Gromov-Witten Invariants and Modular Forms rr - - PowerPoint PPT Presentation

Gromov-Witten Invariants and Modular Forms

❏✐❡ ❩❤♦✉ ❍❛r✈❛r❞ ❯♥✐✈❡rs✐t②

❏♦✐♥t ✇♦r❦ ✇✐t❤ ▼✉r❛❞ ❆❧✐♠✱ ❊♠❛♥✉❡❧ ❙❝❤❡✐❞❡❣❣❡r ❛♥❞ ❙❤✐♥❣✲❚✉♥❣ ❨❛✉

❛r①✐✈✿ ✶✸✵✻✳✵✵✵✷

Overview

• ❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♠♦t✐✈❛t✐♦♥
• ❙♦❧✈✐♥❣ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ❛♠♣❧✐t✉❞❡s ✐♥ t❡r♠s ♦❢ q✉❛s✐

♠♦❞✉❧❛r ❢♦r♠s

• ❊①❛♠♣❧❡✿ ❑P✷
• ❙♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣
• ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❞✐s❝✉ss✐♦♥s

Background

• ✐✈❡♥ ❛ ❈❨ ✸✕❢♦❧❞ ❨ ✱ ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♥t❡r❡st✐♥❣ ♣r♦❜❧❡♠s t♦

❝♦✉♥t t❤❡ ●r♦♠♦✈✲❲✐tt❡♥ ✐♥✈❛r✐❛♥ts ♦❢ ❨ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ♦❢ ❣❡♥✉s ❣ ●r♦♠♦✈✲❲✐tt❡♥ ✐♥✈❛r✐❛♥ts ♦❢ ❨ ❋ ❣

• ❲ (❨ , t) =
• β∈❍✷(❨ ,Z)

❡ω(t)❣,β =

• β∈❍✷(❨ ,Z)

◆●❲

❣,β ❡

• β ω(t) ,

ω✐✶ · · · ω✐❦❣,β =

❦

• ❥=✶

❡✈∗

❥ ω✐❥ ∩ [M❣,❦(❨ , β)]✈✐r ,

❤❡r❡ ω(t) = ❤✶,✶(❨ )

✐=✶

t✐ω✐✱ ✇❤❡r❡ ω✐, ✐ = ✶, ✷ · · · ❤✶,✶(❨ ) ❛r❡ t❤❡ ❣❡♥❡r❛t♦rs ❢♦r t❤❡ ❑❛❤❧❡r ❝♦♥❡ ♦❢ ❨ ✳ ❋♦r s♦♠❡ s♣❡❝✐❛❧ ❈❨ ✸✲❢♦❧❞s✱ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ❋ ❣

• ❲ (❨ , t)

❝♦✉❧❞ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❧♦❝❛❧✐③❛t✐♦♥ t❡❝❤♥✐q✉❡✱ t♦♣♦❧♦❣✐❝❛❧ ✈❡rt❡①✱ ❡t❝✳ ❋♦r ❣❡♥❡r❛❧ ❈❨ ✸✲❢♦❧❞s✱ t❤❡② ❛r❡ ✈❡r② ❞✐✣❝✉❧t t♦ ❝♦♠♣✉t❡✳

Background

• P❤②s✐❝s ✭t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ t❤❡♦r②✮t❡❧❧s t❤❛t ❋ ❣
• ❲ (❨ , t) ✐s t❤❡

❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ♦❢ s♦♠❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ q✉❛♥t✐t② ❝❛❧❧❡❞ t❤❡ ❆ ♠♦❞❡❧ ❣❡♥✉s ❣ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❨ ❋ ❣

• ❲ (❨ , t) = ❧✐♠

▲❱▲ F❣(❨ , t,¯

t) ❚❤❡ ❛❜♦✈❡ ❡①♣r❡ss✐♦♥ ❧✐♠▲❱▲ ♠❡❛♥s t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❜❛s❡❞ ❛t t❤❡ ❧❛r❣❡ ✈♦❧✉♠❡ ❧✐♠✐t t = ✐∞✿ t❤✐♥❦ ♦❢ t,¯ t ❛s ✐♥❞❡♣❡♥❞❡♥t ❝♦♦r❞✐♥❛t❡s✱ ✜① t✱ s❡♥❞ ¯ t t♦ ✐∞✳

• ▼✐rr♦r s②♠♠❡tr② ♣r❡❞✐❝ts t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ♠✐rr♦r ♠❛♥✐❢♦❧❞

❳ ♦❢ ❨ ✐♥ t❤❡ s❡♥s❡ t❤❛t F❣(❨ ) ✭❛♥❞ ✐ts ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❋ ❣(❨ )✮ ✐s ✐❞❡♥t✐❝❛❧ t♦ s♦♠❡ q✉❛♥t✐t② F❣(❳) ✭❛♥❞ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❋ ❣(❳)✮ ❝❛❧❧❡❞ t❤❡ ❇ ♠♦❞❡❧ ❣❡♥✉s ❣ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ♦♥ ❳✱ ✉♥❞❡r t❤❡ ♠✐rr♦r ♠❛♣✳

Background

• ❚❤❡ ❣❡♥✉s ③❡r♦ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ✇❛s

st✉❞✐❡❞ ✐♥t❡♥s✐✈❡❧② s✐♥❝❡ t❤❡ ❝❡❧❡r❛t❡❞ ✇♦r❦ ❈❛♥❞❡❧❛s✱ ❞❡ ▲❛ ❖ss❛✱

• r❡❡♥ ✫ P❛r❦❡s ✭✶✾✾✶✮
• ❚❤❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s F❣(❳), ❣ ≥ ✶ s❛t✐s❢② s♦♠❡ ❞✐✛❡r❡♥t✐❛❧

❡q✉❛t✐♦♥s ❝❛❧❧❡❞ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♥♦♠❛❧② ❡q✉❛t✐♦♥s ❇❡rs❤❛❞s❦②✱

❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮✱ ❛♥❞ ❛r❡ ❡❛s✐❡r t♦ ❝♦♠♣✉t❡ t❤❛♥

F❣(❨ )✳

• ❚❤❛♥❦s t♦ ♠✐rr♦r s②♠♠❡tr②✱ ♦♥❡ ❝❛♥ tr② t♦ ❡①tr❛❝t
• r♦♠♦✈✲❲✐tt❡♥ ✐♥✈❛r✐❛♥ts ♦❢ ❨ ❜② st✉❞②✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ✭t❤❡

♠♦❞✉❧✐ s♣❛❝❡ ♦❢✮ ❳ ❛♥❞ s♦❧✈✐♥❣ F❣(❳) ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥s✳

Motivation

■♥ s♦♠❡ ♥✐❝❡st ❝❛s❡s✱ F❣(❨ ) ✭❛♥❞ ✐ts ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❋ ❣

• ❲ (❨ )✮

❛r❡ ❡①♣❡❝t❡❞ t♦ ❤❛✈❡ s♦♠❡ ♠♦❞✉❧❛r ♣r♦♣❡rt✐❡s✳ ❙♦♠❡ ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡

• ❨ = ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❘✉❞❞ ✭✶✾✾✹✮✱ ❉✐❥❦❣r❛❛❢ ✭✶✾✾✺✮✱ ❑❛♥❡❦♦ ✫ ❩❛❣✐❡r

✭✶✾✾✺✮✳✳✳

❋ ✶

• ❲ (t)

= − ❧♦❣ η(q), q = ❡①♣ ✷π✐t ❋ ✷

• ❲ (t)

= ✶ ✶✵✸✻✽✵(✶✵❊ ✸

✷ − ✻❊✷❊✹ − ✹❊✻)

❋ ❣

• ❲ (t)

✐s ❛ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠ ♦❢ ✇❡✐❣❤t ✻❣ − ✻...

• ❙❚❯ ♠♦❞❡❧✿ ❨ = ❛ s♣❡❝✐❛❧ ❑✸ ✜❜r❛t✐♦♥
• ❋❍❙❱ ♠♦❞❡❧✿ ❨ = ❑✸ × ❚ ✷/Z✷✳

■■❆ − ❍❊ ❞✉❛❧✐t② t❡❧❧s t❤❛t ❋ ❣(❨ ) ❤❛✈❡ ♥✐❝❡ ♠♦❞✉❧❛r ♣r♦♣❡rt✐❡s ❑❛❝❤r✉ ✫ ❱❛❢❛ ✭✶✾✾✺✮✱ ▼❛r✐♥♦ ✫ ▼♦♦r❡ ✭✶✾✾✽✮✱ ❑❧❡♠♠ ✫

▼❛r✐♥♦ ✭✷✵✵✺✮✱ ▼❛✉❧✐❦ ✫ P❛♥❞❤❛r✐♣❛♥❞❡ ✭✷✵✵✻✮✳✳✳

Overview

■♥ t❤✐s t❛❧❦✱ ✇❡ s❤❛❧❧ ✇♦r❦ ♦♥❧② ♦♥ t❤❡ ❇ ♠♦❞❡❧ ♦❢ ❳✳ ❲❡ s❤❛❧❧

• s♦❧✈❡ F❣(❳), ❣ ≥ ✵ ❢r♦♠ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♥♦♠❛❧② ❡q✉❛t✐♦♥s

❢♦r ❝❡rt❛✐♥ ♥♦♥❝♦♠♣❛❝t ❈❨ ✸✕❢♦❧❞s ❳ ❛♥❞ ❡①♣r❡ss t❤❡♠ ✐♥ t❡r♠s ♦❢ t❤❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ❚❤❡ r❡s✉❧ts ✇❡ ♦❜t❛✐♥ ♣r❡❞✐❝t t❤❡ ❝♦rr❡❝t ●❲ ✐♥✈❛r✐❛♥ts ♦❢ t❤❡ ♠✐rr♦r ♠❛♥✐❢♦❧❞ ❨ ✉♥❞❡r t❤❡ ♠✐rr♦r ♠❛♣✳

• ❡①♣❧♦r❡ t❤❡ ❞✉❛❧✐t② ♦❢ F❣(❳) ❢♦r t❤❡s❡ ♣❛rt✐❝✉❧❛r ♥♦♥❝♦♠♣❛❝t

❈❨ ✸✕❢♦❧❞s

• ❝♦♥str✉❝t t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝

♠♦❞✉❧❛r ❢♦r♠s ❢♦r ❣❡♥❡r❛❧ ❈❨ ✸✕❢♦❧❞s ❜② ✉s✐♥❣ q✉❛♥t✐t✐❡s ❝♦♥str✉❝t❡❞ ♦✉t ♦❢ t❤❡ s♣❡❝✐❛❧ ❑❛❤❧❡r ❣❡♦♠❡tr② ♦♥ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ M❝♦♠♣❧❡①(❳) ♦❢ ❝♦♠♣❧❡① str✉❝t✉r❡s ♦❢ ❳

Example

❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ✇❡ ❝♦✉❧❞ ❝♦♠♣✉t❡ ✿ ❨ = ❑P✷ ❆❣❛♥❛❣✐❝✱ ❇♦✉❝❤❛r❞ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮✱ ❬❆❙❨❩❪ ❋ ✶

• ❲ (❨ , t)

= −✶ ✷ ❧♦❣ η(q)η(q✸), q = ❡①♣ ✷π✐τ, t = τ ❋ ✷

• ❲ (❨ , t)

= ❊(✻❆✹ − ✾❆✷❊ + ✺❊ ✷) ✶✼✷✽❇✻ + −✽

✺❆✻ + ✷ ✺❆✸❇✸ + −✽−✸χ ✶✵

❇✻ ✶✼✷✽❇✻ · · · ✇❤❡r❡ ❆, ❇, ❈, ❊ ❛r❡ ❡①♣❧✐❝✐t q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ✭✇✐t❤ ♠✉❧t✐♣❧✐❡r s②st❡♠s✮ ♦❢ ✇❡✐❣❤ts ✶, ✶, ✶, ✷ r❡s♣❡❝t✐✈❡❧②✱ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♠♦❞✉❧❛r ❣r♦✉♣ Γ✵(✸)✳ ■ ✇✐❧❧ ❡①♣❧❛✐♥ ✐♥ ❞❡t❛✐❧ ❤♦✇ t❤✐s ♠♦❞✉❧❛r ❣r♦✉♣ ❝♦♠❡s ♦✉t✳

