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❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ ▲❛♣❧❛❝❡ ❛♥❞ ❉✐r❛❝ ♦♣❡r❛t♦rs ✕ t♦✇❛r❞s s✉♣❡rs②♠♠❡tr✐❡s

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼✐❝❤❡❧ ✭❯♥✐✈❡rs✐té ❞❡ ▲✐è❣❡✱ P❆■ ❉❨●❊❙❚✮ ❋✉♥❞❛♠❡♥t❛❧ ❋r♦♥t✐❡rs ✐♥ P❤②s✐❝s ✷✵✶✹ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧②

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❉❡✜♥✐t✐♦♥ ♦❢ ❤✐❣❤❡r s②♠♠❡tr✐❡s

▲❡t ❉ ∈ D(▼) ❜❡ ❛ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ♦✈❡r ▼✱ ❡✳❣✳ ❉ = ∆ ♦♥ R♥✳ ❚❤r❡❡ t②♣❡s ♦❢ s②♠♠❡tr✐❡s✿ ❳ ∈ ❱❡❝t(▼) s✳t✳ [❉, ❳] = ✵✱ ▲✐❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❡✐❣❡♥s♣❛❝❡s❀ ❆ ∈ D(▼) s✳t✳ [❉, ❆] = ✵✱ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❡✐❣❡♥s♣❛❝❡s❀ ❝♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s✱ ❆ ∈ D(▼) s✳t✳ ❉ ◦ ❆ = ❇ ◦ ❉✱ ❛ss♦❝✐❛t✐✈❡ ❛❧❣❡❜r❛ ✴ ♣r❡s❡r✈❡ ❦❡r ❉❀ ❍✐❣❤❡r ❙②♠♠❡tr✐❡s✱ tr✐✈✐❛❧ ❍❙ ✐❢ ❆ = ❆✵ ◦ ❉✳ ❘❡♠❛r❦✿ ❣❡♥❡r❛❧✐③❡ t♦ ❉ ∈ D(▼, ❊) ❛♥❞ ❉ ∈ D(▼; ❊, ❋)✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✷ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❙❤♦rt r❡✈✐❡✇ ❛♥❞ ❛✐♠

❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥✿ s②♠♠❡tr✐❡s ♦❢ ♦r❞❡r ✷ ← → s❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ♦❢ ∆φ = ❊φ✱

❍❙ ♦❢ ♦r❞❡r ✷ ❝❧❛ss✐✜❡❞ ✐♥ t❤❡ ✢❛t ❝❛s❡ ❬❇♦②❡r✱ ❑❛❧♥✐♥s✱ ▼✐❧❧❡r ✬✼✻❪✱ ❝♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s ♦❢ ♦r❞❡r ✷ ✐♥ t❤❡ ❝✉r✈❡❞ ❝❛s❡ ❬❈❛rt❡r ✬✼✼❪✱

❢✉❧❧ ❛❧❣❡❜r❛ ♦❢ ❍❙ ❞❡t❡r♠✐♥❡❞ ✐♥ t❤❡ ✢❛t ❝❛s❡ ❬❊❛st✇♦♦❞ ✬✵✺❪✱ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❤✐❣❤❡r s♣✐♥ ✜❡❧❞ t❤❡♦r② ❬❱❛s✐❧✐❡✈ ❡t ❛❧✳❪✳ ❍❙ ♦❢ ♦t❤❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❤❛✈❡ ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞✿ ∆❦✱ / ❉✱ ✳✳✳ ❆✐♠ ♦❢ t❤❡ t❛❧❦✿ t♦ ♣r❡s❡♥t ❊❛st✇♦♦❞✬s r❡s✉❧t ✈✐❛ q✉❛♥t✐③❛t✐♦♥ ♠❡t❤♦❞s✱ ❡①t❡♥s✐♦♥ t♦ t❤❡ ❉✐r❛❝ ♦♣❡r❛t♦r ❛♥❞ t♦ t❤❡ s②st❡♠ ∆ ⊕ / ❉✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✸ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❉❡q✉❛♥t✐③❛t✐♦♥ ❛♥❞ s②♠❜♦❧ s♣❛❝❡

