r ss r - - PowerPoint PPT Presentation

SLIDE 1

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s ♦❢ ✐♥t❡❣r❛❜❧❡ ❍❛♠✐❧t♦♥✐❛♥ s②st❡♠s

❏✉❧✐❛ ❇❡r♥❛ts❦❛ ✭❇❡r♥❛ts❦❛❏▼❅✉❦♠❛✳❦✐❡✈✳✉❛✮ P❡tr♦ ❍♦❧♦❞ ✭❍♦❧♦❞❅✉❦♠❛✳❦✐❡✈✳✉❛✮

◆❛t✐♦♥❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❵❑✐❡✈✲▼♦❤②❧❛ ❆❝❛❞❡♠②✬ ❇♦❣♦❧②✉❜♦✈ ■♥st✐t✉t❡ ❢♦r ❚❤♦r❡t✐❝❛❧ P❤②s✐❝s

• ❡♦♠❡tr②✱ ■♥t❡❣r❛❜✐❧✐t② ◗✉❛♥t✐③❛t✐♦♥✱ ✷✵✶✷

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 2

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

❖✉t❧✐♥❡

✶

• ❡♥❡r❛❧ s❝❤❡♠❡

❆❧❣❡❜r❛✐❝ ✐♥t❡❣r❛❜❧❡ s②st❡♠s P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❈❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥

✷

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

✸

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 3

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ ❆❧❣❡❜r❛✐❝ ✐♥t❡❣r❛❜❧❡ s②st❡♠s P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❈❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥

❆❧❣❡❜r❛✐❝ ■♥t❡❣r❛❜❧❡ ❙②st❡♠s

■♥t❡❣r❛❜✐❧✐t② ✐♥ ❑♦✇❛❧❡✇s❦❛ s❡♥s❡ ❆♥② s♦❧✉t✐♦♥ ♦❢ ❛ s②st❡♠ ❛❞♠✐ts ❛ ❤♦❧♦♠♦r♣❤✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ✐♥ t✐♠❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥② s♦❧✉t✐♦♥ ✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡ R✳ ■♥t❡❣r❛❜❧❡ s②st❡♠s ♦♥ ♦r❜✐ts ♦❢ ❛ ❧♦♦♣ ❣r♦✉♣ ♦❜❡② ❡q✉❛t✐♦♥s ♦❢ ▲❛① t②♣❡ ❞▲(λ) ❞t = [❆, ▲(λ)], ▲(λ) = α(λ) β(λ) γ(λ) −α(λ)

• ▲ ∈

g∗,

• g = sl(✷, C) × P(λ, λ−✶)

α(λ) = ◆

❥=✵ α❥λ❥,

β(λ) = ◆

❥=✵ β❥λ❥,

γ(λ) = ◆

❥=✵ γ❥λ❥.

❚❤❡ s♣❡❝tr✉♠ ♦❢ ▲ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ⇒ ❚❤❡r❡ ❡①✐sts t❤❡ s♣❡❝tr❛❧ ❝✉r✈❡ R = {❞❡t(▲(λ) − ✇) = ✵}.

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 4

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ ❆❧❣❡❜r❛✐❝ ✐♥t❡❣r❛❜❧❡ s②st❡♠s P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❈❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥

P❤❛s❡ ❙♣❛❝❡ ❙tr✉❝t✉r❡

❆ ✜♥✐t❡ ❣❛♣ ♣❤❛s❡ s♣❛❝❡ ♦❢ t❤❡ s②st❡♠ ✐s ❛ ❝♦❛❞❥♦✐♥t ♦r❜✐t O◆ ♦❢ t❤❡ ❧♦♦♣ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② g✿ O◆ = {❚r ▲❦(λ) = ❝♦♥st}. ❚❤❡ ❝♦♠♣❧❡① ▲✐♦✉✈✐❧❧❡ t♦r✉s ♦❢ t❤❡ s②st❡♠ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❏❛❝♦❜✐❛♥ ♦❢ ❛ ❘✐❡♠❛♥♥ s✉r❢❛❝❡ R ✭✇❤✐❝❤ ✐s t❤❡ s♣❡❝tr❛❧ ❝✉r✈❡✮✿

