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The order-n Hochschild obstruction to extending a partial star product

Statement

Let $M$ be a smooth manifold (Smooth manifolds and their smooth charts) and $A = C^{\infty} (M, R)$ , and let $n \geq 2$ . Suppose $B_{1}, \dots, B_{n - 1}$ are normalized bidifferential 2-cochains and that the truncated product $f ⋆_{< n} g = f g + \sum_{r = 1}^{n - 1} ℏ^{r} B_{r} (f, g)$ is associative modulo $ℏ^{n}$ . For $i + j = n$ , put $O_{n} (f, g, h) : = \sum_{i = 1}^{n - 1} (B_{i} (B_{n - i} (f, g), h) - B_{i} (f, B_{n - i} (g, h))) .$ Then $O_{n}$ is a normalized multidifferential Hochschild 3-cocycle. The truncated product extends associatively modulo $ℏ^{n + 1}$ by a normalized bidifferential coefficient $B_{n}$ exactly when $[O_{n}] = 0$ in the cohomology $H H_{d i f f, n o r m}^{3} (A, A)$ defined in Hochschild cochains, differential, and Gerstenhaber insertion bracket. Equivalently, $δ B_{n} = O_{n} .$ With the Gerstenhaber bracket convention in that definition, $O_{n} = \frac{1}{2} \sum_{i + j = n} [B_{i}, B_{j}]_{G},$ where both indices range from $1$ to $n - 1$ . In particular, $O_{2} = \frac{1}{2} [B_{1}, B_{1}]_{G}$ . This is a coefficientwise extension criterion at one finite order; it gives no automatic extension at later orders and no general existence or formality theorem.

Facts & Assumptions

Given: A smooth manifold $M$ , $A = C^{\infty} (M, R)$ , $n \geq 2$ , normalized bidifferential cochains $B_{1}, \dots, B_{n - 1}$ , and associativity of $⋆_{< n}$ through all powers below $ℏ^{n}$ .

[A1]

$n$ is an integer at least $2$ ; all coefficients and cochains are over $R$ , so $2$ and $3$ are invertible.

[F1]

Coefficientwise formal associative deformations: the coefficientwise product uses finite sums at each formal order and associativity is equality of each coefficient.

[F2]

Hochschild cochains, differential, and Gerstenhaber insertion bracket: the Hochschild differential $δ$ , normalized multidifferential subcomplex, and Gerstenhaber insertion bracket have the displayed formulas.

[F3]

The shifted Hochschild cochains form a differential graded Lie algebra: with $\partial = [m, -]_{G}$ , the shifted cochains form a differential graded Lie algebra and $\partial B_{r} = - δ B_{r}$ for 2-cochains, while $\partial O_{n} = δ O_{n}$ for 3-cochains.

Proof

1.1givenA1F1F2algebra

Let $m (f, g) = f g$ and $x = \sum_{r = 1}^{n - 1} ℏ^{r} B_{r}$ , so $m + x$ is the truncated product. For arity-two cochains the insertion formula gives $\frac{1}{2} [m + x, m + x]_{G} (f, g, h) = ((f ⋆_{< n} g) ⋆_{< n} h) - f ⋆_{< n} (g ⋆_{< n} h)$ . Thus its coefficient at order $n$ , where there is no $B_{n}$ term, is precisely the displayed $O_{n}$ ; all lower coefficients vanish by the assumed partial associativity. The same insertion formula gives $\frac{1}{2} \sum_{i + j = n} [B_{i}, B_{j}]_{G} = \sum_{i + j = n} B_{i} \circ B_{j} = O_{n}$ , because the bracket of two 2-cochains is symmetric.

2.1A1F3step 1.1algebra

In the shifted Hochschild DGLA put $R = \partial x + \frac{1}{2} [x, x]_{G} = \frac{1}{2} [m + x, m + x]_{G}$ ; the last equality uses $[m, m]_{G} = 0$ . The DGLA identities give the Bianchi identity $\partial R + [x, R]_{G} = \frac{1}{2} \partial [x, x]_{G} + [x, \partial x]_{G} + \frac{1}{2} [x, [x, x]_{G}]_{G} = 0.$ Indeed, the derivation rule and graded skew-symmetry make the first two terms cancel, and graded Jacobi gives $3 [x, [x, x]_{G}]_{G} = 0$ , hence the last term vanishes over $R$ . By step 1.1, $R$ starts at order $ℏ^{n}$ with coefficient $O_{n}$ , while $x$ starts at order $ℏ$ ; therefore the coefficient of $ℏ^{n}$ in the Bianchi identity is $\partial O_{n} = 0$ .

2.2A1F2F3step 1.1algebra

By [F3], $\partial$ on 3-cochains is $δ$ , so $δ O_{n} = 0$ . Each summand of $O_{n}$ is a composition of normalized bidifferential cochains. An input equal to $1$ either makes the inner cochain vanish or remains a unit input of the outer cochain; composition also preserves finite differential order. Thus $O_{n}$ lies in the stated normalized multidifferential subcomplex.

2.3A1F1F2step 1.1algebra

Add $ℏ^{n} B_{n}$ to the truncated product. The order- $n$ terms in its associator that are linear in $B_{n}$ are $B_{n} (f, g) h + B_{n} (f g, h) - f B_{n} (g, h) - B_{n} (f, g h) = - (δ B_{n}) (f, g, h) .$ No term quadratic in $B_{n}$ , or involving $B_{n}$ and a positive-order coefficient, contributes at order $n$ , since those terms have order at least $2 n$ or $n + 1$ . Hence the new order- $n$ associator coefficient is $O_{n} - δ B_{n}$ .

3.1F2step 2.2step 2.3algebra∎

The lower associator coefficients already vanish. Therefore $⋆_{< n} + ℏ^{n} B_{n}$ is associative modulo $ℏ^{n + 1}$ exactly when $δ B_{n} = O_{n}$ . By step 2.2, $O_{n}$ is a cocycle in the normalized multidifferential complex; such a $B_{n}$ exists precisely when its cohomology class is zero. For $n = 2$ , the defining sum has only $(i, j) = (1, 1)$ , yielding $O_{2} = \frac{1}{2} [B_{1}, B_{1}]_{G}$ . This proves the finite-order criterion without asserting that later obstruction classes vanish.

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Sources

Sarah J. Witherspoon, Hochschild Cohomology for Algebras (standard reference, not scraped)
Nima Moshayedi, Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory (standard reference, not scraped)