Wodzicki–Chern–Simons Invariant on the Thurston Example¶

This notebook reproduces the symbolic computation in Section 4 of Diffeomorphism Groups of Circle Bundles over Integral Symplectic Manifolds. It builds the metric, curvature, and lifted curvature tensor and then reduces the Wodzicki–Chern–Simons integrand to a compact rational expression.

The original research program is maintained in EMR-Paper-Computation.

Thurston metric¶

Write $\beta=1+\theta_2-\theta_2^2$. The metric and inverse metric below are expressed in the coordinate frame $(\theta_1,\theta_2,\theta_3,\theta_4)$. Quoting the two recurring polynomial expressions keeps the intermediate tensor calculation compact.

declare symbol θ₁, θ₂, θ₃, θ₄, κ, p

def x~i := [| θ₁, θ₂, θ₃, θ₄ |]~i
def β := `(1 + θ₂ - θ₂^2)

def g_i_j :=
  [|[| 1, 0, 0, 0 |],
    [| 0, 1, 0, 0 |],
    [| 0, 0, κ / sqrt β, (-1 * θ₂ * κ) / sqrt β |],
    [| 0, 0, (-1 * θ₂ * κ) / sqrt β, (`(1 + θ₂) * κ) / sqrt β |]|]

def g~i~j :=
  [|[| 1, 0, 0, 0 |],
    [| 0, 1, 0, 0 |],
    [| 0, 0, `(1 + θ₂) / (κ * sqrt β), θ₂ / (sqrt β * κ) |],
    [| 0, 0, θ₂ / (sqrt β * κ), 1 / (sqrt β * κ) |]|]

g_#_#

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \sqrt{(-θ₂^{2} + θ₂ + 1)}^{-1} κ & -\sqrt{(-θ₂^{2} + θ₂ + 1)}^{-1} θ₂ κ \\ 0 & 0 & -\sqrt{(-θ₂^{2} + θ₂ + 1)}^{-1} θ₂ κ & \sqrt{(-θ₂^{2} + θ₂ + 1)}^{-1} (θ₂ + 1) κ \\ \end{pmatrix}_{\#\#}^{\;\;}$

withSymbols [i, j, k] g_i_j . g~j~k

$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -(-θ₂^{2} + θ₂ + 1)^{-1} θ₂^{2} + (-θ₂^{2} + θ₂ + 1)^{-1} (θ₂ + 1) & 0 \\ 0 & 0 & 0 & -(-θ₂^{2} + θ₂ + 1)^{-1} θ₂^{2} + (-θ₂^{2} + θ₂ + 1)^{-1} (θ₂ + 1) \\ \end{pmatrix}$

Levi-Civita connection and curvature¶

Egison's symbolic tensor indices transcribe the usual formulas

$$ \Gamma^c{}_{ab}=\frac12g^{ce} (\partial_ag_{be}+\partial_bg_{ae}-\partial_eg_{ab}), $$

followed by $R_{ijk}{}^l$. Repeated upper and lower indices are contracted by ..

def Γ~c_a_b := withSymbols [e]
  (1 / 2) * g~c~e . (∂/∂ g_b_e x~a + ∂/∂ g_a_e x~b - ∂/∂ g_a_b x~e)

def R_i_j_k~l := withSymbols [a]
  ∂/∂ Γ~l_j_k x~i - ∂/∂ Γ~l_i_k x~j
    + Γ~l_i_a . Γ~a_j_k - Γ~l_j_a . Γ~a_i_k

def R_i_j_k_l := withSymbols [a] R_i_j_k~a . g_a_l

Γ~1_1_1

$0$

Complex structure and lifted curvature¶

The complex structure $J$ and its covariant derivative determine the curvature $R'$ on the circle bundle. The first coordinate is the fibre direction; the remaining four coordinates belong to the Thurston base.