• ❏❛❝(R) = ❙②♠♠◆ R × R × · · · × R
• ◆

, ◆ > ❣, ✇❤❡r❡ ❣ ✐s t❤❡ ❣❡♥✉s ♦❢ R✳

Pr❡✈✐❛t♦ ❊✳ ❍②♣❡r❡❧❧✐♣t✐❝ q✉❛s✐✲♣❡r✐♦❞✐❝ ❛♥❞ s♦❧✐t♦♥ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♥♦♥❧✐♥❡❛r ❙❝❤r♦❞✐♥❣❡r ❡q✉❛t✐♦♥✱ ❉✉❦❡ ▼❛t❤✳ ❏✳✱ ✺✷ ✭✶✾✽✺✮✱ ✸✷✸✕✸✸✷✳

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 5

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ ❆❧❣❡❜r❛✐❝ ✐♥t❡❣r❛❜❧❡ s②st❡♠s P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❈❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥

❙❡♣❛r❛t✐♦♥ ♦❢ ❱❛r✐❛❜❧❡s

P❤❛s❡ s♣❛❝❡ {γ✵, . . . , γ◆−✶, α✵, . . . , α◆−✶} = ⇒ ❈❛♥♦♥✐❝❛❧❧② ❝♦♥❥✉❣❛t❡❞ ✈❛r✐❛❜❧❡s {λ✶, . . . , λ◆, ✇✶, . . . , ✇◆} ❚❤❡ ❡q✉❛t✐♦♥s ♦❢ ♦r❜✐t ❡❧✐♠✐♥❛t❡ t❤❡ ✈❛r✐❛❜❧❡s {β✵, . . . , β◆−✶}✿ ❢❦(α, β, γ) = ❝❦ ❦ = ✶, . . . , ◆ ⇒ β❥ = β❥(γ, α, ❝) ❥ = ✵, . . . , ◆ − ✶ ⇒ ⇒ ❤❦(γ, α, ❝) = ❤❦(λ, ✇, ❝) ❢♦r ❍❛♠✐❧t♦♥✐❛♥s ❤✶, . . . , ❤◆. ■❢ ♦♥❡ r❡q✉✐r❡s (λ❦, ✇❦) ❜❡ ❛ ♣♦✐♥t ♦❢ t❤❡ s♣❡❝tr❛❧ ❝✉r✈❡ R t❤❡♥ γ(λ❦) = ✵✿ ❞❡t(▲(λ❦) − ✇❦) = ✵ ⇒ γ(λ❦) = ✵.

❇❡r♥❛ts❦❛ ❏✳✱ ❍♦❧♦❞ P✳ ❖♥ ❙❡♣❛r❛t✐♦♥ ♦❢ ❱❛r✐❛❜❧❡s ❢♦r ■♥t❡❣r❛❜❧❡ ❊q✉❛t✐♦♥s ♦❢ ❙♦❧✐t♦♥ ❚②♣❡✱ ❏♦✉r♥❛❧ ♦❢ ◆♦♥❧✐♥❡❛r ▼❛t❤✳ P❤②s✳✱ ✶✹ ✭✷✵✵✼✮✱ ✸✺✸✕✸✼✹✳

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 6

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ ❆❧❣❡❜r❛✐❝ ✐♥t❡❣r❛❜❧❡ s②st❡♠s P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ❈❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥

❈❛♥♦♥✐❝❛❧ ◗✉❛♥t✐③❛t✐♦♥

▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞ ❚❤✐s ✐s ❛ ❤❛❧❢✲❞✐♠❡♥s✐♦♥❛❧ s✉❜♠❛♥✐❢♦❧❞ ✐♥ t❤❡ ♣❤❛s❡ s♣❛❝❡ s✉❝❤ t❤❛t t❤❡ ❡①t❡r✐♦r ❢♦r♠ s♣❡❝✐❢②✐♥❣ t❤❡ s②♠♣❧❡❝t✐❝ str✉❝t✉r❡ ♦♥ t❤❡ ♣❤❛s❡ s♣❛❝❡ ✈❛♥✐s❤❡s ✐❞❡♥t✐❝❛❧❧② ♦♥ ✐t✳ ■♥ t❡r♠s ♦❢ t❤❡ ❝❛♥♦♥✐❝❛❧ ❝♦♦r❞✐♥❛t❡s {λ✶, . . . , λ◆, ✇✶, . . . , ✇◆} ❛ s✉❜♠❛♥✐❢♦❧❞ ♣❛r❛♠❡t❡r✐③❡❞ ❜② {λ✶, . . . , λ◆} ✐s ❛ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞✳ ◗✉❛♥t✐③❛t✐♦♥ ✐♥ t❤❡ ❙❝❤r☎ ♦❞✐♥❣❡r ♣✐❝t✉r❡ ∼ λ❦ → ˆ λ❦, ✇❦ → ˆ ✇❦ = −✐ ∂ ∂λ❦ , {λ❦, ✇❧} = δ❦❧ → [ˆ λ❦, ˆ ✇❧] = ✐δ❦❧ I . ∼ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ ♦❢ ♣❤❛s❡ s♣❛❝❡ s②♠♠❡tr② ❣r♦✉♣ ♦♥ ❛ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞✳

