}^{(i)} v)$ ; this is the stated translation-covariant trivialization, not a consequence of the bare quotient definition.

Proof

1.1A1F1F3F4F5givenalgebra

Put $κ = k + h^{\lor}$ . It is nonzero because $k \geq 0$ and $h^{\lor} > 0$ by [F4]. By the mode formula [F3], on a top vector $v_{i}$ only the normal-ordered terms with current indices $(- 1, 0)$ and $(0, - 1)$ survive in $L_{- 1} v_{i}$ : every other term has a positive current mode acting on the right, which kills $v_{i}$ by [F1]. The two surviving terms agree after relabeling the dual bases, so $L_{- 1}^{(i)} v_{i} = κ^{- 1} \sum_{a} x_{a} (- 1) x^{a} (0) v_{i} .$ The Virasoro action used in the translation convention is valid by [F5].

2.1A1A2F1F2step 1.1algebra∎

Apply [A2] and substitute the identity of step 1.1 into $F (L_{- 1}^{(i)} v_{i})$ . For each $a$ , apply [F2] to the current $x_{a} (- 1)$ with the other top vector $x^{a} (0) v_{i}$ at slot $i$ . The resulting sum over $j \neq i$ has coefficient $(z_{i} - z_{j})^{- 1}$ and inserts $x^{a} (0)$ at $i$ , $x_{a} (0)$ at $j$ . Summing over $a$ is exactly $F \circ Ω_{i j}$ . The contragredient convention in the Statement makes this $Ω_{i j}^{\lor} F$ in column coordinates, yielding the displayed KZ equation.

Remarks

The separate generic integral- $L_{0}$ carrier Coinvariants and conformal blocks on a pointed curve is formulation context only. Its integral-grading hypotheses are not used in the deduction above; no equivalence with the fixed-coordinate current-block family, finite rank, or local freeness is asserted.

KZ equations for WZW conformal blocks on the sphere

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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