Example

❋ ✸

• ❲ (❨ , t)

=

• −✷✺✸✷❆✶✵ + ✸✹✹✹❆✼❇✸ − ✶✶✹✵❆✹❇✻ + ✹✽❆❇✾

❊ ✶✷✹✹✶✻✵❇✶✷ +

• ✸✺✶✻❆✽ − ✸✼✵✽❆✺❇✸ + ✼✸✷❆✷❇✻

❊ ✷ ✶✷✹✹✶✻✵❇✶✷ +

• −✷✻✹✺❆✻ + ✶✾✵✵❆✸❇✸ − ✶✷✵❇✻

❊ ✸ ✶✷✹✹✶✻✵❇✶✷ +

• ✶✷✵✵❆✹ − ✹✷✵❆❇✸

❊ ✹ ✶✷✹✹✶✻✵❇✶✷ − ✷✺❆✷❊ ✺ ✽✷✾✹✹❇✶✷ + ✺❊ ✻ ✽✷✾✹✹❇✶✷ +✺✸✺✾❆✶✷ − ✽✽✻✹❆✾❇✸ + ✹✶✻✵❆✻❇✻ − ✹✾✻❆✸❇✾ + ✷(✽ − ✸χ)❇✶✷ ✽✼✵✾✶✷✵❇✶✷

Solving F❣: Special Kahler metric

❈♦♥s✐❞❡r t❤❡ ❢❛♠✐❧② ♦❢ ❈❨ ✸✲❢♦❧❞s π : X → M✱ ✇❤❡r❡ M ✐s s♦♠❡ ❞❡❢♦r♠❛t✐♦♥ s♣❛❝❡ ✭♦❢ ❝♦♠♣❧❡① str✉❝t✉r❡s✮ ♦❢ t❤❡ ❈❨ ✸✕❢♦❧❞ ❳✳ ❚❤❡ ❜❛s❡ M ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❲❡✐❧✲P❡t❡rss♦♥ ♠❡tr✐❝ ✇❤♦s❡ ❑❛❤❧❡r ♣♦t❡♥t✐❛❧ ✐s ❣✐✈❡♥ ❜② ❡−❑ = ✐

• Ω ∧ ¯

Ω ✭✶✮ ✇❤❡r❡ Ω ✐s ❛ ❤♦❧♦♠♦r♣❤✐❝ s❡❝t✐♦♥ ♦❢ t❤❡ ❍♦❞❣❡ ❧✐♥❡ ❜✉♥❞❧❡ L = ❘✵π∗Ω✸

M|X ✳ ❚❤❡ ❝✉r✈❛t✉r❡ ♦❢ t❤❡ ❲❡✐❧✲P❡t❡rss♦♥ ♠❡tr✐❝ ●✐¯ ❥

s❛t✐s✜❡s t❤❡ s♦✲❝❛❧❧❡❞ s♣❡❝✐❛❧ ❣❡♦♠❡tr② r❡❧❛t✐♦♥ ❙tr♦♠✐♥❣❡r ✭✶✾✾✵✮ −❘

❦ ✐¯  ❧ = ¯

∂¯

Γ❦ ✐❧ = δ❦ ❧ ●✐¯  + δ❦ ✐ ●❧¯  − ❈✐❧♠❈ ❦♠ ¯ 

✭✷✮ ✇❤❡r❡ ❈✐❥❦ = −

• Ω ∧ ∂✐∂❥∂❦Ω ✭❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣ ♦r t❤r❡❡✲♣♦✐♥t

❢✉♥❝t✐♦♥✮ ❛♥❞ ❈

❥❦ ¯ ı = ❡✷❑● ❥¯ ● ❦¯ ❦❈¯ ı¯ ¯ ❦, ✐ = ✶, ✷, · · · ♥ = ❞✐♠M✳

Solving F❣: BCOV holomorphic anomaly equations

❚❤❡ ❣❡♥✉s ❣ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ F❣ ✐s ❛ s❡❝t✐♦♥ ♦❢ L✷−✷❣✳ ❋♦r ❣ = ✶ ❝❛s❡✱ ✐t s❛t✐s✜❡s t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♥♦♠❛❧② ❡q✉❛t✐♦♥ ❇❡rs❤❛❞s❦②✱ ❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮ ∂¯

ı∂❥F✶ = ✶

✷❈❥❦❧❈

❥❦ ¯ ı − ( χ

✷✹ − ✶)●¯

ı❥

✭✸✮ ✇❤❡r❡ t❤❡ q✉❛♥t✐t② χ ✐s t❤❡ ❊✉❧❡r ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ t❤❡ ♠✐rr♦r ♠❛♥✐❢♦❧❞ ❨ ✭♥♦t ♦❢ ❳✮✳

Solving F❣: BCOV holomorphic anomaly equations

❋♦r ❣ ≥ ✷ ❇❡rs❤❛❞s❦②✱ ❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮ ¯ ∂¯

ıF❣ = ✶

✷ ¯ ❈ ❥❦

¯ ı

 

• ❣✶+❣✷=❣, ❣✶,❣✷≥✶

❉❥F❣✶❉❦F❣✷ + ❉❥❉❦F❣−✶   ✭✹✮

× ϕ¯

i

genus = g

=

× × ×

+

genus = g1 genus = g2

1 2C jk ¯ i

Di Dk

× × ×

1 2C jk ¯ i

Dj Dk genus = g − 1

✇❤❡r❡ ❉✐ ✐s t❤❡ s✉♠ ♦❢ t❤❡ ❈❤❡r♥ ❝♦♥♥❡❝t✐♦♥ ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❲❡✐❧✲P❡t❡rss♦♥ ♠❡tr✐❝ ●✐¯

 ❛♥❞ t❤❡ ❑❛❤❧❡r ❝♦♥♥❡❝t✐♦♥ ❑✐ ♦♥ t❤❡

❍♦❞❣❡ ❧✐♥❡ ❜✉♥❞❧❡ L✳

Solving F❣: polynomial recursion

❯s✐♥❣ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② r❡❧❛t✐♦♥ ❛♥❞ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♥♦♠❛❧② ❡q✉❛t✐♦♥ ❢♦r ❣❡♥✉s ✶✱ t❤❡ ❡q✉❛t✐♦♥s ❝♦✉❧❞ ❜❡ s♦❧✈❡❞ r❡❝✉rs✐✈❡❧② ✉s✐♥❣ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❇❡rs❤❛❞s❦②✱ ❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮✱

❨❛♠❛❣✉❝❤✐ ✫ ❨❛✉ ✭✷✵✵✹✮✱ ❆❧✐♠ ✫ ▲ä♥❣❡ ✭✷✵✵✼✮✱ ❢♦r ❡❛❝❤ ❣❡♥✉s ❣✱ t❤❡

t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ F❣ ✐s ❢♦✉♥❞ t♦ t❛❦❡ t❤❡ ❢♦r♠ ¯ ∂¯

ıF❣ = ¯

∂¯

ıP❣, ❣ ≥ ✷

✭✺✮ ✇❤❡r❡ P❣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ✭❝♦❡✣❝✐❡♥ts ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✮ ♦❢ t❤❡ ♣r♦♣❛❣❛t♦rs ❙✐❥, ❙✐, ❙ ❞❡✜♥❡❞ ✈✐❛ ¯ ∂¯

❦❙✐❥ = ❈ ✐❥ ¯ ❦,

¯ ∂¯

❦❙✐ = ●❦¯ ❦❙✐❦,

¯ ∂¯

❦❙ = ●❦¯ ❦❙❦

✭✻✮ ❛♥❞ t❤❡ ❑❛❤❧❡r ❝♦♥♥❡❝t✐♦♥ ❑✐✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❦✐♥❞ ♦❢ ✉♥✐✈❡rs❛❧ ❢r♦♠ r❡❝✉rs✐♦♥✳ ❚❤❡s❡ ❣❡♥❡r❛t♦rs ❡♥❝♦❞❡ ❛❧❧ ♦❢ t❤❡ ❛♥t✐✲❤♦❧♦♠♦r♣❤✐❝ ❞❡♣❡♥❞❡♥❝❡ ♦❢ F❣✳

Solving F❣: differential ring of non-holomorphic generators

▼♦r❡♦✈❡r✱ t❤❡ r✐♥❣ ❣❡♥❡r❛t❡❞ ❜② t❤❡s❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ ❣❡♥❡r❛t♦rs ❙✐❥, ❙✐, ❙, ❑✐ ✐s ❝❧♦s❡❞ ✉♥❞❡r t❤❡ ❝♦✈❛r✐❛♥t ❞❡r✐✈❛t✐✈❡ ❉✐ ❆❧✐♠ ✫ ▲ä♥❣❡

✭✷✵✵✼✮

❉✐❙❥❦ = δ❥

✐ ❙❦ + δ❦ ✐ ❙❥ − ❈✐♠♥❙♠❥❙♥❦ + ❤❥❦ ✐ ,

❉✐❙❥ = ✷δ❥

✐ ❙ − ❈✐♠♥❙♠❙♥❥ + ❤❥❦ ✐ ❑❦ + ❤❥ ✐ ,

❉✐❙ = −✶ ✷❈✐♠♥❙♠❙♥ + ✶ ✷❤♠♥

✐

❑♠❑♥ + ❤❥

✐ ❑❥ + ❤✐,

❉✐❑❥ = −❑✐❑❥ − ❈✐❥❦❙❦ + ❈✐❥❦❙❦❧❑❧ + ❤✐❥, ✭✼✮ ✇❤❡r❡ ❤❥❦

✐ , ❤❥ ✐ , ❤✐, ❤✐❥ ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳

Solving F❣: holomorphic ambiguities

❋r♦♠ ¯ ∂¯

ıF❣ = ¯

∂¯

ıP❣, ❣ ≥ ✷

✭✽✮ ✇❡ ❣❡t F❣ = P❣(❙✐❥, ❙✐, ❙, ❑✐) + ❢ ❣, ❣ ≥ ✷ ✭✾✮ ❚❤❡ ❢✉♥❝t✐♦♥ ❢ ❣ ✐s ♣✉r❡❧② ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♠❜✐❣✉✐t②✳ ■t ❝❛♥ ♥♦t ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② ♦♥❧② ❧♦♦❦✐♥❣ ❛t t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♥♦♠❛❧② ❡q✉❛t✐♦♥s✳ ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ♥❡❡❞❡❞ t♦ ✜① ✐t✳

Solving F❣: boundary condition at LCSL

▼✐rr♦r s②♠♠❡tr② ♣r❡❞✐❝ts t❤❛t ❇❡rs❤❛❞s❦②✱ ❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮ ❋ ✶

❆(❳, t) := ❧✐♠ ▲❈❙▲ F✶(❳, t,¯

t) ⇔ ❋ ✶

• ❲ (❨ , t) = − ✶

✷✹

❤✶,✶(❨ )

• ✐=✶

t✐

• ❝✷(❚❨ )ω✐ + O(❡✷π✐t✐)

❋ ❣

❆(❳, t) := ❧✐♠ ▲❈❙▲ F❣(❳, t,¯

t) ⇔ ❋ ❣

• ❲ (❨ , t) = (−✶)❣ χ

✷ |❇✷❣❇✷❣−✷| ✷❣(✷❣ − ✷)(✷❣ − ✷)! + O(❡✷π✐t✐), ❣ ≥ ✷ ✇❤❡r❡ t = (t✶, t✷, · · · t♥) ❛r❡ t❤❡ ❝❛♥♦♥✐❝❛❧ ❝♦♦r❞✐♥❛t❡s ♦♥ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡s ❛♥❞ ❧✐♠▲❈❙▲ ♠❡❛♥s t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❛t t❤❡ ❧❛r❣❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❧✐♠✐t✳ ❚❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❝♦♠♣✉t✐♥❣ t❤❡ ❝♦♥st❛♥t ♠❛♣ ❝♦♥tr✐❜✉t✐♦♥ t♦ ❋ ❣

• ❲ (❨ , t)✳

Solving F❣: boundary condition at conifold loci

▼♦r❡♦✈❡r✱ ❛t t❤❡ ❝♦♥✐❢♦❧❞ ❧♦❝✐ ❇❡rs❤❛❞s❦②✱ ❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮✱

• ❤♦s❤❛❧ ✫ ❱❛❢❛ ✭✶✾✾✺✮✱ ❍✉❛♥❣ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮

❋ ✶

❝♦♥(t✐ ❝) :

= ❧✐♠

❝♦♥✐ F✶ = − ✶

✶✷ ❧♦❣ ∆✐ + r❡❣✉❧❛r t❡r♠s ❋ ❣

❝♦♥(t✐ ❝) :

= ❧✐♠

❝♦♥✐ F❣ =

❝❣−✶❇✷❣ ✷❣(✷❣ − ✷)(t✐

❝)✷❣−✷ + O((t✐ ❝)✵),

❣ ≥ ✷ ✇❤❡r❡ ❧✐♠❝♦♥✐ ♠❡❛♥s t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❛t t❤❡ ❝♦♥✐❢♦❧❞ ❧♦❝✉s ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❞✐s❝r✐♠✐♥❛♥t ❧♦❝✉s ∆✐ = ✵✱ ✐ = ✶, ✷, · · · , ♠✳ ❚❤❡ q✉❛♥t✐t② t✐

❝ ✐s ❛ s✉✐t❛❜❧② ❝❤♦s❡♥ ❝♦♦r❞✐♥❛t❡ ♥♦r♠❛❧ t♦ t❤❡

❝♦♥✐❢♦❧❞ ❧♦❝✉s ∆✐ = ✵✳ ❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ ❣❛♣ ❝♦♥❞✐t✐♦♥✳

Solving F❣: traditional approach in fixing ❢ ❣

❆❝❝♦r❞✐♥❣ t♦ t❤❡ ❛❜♦✈❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛t t❤❡ ▲❈❙▲ ❛♥❞ ❝♦♥✐❢♦❧❞ ❧♦❝✐✱ ♦♥❡ ❝❛♥ tr② t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥s❛t③ ❢♦r ❢ ❣ ❇❡rs❤❛❞s❦②✱

❈❡❝♦tt✐✱ ❖♦❣✉r✐ ✫ ❱❛❢❛ ✭✶✾✾✸✮✱ ❑❛t③✱ ❑❧❡♠♠ ✫ ❱❛❢❛ ✭✶✾✾✾✮✱ ❨❛♠❛❣✉❝❤✐ ✫ ❨❛✉ ✭✷✵✵✹✮

❢ ❣(③) =

♠

• ✐=✶

❆❣

✐

∆✷❣−✷

✐

✭✶✵✮ ✇❤❡r❡ ❆❣

✐ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ ③✶, · · · ③♥ ♦❢ ❞❡❣r❡❡ (✷❣ − ✷)❞❡❣∆✐

❚❤❡♥ ♦♥❡ ❛✐♠s t♦ s♦❧✈❡ ❢♦r t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ ❆❣

✐ ❢r♦♠ t❤❡ ❜♦✉♥❞❛r②

❝♦♥❞✐t✐♦♥s✳ ❋♦r ♥♦♥❝♦♠♣❛❝t ❈❨ ✸✕❢♦❧❞s✱ ❛ ❞✐♠❡♥s✐♦♥ ❝♦✉♥t✐♥❣

❍❛❣❤✐❣❤❛t✱ ❑❧❡♠♠ ✫ ❘❛✉❝❤ ✭✷✵✵✽✮ s✉❣❣❡sts t❤❛t t❤❡ ♥✉♠❜❡rs ♦❢

✉♥❦♥♦✇♥s ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ❙♦ ✐♥ ♣r✐♥❝✐♣❧❡✱ ❢ ❣ ❝♦✉❧❞ ❜❡ ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡❞✳ ❋♦r ❝♦♠♣❛❝t ❈❨ ✸✕❢♦❧❞s✱ ❢✉rt❤❡r ✐♥♣✉ts✱ ❡✳❣✳✱ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛t t❤❡ ♦r❜✐❢♦❧❞ ❧♦❝✐ ❛r❡ ♥❡❡❞❡❞ ❑❛t③✱ ❑❧❡♠♠ ✫ ❱❛❢❛ ✭✶✾✾✾✮✱ ❍✉❛♥❣ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮✳

Solving F❣: difficulties

• ❚❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❛r❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❞✐✛❡r❡♥t

❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐ts ♦❢ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ F❣✱ ♥❛♠❡❧②✱ t♦ ❋ ❣

❆

❛♥❞ ❋ ❣

❝♦♥✳ ❇✉t t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ t✇♦ ❞✐✛❡r❡♥t

❢✉♥❝t✐♦♥s ✐s ♥♦t q✉✐t❡ ❝❧❡❛r✳❚❤❡② ❛r❡ ♥♦t r❡❧❛t❡❞ ❜② ❛ s✐♠♣❧❡ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ✭r❡❝❛❧❧ t❤❛t t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t r❡q✉✐r❡s ❛ ❝❤♦✐❝❡ ♦❢ t❤❡ ❜❛s❡ ♣♦✐♥t✮✳

• ❚❤❡ ❛♥s❛t③ ❢♦r t❤❡ ❛♠❜✐❣✉✐t② ❢ ❣ ✐s ♠❛❞❡ ❜❛s❡❞ ♦♥ r❡❣✉❧❛r✐t② ❛t

t❤❡ ♦r❜✐❢♦❧❞ ♣♦✐♥ts ✭❧♦❝✐✮✱ ✇❤✐❝❤ ✐s ♥♦t ❡♥s✉r❡❞ ❢♦r ❣❡♥❡r❛❧ ❈❨s✳

• ■♥ ♣r❛❝t✐❝❡✱ t❤❡ ❣❡♥❡r❛t♦rs ❙✐❥, ❙✐, ❙, ❑✐ ❛r❡ ❝♦♠♣✉t❡❞ ❛s

✐♥✜♥✐t❡ s❡r✐❡s ✐♥ t❤❡ ❝♦♠♣❧❡① ❝♦♦r❞✐♥❛t❡s ♦♥ M✱ s♦ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r F❣ ✐s ♥♦t ❝♦♠♣❛❝t✳

• ▼♦❞✉❧❛r✐t② ✐s ♥♦t ♠❛♥✐❢❡st✳

Solving F❣: implement of modularity

■♥ ♦✉r ✇♦r❦✱ ✇❡ ❝♦✉❧❞ ♦✈❡r❝♦♠❡ t❤❡s❡ ❞✐✣❝✉❧t✐❡s ❜② ♠❛❦✐♥❣ ✉s❡ ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡s✱ ❢♦r ❝❡rt❛✐♥ ♥♦♥❝♦♠♣❛❝t ❈❨ ✸✕❢♦❧❞s ❳✳ ❲❡ ❝♦✉❧❞ s♦❧✈❡ F❣ ✐♥ t❡r♠s ♦❢ ♠♦❞✉❧❛r ❢♦r♠s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❝❡❞✉r❡✿

• ✐❞❡♥t✐❢② t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ M ✇✐t❤ ❛ ❝❡rt❛✐♥ ♠♦❞✉❧❛r ❝✉r✈❡

❳Γ = H∗/Γ, Γ ⊆ P❙▲(✷, Z)

• ❝♦♥str✉❝t t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ❛♥❞

❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ❛tt❛❝❤❡❞ t♦ ❳Γ✱ t❤❡ ❧❛tt❡r t✉r♥s ♦✉t t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ❝♦♥str✉❝t❡❞ ♦✉t ♦❢ ♣✉r❡❧② ❣❡♦♠❡tr✐❝ q✉❛♥t✐t✐❡s ✭♣❡r✐♦❞s✱ ❝♦♥♥❡❝t✐♦♥s✱ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣s✳✳✳✮

Solving F❣: implement of modularity

❚❤❡♥ ✇❡ ✶✳ ❡①♣r❡ss t❤❡ q✉❛♥t✐t✐❡s ❙✐❥, ❙✐, ❙, ❑✐ ❛♥❞ F❣, ❣ = ✵, ✶ ✐♥ t❡r♠s ♦❢ t❤❡ ❣❡♥❡r❛t♦rs ♦❢ t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✷✳ s♦❧✈❡ P❣ ✈✐❛ ♣♦❧②♥♦♠✐❛❧ r❡❝✉rs✐♦♥ ✐♥ t❡r♠s ♦❢ t❤❡s❡ ❣❡♥❡r❛t♦rs✱ ♠♦❞✉❧❛r✐t② t❤❡♥ ❣✐✈❡s ❛ ✈❡r② str♦♥❣ ❝♦♥str❛✐♥t ❛♥❞ ❛❧s♦ ❛ ♥❛t✉r❛❧ ❛♥s❛t③ ❢♦r ❢ ❣ ✐♥ t❡r♠s ♦❢ ♠♦❞✉❧❛r ❢♦r♠s✱ ✇✐t❤ ✉♥❞❡r✲❞❡t❡r♠✐♥❡❞ ❝♦❡✣❝✐❡♥ts ✸✳ ❡①♣❧♦r❡ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❋ ❣

❆ ❛♥❞ ❋ ❣ ❝♦♥✱ r❡❛❧✐③❡ t❤❡ ❜♦✉♥❞❛r②

❝♦♥❞✐t✐♦♥s ❛s ❝❡rt❛✐♥ r❡❣✉❧❛r✐t② ❝♦♥❞✐t✐♦♥s ✐♠♣♦s❡❞ ♦♥ t❤❡ ♠♦❞✉❧❛r ♦❜❥❡❝ts F❣ ❛♥❞ ❋ ❣

❆

✹✳ s♦❧✈❡ t❤❡ ✉♥❞❡r✲❞❡t❡r♠✐♥❡❞ ❝♦❡✣❝✐❡♥ts ✐♥ ❢ ❣ ❛♥❞ t❤✉s ❡①♣r❡ss F❣s ✭❋ ❣

❆s✮ ❛s ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ✭q✉❛s✐✮ ♠♦❞✉❧❛r ❢♦r♠s

Local P✷ example: mirror CY 3–fold family

❚❤❡ ♠✐rr♦r ❈❨ ❢❛♠✐❧② ♦❢ ❨ = ❑P✷ ✭❝❛❧❧❡❞ ❧♦❝❛❧ P✷ ❜❡❧♦✇✮ ✐s ❛ ❢❛♠✐❧② ♦❢ ♥♦♥❝♦♠♣❛❝t ❈❨ ✸✲❢♦❧❞s π : X → M ∼ = P✶ ❣✐✈❡♥ ❜② ❈❤✐❛♥❣✱

❑❧❡♠♠✱ ❨❛✉ ✫ ❩❛s❧♦✇ ✭✶✾✾✾✮✱ ❍♦r✐ ✫ ❱❛❢❛ ✭✷✵✵✵✮✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❝❤♦♦s❡

③ ❛s t❤❡ ♣❛r❛♠❡t❡r ❢♦r t❤❡ ❜❛s❡ M✳ ❋♦r ❡❛❝❤ ③✱ t❤❡ ❈❨ ✸✕❢♦❧❞ X③ ✐s ✐ts❡❧❢ ❛ ❝♦♥✐❝ ✜❜r❛t✐♦♥ ✉✈ − ❍(②✐; ③) := ✉✈ − (②✵ + ②✶ + ②✷ + ②✸) = ✵, (✉, ✈) ∈ C✷ ♦✈❡r t❤❡ ❜❛s❡ C✷ ♣❛r❛♠❡tr✐③❡❞ ❜② ②✐, ✐ = ✵, ✶, ✷, ✸ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ ❛✳ t❤❡r❡ ✐s ❛ C∗ ❛❝t✐♦♥✿ ②✐ → λ②✐, λ ∈ C∗, ✐ = ✵, ✶, ✷, ✸❀ ❜✳ ③ = −②✶②✷②✸

②✸

✵

✳ ❙tr❛✐❣❤t❢♦r✇❛r❞ ❝♦♠♣✉t❛t✐♦♥ s❤♦✇s t❤❛t ③ = ✵, ✶/✷✼, ∞ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❧❛r❣❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❧✐♠✐t✱ ❝♦♥✐❢♦❧❞ ♣♦✐♥t✱ ♦r❜✐❢♦❧❞ ♦❢ t❤❡ ❈❨ ✸✕❢♦❧❞ ❢❛♠✐❧②✱ r❡s♣❡❝t✐✈❡❧②✳