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 7

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

❊①❛♠♣❧❡✿ ❈❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ▼♦❞❡❧

■s♦tr♦♣✐❝ ▲❛♥❞❛✉✖▲✐❢s❤✐ts ❡q✉❛t✐♦♥✿ ∂µ ∂t = ✶ ✷❝✵

• µ, ∂✷µ

∂①✷

• + ❝✶

✷❝✵ ∂µ ∂① , µ =

• µ(✵)

✶ , µ(✵) ✷ , µ(✵) ✸

• .

❆s ❛ s②st❡♠ ♦♥ ❛ ❝♦❛❞❥♦✐♥t ♦r❜✐t ♦❢ t❤❡ ❧♦♦♣ ❛❧❣❡❜r❛ su(✷) × P(λ, λ−✶) ▲(λ) =

• ✐µ✸(λ)

µ✶(λ) − ✐µ✷(λ) −µ✶(λ) − ✐µ✷(λ) −✐µ✸(λ)

• ◆ = ✷

µ✶(λ) = µ(✵)

✶

+ µ(✶)

✶ λ,

µ✷(λ) = µ(✵)

✷

+ µ(✶)

✷ λ,

µ✸(λ) = µ(✵)

✸

+ µ(✶)

✸ λ + κλ✷,

κ = ❝♦♥st . ❙♣❡❝tr❛❧ ❝✉r✈❡ R : λ✹✇✷ = −κ✷λ✹ − ❤✸λ✸ − ❤✷λ✷ − ❝✶λ − ❝✵.

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 8

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

P❤❛s❡ ❙♣❛❝❡ ❙tr✉❝t✉r❡

P♦✐ss♦♥ str✉❝t✉r❡ ♦❢ t❤❡ ♣❤❛s❡ s♣❛❝❡✿ {µ(✵)

❦ , µ(✵) ❧

} = ✵, {µ(✵)

❦ , µ(✶) ❧

} = ε❦❧❥µ(✵)

❥

, {µ(✶)

❦ , µ(✶) ❧

} = ε❦❧❥µ(✶)

❥

. µ(✵)

✶ , µ(✵) ✷ , µ(✵) ✸ , µ(✶) ✶ , µ(✶) ✷ , µ(✶) ✸

❢♦r♠ t❤❡ ❛❧❣❡❜r❛ ♦❢ ❊✉❝❧✐❞✐❛♥ ❣r♦✉♣ ❊(✸) = ❙❖(✸) ⋉ ❚✸✱ ✇❤✐❝❤ ✐s t❤❡ ♣❤❛s❡ s♣❛❝❡ s②♠♠❡tr② ❣r♦✉♣✳ ❚❤❡♥ ❞❡♥♦t❡ µ(✵)

✶

= ♣✶, µ(✵)

✷

= ♣✷, µ(✵)

✸

= ♣✸, µ(✶)

✶

= ▲✶, µ(✶)

✷

= ▲✷, µ(✶)

✸

= ▲✸. ❊q✉❛t✐♦♥s ♦❢ t❤❡ ♦r❜✐t ❝✵ = −(♣, ♣) ❝✶ = −✷(♣, ▲) ❍❛♠✐❧t♦♥✐❛♥s ❤✷ = −(▲, ▲) − ✷κ♣✸ ❤✸ = −✷κ▲✸

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 9

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

❙❡♣❛r❛t✐♦♥ ♦❢ ❱❛r✐❛❜❧❡s

λ✶, λ✷ : µ✶(λ❦) + ✐µ✸(λ❦) = ✵, ✇❦ = ✐µ✷(λ❦)/λ✷

❦.