❖r❞❡r✿ ❧♦❝❛❧❧②✱ ❉ ✐s ♦❢ ♦r❞❡r ❦ ✐❢ ❉ = ❉✐✶···✐❦(①)∂✐✶ · · · ∂✐❦ + ❧✳♦✳t✳✳ ■♥❝r❡❛s✐♥❣ ✜❧tr❛t✐♦♥✿ D(▼) = D❦(▼) ✇✐t❤ D❦(▼) · D❧(▼) ⊆ D❦+❧(▼)✳ D(▼) ✐s ❛♥ ❛ss♦❝✐❛t✐✈❡ ✜❧t❡r❡❞ ❛❧❣❡❜r❛ ❙②♠❜♦❧s✿ S(▼) = S❦(▼) ❛♥❞ S❦(▼) := D❦(▼)/D❦−✶(▼) ∼ = Γ(S❦❚▼) ∼ = P♦❧❦(❚ ∗▼)✳ σ❦ : ❉ → σ❦(❉) := ❉✐✶···✐❦(①)∂✐✶ ⊙ · · · ⊙ ∂✐❦ = ❉✐✶···✐❦(①)♣✐✶ · · · ♣✐❦ Pr♦♣❡rt✐❡s✿ ✐❢ {·, ·} = ∂♣✐ ⊗ ∂①✐ − ∂①✐ ⊗ ∂♣✐ t❤❡ P♦✐ss♦♥ ❜r❛❝❦❡t ♦♥ ❚ ∗▼✱ t❤❡♥ σ❦+❧(❆ ◦ ❇) = σ❦(❆)σ❧(❇), σ❦+❧−✶([❆, ❇]) = {σ❦(❆), σ❧(❇)}. S(▼) ✐s ❛ ❣r❛❞❡❞ P♦✐ss♦♥ ❛❧❣❡❜r❛

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✹ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❚❤❡ ▲❛♣❧❛❝✐❛♥✱ ✐ts s②♠❜♦❧ ❛♥❞ s②♠♠❡tr✐❡s

❉❡✜♥✐t✐♦♥✿ ♦♥ (▼, ❣)✱ ✇❡ s❡t ∆ := ∇✐ě✐❥∇❥ + ❝❘✱ ✇✐t❤ ❝ ∈ R ❛♥❞ ❘ t❤❡ s❝❛❧❛r ❝✉r✈❛t✉r❡✳ ❙②♠❜♦❧✿ ❍ = σ✷(∆) = ě✐❥∂✐ ⊙ ∂❥ = ě✐❥♣✐♣❥✳ ❍❛♠✐❧t♦♥✐❛♥ ✢♦✇ ♦❢ ❍ ♦♥ ❚ ∗▼

ě

← → ❣❡♦❞❡s✐❝ s♣r❛② ♦♥ ❚▼ ❙②♠♠❡tr✐❡s✿ [∆, ❳] = ✵ ⇒ {❍, σ✶(❳)} = ✵ = ▲❳ě, ❳ ❑✐❧❧✐♥❣ ✈❡❝t♦r ✜❡❧❞❀ [∆, ❆] = ✵ ⇒ {❍, σ❦(❆)} = ✵, σ❦(❆) ❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞❀ ❤✐❞❞❡♥ s②♠♠❡tr②✱ ✐✳❡✳ s②♠♠❡tr② ♦❢ (❚ ∗▼, ❍) ♥♦t ♦❢ (▼, ě)❀ ∆ ◦ ❆ = ❇ ◦ ∆ ⇒ {❍, σ❦(❆)} ∈ (❍), σ❦(❆) ❝♦♥❢♦r♠❛❧ ❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞❀ ✐❢ ❆ = ❳ + ❢ ∈ D✶(▼)✱ t❤❡♥ ▲❳ě = ❋ě ✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✺ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❈♦♥❝❡♣t ♦❢ q✉❛♥t✐③❛t✐♦♥

