Feynman integrals and Mirror symmetry

Transcription

1 Feynman integrals and Mirror symmetry Pierre Vanhove & Amplitudes and Periods, HIM, Bonn based on , , and work in progress Spencer Bloch, Matt Kerr, Charles Doran, Andrey Novoseltsev Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

2 Feynman integral Feynman integrals with L loops and n propagators in D dimensions I Γ (s, m, ν, D) := R L(1,D 1) V δ( n j=1 ɛ ijq j ) n ((q i) 2 ) ν i L d D l i There are V = 1 + N L vertices (ɛ ij ) is the V N incidence matrix of the directed graph ɛ ij { 1, 0, 1} The momenta q i = L j=1 a ij l j + V e j=1 ρ ijp j mi 2 + iε where l j are the loop momenta, and p j the external momenta. Finally s = (p i p j ) and m = (m 1,..., m n ) and v = (v 1,..., v n ) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

3 Feynman integral : examples I (p 2, m, 1, 1, 1, D) = = 3 d D l i δ(l 1 + l 2 + l 3 = p) 3 i=r (l2 r m 2 r + iε) I (p 2, m, 2, 1, 1, D) = = m 2 1 m 2 1 I (p 2, m, 1, 1, 1, D) I (p 2, m, 1, 0, 1, D) = = d D l 1 d D l 2 (l 2 1 m2 1 + iε)(l2 2 m2 3 + iε) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

4 Master integrals I Integration by Part identities (IBP) one can show that the sunset system satisfies [Caffo, Czyz, Laporta, Remiddi] I (p 2, m, 1, 1, 1, D) I (p 2, m, 1, 1, 1, D) I (p 2, m, 2, 1, 1, D) p 2 I (p 2 I (p 2, m, 2, 1, 1, D), m, 1, 2, 1, D) I (p 2, m, 1, 2, 1, D) p 2 I (p 2, m, 1, 1, 2, D) = A I (p 2, m, 1, 1, 2, D) I (p 2, m, 0, 1, 1, D) I (p 2, m, 0, 1, 1, D) I (p 2, m, 1, 0, 1, D) I (p 2, m, 1, 0, 1, D) I (p 2, m, 1, 1, 0, D) I (p 2, m, 1, 1, 0, D) The 1-form matrix A satisfies the integrability condition da + A A = 0 Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

5 Master integrals II The system can be rewritten as I (1, 1, 1) I (1, 1, 1) p 2 I (2, 1, 1) p 2 I (1, 2, 1) = Â I (2, 1, 1) I (0, 1, 1) I (1, 2, 1) + B I (1, 0, 1) I I (1, 1, 2) I (1, 1, 2) (1, 1, 0) Leads to a 4th order differential operator on I (1, 1, 1) which factorizes into product of two first orders and 2nd order L PF operator The previous system of equation has the geometry interpretation of being a connection associated to a given cohomology: this is the Gauss-Manin connection. But of what cohomology? If A has a nice form we can immediately recognized this leads to MPL [Henn]. What about functions beyond MPL? Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

6 Feynman Integrals: parametric representation Any Feynman integrals has the parametric representation in projective space ν = ν ν n I Γ = Γ(ν LD 2 ) x i 0 U ν (L+1) D 2 Γ Φ ν L D 2 Γ δ(x n = 1) n dx i x 1 ν i i The Symanzik polynomials U Γ and Φ Γ are homogeneous in the x 1,..., x n U Γ is of degree L in P n 1 Φ Γ = U Γ n m2 i x i i,j p i p j w ij of degree L + 1 in P n 1 w ij are homogeneous polynomials of degree L + 1 in P n 1 Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

7 The geometry of a Feynman graph The homogeneous polynomial of n variables and degree L + 1 completely characterises the Feynman graph and its integral Φ Γ = U Γ ( n m 2 i x i) i,j p i p j w ij We can recover both Symanzik polynomials Determines the graph topology the number of propagators is the number of variables n the loop order is the degree minus one L = deg(φ Γ ) 1 Number of vertices v = 1 + n L from Euler characteristic Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