♣✷ = ✐λ✶λ✷ λ✶✇✶ − λ✷✇✷ λ✶ − λ✷ , ♣✶ = ✐ ✷

• κλ✶λ✷ +

❝✵ κλ✶λ✷ − λ✶λ✷(λ✶✇✶ − λ✷✇✷)✷ κ(λ✶ − λ✷)✷

• ,

♣✸ = ✶ ✷

• κλ✶λ✷ −

❝✵ κλ✶λ✷ + λ✶λ✷(λ✶✇✶ − λ✷✇✷)✷ κ(λ✶ − λ✷)✷

• ,

▲✷ = −✐λ✷

✶✇✶ − λ✷ ✷✇✷

λ✶ − λ✷ , ▲✶ = ✐ ✷

• −κ(λ + λ✷) +

❝✶ κλ✶λ✷ + ❝✵(λ✶ + λ✷) κλ✷

✶λ✷ ✷

+ λ✷

✶✇ ✷ ✶ − λ✷ ✷✇ ✷ ✷

κ(λ✶ − λ✷)

• ,

▲✸ = ✶ ✷

• −κ(λ + λ✷) −

❝✶ κλ✶λ✷ − ❝✵(λ✶ + λ✷) κλ✷

✶λ✷ ✷

− λ✷

✶✇ ✷ ✶ − λ✷ ✷✇ ✷ ✷

κ(λ✶ − λ✷)

• .

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 10

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ ❛❧❣❡❜r❛ e(✸) r❡♣r❡s❡♥t❛t✐♦♥

❚❤❡ ❝❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥ λ❦ → ˆ λ❦, ✇❦ → ˆ ✇❦ = −✐∂/∂λ❦ ❣✐✈❡s ❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ e(✸) = {ˆ ▲✸, ˆ ▲± = ˆ ▲✶ ± ✐ˆ ▲✷, ˆ ♣✸, ˆ ♣± = ˆ ♣✶ ± ✐ˆ ♣✷} [ˆ ▲✸, ˆ ▲±] = ±ˆ ▲±, [ˆ ▲+, ˆ ▲−] = ✷ˆ ▲✸, [ˆ ♣✸, ˆ ♣±] = ✵, [ˆ ♣+, ˆ ♣−] = ✵, [ˆ ▲✸, ˆ ♣±] = [ˆ ♣✸, ˆ ▲±] = ±ˆ ♣±, [ˆ ▲+, ˆ ♣−] = [ˆ ♣+, ˆ ▲−] = ✷ˆ ♣✸. ❲✐t❤ ③ = ✷κλ/, ˜ ❝✵ = ✹κ✷❝✵/✹, ˜ ❝✶ = ✷κ❝✶/✸

ˆ ▲✸ = ③✷

✶

③✶ − ③✷ ∂✷ ∂③✷

✶

− ✶ ✹ + ˜ ❝✶ ③✸

✶

+ ˜ ❝✵ ③✹

✶

• −

③✷

✷

③✶ − ③✷ ∂✷ ∂③✷

✷

− ✶ ✹ + ˜ ❝✶ ③✸

✷

+ ˜ ❝✵ ③✹

✷

• ,

ˆ ▲± = ✐③✷

✶

③✶ − ③✷

• − ∂✷

∂③✷

✶

− ✶ ✹ − ˜ ❝✶ ③✸

✶

− ˜ ❝✵ ③✹

✶

∓ ∂ ∂③✶

• −

✐③✷

✷

③✶ − ③✷

• − ∂✷

∂③✷

✷

− ✶ ✹ − ˜ ❝✶ ③✸

✷

− ˜ ❝✵ ③✹

✷

∓ ∂ ∂③✷

• ,

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 11

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ ❛❧❣❡❜r❛ e(✸) r❡♣r❡s❡♥t❛t✐♦♥