❉❡✜♥✐t✐♦♥✿ ❛ q✉❛♥t✐③❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ Q : S(▼) → D(▼), s✉❝❤ t❤❛t σ❦ ◦ Q(P) = P ❢♦r ❛❧❧ P ∈ S❦(▼)✳ ❋❛❝ts✿ ♥♦ q✉❛♥t✐③❛t✐♦♥ s✳t✳ [Q(P✶), Q(P✷)] = Q({P✶, P✷}) ❢♦r ❛❧❧ P✶, P✷ ∈ S(▼)✱ ✐❢ [Q(❳), Q(❨ )] = Q({❳, ❨ }) ❢♦r ❛❧❧ ❳, ❨ ∈ S✶(▼)✱ t❤❡♥ Q(❳) = ▲❳ = ❳ + λ❞✐✈❳ ✐s t❤❡ ▲✐❡ ❞❡r✐✈❛t✐✈❡ ♦♥ λ✲❞❡♥s✐t✐❡s✱ λ ∈ R✱ ♥♦ q✉❛♥t✐③❛t✐♦♥ s✳t✳ [Q(❳), Q(P)] = Q({❳, P}) ❢♦r ❛❧❧ ❳ ∈ S✶(▼)✱ P ∈ S(▼)✳ ❊①❛♠♣❧❡s ♦♥ ❚ ∗R♥✿ ♥♦r♠❛❧ ♦r❞❡r✐♥❣ N : P✐✶···✐❦(①)♣✐✶ · · · ♣✐❦ → P✐✶···✐❦(①)∂✐✶ · · · ∂✐❦✱ N(❳) = ▲❳ ♦♥ ✵✲❞❡♥s✐t✐❡s✱ ❡q✉✐✈✳ ✉♥❞❡r gl(♥) ⋉ heis(✷♥) ∼ = ✶, ①✐, ♣✐, ①✐♣❥❀ ❲❡②❧ q✉❛♥t✐③❛t✐♦♥ Q❲ = N ◦ ❡①♣ ✶

✷❞✐✈

• ✱ ✇✐t❤ ❞✐✈ = ∂✐∂♣✐ t❤❡ ❞✐✈❡r❣❡♥❝❡✱

Q❲ (❳) = ▲❳ ♦♥ ✶

✷✲❞❡♥s✐t✐❡s✱ ❡q✉✐✈✳ ✉♥❞❡r

sp(✷♥) ⋉ heis(✷♥) ∼ = ✶, ①✐, ♣✐, ①✐①❥, ①✐♣❥, ♣✐♣❥✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✻ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❙②♠♠❡tr✐❡s ♦❢ ∆ ♦♥ R♣,q ✭■✮

❈♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s ♦❢ ∆ ✿ [∆, Q❲ (P)] = ✵ ⇔ {❍, P} = ✵❀ {❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞} ∼ = s♣❛♥(e(♣, q)) ✐♥ S(R♥)✱ ✇✐t❤ e(♣, q) t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ❑✐❧❧✐♥❣ ✈❡❝t♦r ✜❡❧❞s✳ ❍❙ ♦❢ ♦r❞❡r ✶ ♦❢ ∆ ✿ ▲❳ě = ❋ě ⇒ ∆ ◦ (❳ ✐∂✐ + ♥−✷

✷♥ ∂✐❳ ✐) = (❳ ✐∂✐ + ♥+✷ ✷♥ ∂✐❳ ✐) ◦ ∆✱

❤❡♥❝❡ ∆ : Γ(❱♦❧⊗ ♥−✷

✷♥ ) → Γ(❱♦❧⊗ ♥+✷ ✷♥ ) ✐s ❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t✳

conf(R♣,q) = o(♣ + ✶, q + ✶) ∼ = ♣✐, ①✐♣❥ − ①❥♣✐, ①✐①❥♣❥ − ✶

✷①✷♣✐✳

❚❤❡♦r❡♠ ✭❉✉✈❛❧✲▲❡❝♦♠t❡✲❖✈s✐❡♥❦♦✱ ✬✾✾✮

t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❧✐♥❡❛r ❜✐❥❡❝t✐♦♥ Q : S(R♣,q) → D(R♣,q) ✇❤✐❝❤ ✐s conf(R♣,q)✲❡q✉✐✈❛r✐❛♥t ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ♣r✐♥❝✐♣❛❧ s②♠❜♦❧ ♠❛♣s✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✼ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❙②♠♠❡tr✐❡s ♦❢ ∆ ♦♥ R♣,q ✭■✮