def J_a_b :=
  [|[| 0, 1, 0, 0 |],
    [| -1, 0, 0, 0 |],
    [| 0, 0, 0, κ |],
    [| 0, 0, -1 * κ, 0 |]|]

def J_a~c := withSymbols [b] J_a_b . g~b~c

def ∇J_m_a_b := withSymbols [n]
  ∂/∂ J_a_b x~m + Γ~n_m_a . J_n_b + Γ~n_m_b . J_a_n

def ∇J~m_a_b := withSymbols [t] ∇J_t_a_b . g~t~m
def ∇J_m~a_b := withSymbols [t] ∇J_m_t_b . g~t~a
def ∇J_m_a~b := withSymbols [t] ∇J_m_a_t . g~t~b

def δ :=
  generateTensor
    (\match as list integer with
       | [$n, #n] -> 1
       | [_, _] -> 0)
    [5, 5]

def R'{_i_j}_k~l : Tensor MathValue :=
  generateTensor
    (\match as list integer with
       | [#1, #1, _, _] -> 0
       | [_, _, #1, #1] -> 0
       | [#1, $b, #1, $d] -> -1 * p^2 * δ~(b - 1)_(d - 1)
       | [$a, #1, #1, $d] ->      p^2 * δ~(a - 1)_(d - 1)
       | [#1, $b, $c, #1] ->      p^2 * g_(b - 1)_(c - 1)
       | [$a, #1, $c, #1] -> -1 * p^2 * g_(a - 1)_(c - 1)
       | [#1, $b, $c, $d] -> -1 * p * ∇J_(b - 1)_(c - 1)~(d - 1)
       | [$a, #1, $c, $d] ->      p * ∇J_(a - 1)_(c - 1)~(d - 1)
       | [$a, $b, #1, $d] -> -1 * p * ∇J~(d - 1)_(a - 1)_(b - 1)
       | [$a, $b, $c, #1] ->      p * ∇J_(c - 1)_(a - 1)_(b - 1)
       | [$a, $b, $c, $d] -> R_(a - 1)_(b - 1)_(c - 1)~(d - 1)
                             - p^2 * J_(b - 1)_(c - 1) * J_(a - 1)~(d - 1)
                             + p^2 * J_(a - 1)_(c - 1) * J_(b - 1)~(d - 1)
                             + 2 * p^2 * J_(a - 1)_(b - 1) * J_(c - 1)~(d - 1))
    [5, 5, 5, 5]

Wodzicki–Chern–Simons contraction¶

The alternating contraction contains three copies of $R'$. Its raw result uses negative powers of the quoted atom $\beta$. Multiplying by $16\beta^8$ clears those Laurent denominators. A Gröbner basis for the defining quote relations then gives a canonical polynomial normal form, after which the denominator is restored.

def S := withSymbols [i, j, k]
  let (es, os) := evenAndOddPermutations 5 in
    sum (map (\σ -> R'_(σ 1)_j_1~i . R'_(σ 2)_(σ 3)_k~j . R'_(σ 4)_(σ 5)_i~k) es) -
    sum (map (\σ -> R'_(σ 1)_j_1~i . R'_(σ 2)_(σ 3)_k~j . R'_(σ 4)_(σ 5)_i~k) os)

def quoteGb :=
  groebnerBasis ['(1 + θ₂ - θ₂^2 - β), '(1 + θ₂ - `(1 + θ₂))]

def sSimplified := polyNF quoteGb (16 * β^8 * S) / (16 * β^8)

sSimplified

$192 p^{6} κ - 40 (-θ₂^{2} + θ₂ + 1)^{-2} p^{4} κ + \frac{-25}{16} (-θ₂^{2} + θ₂ + 1)^{-4} p^{2} κ$

def sClosedForm :=
  p^2 * κ * (-25 - 640 * p^2 * β^2 + 3072 * p^4 * β^4) / (16 * β^4)

sSimplified = sClosedForm

$True$

Result¶

Egison reduces the full contraction to

$$ S=192p^6\kappa-\frac{40p^4\kappa}{\beta^2} -\frac{25p^2\kappa}{16\beta^4} =\frac{p^2\kappa(-25-640p^2\beta^2+3072p^4\beta^4)} {16\beta^4}. $$

Thus the expression previously simplified with an external computer algebra system is now calculated and normalized entirely by Egison.