ˆ ♣✸ = − ✷κ

• ③✶③✷

(③✶ − ③✷)✷

• ③✷

✶

∂✷ ∂③✷

✶

+ ③✷

✷

∂✷ ∂③✷

✷

− ✷③✶③✷ ∂✷ ∂③✶∂③✷

• − ③✶③✷

✹ + ˜ ❝✵ ③✶③✷ − − ✷③✷

✶③✷ ✷

(③✶ − ③✷)✸ ∂ ∂③✶ − ∂ ∂③✷

• ,

ˆ ♣± = ✐ ✷κ

• ③✶③✷

(③✶ − ③✷)✷

• ③✷

✶

∂✷ ∂③✷

✶

+ ③✷

✷

∂✷ ∂③✷

✷

− ✷③✶③✷ ∂✷ ∂③✶∂③✷

• + ③✶③✷

✹ + ˜ ❝✵ ③✶③✷ − − ✷③✷

✶③✷ ✷

(③✶ − ③✷)✸ ∂ ∂③✶ − ∂ ∂③✷

• ±

③✶③✷ ③✶ − ③✷

• ③✶

∂ ∂③✶ − ③✷ ∂ ∂③✷

• .

❍❛♠✐❧t♦♥✐❛♥s✿

ˆ ❤✷ = ✷③✷

✶③✷

③✶ − ③✷ ∂✷ ∂③✷

✶

− ✶ ✹ + ˜ ❝✵ ③✹

✶

+ ˜ ❝✶ ③✸

✶

• − ✷③✶③✷

✷

③✶ − ③✷ ∂✷ ∂③✷

✷

− ✶ ✹ + ˜ ❝✵ ③✹

✷

+ ˜ ❝✶ ③✸

✷

• ,

ˆ ❤✸ = − ✷κ③✷

✶

③✶ − ③✷ ∂✷ ∂③✷

✶

− ✶ ✹ + ˜ ❝✵ ③✹

✶

+ ˜ ❝✶ ③✸

✶

• + ✷κ③✷

✷

③✶ − ③✷ ∂✷ ∂③✷

✷

− ✶ ✹ + ˜ ❝✵ ③✹

✷

+ ˜ ❝✶ ③✸

✷

• .

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 12

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

❍❛r♠♦♥✐❝ ❆♥❛❧②s✐s✿ t❤❡ ❝❛s❡ ♦❢ ❝✵ = ✵✱ ❝✶ = ✵

❈♦♥s✐❞❡r t❤❡ ♦r❜✐t O✵ ✇✐t❤ ❝✵ = ✵✱ ❝✶ = ✵✿ (♣, ♣) = ✵, (♣, ▲) = ✵. ❚❤❡ s♣❡❝tr❛❧ ❝✉r✈❡ R : λ✷✇✷ = −κ✷λ✷ − ❤✸λ − ❤✷λ. ❖♥ t❤❡ ♦r❜✐t O✵ t❤❡ ❛❧❣❡❜r❛ sl(✷) = {ˆ ▲+, ˆ ▲−, ˆ ▲✸} ❛❝ts✳ ❘❡♣r❡s❡♥t❛t✐♦♥ s♣❛❝❡ ˆ ▲✸❢ (③✶, ③✷) = ♠❢ (③✶, ③✷), ❢ (③✶, ③✷) = ❲ (③✶)❲ (③✷) ❲ ′′ +

• −✶

✹ − ♠ ③ − ❈ ③✷

• ❲ = ✵,

❈ = µ✷ − ✶/✹ ✖ t❤❡ ❲❤✐tt❛❦❡r ❡q✉❛t✐♦♥ ✇✐t❤ s♦❧✉t✐♦♥s ❲−♠,µ(③)✳ ˆ ▲✷❢ (③✶, ③✷) = ❏(❏ + ✶)❢ (③✶, ③✷) ⇒ ❈ = ❏(❏ + ✶).