❈♦♠♠✉t✐♥❣ s②♠♠❡tr✐❡s ♦❢ ∆ ✿ [∆, Q❲ (P)] = ✵ ⇔ {❍, P} = ✵❀ {❑✐❧❧✐♥❣ t❡♥s♦r ✜❡❧❞} ∼ = s♣❛♥(e(♣, q)) ✐♥ S(R♥)✱ ✇✐t❤ e(♣, q) t❤❡ ▲✐❡ ❛❧❣❡❜r❛ ♦❢ ❑✐❧❧✐♥❣ ✈❡❝t♦r ✜❡❧❞s✳ ❍❙ ♦❢ ♦r❞❡r ✶ ♦❢ ∆ ✿ ▲❳ě = ❋ě ⇒ ∆ ◦ (❳ ✐∂✐ + ♥−✷

✷♥ ∂✐❳ ✐) = (❳ ✐∂✐ + ♥+✷ ✷♥ ∂✐❳ ✐) ◦ ∆✱

❤❡♥❝❡ ∆ : Γ(❱♦❧⊗ ♥−✷

✷♥ ) → Γ(❱♦❧⊗ ♥+✷ ✷♥ ) ✐s ❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t✳

conf(R♣,q) = o(♣ + ✶, q + ✶) ∼ = ♣✐, ①✐♣❥ − ①❥♣✐, ①✐①❥♣❥ − ✶

✷①✷♣✐✳

❚❤❡♦r❡♠ ✭❉✉✈❛❧✲▲❡❝♦♠t❡✲❖✈s✐❡♥❦♦✱ ✬✾✾✮

❋♦r λ, µ ❣❡♥❡r✐❝ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❧✐♥❡❛r ❜✐❥❡❝t✐♦♥ Qλ,µ : S(R♣,q) ⊗ Γ(❱♦❧⊗(µ−λ)) → D(R♣,q; ❱♦❧⊗λ, ❱♦❧⊗µ) ✇❤✐❝❤ ✐s conf(R♣,q)✲❡q✉✐✈❛r✐❛♥t ❛♥❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ♣r✐♥❝✐♣❛❧ s②♠❜♦❧ ♠❛♣s✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✼ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❙②♠♠❡tr✐❡s ♦❢ ∆ ♦♥ R♣,q ✭■■✮

❚❤❡♦r❡♠ ✭❊❛st✇♦♦❞ ✬✵✺✱ ▼✳✬✶✹✮

Q ♥−✷

✷♥ , ♥−✷ ✷♥

: {❈❑✲t❡♥s♦rs}

∼

− → {❍❙ ♦❢ ∆}. {❍❙ ♦❢ ∆}/{❚r✐✈✐❛❧ ❍❙} ∼ = U(g)/J ✱ ✇✐t❤ g = o(♣ + q + ✷, C) ❛♥❞ J ✐ts ❏♦s❡♣❤ ✐❞❡❛❧✳ ❯♥✐q✉❡ ✐♥✈❛r✐❛♥t st❛r✲❞❡❢♦r♠❛t✐♦♥ ♦❢ ❘[ O♠✐♥ ]✳ ❘❡♠❛r❦✿ ▲❡t ∆❨ = ∇✐ě✐❥∇❥ +

♥−✷ ✹(♥−✶)❘ ♦♥ S♣ × Sq✳ ■❢ ♣ + q ✐s ❡✈❡♥ ❛♥❞ ≥ ✻✱

❦❡r ∆❨ ✐s t❤❡ ♠✐♥✐♠❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❖(♣ + ✶, q + ✶)✱ ❛♥❞ J ✐s ✐ts ✈❛♥✐s❤✐♥❣ ✐❞❡❛❧ ❬❑♦❜❛②❛s❤✐✲❖rst❡❞✲✳✳✳✱ ✬✵✸✲✳✳✳❪✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✽ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ ❉✐r❛❝ ♦♣❡r❛t♦r ♦♥ R♣,q

