Calabi-Yaus and Other Animals [arXiv:1805.09326] with J. Bourjaily, Y.-H. He, A. Mcleod, and M. Wilhelm [arXiv:1810.07689] with J. Bourjaily, A. Mcleod, and M. Wilhelm Matt von Hippel (Niels Bohr International Academy) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 1 / 24
Introduction Multiple Polylogarithms Integrals over rational factors: � z 1 G ( w 1 , w 2 , . . . ; z ) = G ( w 2 , . . . ; x ) dx x − w 1 0 Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 2 / 24
Introduction Elliptic Multiple Polylogarithms Integrals over an elliptic curve: � z 1 � � 0 n 2 ... y ( x )E( n 2 ... E 0 c 2 ... ; z = c 2 ... ; x ) dx 0 where y 2 ∼ ( x 4 ) + x 3 + . . . Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 3 / 24
Introduction ??? Multiple Polylogarithms Integrals over a higher-dimensional manifold: 1 � F (? ? ? ) = y ( x 1 , x 2 , . . . ) F (? ? ? ; x 1 , x 2 , . . . ) dx 1 dx 2 . . . where y 2 ∼ P ( x 1 , x 2 , . . . ) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 4 / 24
Introduction Known Examples are Calabi-Yau Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Known Examples are Calabi-Yau Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Known Examples are Calabi-Yau Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Known Examples are Calabi-Yau Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Known Examples are Calabi-Yau Eight-loop φ 4 vacuum graph with a K3 (CY 2 ) [Brown, Schnetz] Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Known Examples are Calabi-Yau Eight-loop φ 4 vacuum graph with a K3 (CY 2 ) [Brown, Schnetz] L -loop “traintracks” appear to be CY L − 1 [Bourjaily, He, Mcleod, MvH, Wilhelm] Known to be CY L − 1 at L loops [Bloch, Kerr, Vanhove; Broadhurst] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 5 / 24
Introduction Questions: Why are these examples Calabi-Yau? Are more Feynman integrals Calabi-Yau? (All?) How bad can it get? (Dimensions vs. loop order) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 6 / 24
Introduction Questions: Why are these examples Calabi-Yau? Are more Feynman integrals Calabi-Yau? (All?) How bad can it get? (Dimensions vs. loop order) My Goals Today: Make what definite statements I can Inspire further investigation! Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 6 / 24
Introduction Introduction 1 Direct Integration and Rigidity 2 Marginal Integrals are Calabi-Yau 3 A Calabi-Yau Bestiary 4 Traintracks 5 Conclusions 6 Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 7 / 24
Direct Integration and Rigidity Symanzik Form Introduce “alpha parameters” for each propagator: � ∞ 1 e i ( p 2 − m 2 ) α d α p 2 − m 2 = 0 Get well-known form, projective integral over one variable per edge: [ d E − 1 x i ] U E − ( L +1) D / 2 � Γ( E − LD / 2) F E − LD / 2 x i ≥ 0 Graph polynomials U and F defined by: � �� � � � � � � x i m 2 U ≡ x i , F ≡ + U s T 1 x i i e i { T }∈ T 1 e i / ∈ T { T 1 , T 2 }∈ T 2 e i / ∈ T 1 ∪ T 2 (Neglecting numerators, higher propagator powers) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 8 / 24
Direct Integration and Rigidity Symanzik Form: Special Cases Two cases where things simplify, both for even dimensions: E = LD / 2: Explored by mathematicians. Superficial divergence from gamma function, if there are no subdivergences can strip this off, no need for dim reg. Only U contributes. 1 � [ d E − 1 x i ] U D / 2 x i ≥ 0 E = ( L + 1) D / 2: Marginal . If finite, can again avoid dim reg. Only F contributes. 1 � [ d E − 1 x i ] F D / 2 x i ≥ 0 In D = 2, these are the sunrise/banana graphs! Many more cases in D = 4 Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 9 / 24
Direct Integration and Rigidity Direct Integration We can attempt to integrate these with direct integration: Start with a rational function. Can partial-fraction in some variable x , getting P ( z ) R ( z ) � x − Q ( z ) + ( x − S ( z )) 2 + . . . x ≥ 0 where z represents the other variables. Linear denominators integrate to logarithms, double poles and higher stay rational If P , Q , . . . rational in another variable, repeat: get polylogarithms Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 10 / 24