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 13

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ sl(✷)

❋♦r µ = −❏ − ✶/✷ ❲❤✐tt❛❦❡r ❢✉♥❝t✐♦♥s ❲−♠,µ ❛r❡ ❝♦♥♥❡❝t❡❞ ✇✐t❤ ❛ss♦❝✐❛t❡❞ ▲❛❣✉❡rr❡ ♣♦❧②♥♦♠✐❛❧s ▲α

♥✳

❇❛s✐s ❢✉♥❝t✐♦♥s ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ s♣❛❝❡ ❢❏♠(③✶, ③✷) ∼ (③✶③✷)−❏❡−(③✶+③✷)/✷▲−✷❏−✶

❏−♠

(③✶ + ③✷). ❊✈❡r② ❢✉♥❝t✐♦♥ ❢❏❏(③✶, ③✷) = (③✶③✷)−❏❡−(③✶+③✷)/✷✱ ❏ = ✵, ✶, . . . ✱ ❣✐✈❡s r✐s❡ t♦ t❤❡ sl(✷) ❱❡r♠❛ ♠♦❞✉❧❡ {❢❏♠ = ˆ ▲❏−♠

−

❢❏❏, ♠ = ❏, ❏ − ✶, . . . }✿ ˆ ▲✸❢❏♠ = ♠❢❏♠, ˆ ▲−❢❏♠ = ❢❏,♠−✶, ˆ ▲+❢❏♠ = (❏ − ♠)(❏ + ♠ + ✶)❢❏,♠+✶. ❚❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ♥♦t st❛♥❞❛r❞ ˆ ▲±˜ ❢❏♠ =

• (❏ ∓ ♠)(❏ ± ♠ + ✶) ˜

❢❏,♠±✶, ˆ ▲✸˜ ❢❏♠ = ♠˜ ❢❏♠.

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 14

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥ P❤❛s❡ s♣❛❝❡ str✉❝t✉r❡ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ◗✉❛♥t✐③❛t✐♦♥ ∼ t❤❡ s②♠♠❡tr② ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s

✬❯♥✐t❛r✐③❛t✐♦♥✬ ♦❢ sl(✷) ❘❡♣r❡s❡♥t❛t✐♦♥

❚❤❡ st❛♥❞❛r❞ ✭❝❛♥♦♥✐❝❛❧✮ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❝♦♥str✉❝t❡❞ ❜② ♠❡❛♥s ♦❢ t❤❡ ✐♥t❡rt✇✐♥✐♥❣ ♦♣❡r❛t♦r ˆ ❆ ˜ ❢❏♠ ≡ ˆ ❆❢❏♠ =

• Γ(❏ + ♠ + ✶)

Γ(❏ − ♠ + ✶)❢❏♠ = ✐❏−♠ Γ(❏ + ♠ + ✶)Γ(❏ − ♠ + ✶)× ×(③✶③✷)−❏❡−(③✶+③✷)/✷▲−✷❏−✶

❏−♠

(③✶ + ③✷)✳ ❚❤❡ ✐♥♥❡r ♣r♦❞✉❝t ˜ ❢❏♠, ˜ ❢❏♥ = ∞

✵

∞

✵

˜ ❢ ∗

❏♠(③✶, ③✷)˜

❢❏♥(③✶, ③✷) ❏−♥

✐=✵ Γ(−❏+✐) ✐! Γ(−♥−✐) (❏−♥−✐)!

❞③✶❞③✷ ③✶−❏

✶

③✶−❏

✷

= δ♥♠. ❍❡r❡ t❤❡ s✉♠♠❛t✐♦♥ t❤❡♦r❡♠ ❛♥❞ t❤❡ ✐♥t❡❣r❛❧ ✭❞✐✈❡r❣❡♥t✮ ❛r❡ ✉s❡❞✿ ∞

✵

❡−①①α▲α

♥(①)▲α ♠(①) ❞① = Γ(α + ♥ + ✶)

♥! δ♥♠, α = −✷❏ − ✶.

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 15

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

❆ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❛❧❣❡❜r❛✐❝ ❣❡♦♠❡tr② ♠❡t❤♦❞s ✇✐t❤ ♠❡t❤♦❞s ♦❢ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ❢♦r ▲✐❡ ❛❧❣❡❜r❛s ❣✐✈❡s ❛ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❤❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ❛ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞✳ ❚❤❡ ❝❛s❡ ✐s✿ ❛♥ ❛❧❣❡❜r❛ r❡♣r❡s❡♥t❛t✐♦♥ ✐s r❡❛❧✐③❡❞ ❜② ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ♦❢ ❤✐❣❤ ♦r❞❡r✱ ❛♥❞ ❝❛♥ ♥♦t ❜❡ r✐s❡♥ t♦ ❛ ❣r♦✉♣✳ ❚❤❡r❡ ❛r❡ ❛ ❧♦t ♦❢ ✐♥t❡❣r❛❜❧❡ s②st❡♠s✱ ❢♦r ❡①❛♠♣❧❡ ●❛✉❞✐♥✬s ♠♦❞❡❧✱ ✇❤❡r❡ t❤❡ ♣r♦♣♦s❡❞ s❝❤❡♠❡ ♣r♦✈✐❞❡s ❛ ❜❛s✐s ✐♥ t❤❡ ♣❤❛s❡ s♣❛❝❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐t ❣✐✈❡s ❛♥ ❛♣♣r♦♣r✐❛t❡ ❜❛s✐s ❢♦r ❇❡t❤❡ ❛♥③❛t③ ♣r♦❝❡❞✉r❡✳

