D ℋ-Space Metric

Update

In the following the derivation of the ℋ-space metric, is given.

We begin with the cut function, -- -- uB = Z (ξa(τ),ζ,ζ) = X (τ,ζ,ζ) that satisfies the good cut eqation -- ∂2Z = σ(Z, ζ,ζ). The (-- ζ,ζ) are (for the time being) completely independent of each other though -- ζ is to be treated as being “close” the complex conjugate of ζ.

(Later we will introduce ∗ ζ instead of ζ via

ζ∗ = -ζ-+-W--, (379 ) 1 − W ζ
for the purpose of simplifying an integration.)

Taking the gradient of a -- Z (z ,ζ,ζ), multiplied by an arbitrary four vector a v, (i.e., a V = v Z,a), we see that it satisfies the linear Good Cut equation,

∂2Z, = σ, Z, (380 ) 2a Z a ∂ V = σ,Z V.
Let V0 be a particular solution, and assume for the moment that the general solution can be written as
∗ Z,a = V0la (381 )
with the four components of l∗a to be determined. Substituting Equation (381View Equation) into the linearized GCE equation we have
2 ∗ ∗ ∂ (V0la) = σ,Z V la, ∂(l∗a∂ (V0) + V0 ∂l∗a) = σ,Z V l∗a, ∗ 2 ∗ 2∗ ∗ la∂ (V0) + 2∂V0 ∂la + V0 ∂ la = σ,Z V0la, 2∂V0 ∂l∗a + V0 ∂2l∗a = 0, ∗ 2 2∗ 2V0 ∂V0∂la + V0 ∂ la = 0, ∂V 20 ∂l∗a + V02∂2l∗a = 0, ∂(V 2∂l∗) = 0, 0 a
which integrates immediately to
V 20 ∂l∗a = m ∗a (382 )

The ∗ m a are three independent l = 1, s = 1 functions. By taking linear combinations they can be written as

∗ b b m a = Tamˆb = Ta ∂ˆlb

where ˆla is our usual √- ( - --- -) ˆla = 22- 1,− ζ1++ζζζ,− i(ζ1+−ζζζ), 11−+ζζζζ. The coefficients Tab are functions only of the coordinates, za.

Assuming that the monople term in 2 V is sufficiently large so that it has no zeros and then by rescaling V we can write V −2 as a monopole plus higher harmonics in the form

V0− 2= 1 + ∂W,

where W is a spin-wt s = − 1 quantity. From Equation (382View Equation), we obtain

∗ −2 ∗ ∗ ∂la = V0 m a = (1 + ∂W )m a, ∂l∗a = m ∗a + ∂W m ∗a, ∗ ∗ ∗ ∂la = m a + ∂ (W m a), ∂l∗a = Tba∂ˆlb + Tba∂(W ˆmb), ∗ b ˆ ∂la = Ta∂(lb + W ˆmb ),
which integrates to
∗ b la = Ta (ˆla + W mˆa ). (383 )

The general solution to the linearized GCE is thus

∗ b ˆ Z,a = V0la = V0T a(la + W mˆa ), (384 ) V = vaZ,a = V0vaT ba(ˆla + W ˆma ).

We now demonstrate that

∫ (gabvavb)−1 = (8π)− 1 V− 2d Ω. (385 ) -- dζ∧d ζ dΩ = 4i--------2. (386 ) (1 + ζζ)

In the integral, (385View Equation), we replace the independent variables -- (ζ,ζ) by

-- -- ζ∗ = ζ-+-W--, ζ∗ = ζ (387 ) 1 − W ζ
after some algebraical manipulation we obtain
dΩ ∗ = V − 2d Ω, (388 ) 0
and (surprisingly)
√ --( ∗ -- -- ∗ ∗-) (ˆla + W ˆma) = L ∗≡ --2- 1, − ζ--+-ζ-,− i(ζ-−-ζ-)-, 1 −-ζ-ζ , (389 ) a 2 1 + ζ ∗ζ 1 + ζ ∗ζ 1 + ζ∗ζ
so that
a b ∗ V = V0v TaL b. (390 )

Inserting Equations (387View Equation), (388View Equation) and (390View Equation) into (385View Equation) we obtain

∫ (gabvavb)−1 = (8π)−1 (V0vaTabL ∗b)−2V02dΩ∗, (391 ) ∫ −1 a b ∗− 2 ∗ = (8π) (v TaL b) dΩ , ∫ = (8π)−1 (v∗bL ∗b)−2dΩ ∗.
Using the form Equation (389View Equation) the last integral can be easily evaluated (most easily done using 𝜃 and φ) leading to
a b− 1 ∗a ∗b −1 c d a b− 1 (gabv v ) = (ηabv v ) = (TaTb ηcdv v) , (392 ) gab = T caTbd ηycd,
our sort for relationship.

We can go a step further. By taking the derivative of Equation (391View Equation) with respect to va, we easily find the covariant form of v, namely

v g vb ∫ -----a---- = ----ab---- = (8π)−1 (vaTabL ∗b)−3TabL ∗bdΩ ∗. (gabvavb)2 (gabvavb)2


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