Local P✷ example: mirror curve family

❚❤❡ ❞❡❣❡♥❡r❛t✐♦♥ ❧♦❝✉s ♦❢ t❤✐s ❝♦♥✐❝ ✜❜r❛t✐♦♥ ✐s ❛ ❝✉r✈❡ E③ ✭❝❛❧❧❡❞ t❤❡ ♠✐rr♦r ❝✉r✈❡✮ s✐tt✐♥❣ ✐♥s✐❞❡ X③ ✿ ❍(②✐; ③) = ②✵ + ②✶ + ②✷ + ②✸ = ✵, ③ = −②✶②✷②✸ ②✸

✵

. ❚❤✐s ✇❛②✱ ✇❡ ❣❡t t❤❡ ♠✐rr♦r ❝✉r✈❡ ❢❛♠✐❧② π : E → M✳ ■t ✐s ❡q✉✐✈❛❧❡♥t t♦ ②✷

✸ − (②✵ + ✶)②✸ = ③②✸ ✵

✭✶✶✮ ✇✐t❤ ∆ = −✷✼ + ✶ ③ , ❥(③) = (✶ − ✷✹③)✸ ③✸(✶ − ✷✼③)

Local P✷ example: arithmetic of moduli space

❈♦♠♣❛r✐♥❣ ✐t ✇✐t❤ t❤❡ ❡❧❧✐♣t✐❝ ♠♦❞✉❧❛r s✉r❢❛❝❡ ❛ss♦❝✐❛t❡❞ t♦ Γ✵(✸)✱ ✐✳❡✳✱ t❤❡ ❍❡ss❡ ❢❛♠✐❧② πΓ✵(✸) : EΓ✵(✸) → ❳Γ✵(✸) = H∗/Γ✵(✸) ①✸

✶ + ①✸ ✷ + ①✸ ✸ − ③− ✶

✸ ①✶①✷①✸ = ✵,

❥(③) = (✶ + ✷✶✻③) ③(✶ − ✷✼③)✸ ❖♥❡ ❝❛♥ s❡❡ t❤❛t ❆❣❛♥❛❣✐❝✱ ❇♦✉❝❤❛r❞ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮✱ ❍❛❣❤✐❤❛t✱ ❑❧❡♠♠

✫ ❘❛✉❝❤ ✭✷✵✵✽✮✱ ❬❆❙❨❩❪

M ∼ = ❳✵(✸) := H∗/Γ✵(✸) ■♥ ❢❛❝t t❤❡s❡ t✇♦ ❢❛♠✐❧✐❡s ❛r❡ r❡❧❛t❡❞ ❜② ❛ ✸✕✐s♦❣❡♥②✱ s❡❡ ❡✳❣✳✱ ❊❧❧✐♣t✐❝

❈✉r✈❡s ✭●❚▼✮ ❍✉s❡♠¨ ♦❧❧❡r✳ ✭✷✵✵✹✮✳

❚❛❦✐♥❣ t❤❡ ❣❡♥❡r❛t♦r ♦❢ t❤❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥❛❧ ✜❡❧❞ ✭❍❛✉♣t♠♦❞✉❧✮ ♦❢ ❳✵(✸) t♦ ❜❡ α = ✷✼③✱ t❤❡♥ t❤❡ ♣♦✐♥ts α = ✵, ✶, ∞ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❧❛r❣❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❧✐♠✐t✱ ❝♦♥✐❢♦❧❞ ♣♦✐♥t✱ ♦r❜✐❢♦❧❞ ♦❢ t❤❡ ❈❨ ✸✕❢♦❧❞ ❢❛♠✐❧② r❡s♣❡❝t✐✈❡❧②✳

Local P✷ example: periods of elliptic curve family

❚❤❡ P✐❝❛r❞✲❋✉❝❤s ♦♣❡r❛t♦r ❛tt❛❝❤❡❞ t♦ t❤❡ ❍❡ss❡ ❡❧❧✐♣t✐❝ ❝✉r✈❡ ❢❛♠✐❧② ✐s t❤❡ ❤②♣❡r❣❡♦♠❡tr✐❝ ♦♣❡r❛t♦r ❑❧❡♠♠✱ ▲✐❛♥✱ ❘♦❛♥ ✫ ❨❛✉ ✭✶✾✾✹✮✱

▲✐❛♥ ✫ ❨❛✉ ✭✶✾✾✹✮

L❡❧❧✐♣t✐❝ = θ✷ − α(θ + ✶ ✸)(θ + ✷ ✸) ✭✶✷✮ ✇❤❡r❡ θ = α ∂

∂α✳ ❆ ❜❛s✐s ♦❢ t❤❡ s♣❛❝❡ ♦❢ ♣❡r✐♦❞s ❝♦✉❧❞ ❜❡ ❝❤♦s❡♥ t♦

❜❡ ω✵ =

✷❋✶(✶

✸, ✷ ✸, ✶; α), ω✶ = ✐ √ ✸

✷❋✶(✶

✸, ✷ ✸, ✶; ✶ − α) ✭✶✸✮ ❚❤❡♥ ✇❡ ❣❡t τ = ω✶ ω✵ = ✐ √ ✸

✷❋✶(✶ ✸, ✷ ✸, ✶; ✶ − α) ✷❋✶(✶ ✸, ✷ ✸, ✶; α)

✭✶✹✮ ■t ❢♦❧❧♦✇s t❤❡♥ t❤❡ ♣♦✐♥ts α = ✵, ✶, ∞ ❝♦rr❡s♣♦♥❞ t♦ [τ] = [✐∞], [✵], [❡①♣ ✷π✐/✸]✱ r❡s♣❡❝t✐✈❡❧②✳

Local P✷ example: periods of elliptic curve family

❉❡✜♥❡ ▼❛✐❡r ✭✷✵✵✻✮ ❆ = ω✵, ❇ = (✶ − α)

✶ ✸ ❆,

❈ = α

✶ ✸ ❆

✭✶✺✮ t❤❡♥ ❆✸ = ❇✸ + ❈ ✸ ▼♦r❡♦✈❡r✱ ♦♥❡ ❝❛♥ s❤♦✇ ❉✐✈ ❆ = ✶ ✸(α = ∞), ❉✐✈ ❇ = ✶ ✸(α = ✶), ❉✐✈ ❆ = ✶ ✸(α = ✵). ✭✶✻✮

Local P✷ example: modular forms

■t t✉r♥s ♦✉t t❤❛t t❤❡ ❆, ❇, ❈ ❞❡✜♥❡❞ ♦✉t ♦❢ ♣❡r✐♦❞s ❛r❡ ♠♦❞✉❧❛r ❢♦r♠s ✭✇✐t❤ ♠✉❧t✐♣❧✐❡r s②st❡♠s✮ ✇✐t❤ r❡s♣❡❝t t♦ Γ✵(✸) ✭s❡❡ ❡✳❣✳✱

▼❛✐❡r ✭✷✵✵✻✮✱ ❚❤❡ ✶✲✷✲✸ ♦❢ ▼♦❞✉❧❛r ❋♦r♠s ✭✷✵✵✻✮ ❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮✳

▼♦r❡♦✈❡r✱ t❤❡② ❤❛✈❡ ✈❡r② ♥✐❝❡ θ ♦r η ❡①♣❛♥s✐♦♥s ✭q = ❡①♣ ✷π✐τ✮ ❆(τ) = θ✷(✷τ)θ✷(✻τ) + θ✸(✷τ)θ✸(✻τ) =

• (♠,♥)∈Z✷

q♠✷−♠♥+♥✷ ❇(τ) = η(τ)✸ η(✸τ) =

• (♠,♥)∈Z✷

(❡①♣(✷π✐ ✸ ))♠+♥q♠✷−♠♥+♥✷ ❈(τ) = ✸η(✸τ)✸ η(τ) = q

✶ ✸

• (♠,♥)∈Z✷

q♠✷−♠♥+♥✷+♠+♥ = ✶ ✷(❆(τ ✸) − ❆(τ))

Local P✷ example: quasi modular forms

◆♦✇ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ str✉❝t✉r❡✳ ❖♥❡ ❝❛♥ s❤♦✇ t❤❛t ❉α = αβ❆✷ ✭✶✼✮ ✇❤❡r❡ ❉ =

✶ ✷π✐ ∂ ∂τ , β := ✶ − α✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦

❆✷ = ❉ ❧♦❣ ❈ ✸ ❇✸ (= ✸❊✷(✸τ) − ❊✷(τ) ✷ ) ✭✶✽✮ ✇❤✐❝❤ tr❛♥s❢♦r♠s ❛s ❛♥ ❤♦♥❡st ♠♦❞✉❧❛r ❢♦r♠ ✉♥❞❡r Γ✵(✸)✳ ◆♦✇ ✇❡ ❞❡✜♥❡ t❤❡ ❛♥❛❧♦❣✉❡ ♦❢ t❤❡ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠ ❊✷ ❜② ❊ = ❉ ❧♦❣ ❈ ✸❇✸(= ✸❊✷(✸τ) + ❊✷(τ) ✹ ) ✭✶✾✮ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐t tr❛♥s❢♦r♠s ❛s ❛ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠ ✉♥❞❡r Γ✵(✸)✳ ❲❡ ❞❡♥♦t❡ ✐ts ♠♦❞✉❧❛r ❝♦♠♣❧❡t✐♦♥ ❊ +

−✸ π■♠τ (✸ ✹ · ✶ ✸ + ✶ ✹) ❜② ˆ

❊✳

Local P✷ example: ring of quasi modular forms

■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥s ❛♥❞ t❤❡ P✐❝❛r❞✲❋✉❝❤s ❡q✉❛t✐♦♥ t❤❛t t❤❡ r✐♥❣ ❣❡♥❡r❛t❡❞ ❜② ❆, ❇, ❈, ❊ ✐s ❝❧♦s❡❞ ✉♣♦♥ t❛❦✐♥❣ ❞❡r✐✈❛t✐✈❡ ❉ =

✶ ✷π✐ ∂ ∂τ ❬❆❙❨❩❪

❉❆ = ✶ ✻❆(❊ + ❈ ✸ − ❇✸ ❆ ) ❉❇ = ✶ ✻❇(❊ − ❆✷) ❉❈ = ✶ ✻❆(❊ + ❆✷) ❉❊ = ✶ ✻(❊ ✷ − ❆✹) ✭✷✵✮ ❚❤✐s ✐s t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ✭✇✐t❤ ♠✉❧t✐♣❧✐❡r s②st❡♠s✮ ❢♦r Γ✵(✸)✳ ❚❤❡ r✐♥❣ ❣❡♥❡r❛t❡❞ ❜② ˆ ❊, ❆, ❇, ❈ ✐s t❤❡♥ t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s ✭✇✐t❤ ♠✉❧t✐♣❧✐❡r s②st❡♠s✮ ✳

Local P✷ example: Fricke involution on modular curve

❚❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ❛✉t♦♠♦r♣❤✐s♠✱ ❝❛❧❧❡❞ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ❲◆✱ ♦❢ t❤❡ ❢❛♠✐❧② πΓ✵(✸) : EΓ✵(✸) → ❳Γ✵(✸)✳ ❍❡r❡ ◆ = ✸✳ ■♥ t❡r♠s ♦❢ ❝♦♦r❞✐♥❛t❡s✱ ✐t ✐s ❞❡s❝r✐❜❡❞ ❜② ❲◆ : τ → − ✶ ◆τ , α → β := ✶ − α ✭✷✶✮ ❯s✐♥❣ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ♠♦❞✉❧❛r ❝✉r✈❡ ❛s ❛ ♠♦❞✉❧✐ s♣❛❝❡ ❳✵(✸) = {(❊, ❈)| ❈ < ❊◆ ∼ = Z✷