❙❦❧②❛♥✐♥ ❊✳✱ ❙❡♣❛r❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ●❛✉❞✐♥ ♠♦❞❡❧ ❏✳ ❙♦✈✳ ▼❛t❤✳ ✹✼ ✭✶✾✽✾✮✱ ✷✹✼✸✕✷✹✽✽✳ ❋❡✐❣✐♥ ❇✳✱ ❋r❡♥❦❡❧ ❊✳✱ ❘❡s❤❡t✐❦❤✐♥ ◆✳✱ ●❛✉❞✐♥ ▼♦❞❡❧✱ ❇❡t❤❡ ❆♥③❛t③ ❛♥❞ ❈r✐t✐❝❛❧ ▲❡✈❡❧ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✶✻✻ ✭✶✾✾✹✮✱ ✷✼✕✻✷✳ ❚❤❡ ❡♥❞

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s

SLIDE 16

• ❡♥❡r❛❧ s❝❤❡♠❡

❊①❛♠♣❧❡✿ ❝❧❛ss✐❝❛❧ ❍❡✐s❡♥❜❡r❣ ♠♦❞❡❧ ❈♦♥❝❧✉s✐♦♥ ❛♥❞ ❞✐s❝✉ss✐♦♥

◗✉❛♥t✐③❛t✐♦♥✿ λ❦ → ˆ λ❦, ✇❦ → ˆ ✇❦ = −✐∂/∂λ❦

❚❤❡ ❛❧❣❡❜r❛ e(✸) r❡♣r❡s❡♥t❛t✐♦♥✿

ˆ ♣✷ = λ✶λ✷ λ✶ − λ✷

• λ✶

∂ ∂λ✶ − λ✷ ∂ ∂λ✷

• ,

ˆ ♣✶ = ✐ ✷κ

• κ✷λ✶λ✷ + ❝✵ + ˆ

♣✷

✷

λ✶λ✷

• ,

ˆ ❉ = ✷ˆ ▲✷ˆ ♣✷ λ✶λ✷ + ˆ ♣✷

✷(λ✶ + λ✷)

λ✷

✶λ✷ ✷

ˆ ♣✸ = ✶ ✷κ

• κ✷λ✶λ✷ − ❝✵ + ˆ

♣✷

✷

λ✶λ✷

• ,

= −✷ λ✶ − λ✷

• λ✷

✶

∂✷ ∂λ✷

✶

− λ✷

✷

∂✷ ∂λ✷

✷

• ˆ

▲✷ = −✶ λ✶ − λ✷

• λ✷

✶

∂ ∂λ✶ − λ✷

✷

∂ ∂λ✷

• ,

ˆ ▲✶ = ✐ ✷κ

• −κ✷(λ✶ + λ✷) +

❝✶ λ✶λ✷ + ❝✵(λ✶ + λ✷) λ✷

✶λ✷ ✷

+ ˆ ❉

• ,

ˆ ▲✸ = ✶ ✷κ

• −κ✷(λ✶ + λ✷) −

❝✶ λ✶λ✷ − ❝✵(λ✶ + λ✷) λ✷

✶λ✷ ✷

− ˆ ❉

• .

❏✉❧✐❛ ❇❡r♥❛ts❦❛✱ P❡tr♦ ❍♦❧♦❞ ❍❛r♠♦♥✐❝ ❛♥❛❧②s✐s ♦♥ ▲❛❣r❛♥❣✐❛♥ ♠❛♥✐❢♦❧❞s