8 From parametric representation to graph The most general quadric polynomial in P 2 W 2,3 (x 1, x 2, x 3 ) = w i1,i 2,i 3 x i 1 1 x i 2 2 x i 3 3 i 1 +i 2 +i 3 =2 ir 0 The graph has n = 3 propagators, L = 1 loop, v = 3 vertices This can only be a triangle graph p 1 + p 2 + p 3 = 0; p 2 i 0 Φ = (x 1 + x 2 + x 3 )(m 2 1 x 1 + m 2 2 x 2 + m 2 3 x 3) (p 2 1 x 2x 3 + p 2 2 x 1x 3 + p 2 3 x 1x 2 ) This is the most general quadric Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

9 From parametric representation to graph The most general cubic in P 2 W 3,3 = i 1 +i 2 +i 3 =3 ir 0 w i1,i 2,i 3 x i 1 1 x i 2 2 x i 3 3 The graph has n = 3 propagators, L = 2 loops, v = 2 vertices This can only be a sunset graph Φ = (x 1 x 2 + x 1 x 3 + x 2 x 3 )(m 2 1 x 1 + m 2 2 x 2 + m 2 3 x 3) p 2 x 1 x 2 x 3 This is not the most general cubic : important restriction on the parameters Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

10 From parametric representation to graph The most general polynomial of degree n in P n 1 W n,n = i 1 + +in=n ir 0 w i1,...,i n x i 1 1 x i n n The graph has n propagators, L = n 1 loops, v = 2 vertices This can only be a n-loop sunset graphs Φ n = n x i n x 1 i n mi 2 x i p 2 Notice that the kinematics parameters always enter linearly Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41 n x i

11 The toric approach The number of master integrals are given by the Euler characteristic of the complement of the graph polynomials [Bitoun, Bogner, Klausen, Panzer] #master-integrals = ( 1) n χ(c n \{x 1 x n (U + Φ Γ = 0)}) For the sunset we get 7 : 4 non-degenerate integrals and 3 reduced topologies Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

12 The toric approach : review Consider the homogeneous polynomial of degree L + 1 in P n 1 P(z 1,..., z r ) = a 1,...,a n 1 a 1 + +a n 1 =L+1 z a1,,a n 1 n 1 with {z r } = {z a1,,a n 1 } and a = (a 1,..., a n 1 ) and A = (a 1,, a r ) finite subset of Z r For every vector l L such that L := {(l 1,..., l r ) Z r, l l r = 0, l 1 a l r a r = 0} then there are the following differential operators l := l i z i l i z i l i >0 l i <0 and a system of n differential operator (including the Euler operator) x a i i E a,c := a 1 z a r z r c (c C n ) z 1 z r Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

13 The GZK approach : review GZK have shown that the functions Φ := x 1 = = x n 1 =1 n 1 P(z 1,..., z r ) m i i satisfy l Φ = 0 for all l = (l 1,..., l r ) L and E a,c Φ = 0 The generic solution of GZK system are the hypergeometric series r z γ j +l j j Φ L,γ (z 1,, z r ) = Γ(γ j + l j + 1) (l 1,...,l r ) L j=1 with (γ 1,..., γ r ) C r In general for a well choosen l L the differential operator factorizes a piece giving the Picard-Fuchs operator x β i dx i x i Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

14 The toric approach: consequences The following period integral π P = is the maximal cut integral Max-Cut(I Γ (1,..., 1)) := is actually π Γ = Γ(ν LD 2 ) x 1 = = x n 1 =1 R L(1,D 1) x i =1 n 1 1 dx i P(z 1,..., z r ) x i V n δ( ɛ ij q j ) U ν (L+1) D 2 Γ Φ ν L D 2 Γ j=1 n δ((q i ) 2 ) δ(x n = 1) n dx i x 1 ν i i L d D l i One can easily derive the Picard-Fuchs operator from the graph polynomials Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