◆, |❈| = ◆}

t❤❡♥ t❤❡ ❋✐r❝❦❡ ✐♥✈♦❧✉t✐♦♥ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡s❝r✐♣t✐♦♥ ❲◆ : (❊, ❈) → (❊/❈, ❊◆/❈) ❚❤❡ ♠✐rr♦r ❝✉r✈❡ ❢❛♠✐❧② X ✐s r❡❧❛t❡❞ t♦ EΓ✵(✸) ❜② ❛ ✸✕✐s♦❣❡♥② ❛♥❞ t❤❡ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ✐s ❛❧s♦ ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ X ✳

Local P✷ example: Fricke involution on modular curve

Γ✵(◆) = ❛ ❜ ❝ ❞

• ❝ ≡ ✵ ♠♦❞ ◆
• ⊆ P❙▲(✷, Z)

Γ✵(✸)

❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥

❲◆ : τ → − ✶ ◆τ ❋♦r ❳✵(◆) = H∗/Γ✵(◆), ◆ = ✹, ✸, ✷, ✶∗ ✇❤✐❝❤ ❤❛s t❤r❡❡ s✐♥❣✉❧❛r ♣♦✐♥ts✱ t❤❡ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ❡①❝❤❛♥❣❡s t❤❡ t✇♦ ❞✐st✐♥❣✉✐s❤❡❞ ❝✉s♣s [✐∞] = [✶/◆] ❛♥❞ [✵] ♦♥ t❤❡ ♠♦❞✉❧❛r ❝✉r✈❡✱ ❛♥❞ ✜①❡s t❤❡ ❧❛st ♦❢ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦♥ ❳✵(◆)✳

Local P✷: Fricke involution on modular forms

❯♥❞❡r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥✱ ♦♥❡ ❤❛s ▼❛✐❡r ✭✷✵✵✻✮✱ ❬❆❙❨❩❪ ❆(τ) → √ ◆ ✐ τ❆(τ) ❇(τ) → √ ◆ ✐ τ❈(τ) ❈(τ) → √ ◆ ✐ τ❇(τ) ˆ ❊(τ, ¯ τ) → −( √ ◆ ✐ τ)✷ ˆ ❊(τ, ¯ τ) ✭✷✷✮ ❚❤✐s ✐♥✈♦❧✉t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ✧❞✉❛❧✐t②✧ ❢♦r t❤❡ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s F❣✱ ❛s ■ s❤❛❧❧ ❡①♣❧❛✐♥ ❜❡❧♦✇✳

Local P✷ example: Picard-Fuchs and periods of the CY 3-fold

❚❤❡ P✐❝❛r❞✲❋✉❝❤s ♦♣❡r❛t♦r ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❈❨ ✸✲❢♦❧❞ ❳ : ✉✈ − ❍(②✐, ③) = ✵ ✐s ▲❡r❝❤❡✱ ▼❛②r✱ ✫ ❲❛r♥❡r ✭✶✾✾✻✮✱ ❈❤✐❛♥❣✱ ❑❧❡♠♠✱

❨❛✉ ✫ ❩❛s❧♦✇ ✭✶✾✾✾✮✱ ❍❛❣❤✐❣❤❛t✱ ❑❧❡♠♠ ✫ ❘❛✉❝❤ ✭✷✵✵✽✮

L❈❨ = L❡❧❧✐♣t✐❝ ◦ θ ✭✷✸✮ ❚❤❡ ♣❡r✐♦❞s ❛r❡ ❣✐✈❡♥ ❜② ❳ ✵ = ✶, t, t❝ = κ−✶❋t✱ ✇❤❡r❡ t ∼ ❧♦❣ α + · · · ♥❡❛r t❤❡ ▲❈❙▲ α = ✵✱ t❝ ✐s t❤❡ ✈❛♥✐s❤✐♥❣ ♣❡r✐♦❞ ❛t t❤❡ ❝♦♥✐❢♦❧❞ ♣♦✐♥t α = ✶✱ ❛♥❞ κ ✐s t❤❡ ❝❧❛ss✐❝❛❧ tr✐♣❧❡ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡r ♦❢ t❤❡ ♠✐rr♦r ❨ = ❑P✷ ♦❢ ❳✳ ❍❡r❡ ✇❡ ❤❛✈❡ ❝❤♦s❡♥ t❤❡ ♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ t❝ s♦ t❤❛t θt = ω✵, θt❝ = ω✶ ✭✷✹✮ ❚❤❡♥ τ = ω✶ ω✵ = θt❝ θt = κ−✶❋tt ✭✷✺✮

Local P✷ example: holomorphic limit at the LCSL

❘❡❝❛❧❧ t❤❡ ❧❛r❣❡ ❝♦♠♣❧❡① str✉❝t✉r❡ ❧✐♠✐t ✐s ❣✐✈❡♥ ❜② α = ✵ ♦r ❡q✉✐✈❛❧❡♥t❧② [τ] = [✐∞] ♦♥ ❳✵(✸)✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡ s❤❛❧❧ ❝♦♠♣✉t❡ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② q✉❛♥t✐t✐❡s ✭❝♦♥♥❡❝t✐♦♥s✱ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣· · · ✮ ✐♥ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❛t t❤✐s ♣❛rt✐❝✉❧❛r ♣♦✐♥t✳ ■t ✐s ✐♥ t❤✐s ❧✐♠✐t t❤❛t F❣ ❜❡❝♦♠❡s ❋ ❣

❆(❳) ❛♥❞ ✐s

♠✐rr♦r t♦ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ❋ ❣

• ❲ (❨ ) ♦❢ t❤❡ ●❲ ✐♥✈❛r✐❛♥ts ♦♥

❨ ✳ ❯s✐♥❣ t❤❡ ♠♦❞✉❧❛r✐t②✳ ✇❡ s❤❛❧❧ s❡❡ t❤❛t t❤❡ ❢✉❧❧ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ F❣ ❝♦✉❧❞ ❜❡ r❡❝♦✈❡r❡❞ ❜② ✐ts ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❋ ❣

❆ ❍✉❛♥❣ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮✱ ❆❣❛♥❛❣✐❝✱ ❇♦✉❝❤❛r❞ ✫ ❑❧❡♠♠ ✭✷✵✵✻✮

Local P✷ example: special geometry quantities in the holomorphic limit

❚❛❦❡ t❤❡ ❝♦♦r❞✐♥❛t❡ ① = ❧♥ α ♥❡❛r t❤❡ ▲❈❙▲✳ ■♥ t❤✐s ❝♦♦r❞✐♥❛t❡ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ♦❢ t❤❡ ❝♦♥♥❡❝t✐♦♥s ❛r❡ ❧✐♠ ❑① = ✵, ❧✐♠ Γ①

①① = ∂① ❧♦❣ ∂t

∂① ❚❤❡ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣ ✐s ❈①①① = κ β ✇❤❡r❡ κ ✐s t❤❡ ❝❧❛ss✐❝❛❧ tr✐♣❧❡ ✐♥t❡rs❡❝t✐♦♥ ♥✉♠❜❡r✱ ✐t ✐s −✶

✸ ✐♥ t❤❡

❧♦❝❛❧ P✷ ❝❛s❡✳ ❯s✐♥❣ t❤❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❢♦r F✶ ❛t ▲❈❙▲ ❛♥❞ ❛t ❝♦♥✐❢♦❧❞✱ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ♦❢ t❤❡ ❣❡♥✉s ♦♥❡ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ ✐s s♦❧✈❡❞ t♦ ❜❡ ❋ ✶

❆ = −✶

✷ ❧♦❣ θt + ❧♦❣ β❛α❜+ ✶

✷

✇❤❡r❡ ❛ = − ✶

✶✷, ❜ + ✶ ✷ = − ✶ ✷✹

• ❝✷(❚❨ )❏ = − ✷

✷✹✳

Local P✷ example: special geometry quantities in terms of modular forms

❘❡❝❛❧❧ α = ❈ ✸

❆✸ , β = ❇✸ ❆✸ ✱ ✉s✐♥❣ t❤❡ η ❡①♣❛♥s✐♦♥s

❈ = ✸η(✸τ)✸ η(τ) , ❇ = η(τ)✸ η(✸τ) ❛♥❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❆ ❜② ❆ = ω✵ = θt✱ ✇❡ t❤❡♥ ❣❡t ❈ttt = ✶ (❳ ✵)✷ ❈①①①(∂① ∂t )✸ = κ ❇✸ = −✶ ✸ η(✸τ)✸ η(τ)✾ ❋ ✶

❆ = − ✶

✶✷ ❧♦❣ ❇✸❈ ✸ = −✶ ✷ ❧♦❣ η(τ)η(✸τ), ❉❋ ✶

❆ = − ✶

✶✷❊ ❤❡r❡ ❛s ❜❡❢♦r❡ ❉ =

✶ ✷π✐ ∂ ∂τ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱

F✶ = −✶ ✷ ❧♦❣ √ ■♠τ √ ■♠✸τη(τ)η(τ)η(✸τ)η(✸τ), ❉F✶ = − ✶ ✶✷ ˆ ❊

Local P✷ example: generators in terms of modular forms

❚❤❡ ❣❡♥❡r❛t♦rs ❙①, ❙ ❛r❡ ❝❤♦s❡♥ s♦ t❤❛t✿ ❧✐♠ ❙① = ❧✐♠ ❙ = ✵, ❧✐♠ ❑① = ✵ ✇❤✐❧❡ ❧✐♠ ❙①① ✐s s♦❧✈❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✐♥t❡❣r❛t❡❞ s♣❡❝✐❛❧ ❣❡♦♠❡tr② r❡❧❛t✐♦♥ Γ❦

✐❥ = δ❦ ✐ ❑❥ + δ❦ ❥ ❑✐ − ❈✐❥♠❙♠❦ + s♠ ✐❥

■♥ t❤✐s ❝❛s❡✱ ✐t s✐♠♣❧✐✜❡s t♦ ❧✐♠ Γ①

①① = ❧✐♠ ✷❑① − ❈①①① ❧✐♠ ❙①① + s① ①①

❚❤❛t ✐s✱ θ ❧♦❣ ❆ = −κ β ❧✐♠ ❙①① + s①

①①

Local P✷ example: generators in terms of modular forms

❘❡❝❛❧❧ t❤❛t ❉α = αβ❆✷ ✇✐t❤ ❉ =

✶ ✷π✐ ∂ ∂τ ✱ ♦♥❡ ❤❛s

θ = α ∂ ∂α = ✷π✐α ∂τ ∂α❉ = ✶ β❆✷ ❉ ❚❤❡♥ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜❡❝♦♠❡s ✶ β❆✷ ❉ ❧♦❣ ❆ = ✶ β❆✷ ✶ ✻(❊ + ❈ ✸ − ❇✸ ❆ ) = −κ β ❧✐♠ ❙①① + s①

①①

❆ ♥❛t✉r❛❧ ❝❤♦✐❝❡ ❢♦r s①

①① ✐s

s①

①① = ❈ ✸ − ❇✸

✻β❆✸ = ❈ ✸ − ❇✸ ✻❇✸ s♦ t❤❛t ❧✐♠ ❙①① = ❊ −✻κ❆✷ = ✶ ✷ ❊ ❆✷

Local P✷ example: differential ring of generators

❘❡❝❛❧❧ t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ ♦❢ ❣❡♥❡r❛t♦rs ❉✐❙❥❦ = δ❥

✐ ❙❦ + δ❦ ✐ ❙❥ − ❈✐♠♥❙♠❥❙♥❦ + ❤❥❦ ✐ ,

❉✐❙❥ = ✷δ❥

✐ ❙ − ❈✐♠♥❙♠❙♥❥ + ❤❥❦ ✐ ❑❦ + ❤❥ ✐ ,

❉✐❙ = −✶ ✷❈✐♠♥❙♠❙♥ + ✶ ✷❤♠♥

✐

❑♠❑♥ + ❤❥

✐ ❑❥ + ❤✐,

❉✐❑❥ = −❑✐❑❥ − ❈✐❥❦❙❦ + ❈✐❥❦❙❦❧❑❧ + ❤✐❥, ✭✷✻✮ ✇❤❡r❡ ❤❥❦

✐ , ❤❥ ✐ , ❤✐, ❤✐❥ ❛r❡ ❤♦❧♦♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥s✳