Direct Integration and Rigidity Rigidity What if some of P , Q , . . . aren’t rational? Square root of a quadratic: this is expected to still be polylogarithmic. Sometimes possible to manifestly rationalize with a change of variables, see e.g. [Besier, Van Straten, Weinzierl] Square root of cubic or higher: in general, cannot be rationalized, sign of non-polylogarithmicity Try all possible integration orders. We define the rigidity of an integral as the minimum number of variables left in the root. N.B.: This does not rule out more unusual changes of variables/re-parametrizations! To do that, would need a “more invariant” picture (differential equations?) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 11 / 24
Marginal Integrals are Calabi-Yau What is a Calabi-Yau? Compact K¨ ahler manifold with vanishing first Chern class Ricci-flat Preserves N=1 supersymmetry of compactifications Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 12 / 24
Marginal Integrals are Calabi-Yau What is a Calabi-Yau? Compact K¨ ahler manifold with vanishing first Chern class Ricci-flat Preserves N=1 supersymmetry of compactifications . . . not helpful! Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 12 / 24
Marginal Integrals are Calabi-Yau How do you diagnose a Calabi-Yau? Embed the patient in a weighted projective space! Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 13 / 24
Marginal Integrals are Calabi-Yau How do you diagnose a Calabi-Yau? Embed the patient in a weighted projective space! projective space: ( x 1 , x 2 , . . . ) ∼ ( λ x 1 , λ x 2 , . . . ) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 13 / 24
Marginal Integrals are Calabi-Yau How do you diagnose a Calabi-Yau? Embed the patient in a weighted projective space! weighted projective space: ( x 1 , x 2 , . . . ) ∼ ( λ w 1 x 1 , λ w 2 x 2 , . . . ) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 13 / 24
Marginal Integrals are Calabi-Yau How do you diagnose a Calabi-Yau? Embed the patient in a weighted projective space! weighted projective space: ( x 1 , x 2 , . . . ) ∼ ( λ w 1 x 1 , λ w 2 x 2 , . . . ) Curve should scale uniformly in λ (homogeneous polynomial) Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 13 / 24
Marginal Integrals are Calabi-Yau How do you diagnose a Calabi-Yau? Embed the patient in a weighted projective space! weighted projective space: ( x 1 , x 2 , . . . ) ∼ ( λ w 1 x 1 , λ w 2 x 2 , . . . ) Curve should scale uniformly in λ (homogeneous polynomial) If the sum of the coordinate weights equals the overall scaling (degree), your curve is Calabi-Yau! Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 13 / 24
Marginal Integrals are Calabi-Yau Did you check the patient for singularities? Strictly, this only works if the Calabi-Yau is not singular F is singular ≡ points where ∇ F = 0 Generically, our manifolds are singular! Can blow up to smooth singularities – we usually skip this part Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 14 / 24
Marginal Integrals are Calabi-Yau Did you check the patient for singularities? Excuses: All cases we’ve checked in detail work [ongoing with Candelas, Elmi, Schafer-Nameki, Wang] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 14 / 24
Marginal Integrals are Calabi-Yau Did you check the patient for singularities? Excuses: All cases we’ve checked in detail work [ongoing with Candelas, Elmi, Schafer-Nameki, Wang] Even mathematicians assume this will work [Brown 0910.0114] Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 14 / 24
Marginal Integrals are Calabi-Yau Did you check the patient for singularities? Excuses: All cases we’ve checked in detail work [ongoing with Candelas, Elmi, Schafer-Nameki, Wang] Even mathematicians assume this will work [Brown 0910.0114] Charles Doran: “A Calabi-Yau is whatever you want it to be” Matt von Hippel (NBIA) Calabi-Yaus and Other Animals November 7, 2018 14 / 24
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