Quadratic relations for periods of connections Claude Sabbah Joint work with Javier Fresán (CMLS, Palaiseau) and Jeng-Daw Yu (NTU, Taipei) Centre de mathématiques Laurent Schwartz CNRS, École polytechnique, Institut polytechnique de Paris Palaiseau, France
Quadratic relations for periods: the classical case Example: the case of a curve. ∙ 푋 : curve of genus 푔 ⩾ 1 . ∙ 푋 connected smooth complex projective mfld of dim. 푛 . ∙ Pairing 햲 : ∙ ( 훾 푖 ) 푖 : basis of 퐻 푚 ( 푋 an , ℚ ) , ( 휔 푗 ) 푗 : basis of 퐻 푚 dR ( 푋 ) . ∙ 휔 1 , … 휔 푔 basis of 퐻 0 ( 푋, Ω 1 푋 ) , ∙ Period matrix 햯 푚 = ( 햯 푚 ; 푖푗 ) , with 햯 푚 ; 푖푗 ∶= ∫ 훾 푖 ∙ 푓 1 , … , 푓 푔 ∈ 퐻 1 ( 푋, 풪 푋 ) , 휔 푗 . Res ∙ ̌ 퐻 1 ( 푋, 풪 푋 ) ⊗ ̌ 푋 ) → ̌ 퐻 0 ( 푋, Ω 1 퐻 1 ( 푋, Ω 1 ≃ ℂ . 푋 ) ∙ De Rham duality pairing ∫ 푋 ∙ 햲 ( 푓 푖 , 휔 푗 ) = − 햲 ( 휔 푗 , 푓 푖 ) = [ 푓 푖 휔 푗 ] ∈ 퐻 1 ( 푋, Ω 1 푋 ) = ℂ . 햰 푚 ∶ 퐻 푚 dR ( 푋 ) ⊗ 퐻 2 푛 − 푚 ( 푋 ) ⟶ 퐻 2 푛 dR ( 푋 ) ∼ ℂ . ← ← ← ← ← ← ← ← → ← ← ∙ Can also represent 퐻 1 ( 푋, 풪 푋 ) by (0 , 1) forms 휂 푖 and dR 햲 ( 휂 푖 , 휔 푗 ) = − 햲 ( 휔 푗 , 휂 푖 ) = 1 2 휋 헂 ∫ 푋 ∙ Betti intersection pairing 휂 푖 ∧ 휔 푗 . 햡 푚 ∶ 퐻 푚 ( 푋 an , ℚ ) ⊗ 퐻 2 푛 − 푚 ( 푋 an , ℚ ) ⟶ 퐻 0 ( 푋 an , ℚ ) ≃ ℚ . ∙ Quadratic relations for the associated matrices, e.g. 푚 = 푛 : ∙ Pairing 햡 : ∙ ( 훼 푖 , 훼 푔 + 푖 ) 푖 =1 , … ,푔 symplectic basis of 퐻 1 ( 푋, ℤ ) . (−1) 푛 햡 푛 = 햯 푛 ⋅ ( 햰 푛 ) −1 ⋅ 푡 햯 푛 ∙ ⟿ 햡 푖,푔 + 푖 = − 햡 푔 + 푖,푖 = 1 and 햡 푘, 퓁 = 0 otherwise. ∙ Set 햲 푚 ∶= (2 휋 헂 ) − 푛 햰 푚 , ∀ 푚 . (−2 휋 헂 ) 푛 햡 푛 = 햯 푛 ⋅ ( 햲 푛 ) −1 ⋅ 푡 햯 푛 ∙ ⟿ Bilinear relations.