❆❝❝♦r❞✐♥❣ t♦ ♦✉r ❝❤♦✐❝❡s✱ t❤❡ ♦♥❧② ♥♦♥tr✐✈✐❛❧ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❛❜♦✈❡ ✐s ❉❙①① = −❈①①①❙①①❙①① + ❤①

①①

Local P✷ example: differential ring of generators

❚❛❦✐♥❣ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t✱ ♦♥❡ t❤❡♥ ❣❡ts ∂① ❧✐♠ ❙①① + ✷ ❧✐♠ Γ①① ❧✐♠ ❙①① = −❈①①① ❧✐♠ ❙①①❙①① + ❤①①

①

■t ❢♦❧❧♦✇s t❤❛t ❤①①

① = − ✶

✶✷ ❆✸ ❇✸ ◆♦t❡ t❤❛t t❤❡ q✉❛♥t✐t✐❡s s = s①

①① = ❈ ✸−❇✸ ✻❇✸ , ❤ = ❤①① ① = − ✶ ✶✷ ❆✸ ❇✸ ❛r❡

♠♦❞✉❧❛r ❛♥❞ t❤✉s ❤♦♥❡st ❤♦❧♦♠♦r♣❤✐❝ ♦❜❥❡❝ts ✐♥ t❤❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ ❝♦♠♣❧❡t✐♦♥ ✭❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ ❝♦♠♣❧❡t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❝♦♠♣❧❡t✐♦♥ t♦ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ♦❜❥❡❝ts✳✮ ❚❤❛t ✐s✱ t❤❡s❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♠❜✐❣✉✐t✐❡s ❛r❡ r❡❛❧❧② ❤♦❧♦♠♦r♣❤✐❝ ❛s t❤❡② s❤♦✉❧❞ ❜❡✳

Local P✷ example: polynomial part P❣

P♦❧②♥♦♠✐❛❧ r❡❝✉rs✐♦♥ ❣✐✈❡s P✷ = ✺ ✷✹❈ ✷❙✸ + ✶ ✹❈❤❙ − ✸ ✽❈❙✷s + ✶ ✽❙✷∂❈ ✭✷✼✮ ❙tr❛✐❣❤t❢♦r✇❛r❞ ❝♦♠♣✉t❛t✐♦♥s s❤♦✇ t❤❛t ✐ts ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❛t ▲❈❙▲ ✐s P✷

❆ = ❊(✻❆✹ − ✾❆✷❊ + ✺❊ ✷)

✶✼✷✽❇✻ ✭✷✽✮ ❚❤✐s ✐♠♣❧✐❡s ✐♥ r❡t✉r♥ t❤❛t t❤❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ q✉❛♥t✐t② P✷ ✐s P✷ = ˆ ❊(✻❆✹ − ✾❆✷ ˆ ❊ + ✺ˆ ❊ ✷) ✶✼✷✽❇✻ ✭✷✾✮ ❚❤❛t ✐s✱ t❤❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ q✉❛♥t✐t② P✷ ❝♦✉❧❞ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ✐ts ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t P✷

❆✱ t❤❛♥❦s t♦ ✭q✉❛s✐✮ ♠♦❞✉❧❛r✐t②✳

Local P✷ example: holomorphic ambiguity in terms

• f modular forms

❇② ✐♥❞✉❝t✐♦♥✱ ♦♥❡ ❝❛♥ ♣r♦✈❡ t❤❛t P❣ ❛♥❞ F❣ ❛r❡ ♥♦♥✲❤♦❧♦♠♦r♣❤✐❝ ❝♦♠♣❧❡t✐♦♥s ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s ✭t❤❛t ✐s✱ ♠♦❞✉❧❛r ✇❡✐❣❤ts ❛r❡ ✵✮ ❢♦r ❣ ≥ ✷✳ ❚❤❛t ✐s✱ t❤❡② ❛r❡ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢✉♥❝t✐♦♥s✳ ❚❤❡ ❛❜♦✈❡ r❡s✉❧t ❢♦r P✷ s✉❣❣❡sts t❤❡ ❢♦❧❧♦✇✐♥❣ ❛♥s❛t③ ❢♦r t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❛♠❜✐❣✉✐t② ❢ ✷ ❢ ✷ = ❝✶❆✻ + ❝✷❆✸❇✸ + ❝✸❇✻ ✶✼✷✽❇✻ ✭✸✵✮ ✭■♥ ❢❛❝t✱ t❤✐s ❢♦r♠ ❝♦✉❧❞ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ F❣ ♦♥ t❤❡ ❞❡❢♦r♠❛t✐♦♥ s♣❛❝❡ M✳✮

Local P✷ example: boundary condition at LCSL

❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ❛t t❤❡ ▲❈❙▲ ❣✐✈❡♥ ❜② α = ✵ ♦r ❡q✉✐✈❛❧❡♥t❧② t = ✐∞ ✐s ❋ ❣

❆ = (−✶)❣ χ

✷ |❇✷❣❇✷❣−✷| ✷❣(✷❣ − ✷)(✷❣ − ✷)! + O(❡✷π✐t), ❣ ≥ ✷ ■t ✐s ❡❛s② t♦ ❛♣♣❧② t❤✐s ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ s✐♥❝❡ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐ts ♦❢ t❤❡ q✉❛♥t✐t✐❡s ❆, ❇, ❈, ˆ ❊ ❜❛s❡❞ ❛t t❤❡ ▲❈❙▲ ❛r❡ ✈❡r② ❡❛s② t♦ ❝♦♠♣✉t❡ ❢r♦♠ t❤❡✐r ❡①♣r❡ss✐♦♥s ✐♥ t❡r♠s ♦❢ ❤②♣❡r❣❡♦♠❡tr✐❝ ❢✉♥❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ❆(α) = ✷❋✶(✶ ✸, ✷ ✸, ✶; α) = ✶+✷α ✾ +✶✵α✷ ✽✶ +✺✻✵α✸ ✻✺✻✶ +✸✽✺✵α✹ ✺✾✵✹✾ +✷✽✵✷✽α✺ ✺✸✶✹✹✶ +· · · ❇(α) = (✶−α)

✶ ✸ ✷❋✶(✶

✸, ✷ ✸, ✶; α) = ✶−α ✾ −✺α✷ ✽✶ −✷✼✼α✸ ✻✺✻✶ −✶✽✽✵α✹ ✺✾✵✹✾ +· · · · · ·

Local P✷ example: gap condition at the conifold point

❚❤❡ ❡①♣r❡ss✐♦♥ t❝(β) ✐s ❝❧❡❛r ♥❡❛r t❤❡ ❝♦♥✐❢♦❧❞ ♣♦✐♥t α = ✶ ♦r ❡q✉✐✈❛❧❡♥t❧② β = ✵ s✐♥❝❡ t❝ ✐s ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ✈❛♥✐s❤✐♥❣ ♣❡r✐♦❞ ♦❢ t❤❡ P✐❝❛r❞✲❋✉❝❤s ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❈❨ ✸✕❢♦❧❞ ❛t t❤❡ t❤✐s ♣♦✐♥t✳ ❋r♦♠ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ♣❡r✐♦❞ t❝(β) = β + ✶✶β✷ ✶✽ + ✶✵✾β✸ ✷✹✸ + ✾✸✽✾β✹ ✷✻✷✹✹ + ✽✽✸✺✶β✺ ✷✾✺✷✹✺ + · · · s♦❧✈❡❞ ❛s t❤❡ ✈❛♥✐s❤✐♥❣ ♣❡r✐♦❞ ♦❢ L❈❨ = L❡❧❧✐♣t✐❝ ◦ θ✱ ✇❡ ❝❛♥ ✐♥✈❡rt t❤❡ s❡r✐❡s t♦ ❣❡t β = β(t❝)✿ β(t❝) = t❝ − ✶✶t✷

❝

✶✽ + ✶✹✺t✸

❝

✹✽✻ − ✻✼✸✸t✹

❝

✺✷✹✽✽ + ✶✷✵✶✷✼t✺

❝

✷✸✻✶✾✻✵ + · · ·

Local P✷ example: gap condition at the conifold point

❚♦ ❛♣♣❧② t❤❡ ❣❛♣ ❝♦♥❞✐t✐♦♥ ❋ ❣

❝♦♥(t❝) =

❝❣−✶❇✷❣ ✷❣(✷❣ − ✷)(t❝)✷❣−✷ + O(t✵

❝ ),

❣ ≥ ✷, ♦♥❡ ♥❡❡❞s t♦ ❡✈❛❧✉❛t❡ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❋ ❣

❝♦♥ ❢r♦♠

F❣ = P❣ + ❢ ❣✳ ■♥ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ❜❛s❡❞ ❛t t❤❡ ❝♦♥✐❢♦❧❞✱ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ♦❢ ❢ ❣ ✐s ✐ts❡❧❢✱ ✇❤✐❧❡ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t P❣

❝♦♥ ♦❢

P❣ ✐s ❞✐✛❡r❡♥t ❢r♦♠ ✭t❤❡ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢✮ P❣

❆ s✐♥❝❡ ❛

❞✐✛❡r❡♥t ❜❛s❡ ♣♦✐♥t ✐s t❛❦❡♥✳ ❚❤❡♥ ♦♥❡ ♥❡❡❞s t♦ ❝♦♠♣✉t❡ t❤❡ β ♦r t❝ ❡①♣❛♥s✐♦♥ ♦❢ ❋ ❣

❝♦♥✳

Local P✷ example: series expansion in dual coordinates

❲❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ α, ¯ α ❡①♣❛♥s✐♦♥s ♦❢ F❣(α, ¯ α)✱ ✇❡ ❝♦✉❧❞ tr② t♦ ❞♦ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ ❣❡t (F❣ ◦ (α, ¯ α))(β, ¯ β)✳ ❇✉t ✐t ✐s ❡①tr❡♠❛❧❧② ❝♦♠♣❧✐❝❛t❡❞ t♦ ❞♦ t❤✐s ❞✐r❡❝t❧②✳

Local P✷ example: series expansion in dual coordinates

■♥st❡❛❞ ♦❢ ❞♦✐♥❣ ❛♥❛❧②t✐❝ ❝♦♥t✐♥✉❛t✐♦♥ t♦ ❣❡t (F❣ ◦ (α, ¯ α))(β, ¯ β)✱ ✇❡ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ❛s ❢♦❧❧♦✇s ❆(α) = ✶

√ ◆ ✐ τ

❆(β) ❇(α) = (✶ − α)

✶ ✸ ❆(α) = β ✶ ✸

✶

√ ◆ ✐ τ

❆(β) = ✶

√ ◆ ✐ τ

❈(β) ❈(α) = α

✶ ✸ ❆(α) =

✶

√ ◆ ✐ τ

❇(β) ˆ ❊(α, ¯ α) = −( ✶

√ ◆ ✐ τ

)✷ ˆ ❊(β, ¯ β) ❤❡r❡ ✇❡ tr❡❛t ❆, ❇, ❈, ❊ ❛s ❢✉♥❝t✐♦♥s ❆(•), ❇(•), · · · ✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❆(•) = ✷❋✶(✶

✸, ✷ ✸, ✶; •), · · ·

Local P✷ example: Fricke involution

❆t ✜rst ❣❧❛♥❝❡✱ ✐t s❡❡♠s t❤❛t ✇❡ ♦♥❧② ✉s❡❞ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ τ ✐♥ t❡r♠s ♦❢ ❆(α), ❆(β) t♦ ❣❡t t❤❡ s❡r✐❡s ❡①♣❛♥s✐♦♥ ✐♥ t❤❡ ❞✉❛❧ β ❝♦♦r❞✐♥❛t❡ ❢♦r ❛❧❧ ♦❢ t❤❡ ❣❡♥❡r❛t♦rs✳ ❍♦✇❡✈❡r✱ ✇❤❛t ✐s r❡❛❧❧② ✇♦r❦✐♥❣ ✐s t❤❡ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥✿ ❲◆ : τ ↔ − ✶ ◆τ α ↔ β

✷❋✶(✶

✸, ✷ ✸, ✶; α) ↔ ✷❋✶(✶ ✸, ✷ ✸, ✶; β) ❆(τ) ↔ ❆(❲◆τ) F❣(τ, ¯ τ) ↔ F❣|❲◆(❲◆τ, ❲◆τ) ✇✐t❤ √ ◆ ✐ τ = ❆(❲◆τ) ❆(τ) =