❊①❛♠♣❧❡s✿ ✜rst ♦r❞❡r ❍❙ ❛r❡ ❝❧❛ss✐✜❡❞ ❬❇❡♥♥✲❑r❡ss✱ ✬✵✹❪✿ ě✐❥γ(ι❡✐ ❑)∇❡❥ − κ κ + ✶γ(❞❑) + ♥ − κ ✷(♥ + ✶ − κ)γ (δ❑) ✇❤❡r❡ ❑ ❜❡❧♦♥❣s t♦ {❝♦♥❢♦r♠❛❧ ❑✐❧❧✐♥❣ κ✲❢♦r♠} ∼ = κ+✶ C♣+q+✷✳ ❙❛♠❡ str❛t❡❣② ❛s ❢♦r ∆ ❛♣♣❧✐❡s✿ s②♠❜♦❧ s♣❛❝❡ ✐s ❛ ❣r❛❞❡❞ P♦✐ss♦♥ s✉♣❡r❛❧❣❡❜r❛ ✭∼ = Γ(S❚▼ ⊗ ❚ ∗▼)✮✱ / ❉ : Γ(❙ ⊗ ❱♦❧⊗ ♥−✶

✷♥ ) → Γ(❙ ⊗ ❱♦❧⊗ ♥+✶ ✷♥ ) ✐s ❝♦♥❢♦r♠❛❧❧② ✐♥✈❛r✐❛♥t✱

❝♦♥❢♦r♠❛❧❧② ❡q✉✐✈❛r✐❛♥t q✉❛♥t✐③❛t✐♦♥ ❡①✐sts ❛♥❞ ✐s ✉♥✐q✉❡ ❬▼✳✱ ✬✵✾❪✳

❚❤❡♦r❡♠ ✭ ❊❛st✇✇♦❞✲❙♦♠❜❡r❣✲❙♦✉↔❡❦❀ ▼✳✱ ❙✐❧❤❛♥✮

Q ♥−✶

✷♥ , ♥−✶ ✷♥

: {❈❑ ❤♦♦❦✲t❡♥s♦rs}

∼

− → {❍❙ ♦❢ / ❉}✳ {❍❙ ♦❢ / ❉}/{❚r✐✈✐❛❧ ❍❙} ∼ = U(g)/J /

❉✱ ✐❢ ♣ + q ✐s ♦❞❞ ♦r ♣ + q ✐s ❡✈❡♥ ❛♥❞

/ ❉ : ❙+ → ❙−✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✾ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

Pr♦❜❧❡♠❛t✐❝

❉❡t❡r♠✐♥❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❍❙ ♦❢ t❤❡ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs

C∞(R♣,q) ⊕ Γ(❙) → C∞(R♣,q) ⊕ Γ(❙) ❢ φ

• →

∆❢ / ❉φ

• ❚❤❡ ❍❙ r❡❛❞ ❛s

✵ ✵ ❉ ❛ ❆ ❜ ❇ ✵ ✵ ❉ ✇✐t❤ ♥❡✇ s②♠♠❡tr✐❡s✿ ❉ ♦♥ ❙ ❛♥❞ ❉ ♦♥

♣ q

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✵ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

Pr♦❜❧❡♠❛t✐❝

❉❡t❡r♠✐♥❡ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❍❙ ♦❢ t❤❡ s②st❡♠ ♦❢ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs

C∞(R♣,q) ⊕ Γ(❙) → C∞(R♣,q) ⊕ Γ(❙) ❢ φ

• →

∆❢ / ❉φ

• ❚❤❡ ❍❙ r❡❛❞ ❛s

∆ ✵ ✵ / ❉ ❛ α− α+ ❆

• =
• ❜

β+ β− ❇ ∆ ✵ ✵ / ❉

• ✇✐t❤ ♥❡✇ s②♠♠❡tr✐❡s✿

∆α− = β+ / ❉ ♦♥ Γ(❙) ❛♥❞ / ❉α+ = β−∆ ♦♥ C∞(R♣,q).