� � � � Sketch of proof. Quadratic relations for periods of vector bundles 햰 푛 with log connection 퐻 푛 dR ( 푋 an ) ⊗ 퐻 푛 dR ( 푋 an ) ℂ Vector bundles with log connection. P 푛 ≀ ∙ 푋 connected smooth quasi-projective, ( 푉 , ∇) : alg. vect. bdle 햯 푛 퐻 푛 ( 푋 an ) ⊗ 퐻 푛 dR ( 푋 an ) ℂ on 푋 with flat connection having reg. sing. at ∞ on 푋 . ∙ 퐻 푘 dR ( 푋, ( 푉 , ∇)) , 퐻 푘 dR , c ( 푋, ( 푉 , ∇)) : ≀ P 푛 ∙ Choose ( 푋, 퐷 ) smooth proj. 퐷 = ncd, 푋 = 푋 ∖ 퐷 . 햡 푛 퐻 푛 ( 푋 an ) ⊗ 퐻 푛 ( 푋 an ) � ℂ ∙ Deligne’s canonical extension ( 푉 0 , ∇) : ∗ 푉 0 : vect. bdle on 푋 extending 푉 : ∙ P 푛 ∶= Poincaré isomorphism. ∙ Compatibility proved by de Rham by realizing 퐻 푛 ( 푋 an ) as ∗ ∇ ∶ 푉 0 → Ω 1 푋 (log 퐷 ) ⊗ 푉 0 extending ∇ currents. ∗ eigenvalues of res 퐷 푖 ∇ have real part in [0 , 1) . ∙ In term of matrices (e.g. 햰 푛 ( 휔, 휔 ′ ) = 푡 휔 ⋅ 햰 푛 ⋅ 휔 ′ ): dR ( 푋, ( 푉 , ∇)) ≃ 푯 푘 ( 푋, (Ω∙ ∙ 퐻 푘 푋 (log 퐷 ) ⊗ 푉 0 , ∇)) , 푡 P 푛 ⋅ 햯 푛 = 햰 푛 , 햡 푛 ⋅ P 푛 = 햯 푛 . dR , c ( 푋, ( 푉 , ∇)) ≃ 푯 푘 ( 푋, (Ω∙ ∙ 퐻 푘 푋 (log 퐷 ) ⊗ 푉 0 (− 퐷 ) , ∇)) . ⟹ ∙ Assume given pairing ⟨ ∙ , ∙ ⟩ ∶ 푉 ⊗ 푉 → 풪 푋 s.t. 햡 푛 = 햯 푛 ⋅ ( P 푛 ) −1 = 햯 푛 ⋅ ( 푡 햰 푛 ) −1 ⋅ 푡 햯 푛 . ∼ ∙ nondegener. i.e., induces 푉 ⟶ 푉 ∨ , ∙ ± -symmetric, i.e., ⟨ 푤, 푣 ⟩ = ± ⟨ 푣, 푤 ⟩ , ∙ Use 푡 햰 푛 = (−1) 푛 햰 푛 . � ∙ compatible with ∇ , i.e., d ⟨ 푣, 푤 ⟩ = ⟨ ∇ 푣, 푤 ⟩ + ⟨ 푣, ∇ 푤 ⟩ . ∙ ⟿ Tr 햲 푚 ∶ 퐻 푚 dR , c ( 푋, ( 푉 , ∇)) ⊗퐻 2 푛 − 푚 ( 푋, ( 푉 , ∇)) ⟶ 퐻 2 푛 dR , c ( 푋, ( 풪 푋 , d)) ≃ ℂ dR
Intersection pairings between flat sections. Middle quadratic relations. [ ] ∙ 풱 = 푉 an , ∇ loc. cst. sheaf of horiz. sections. ∙ 퐻 푚 퐻 푚 dR , c ( 푋, ( 푉 , ∇)) → 퐻 푚 , dR , mid ( 푋, ( 푉 , ∇)) ∶= im dR ( 푋, ( 푉 , ∇)) [ ] ∙ ⟿ ± -sym. nondeg. pairing ⟨ ∙ , ∙ ⟩ ∶ 풱 ⊗ 풱 → ℂ 푋 . ∙ 퐻 mid 푚 ( 푋 an , 풱 퐻 푚 ( 푋 an , 풱 푚 ( 푋 an , 풱 ℚ ) → 퐻 BM ℚ ) ∶= im ℚ ) ∙ Assume defined over ℚ : ∙ ⟿ Nondeg. ± -sym. pairings, e.g. for 푚 = 푛 : ℚ , ∙ 풱 = ℂ ⊗ ℚ 풱 햲 mid ∶ 퐻 푛 dR , mid ( 푋, ( 푉 , ∇)) ⊗ 퐻 푛 dR , mid ( 푋, ( 푉 , ∇)) ⟶ ℂ , ℚ → ℚ 푋 . ∙ ⟨ ∙ , ∙ ⟩ ∶ 풱 ℚ ⊗ 풱 햡 mid ∶ 퐻 mid ( 푋 an , 풱 ℚ ) ⊗ 퐻 mid ( 푋 an , 풱 ℚ ) ⟶ ℚ , 푛 푛 ∙ ⟿ 퐻 푚 ( 푋 an , 풱 ℚ ) , 퐻 BM 푚 ( 푋 an , 풱 ℚ ) , 햯 mid ∶ 퐻 mid ( 푋 an , 풱 ℚ ) ⊗ 퐻 푛 dR , mid ( 푋, ( 푉 , ∇)) ⟶ ℂ . 푛 ∙ ⟿ 햡 푚 ∶ 퐻 푚 ( 푋 an , 풱 2 푛 − 푚 ( 푋 an , 풱 ℚ ) → ℚ . ℚ ) ⊗ 퐻 BM Period pairings. Corollary (Quadratic relations) . ∙ Two period pairings (by using ⟨ ∙ , ∙ ⟩ ): ±(−2 휋 헂 ) 푛 햡 mid = 햯 mid ⋅ ( 햲 mid ) −1 ⋅ 푡 햯 mid 햯 푚 ∶ 퐻 푚 ( 푋 an , 풱 ℚ ) ⊗ 퐻 2 푛 − 푚 ( 푋, ( 푉 , ∇)) ⟶ ℂ dR 푚 ( 푋 an , 풱 ℚ ) ⊗ 퐻 2 푛 − 푚 햯 BM 푚 ∶ 퐻 BM dR , c ( 푋, ( 푉 , ∇)) ⟶ ℂ Example (Matsumoto, 1994). ∙ Quadratic relations for generalized hypergeometric functions Theorem (Matsumoto & al., 1994) . (Appell, Lauricella...). ∙ 햯 푚 and 햯 BM 푚 are nondeg. ∙ “Quadratic relations” e.g. for 푚 = 푛 : ±(−2 휋 헂 ) 푛 햡 푛 = 햯 푛 ⋅ ( 햲 푛 ) −1 ⋅ 푡 햯 BM 푛 .
