Highest-Power Coefficient in the EMR Curvature Formula

This notebook reproduces the finite-dimensional computation behind Theorem 3.4 of Diffeomorphism Groups of Circle Bundles over Integral Symplectic Manifolds. The program alternates over all permutations and contracts copies of the standard complex structure.

The complete research code is available in EMR-Paper-Computation.

Standard complex structure

On a real vector space of dimension $2k$, let

$$ J=\begin{pmatrix}0&I_k\\-I_k&0\end{pmatrix}. $$

The notebook uses $k=2$ for a quick interactive calculation. The final section records the larger computations with exactly the same definitions.

def k := 2

def J :=
  generateTensor
    (\match as list integer with
       | [$i, #(i + k)] -> 1
       | [$i, #(i - k)] -> -1
       | _ -> 0)
    [2 * k, 2 * k]

J
$\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ \end{pmatrix}$
withSymbols [a, b, c] J_a~c . J_c~b
$\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix}$

Alternating product

Let $S_k$ be the signed sum of products

$$ S_k=\sum_{\sigma\in S_{2k}}\operatorname{sgn}(\sigma) \prod_{i=1}^{k}J_{\sigma(2i-1),\sigma(2i)}. $$

evenAndOddPermutations supplies the two signs separately.

def S :=
  let (es, os) := evenAndOddPermutations (2 * k) in
    sum (map (\σ -> product (map (\i -> J_(σ (2 * i - 1))_(σ (2 * i))) (between 1 k))) es) -
    sum (map (\σ -> product (map (\i -> J_(σ (2 * i - 1))_(σ (2 * i))) (between 1 k))) os)

S
$-8$

Curvature coefficient

The highest power of the bundle parameter $p$ is built from

$$ T_{abc}{}^d=-J_{bc}J_a{}^d+J_{ac}J_b{}^d +2J_{ab}J_c{}^d. $$

The indices $a_0,\ldots,a_{k-1}$ form a cyclic chain. Tensor product followed by . contracts that chain, while the outer fold adds the even and odd permutation contributions.

def T_a_b_c~d :=
  -1 * J_b_c . J_a~d +
  J_a_c . J_b~d +
  2 * J_a_b . J_c~d

def S' :=
  withSymbols [a]
    let (es, os) := evenAndOddPermutations (2 * k) in
      (\xs -> foldl (+) (head xs) (tail xs))
        (map
          (\σ ->
            (\xs -> foldl (.) (head xs) (tail xs))
              (map (\i -> T_(σ (2 * i - 1))_(σ (2 * i))_(a_(modulo i k))~(a_(i - 1))) (between 1 k)))
          es) -
      (\xs -> foldl (+) (head xs) (tail xs))
        (map
          (\σ ->
            (\xs -> foldl (.) (head xs) (tail xs))
              (map (\i -> T_(σ (2 * i - 1))_(σ (2 * i))_(a_(modulo i k))~(a_(i - 1))) (between 1 k)))
          os)

S'
$192$

Higher dimensions

The same program gives

$$ \begin{array}{c|rr} k&S_k&S'_k\\ \hline 2&-8&192\\ 3&-48&0\\ 4&384&61440 \end{array} $$

On Egison 5.1.0 the full $k=4$ run takes about 25 minutes. Keeping the interactive notebook at $k=2$ makes every cell quick to rerun; changing the first definition to def k := 4 performs the complete Theorem 3.4 computation without any other code changes.

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