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✵ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❍❙

❊①❛♠♣❧❡s✿ ✐❢ Λ ∈ Γ(❙) ✐s ❛ t✇✐st♦r s♣✐♥♦r✱ ∇✐Λ = − ✶

♥γ✐( /

❉Λ)✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ s②♠♠❡tr✐❡s ❬❲❡ss✲❩✉♠✐♥♦✱ ◆✉❝❧✳ P❤②s✳ ❇ ✶✾✼✹❪✿ ∆α−

Λ = β+ Λ /

❉,

• α−

Λ (φ) = ε(Λ, φ),

β+

Λ ( /

❉φ) = ε(Λ, / ❉

✷φ) + ✷ ♥ε( /

❉Λ, / ❉φ), φ ∈ Γ(❙); / ❉α+

Λ = β− Λ ∆,

• α+

Λ (❢ ) = γ✐(Λ)∂✐❢ + ♥−✷ ♥ ( /

❉Λ) · ❢ , β−

Λ (∆❢ ) = Λ · ∆❢ ,

❢ ∈ C∞(R♣,q).

Pr♦♣♦s✐t✐♦♥ ✭▼✳✱ ❙✐❧❤❛♥✮

❚❤❡ ♠❛tr✐① ♦❢ ♦♣❡r❛t♦rs

• ❛

α− α+ ❆

• ✐s ❛ ❍❙ ✐✛

❛ ✐s ❛ ❍❙ ♦❢ ∆ ❛♥❞ ❆ ✐s ❛ ❍❙ ♦❢ / ❉✱ α− =

✐ ❛✐ ◦ α− Λ✐ ✱ ✇✐t❤ ❛✐ ❍❙ ♦❢ ∆ ❛♥❞ α− Λ✐ ❛s ❛❜♦✈❡✱

α+ =

✐ α+ Λ✐ ◦ ❛✐✱ ✇✐t❤ ❛✐ ❍❙ ♦❢ ∆ ❛♥❞ α+ Λ✐ ❛s ❛❜♦✈❡✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✶ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❈♦♠♣♦s✐t✐♦♥ ♦❢ t✇✐st♦r s♣✐♥♦rs ❛❝t✐♦♥s

▲✐❡ ✭s✉♣❡r✲✮❛❧❣❡❜r❛ ❄

❈❛♥❞✐❞❛t❡✿ ❈♦♥❢✳ ❑✐❧❧✐♥❣ ✈❡❝t♦r ✜❡❧❞s ⊕ ❚✇✐st♦r✲s♣✐♥♦rs✳ ❍✐♥t ❢r♦♠ ❘❡♣✳ ❚❤❡♦r②✿ ✐♥ ♦❞❞ ❞✐♠❡♥s✐♦♥✱ ❚✇❙♣ ⊗ ❚✇❙♣ ∼ = ∧+C♥+✷ ∼ = s♣❛❝❡ ♦❢ ❝♦♥❢✳ ❑✐❧❧✐♥❣ ♦❞❞ ❢♦r♠s❀ ✐♥ ❡✈❡♥ ❞✐♠❡♥s✐♦♥✱ ❚✇❙♣ ⊗ ❚✇❙♣ ∼ = ∧C♥+✷ ∼ = s♣❛❝❡ ♦❢ ❛❧❧ ❝♦♥❢✳ ❑✐❧❧✐♥❣ ❢♦r♠s✳ ❋❛❝t✿ t❤❡ ❝♦♠♣♦s✐t✐♦♥ α+

Λ′ ◦ α− Λ ❣✐✈❡s ✐♥❞❡❡❞ r✐s❡ t♦ ❛❧❧ ❍❙ ♦❢ ✶st ♦r❞❡r ♦❢ /

❉✳

Pr♦♣♦s✐t✐♦♥ ✭▼✳✱ ❙✐❧❤❛♥✮

■❢ ♥ ≥ ✺✱ t❤❡r❡ ❡①✐sts ♥♦ ▲✐❡ ✭s✉♣❡r✲✮❛❧❣❡❜r❛ ❝♦♥t❛✐♥✐♥❣ o(♥ + ✷, C) ❛♥❞ ❣❡♥❡r❛t✐♥❣ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❍❙ ♦❢ ∆ ⊕ / ❉✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✷ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❙✉♣❡r❣❡♦♠❡tr✐❝ r❡❢♦r♠✉❧❛t✐♦♥ ✐♥ ❞✐♠❡♥s✐♦♥ ✸