✷❋✶(✶ ✸, ✷ ✸, ✶; β) ✷❋✶(✶ ✸, ✷ ✸, ✶; α)

Local P✷ example: Fricke involution

❙✐♥❝❡ F❣ ❤❛s ✇❡✐❣❤t ✵✱ t❤❡

√ ◆ ✐ τ ❢❛❝t♦rs ❛r❡ ❝❛♥❝❡❧❧❡❞ ♦✉t✳ ❋r♦♠

P✷(α, ¯ α) = ˆ ❊(α, ¯ α)(✻❆(α)✹ − ✾❆(α)✷ ˆ ❊(α, ¯ α) + ✹ˆ ❊(α, ¯ α)✷) ✶✼✷✽❇✻(α) ❢ ✷(α) = ❝✶❆✻(α) + ❝✷❆✸(α)❇✸(α) + ❝✸❇✻(α) ✶✼✷✽❇✻(α) ✇❡ t❤❡♥ ❣❡t P✷(α(β, ¯ β), ¯ α(β, ¯ β)) = (−ˆ ❊(β, ¯ β))(✻❆(β)✹ − ✾❆(β)✷(−ˆ ❊(β, ¯ β)) + ✹(−ˆ ❊(β, ¯ β)✷)) ✶✼✷✽❈ ✻(β) ❢ ✷(α(β)) = ❝✶❆✻(β) + ❝✷❆✸(β)❈ ✸(β) + ❝✸❈ ✻(β) ✶✼✷✽❈ ✻(β)

Local P✷ example: solving the unknowns

❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❡❛s✐❧② ❣❡t t❤❡ β ❡①♣❛♥s✐♦♥ ♦❢ ❋ ✷

❝♦♥(β)

= ❧✐♠(P✷(α(β, ¯ β), ¯ α(β, ¯ β)) + ❢ ✷(α(β)) = (−❊(β)(✻❆(β)✹ − ✾❆(β)✷(−❊(β) + ✹(−❊(β)✷)) ✶✼✷✽❈ ✻(β) +❝✶❆✻(β) + ❝✷❆✸(β)❈ ✸(β) + ❝✸❈ ✻(β) ✶✼✷✽❈ ✻(β) ❋r♦♠ t❤❡ ❡①♣❛♥s✐♦♥ β(t❝) = t❝ − ✶✶t✷

❝

✶✽ + ✶✹✺t✸

❝

✹✽✻ − ✻✼✸✸t✹

❝

✺✷✹✽✽ + ✶✷✵✶✷✼t✺

❝

✷✸✻✶✾✻✵ + · · · ✇❡ ❝❛♥ ❡①♣r❡ss ❋ ✷

❝♦♥(β) ✐♥ t❡r♠s ♦❢ t❝ s❡r✐❡s✳ ❲❡ ❝❛♥ t❤❡♥ ♠❛❦❡ ✉s❡

♦❢ t❤❡ ❣❛♣ ❝♦♥❛t✐♦♥ t♦ ❣❡t ❧✐♥❡❛r ❡q✉❛t✐♦♥s s❛t✐s✜❡❞ ❜② ❝✶, ❝✷, ❝✸✳

Local P✷ example: predicting GW (GV) invariants

■t t✉r♥s ♦✉t t❤❛t ❝✶ = −✽ ✺, ❝✷ = ✷ ✺, ❝✸ = −✽ − ✸χ ✶✵ ✭✸✶✮ ❇② ✉s✐♥❣ t❤❡ ♠✐rr♦r ♠❛♣ α = −✷✼q − ✶✻✷q✷ − ✷✹✸q✸ − ✶✺✶✷q✹ + ✽✶✵✵q✺ + · · · , q = ❡①♣ ✷π✐t ✇❡ t❤❡♥ ❣❡t t❤❡ q = ❡①♣ ✷π✐t ❡①♣❛♥s✐♦♥ ♦❢ ❋ ✷

• ❲ (❨ , t) = ❊(✻❆✹ − ✾❆✷❊ + ✺❊ ✷)

✶✼✷✽❇✻ +−✽

✺❆✻ + ✷ ✺❆✸❇✸ + −✽−✸χ ✶✵

❇✻ ✶✼✷✽❇✻ ❚❤✐s ❣✐✈❡s ❡①❛❝t❧② t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ ❣❡♥✉s ✷ ●❲ ✭●❱✮ ✐♥✈❛r✐❛♥ts ❧✐st❡❞ ✐♥ ❑❛t③✱ ❑❧❡♠♠✱ ✫ ❱❛❢❛ ✭✶✾✾✾✮✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ✜rst ❢❡✇ ●❱ ✐♥✈❛r✐❛♥ts ♥❣=✷

❞

, ❞ = ✶, ✷, · · · ❛r❡ ✵, ✵, ✵, −✶✵✷, ✺✹✸✵, −✶✾✹✵✷✷, ✺✼✽✹✽✸✼, −✶✺✺✸✷✷✷✸✹, ✸✽✾✹✹✺✺✹✺✼, · · ·

Other examples: local ❞P♥, ♥ = ✺, ✻, ✼, ✽

❖♥❡ ❝❛♥ ❡❛s✐❧② ✇♦r❦ ♦✉t t❤❡ ❤✐❣❤❡r ❣❡♥✉s ❝❛s❡s ❢♦r ❧♦❝❛❧ P✷ ✉s✐♥❣ t❤❡ s❛♠❡ ❛♣♣r♦❛❝❤✳ ❋♦r t❤❡ ❧♦❝❛❧ ❞❡❧ P❡③③♦ ❣❡♦♠❡tr✐❡s ❑❞P♥, ♥ = ✺, ✻, ✼, ✽ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ♠♦❞✉❧❛r ❣r♦✉♣s ❜❡✐♥❣ Γ✵(◆) ✇✐t❤ ◆ = ✹, ✸, ✷, ✶∗ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ ✇❡ ♦✉t❧✐♥❡❞ ❛❜♦✈❡ ❝♦♥str✉❝ts t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ❢r♦♠ t❤❡ ♣❡r✐♦❞s✱ ❛♥❞ s♦❧✈❡s F❣✳ ❚❤❡ F❣s ❛❧s♦ ♣r❡❞✐❝t t❤❡ ❝♦rr❡❝t ●❱ ✐♥✈❛r✐❛♥t✳ ❚❤✐s ❛♣♣r♦❛❝❤ ♠❛❦✐♥❣ ✉s❡ ♦❢ ♠♦❞✉❧❛r✐t② ✇♦r❦s ❢♦r ♥♦♥❝♦♠♣❛❝t ❈❨ ✸✲❢♦❧❞ ❢❛♠✐❧✐❡s ✇❤♦s❡ ♠✐rr♦r ❝✉r✈❡s ❛r❡ ♦❢ ❣❡♥✉s ♦♥❡✱ ❛♥❞ ✇❤♦s❡ ❜❛s❡ M ❝♦✉❧❞ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ s♦♠❡ ♠♦❞✉❧❛r ❝✉r✈❡s✳

Local CY examples: differential ring of special geometry generators vs differential ring of almost-holomorphic modular forms

■♥ t❤❡s❡ ❡①❛♠♣❧❡s✱ ✜rst ✇❡ ❝♦♥str✉❝t❡❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✱ t❤❡♥ ✇❡ ❡①♣r❡ss❡❞ t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ ♦❢ ❣❡♥❡r❛t♦rs ❙✐❥, ❙✐, ❙, ❑✐ ✐♥ t❡r♠s ♦❢ t❤❡s❡ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ❆❢t❡r t❤❛t ✇❡ ❞✐❞ t❤❡ ♣♦❧②♥♦♠✐❛❧ r❡❝✉rs✐♦♥✳ ❲❡ ❝♦✉❧❞ ❤❛✈❡ st❛rt❡❞ ❢r♦♠ t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ ♦❢ ❣❡♥❡r❛t♦rs ❙✐❥, ❙✐, ❙, ❑✐ ❝♦♥str✉❝t❡❞ ❢r♦♠ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ✇✐t❤♦✉t ❦♥♦✇✐♥❣ t❤❡✐r r❡❧❛t✐♦♥s t♦ t❤❡ ❣❡♥❡r❛t♦rs ❆, ❇, ❈, ❊✱ ♣r♦✈✐❞❡❞ t❤❛t ✇❡ ❦♥♦✇ t❤❡ ❝♦rr❡❝t ♥♦t✐♦♥ ♦❢ τ ❛♥❞ t❤❡✐r ❣r❛❞✐♥❣s ❛s ✧♠♦❞✉❧❛r ✇❡✐❣❤ts✧ ✇❤✐❝❤ t❡❧❧ ❤♦✇ t❤❡② tr❛♥s❢♦r♠✳

Local CY examples: special geometry polynomial ring

❚❤✐s r✐♥❣ ♦❢ ❣❡♥❡r❛t♦rs ❙✐❥, ❙✐, ❙, ❑✐ ❝♦♥str✉❝t❡❞ ❢r♦♠ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ✭❝❛❧❧❡❞ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ❜❡❧♦✇✮ ✐s ❛s ❢♦❧❧♦✇s ✭❢♦r ❑P✷✱ κ = −✶

✸✮✿

❧✐♠ ❙tt(= ✶ ✷❊), θt(= ❆), ❈ −✶

ttt = κ−✶ ∂t

∂τ (= κ−✶❇✸) ❉❙tt = −❙tt❙tt − ✶ ✶✷κ(θt)✹ ❉θt = ❈ −✶

ttt θ✷t = −❙ttθt + ❈ −✶ ttt s① ①①

❉❈ −✶

ttt = −✸❈ −✶ ttt ❙tt + ❈ −✶ ttt (θt)✷(∂① ❧♦❣ ❈ −✶ ①①① + ✸s① ①①)

✇❤❡r❡ s①

①① = ✶ ✻ α β − ✶ ✸, ∂① ❧♦❣ ❈ −✶ ①①① = −α β ✳ ❚♦ ♠❛❦❡ t❤❡ r✐♥❣ ❝❧♦s❡❞✱

✇❡ ❛❞❞ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ q✉❛♥t✐t② ∂① ❧♦❣ ❈①①① = α

β = ❈ ✸ ❇✸ ✳ ◆♦✇ ✐ts

❞❡r✐✈❛t✐✈❡ ❧✐❡s ✐♥ t❤❡ r✐♥❣ ♦❢ t❤❡ ❛❜♦✈❡ ❣❡♥❡r❛t♦rs✿ ❉∂① ❧♦❣ ❈①①① = (∂① ❧♦❣ ❈①①①)❉ ❧♦❣ ∂① ❧♦❣ ❈①①① = (∂① ❧♦❣ ❈①①①)(θt)✷

Local CY examples: special geometry polynomial ring vs ring of quasi modular forms

❚❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ♦❢ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐s ❡ss❡♥t✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s

❬❆❙❨❩❪

❉❆ = ✶ ✷r ❆(❊ + ❈ r − ❇r ❆r ❆✷) ❉❇ = ✶ ✷r ❇(❊ − ❆✷) ❉❈ = ✶ ✷r ❈(❊ + ❆✷) ❉❊ = ✶ ✷r (❊ ✷ − ❆✹) ✭✸✷✮ ✇❤❡r❡ ❉ =