15 Feynman integral and periods I Γ = Γ(ν LD 2 ) n Ω Γ ; Ω Γ := Uν (L+1) D 2 Φ Γ (x i ) ν L D 2 n 1 dx i x 1 ν i i Ω Γ algebraic differential form on the complement of the graph hypersurface Ω Γ H n 1 (P n 1 \X Γ ) X Γ := {Φ Γ (x i ) = 0, x i P n 1 } The domain of integration is the simplex n n := {x 1 0,..., x n 0 [x 1,..., x n ] P n 1 } Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

16 Feynman integral and periods D n and X Γ are separated by performing a series of iterated blowups of the complement of the graph hypersurface [Bloch, Esnault, Kreimer] The Feynman integral are periods of the relative cohomology after performing the appropriate blow-ups H n 1 ( P n 1 \ X F ; Dn \ Dn X Γ ) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

17 Picard-Fuchs equation M(s ij, m i ) := H ( P n 1 \ X F ; Dn \ Dn X Γ ) Since Ω Γ varies when one changes the kinematic variables s ij one needs to study a variation of (mixed) Hodge structure Consequently the Feynman integral will satisfy a differential equation L PF I Γ = S Γ The Picard-Fuchs operator will arise from the study of the variation of the differential in the cohomology when kinematic variables change Generically there is an inhomogeneous term S Γ 0 Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

18 The differential operator: from the period The analytic period of the elliptic curve around p 2 has the same integrand as the Feynman integral but we have just changed the domain of integration π 0 (p 2 ) := Ω x = y =1 This is the imaginary part or the maximal cut of the amplitude C Im(I (p 2 )) = 3 δ(l 2 i m2 i ) δ(l 1+l 2 +p) d 2 l 1 d 2 l 2 The other period is π 1 (s) = log(s) π 0 (s) + ϖ 1 (s) with ϖ 1 (s) analytic is obtained by looking at different unitarity cut cutting less lines Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

19 The differential operator: from the period The integral is the analytic period of the elliptic curve around p 2 π 0 (p 2 ) := 1 ( ) n! 2 3 (p 2 ) n+1 m 2n i n 1!n 2!n 3! i n 0 n 1 +n 2 +n 3 =n From the series expansion we can deduce the Picard-Fuch differential operator (the system has maximal unipotent monodromy) L π 0 (p 2 ) = 0 With this method one easily derives the PF at all loop order for the all equal mass sunset and show the order(pf)=loop [Vanhove] Gives for the 3-loop sunset the PF has order 2 + #( masses) = 3, 4, 5, 6. The geometry is a K 3 of Picard rank 20 #( masses) = 19, 17, 18, 16 [Doran, Novoseltsev, Vanhove; to appear] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

20 The differential equation By general consideration we know that since the integrand is a top form we have L Γ I Γ = dβ Γ = β Γ = S Γ 0 n n Writing the differential equation as δ s := s d ds with s = 1/p2 ( ) ( ) δ 2 1 s + q 1 (s)δ s + q 0 (s) s I (s) = Y + 3 log(mi 2 )c i(s) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

21 The differential equation Using works from [del Angel,Müller-Stach] and [Doran, Kerr] we know that when rank of the D-module system of differential equations that Y is the Yukawa coupling Y := E(p 2 ) Ω s d ds Ω = 2s2 4 µ i 4s i m2 i (µ2 i s 1) The Yukawa coupling is the Wronskian of the Picard-Fuchs operator and only depends on the form of the Picard-Fuchs operator ( π0 (s) π Y = s det 1 (s) d ds π 0(s) d ds π 1(s) So far all we got can be deduced from the graph polynomial, and the associated Picard-Fuchs operator. ) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