A conjecture of Broadhurst and Roberts Generalization of the quadratic relations (F-S-Y). ∙ Since 퐼 0 , 퐾 0 are sols of a diff. eq. with irreg. sing. need to Bessel moments and Bernoulli matrices. extend quadratic relations to this case. ∙ Bessel moments : ∙ ⟿ Consider (Kl 2 , ∇) rk 2 vect. bdle on 픾 m ⟷ “modified ∙ Special values of some Feynman integrals expressed as pe- Bessel diff. eq.” and (Sym 푘 Kl 2 , ∇) . riod of Laurent polynomials. E.g. 푓 ( 푥, 푦, 푧 ) = (1 + 푥 + 푦 + 푧 )(1 + 푥 −1 + 푦 −1 + 푧 −1 ) . ∙ ⟿ Nondegen. de Rham pairing dR , c ( 픾 m , Sym 푘 Kl 2 ) ⊗ 퐻 1 dR ( 픾 m , Sym 푘 Kl 2 ) ⟶ ℂ . 햲 푘 ∶ 퐻 1 ∙ These periods are also expressed as 푘 -moments of the “mod- ified Bessel functions” 퐼 0 ( 푡 ) , 퐾 0 ( 푡 ) (e.g. 푘 = odd integer): BM 푘 ( 푖, 푗 ) = ⋆ ∫ ∙ ⟿ Rapid decay and moderate twisted homology and ∞ ( 푡 ) ⋅ 푡 2 푗 d 푡 [ ] 퐼 푖 0 ( 푡 ) 퐾 푘 − 푖 푡 . ( 픾 m , Sym 푘 Kl 2 ) ∶= im 1 ( 픾 m , Sym 푘 Kl 2 ) → 퐻 mod 0 퐻 mid 퐻 rd (… ) . 0 1 1 ∙ Bernoulli matrix ( 퐵 푛 ∶= 푛 th Bernoulli nbr): ∙ ⟿ Nondegen. Betti intersection pairing: 퐵 푘 − 푖 − 푗 −1 B 푘 ( 푖, 푗 ) = (−1) 푘 − 푖 ( 푘 − 푖 )!( 푘 − 푗 )!) ( 푘 − 푖 − 푗 − 1)! . 1 ( 픾 m , Sym 푘 Kl 2 ) ⊗ 퐻 mod ( 픾 m , Sym 푘 Kl 2 ) ⟶ ℚ . ⋅ 햡 푘 ∶ 퐻 rd 푘 ! 1 Conjecture (B-R, by computation, e.g. 푘 odd) . Set 푘 ′ = ( 푘 −1)∕2 . ∙ ⟿ Nondegen. Period pairings Consider the 푘 ′ × 푘 ′ matrices 1 ( 픾 m , Sym 푘 Kl 2 ) ⊗ 퐻 1 dR ( 픾 m , Sym 푘 Kl 2 ) ⟶ ℂ 햯 rd , mod ∶ 퐻 rd 푘 ( 픾 m , Sym 푘 Kl 2 ) ⊗ 퐻 1 dR , c ( 픾 m , Sym 푘 Kl 2 ) ⟶ ℂ . 햯 mod , rd ∶ 퐻 mod BM 푘 = (BM 푘 ( 푖, 푗 )) 1 ⩽ 푖,푗 ⩽ 푘 ′ B 푘 = ( B 푘 ( 푖, 푗 )) 1 ⩽ 푖,푗 ⩽ 푘 ′ . and 푘 1 There exists D 푘 ∈ GL 푘 ′ ( ℚ ) defined by an explicit algorithm s.t. ∙ ⟿ Middle quadratic relations: (−2 휋 헂 ) 푘 +1 B 푘 = BM 푘 ⋅ D 푘 ⋅ 푡 BM 푘 . 푘 ) −1 ⋅ 푡 햯 mid (−2 휋 헂 ) 푘 +1 햡 mid = 햯 mid ⋅ ( 햲 mid 푘 푘 푘 ¿ Interpret the conj. in terms of quadratic relations for periods ?
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