✶ ✷π✐ ∂ ∂τ ❛♥❞ r = ✷, ✸, ✹, ✻ ❢♦r ◆ = ✹, ✸, ✷, ✶∗ r❡s♣❡❝t✐✈❡❧②✳

Special geometry polynomial ring

❚❤✐s s❡❡♠s t♦ s✉❣❣❡st t❤❛t t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ ❝♦♥♥❡❝t✐♦♥s✱ ❨✉❦❛✇❛ ❝♦✉♣❧✐♥❣s✱ ❡t❝✳ ✐s ❛ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ♦❢ t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✱ ❡✈❡♥ ❢♦r t❤❡ ❝❛s❡s ✐♥ ✇❤✐❝❤ t❤❡ ❛r✐t❤♠❡t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ ✐s ✉♥❦♥♦✇♥✳ ❚❤✐s ❧❡❛❞s ✉s t♦ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✧s♣❡❝✐❛❧✧ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ♦♥ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ ✭❞✐♠M❂✶✮❬❆❙❨❩❪ ❑✵ = κ ❈ −✶

ttt (θt)−✸ ,

• ✶ = θt ,

❑✷ = κ ❈ −✶

ttt ❑t

❚✷ = ❙tt , ❚✹ = ❈ −✶

ttt ˜

❙t , ❚✻ = ❈ −✷

ttt ˜

❙✵ , ✭✸✸✮ ✇✐t❤ θ = ③ ❞

❞③ ✱ ③ ✐s t❤❡ ❛❧❣❡❜r❛✐❝ ❝♦♦r❞✐♥❛t❡✳ ❲❡ ❢✉rt❤❡r♠♦r❡ ♥❡❡❞ ❛

❣❡♥❡r❛t♦r ❈✵ = θ ❧♦❣ ③✸❈③③③ ❢♦r t❤❡ ❝♦❡✣❝✐❡♥ts ❢r♦♠ t❤❡✐r ❞❡r✐✈❛t✐✈❡s✳

Special geometry polynomial ring

❚❤✐s ❞✐✛❡r❡♥t✐❛❧ r✐♥❣✱ ✇❤✐❝❤ ❤❛s ❛ ♥✐❝❡ ❣r❛❞✐♥❣ ❝❛❧❧❡❞ t❤❡ ✇❡✐❣❤t✱ ✐s ❣✐✈❡♥ ❜②

∂τ❑✵ = −✷❑✵ ❑✷ − ❑ ✷

✵ ● ✷ ✶ (˜

❤③

③③③ + ✸(s③ ③③ + ✶)) ,

∂τ●✶ = ✷●✶ ❑✷ − κ●✶ ❚✷ + ❑✵● ✸

✶ (s③ ③③ + ✶) ,

∂τ❑✷ = ✸❑ ✷

✷ − ✸κ❑✷ ❚✷ − κ✷❚✹ + ❑ ✷ ✵ ● ✹ ✶ ❦③③ − ❑✵ ● ✷ ✶ ❑✷ ˜

❤③

③③③ ,

∂τ❚✷ = ✷❑✷ ❚✷ − κ❚ ✷

✷ + ✷κ❚✹ + ✶

κ ❑ ✷

✵ ● ✹ ✶ ˜

❤③

③③ ,

∂τ❚✹ = ✹❑✷❚✹ − ✸κ❚✷ ❚✹ + ✷κ❚✻ − ❑✵ ● ✷

✶ ❚✹˜

❤③

③③③ − ✶

κ❑ ✷

✵ ● ✹ ✶ ❚✷❦③③

+ ✶ κ✷ ❑ ✸

✵ ● ✻ ✶ ˜

❤③③ , ∂τ❚✻ = ✻❑✷ ❚✻ − ✻κ❚✷ ❚✻ + κ ✷ ❚ ✷

✹ − ✶

κ❑ ✷

✵ ● ✹ ✶ ❚✹ ❦③③ + ✶

κ✸ ❑ ✹

✵ ● ✽ ✶ ˜

❤③ − ✷ ❑✵ ● ✷

✶ ❚✻˜

❤③

③③③ ,

✭✸✹✮

✇❤❡r❡ τ = κ−✶❋tt ❛♥❞ t❤❡ s✉❜✲✐♥❞✐❝❡s ❛r❡ t❤❡ ✇❡✐❣❤ts ❬❆❙❨❩❪✳ ❋♦r ❡❛❝❤ ❣ ≥ ✷✱ t❤❡ ♥♦r♠❛❧✐③❡❞ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥ (❳ ✵)✷−✷❣F❣ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ ✇❡✐❣❤t ✵ ✐♥ t❤❡s❡ ❣❡♥❡r❛t♦rs✳

Candidate of the ring of almost-holomorphic modular forms

❆s ✇❡ ❤❛✈❡ ❞✐s❝✉ss❡❞✱ ❢♦r ❧♦❝❛❧ P✷ ❛♥❞ ❧♦❝❛❧ ❞❡❧ P❡③③♦ ❝❛s❡s✱ t❤✐s r✐♥❣✱ ✐♥ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t✱ ✐s ❡ss❡♥t✐❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s✳ ❋♦r ❛ ❣❡♥❡r❛❧ ❈❨ ✸✕❢♦❧❞ ❳ ✭❝♦♠♣❛❝t ♦r ♥♦♥❝♦♠♣❛❝t✮✱ ✇❡ ❝♦✉❧❞ ✉s❡ t❤✐s r✐♥❣✱ ❛s ❛ ❣✉✐❞❛♥❝❡ ❢♦r t❤❡ st✉❞② ♦❢ ♠♦❞✉❧❛r ❢♦r♠s✳

Candidate of the ring of almost-holomorphic modular forms

❋♦r ❡①❛♠♣❧❡✱ ❢♦r t❤❡ ♠✐rr♦r q✉✐♥t✐❝ ❢❛♠✐❧②✱ ♦♥❡ ❣❡ts

∂τ❈✵ = ❈✵ (✶ + ❈✵) ❑✵ ● ✷

✶ ,

∂τ❑✵ = −✷❑✵ ❑✷ − ❈✵❑ ✷

✵ ● ✷ ✶ ,

∂τ●✶ = ✷●✶ ❑✷ − ✺●✶ ❚✷ − ✸ ✺❑✵● ✸

✶ ,

∂τ❑✷ = ✸❑ ✷

✷ − ✶✺❑✷ ❚✷ − ✷✺❚✹ + ✷

✷✺ ❑ ✷

✵ ● ✹ ✶ −

✾ ✺ + ❈✵

• ❑✵ ● ✷

✶ ❑✷ ,

∂τ❚✷ = ✷❑✷ ❚✷ − ✺❚ ✷

✷ + ✶✵❚✹ + ✶

✷✺(✶ + ❈✵) ❑ ✷

✵ ● ✹ ✶ ,

∂τ❚✹ = ✹❑✷❚✹ − ✶✺❚✷ ❚✹ + ✶✵❚✻ − ✾ ✺ + ❈✵

• ❑✵ ● ✷

✶ ❚✹ −

✷ ✶✷✺❑ ✷

✵ ● ✹ ✶ ❚✷

− ✶ ✻✷✺(✶ + ❈✵)❑ ✸

✵ ● ✻ ✶ ,

∂τ❚✻ = ✻❑✷ ❚✻ − ✸✵❚✷ ❚✻ + ✺ ✷❚ ✷

✹ −

✷ ✶✷✺❑ ✷

✵ ● ✹ ✶ ❚✹ +

✷ ✼✽✶✷✺(✶ + ❈✵)❑ ✹

✵ ● ✽ ✶

− ✷ ✾ ✺ + ❈✵

• ❑✵ ● ✷

✶ ❚✻ .

✭✸✺✮

Candidate of the ring of almost-holomorphic modular forms

❙✐♥❝❡ ✇❡ ❦♥♦✇ ❤♦✇ t♦ t❛❦❡ t❤❡ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐t ✐♥ t❤✐s s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣✱ ✇❡ ❝❛♥ ❣❡t t❤❡ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ❢r♦♠ ✐t✳ ❇✉t ✐❢ t❤❡ ❛r✐t❤♠❡t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡ ✐s ♥♦t ❝❧❡❛r✱ ✇❡ ❝❛♥♥✬t r❡❛❧❧② s❛② t❤❛t t❤❡ s♣❡❝✐❛❧ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✐s t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳ ❆❧s♦ ✐❢ t❤❡r❡ ✐s ♥♦ s✉✐t❛❜❧❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠♦❞✉❧❛r ❣r♦✉♣✱ ✐t ✐s ♥♦t ❝❧❡❛r ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ❛s ❛♥ ❞✉❛❧✐t② ♦❢ t❤❡ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s✳

Conclusions

■♥ s✉♠♠❛r②✱ ✇❡ ❤❛✈❡

• ❝♦♠♣✉t❡❞ F❣s ✐♥ t❡r♠s ♦❢ ♠♦❞✉❧❛r ❢♦r♠s ❢♦r ❝❡rt❛✐♥ ❈❨

♠❛♥✐❢♦❧❞s ✇❤♦s❡ ♠♦❞✉❧✐ s♣❛❝❡s ❤❛✈❡ ♥✐❝❡ ❛r✐t❤♠❡t✐❝ ❞❡s❝r✐♣t✐♦♥s✿ M ∼ = H∗/Γ✵(◆)

• ❢♦✉♥❞ ❛ ❞✉❛❧✐t② ❛❝t✐♥❣ ♦♥ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s

❢♦r t❤❡s❡ ❡①❛♠♣❧❡s✿ F❣

❝♦♥ = F❣ ❆|❲◆

• ❝♦♥str✉❝t❡❞ t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ ✇❤✐❝❤ ❤❛s ❛

♥✐❝❡ ❣r❛❞✐♥❣✳ ❚❤❡ ♥♦r♠❛❧✐③❡❞ t♦♣♦❧♦❣✐❝❛❧ str✐♥❣ ♣❛rt✐t✐♦♥ ❢✉♥❝t✐♦♥s ❛r❡ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s ♦❢ ❞❡❣r❡❡ ③❡r♦ ✐♥ t❤❡s❡ ❣❡♥❡r❛t♦rs✳ ❚❤✐s r✐♥❣ ✐s ❛ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r t❤❡ r✐♥❣ ♦❢ ❛❧♠♦st✲❤♦❧♦♠♦r♣❤✐❝ ♠♦❞✉❧❛r ❢♦r♠s✳

Discussions and future directions

❚❤❡r❡ ❛r❡ s♦♠❡ ✐♥t❡r❡st✐♥❣ q✉❡st✐♦♥s

• ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ τ ❝♦♦r❞✐♥❛t❡ ✐♥ ❣❡♥❡r❛❧✐t②✱ ❡✳❣✳ ❢♦r ❝♦♠♣❛❝t

❈❨✱ ♠✉❧t✐✲♠♦❞✉❧✐ ❝❛s❡s❄

• ❡♥✉♠❡r❛t✐✈❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ qτ = ❡①♣ ✷π✐τ ❡①♣❛♥s✐♦♥s❄

✭❡①❛♠♣❧❡✿ q✉✐♥t✐❝✱ ✇❤❡r❡ qt = ❡①♣(✷π✐t) ✐s ✉s❡❞ ✐♥ ●❲ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s✮ qt = qτ − ✺✼✺q✷

τ + ✽✷✺✵q✸ τ + ✹✸✼✺✶✷✺✵q✹ τ + . . .

• ❡♥✉♠❡r❛t✐✈❡ ♠❡❛♥✐♥❣ ♦❢ ❤♦❧♦♠♦r♣❤✐❝ ❧✐♠✐ts ♦❢ F❣ ❛t s♦♠❡

♦t❤❡r ♣♦✐♥ts ♦♥ t❤❡ ♠♦❞✉❧✐ s♣❛❝❡✳

• ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❣❛♣ ❝♦♥❞✐t✐♦♥ ✐♥ ♠❛t❤❡♠❛t✐❝s
• ❋r✐❝❦❡ ✐♥✈♦❧✉t✐♦♥ ♦♥ t❤❡ ❧❡✈❡❧ ♦❢ ♠♦❞✉❧✐ s♣❛❝❡ ♦❢ ❈❨s
• ❍♦✇ ❡①❛❝t❧② ✐s t❤❡ s♣❡❝✐❛❧ ❣❡♦♠❡tr② ♣♦❧②♥♦♠✐❛❧ r✐♥❣ r❡❧❛t❡❞ t♦

t❤❡ r✐♥❣ ♦❢ q✉❛s✐ ♠♦❞✉❧❛r ❢♦r♠s ❢♦r ❣❡♥❡r❛❧ ❈❨s✱ ❡✳❣✱ ♠✐rr♦r q✉✐♥t✐❝❄

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• ❝♦❧❧❛❜♦r❛t♦rs ▼✉r❛❞ ❆❧✐♠✱ ❊♠❛♥✉❡❧ ❙❝❤❡✐❞❡❣❣❡r

❛♥❞ ❙❤✐♥❣✲❚✉♥❣ ❨❛✉

• ❨♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