22 The differential equation v 3 q 3 v 4 v2 p 1 q 2 p p 2 3 v 5 q 1 v 1 v 6 E s ( δ 2 s + q 1 (s)δ s + q 0 (s) ) ( ) 1 s I (s) = Y + 3 log(mi 2 )c i (s) The mass dependent log-terms come from derivative of partial elliptic integrals on globally well-defined algebraic 0-cycles arising from the punctures on the elliptic curve [Bloch, Kerr,Vanhove] c 1 (s) = d ds q3 Ω q 2 They are rational function by construction. Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

23 Sunset as an elliptic dilogarithm Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

24 The two-loop sunset integral We consider the sunset integral in two Euclidean dimensions I 2 = Ω ; 3 := {[x : y : z] P 2 x 0, y 0, z 0} 3 The sunset integral is the integration of the 2-form Ω = zdx dy + xdy dz + ydz dx (m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) p2 xyz H2 (P 2 E p 2) The sunset family of open elliptic curve E p 2 = {(m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) p2 xyz = 0} For m 1 = m 2 = m 3 we have a modular curve E p 2 X 1 (6) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

25 The 2-loop sunset integral as elliptic dilogarithm The integral divided by a period of the elliptic curve is a function defined on the punctured torus [Bloch, Kerr,Vanhove] I iϖ r π ( { X L 2 Z, Y } { Z + L 2 Z X, Y } { X + L 2 X Y, Z }) Y mod period ϖ r is the elliptic curve period which is real on the line 0 < p 2 < (m 1 + m 2 + m 3 ) 2 The sunset integral is the regulator period (with tame Milnor symbol) in the K 2 of the elliptic curve [Bloch, Vanhove] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

26 The 2-loop sunset integral as elliptic dilogarithm 4 ϕ 0 q3 3 ϕ P ϕ 0 P 23 q 1 6 ϕ 0 ϕ 2 0 ϕ 1 ϕ1 0 P 12 q 2 E s P 1 = [1, 0, 0]; Q 1 = [0, m 2 3, m2 2 ]; x(p 1)x(Q 1 ) = 1 P 2 = [0, 1, 0]; Q 2 = [ m 2 3, 0, m2 1 ]; x(p 2)x(Q 2 ) = 1 P 3 = [0, 0, 1]; Q 3 = [ m 2 2, m2 1, 0]; x(p 3)x(Q 3 ) = 1 Representing the ratio of the coordinates on the sunset cubic curve as functions on E C /q Z X Z (x) = θ 1(x/x(Q 1 ))θ 1 (x/x(p 3 )) θ 1 (x/x(p 1 ))θ 1 (x/x(q 3 )) Y Z (x) = θ 1(x/x(Q 2 ))θ 1 (x/x(p 3 )) θ 1 (x/x(p 2 ))θ 1 (x/x(q 3 )) θ 1 (x) is the Jacobi theta function θ 1 (x) = q 1 8 x 1/2 x 1/2 i (1 q n )(1 q n x)(1 q n /x). n 1 Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

27 The 2-loop sunset integral as elliptic dilogarithm Since L 2 { X Z, Y Z } x ( ) X = log x 0 Z (y) d log y log(θ 1 (x)) d log x = (Li 1 (q n x) + Li 1 (q n /x) + cste) d log(x) n 1 = n 1(Li 2 (q n x) Li 2 (q n /x)) + cste log(x) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

28 The 2-loop sunset integral as elliptic dilogarithm We find I (s) iϖ r π ( ( ) x(p1 ) Ê 2 + x(p 2 ) Ê2 ( ) x(p2 ) + x(p 3 ) Ê2 ( )) x(p3 ) x(p 1 ) mod periods where Ê 2 (x) = n 0 (Li 2 (q n x) Li 2 ( q n x)) n 1 (Li 2 (q n /x) Li 2 ( q n /x)). Close to the form given by [Brown, Levin]. See as well [Adams, Bogner, Weinzeirl] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

