Manuscript IX-Free-2@1 · IX-Free-2.tex · section · abb2397a

Actual spaces, declared domain, and coefficient assignment

Actual spaces, declared domain, and coefficient assignment · lines 585–693

IX-Free-2@1#eq:Dfirst, IX-Free-2@1#eq:Dsecond, IX-Free-2@1#eq:Dnewold, IX-Free-2@1#eq:Ddecl

All spans and tensor products in this paper are algebraic over $C$ . Define

B = span {B_{f} : f \in T}, W = span {W_{g} : g \in T},

L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},

R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}},

and

L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .

These last sums are direct by total field degree. The actual spaces $L_{4}, R_{4}$ coincide, but their ordered roles remain distinct.

Let

E_{L} = T \otimes T, E_{R} = T \otimes T,

with ordered presentation maps

p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},

p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .

Inside $P_{Σ} \otimes P_{Σ}$ , the declared domains are

D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, = D_{new} + D_{old} . D_{old} = P_{mc} \otimes P_{mc},

Thus the full inherited old domain is used, not a smaller regular subdomain.

We will define coefficient maps

C_{j} : D_{decl} ⟶ P_{Σ}, 0 \leq j \leq 3.

Their prescribed restrictions are displayed below. On the degree-four rows, $L_{j}, R_{j}$ initially denote the labelled prescriptions; their actual descent is proved in Section sec:descent.

{}1.2

Domain	$C_{1}$	$C_{2}$	$C_{3}$
$B \otimes W$	$Q_{f, g}$	$0$	$0$
$W \otimes B$	$Q^{R}_{g, f^{'}}$	$0$	$0$
$L_{4} \otimes B$	$L_{1}$	$L_{2}$	$0$
$B \otimes R_{4}$	$R_{1}$	$R_{2}$	$0$
$L_{2} \otimes B$	$G_{12, 23} + G_{12, 13}$	$N_{c} (f, g, f^{'}) 1$	$0$
$B \otimes R_{2}$	$G_{12, 23} + G_{13, 23}$	$N_{c} (f, g, f^{'}) 1$	$0$
$D_{old}$	$C_{1}^{H}$	$C_{2}^{H}$	$C_{3}^{H}$

On every row,

C_{0} (F \otimes G) = F G .

The explicit zero entries are part of the selected contraction prescription. They are not a classification excluding every alternative contact term.

Displayed formulas in this passage

IX-Free-2@1#eq:Dfirst · line 627
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, = D_{new} + D_{old} . D_{old} = P_{mc} \otimes P_{mc},$
IX-Free-2@1#eq:Dsecond · line 627
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, = D_{new} + D_{old} . D_{old} = P_{mc} \otimes P_{mc},$
IX-Free-2@1#eq:Dnewold · line 627
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, = D_{new} + D_{old} . D_{old} = P_{mc} \otimes P_{mc},$
IX-Free-2@1#eq:Ddecl · line 627
$D_{first} D_{second} D_{new} D_{decl} = (B \otimes W) + (W \otimes B), = (L \otimes B) + (B \otimes R), = D_{first} + D_{second}, = D_{new} + D_{old} . D_{old} = P_{mc} \otimes P_{mc},$
unlabeled · line 590
$B = span {B_{f} : f \in T}, W = span {W_{g} : g \in T},$
unlabeled · line 594
$L_{4} = span {B_{f} W_{g}}, L_{2} = span {Q_{f, g}},$
unlabeled · line 598
$R_{4} = span {W_{g} B_{f^{'}}}, R_{2} = span {Q^{R}_{g, f^{'}}},$
unlabeled · line 603
$L = L_{4} \oplus L_{2}, R = R_{4} \oplus R_{2} .$
unlabeled · line 612
$E_{L} = T \otimes T, E_{R} = T \otimes T,$
unlabeled · line 617
$p_{L} (f \otimes g) = B_{f} W_{g}, q_{L} (f \otimes g) = Q_{f, g},$
unlabeled · line 621
$p_{R} (g \otimes f^{'}) = W_{g} B_{f^{'}}, q_{R} (g \otimes f^{'}) = Q^{R}_{g, f^{'}} .$
unlabeled · line 645
$C_{j} : D_{decl} ⟶ P_{Σ}, 0 \leq j \leq 3.$
unlabeled · line 686
$C_{0} (F \otimes G) = F G .$