Π❙∗ ∼ = R✸|✷ ✐s ❛ s✉♣❡r♠❛♥✐❢♦❧❞ ✇✐t❤ s❤❡❛❢ ♦❢ ❢✉♥❝t✐♦♥s O(Π❙∗) = C∞(R✸) ⊕ Γ(❙) ⊕ Γ(∧✷❙). ❚❤❡ ♣❛✐r✐♥❣ ε ✐s ❛ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥t ♦❢ Γ(∧✷❙)✳ ❲❡ ❞❡✜♥❡ : O

• Π❙∗

→ O

• Π❙∗

❜② t❤❡ ❢♦r♠✉❧❛ := ε∆ + / ❉ + ε∗ =   ✵ ✵ ✶ ✵ / ❉ ✵ ∆ ✵ ✵  ✳

Pr♦♣♦s✐t✐♦♥

❲❡ ❤❛✈❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ❛❧❣❡❜r❛s {❍❙ ♦❢ }/{tr✐✈✐❛❧ ❍❙} ∼ = {❍❙ ♦❢ ∆ ✵ ✵ / ❉

• }/{tr✐✈✐❛❧ ❍❙}.

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✸ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

❚✇✐st♦r✲s♣✐♥♦rs ❛s ♦❞❞ ✈❡❝t♦r ✜❡❧❞s

▲❡t (①✐, θ❛) ❜❡ ❝♦♦r❞✐♥❛t❡s ♦♥ R✸|✷ ❛♥❞ (∂✐, ∂θ❛) t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡r✐✈❛t✐✈❡s✳ ❋♦r ❛❧❧ λ ∈ R✱ ✇❡ ❞❡✜♥❡ ♦♥ O

• Π❙∗

▲❳ = ❳ ✐∂✐ − ✶

✷γ(❞❳ ♭)❜ ❛θ❛∂θ❜ + (λ + ✶ ✷♥θ❛∂θ❛)(∂✐❳ ✐)✱

▲+

Λ = γ✐ ❛ ❜Λ❛θ❜∂✐ + ✷(λ − ✶ ♥θ❛∂θ❛)( /

❉Λ), ▲−

Λ = ε❛❜Λ❛∂θ❜✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ s♣❛❝❡ ❝, ▲❳ ⊕ ▲+

Λ , ▲− Λ ✐s st❛❜❧❡ ✉♥❞❡r t❤❡ ❝♦♠♠✉t❛t♦r ✐♥ D(R✸|✷) ❛♥❞

✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ▲✐❡ s✉♣❡r❛❧❣❡❜r❛ s♣♦(✹|✷)✳ ❋♦r λ = ♥−✷

✷♥ ✱ ✇❡ ❤❛✈❡

▲❳ = ▲❳, ▲+

Λ = ▲− Λ ,

▲−

Λ = ▲+ Λ .

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✹ ✴ ✶✺

■♥tr♦❞✉❝t✐♦♥ ❍✐❣❤❡r s②♠♠❡tr✐❡s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ❍❙ ♦❢ ∆ ⊕ / ❉

▼❛✐♥ r❡s✉❧t

❚❤❡♦r❡♠ ✭▼✳✱ ➆✐❧❤❛♥ ✮

■♥ ❞✐♠ ✸✱ {❍❙ ♦❢ ∆ ⊕ / ❉}/{❚r✐✈✐❛❧ ❍❙} ∼ = U(spo(✹|✷)/J ✱ ✇✐t❤ J ❛ ❏♦s❡♣❤✲❧✐❦❡ ✐❞❡❛❧✳ ■♥ ❞✐♠ ✹✱ {❍❙ ♦❢ ∆ ⊕ / ❉}/{❚r✐✈✐❛❧ ❍❙} ∼ = U(sl(✹|✶)/J ✳

❏❡❛♥✲P❤✐❧✐♣♣❡ ▼■❈❍❊▲ ✭❯▲❣✮ ▼❛rs❡✐❧❧❡✱ ✶✼t❤ ❏✉❧② ✶✺ ✴ ✶✺