29 The sunset integral and the motive The integral is given by I 2 dxdy (t) = (x + y + 1)(x + y + xy) txy 0 0 The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 Character ψ : Lattice(E t ) S 1. Pairing ɛ 1, ɛ 2 = ɛ 2, ɛ 1 = 2iπ The amplitude integral is not the regulator map which involves a real projection r : K 2 (E t ) H 1 (E t, R) The amplitude is multivalued in t whereas the regulator is single-valued [Bloch, Vanhove] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

30 The sunset integral and the motive The integral is given by I 2 (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 The regulator is an Eichler integral i I 2 (t) = periods + ϖ r τ (m,n) (0,0) ψ 2 (n)(τ x) (m + nx) 3 dx Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

31 The sunset integral and the motive The integral is given by I 2 (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 The regulator is an Eichler integral I 2 (t) = periods + ϖ 1 (m,n) (0,0) ψ 2 (n) n 2 (m + nτ) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

32 The three-loop sunset as an elliptic trilogarithm Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

33 The three-loop sunset graph: integral We look at the 3-loop sunset graph in D = 2 dimensions The Feynman parametrisation is given by I 2 (m Q i; K 2 ) = x i 0 1 (m m2 i x i)(1 + 3 x 1 i ) K 2 3 dx i x i Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

34 three-loop sunset graph: differential equation For the all equal mass case the geometry of the 3-loop sunset graph is a K 3 surface (Shioda-Inose family for Γ 1 (6) +3 ) with Picard number 19 and discriminant of Picard lattice is 6 (m m 2 x i )(1 + 3 x 1 i ) The t = p 2 /m 2 Picard-Fuchs equation (t 2 (t 4)(t 16) d 3 3 x i p 2 dt 3 + 6t(t2 15t + 32) d 2 dt 2 3 x i = 0 + (7t 2 68t + 64) d )J dt + t 4 2 (t) = 4! Q One miracle is that this picard-fuchs operator is the symmetric square of the picard-fuchs operator for the sunset graph [Verrill] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

35 three-loop sunset graph: solution It is immediate to use the Wronskian method to solve the differential equation [Bloch, Kerr, Vanhove] m 2 I 2 (t) = Q 40π2 log(q) ϖ 1 (τ) ( ) 48ϖ 1 (τ) 24Li 3 (τ, ζ 6 ) + 21 Li 3 (τ, ζ 2 6 ) + 8Li 3(τ, ζ 3 6 ) + 7Li 3(τ, 1) with Li 3 (τ, z) [Zagier; Beilinson, Levin] Li 3 (τ, z) := Li 3 (z) + (Li 3 (q n z) + Li 3 (q n z 1) ) n 1 ( 1 12 log(z) log(q) log(z)2 1 ) 720 (log(q))3. The 3-loop sunset integral is a regulator period of a motivic class of the K 3 of the the K 3 surface [Bloch, Kerr, Vanhove] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

36 Elliptic polylogarithms or not? Another way of writting this is (c I (s) = s 2 ijk log(mi 2 π 0 (s) Q) log(m2 j Q) log(m2 k Q) 3! + c ij 2! log(m2 i Q) log(m2 j Q) + c i log(mi 2 Q) + c + (β 1,β 2,β 3 ) H 2 (M,Z) n β 0 Li 3( 3 ) Q β i i ) This form generalizes for the higher-loop banana graphs and involve Li r with r 4 Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

37 Mirror Symmetry sunset Pierre Vanhove (IPhT & HSE) sunrise Feynman integrals & Mirror symmetry 1/03/ / 41

38 The sunset Gromov-Witten invariants Around 1/s = p 2 = the sunset Feynman has the expansion I (s) = π 0 3R3 0 + l 1 +l 2 +l 3 =l>0 (l 1,l 2,l 3 ) N 3 \(0,0,0) l(1 lr 0 )N l1,l 2,l 3 3 m 2l i i e lr 0. where the Kähler parameters are Q i = mi 2eR 0 and R 0 is the logarithmic Mahler measure defined by d log xd log y R 0 := iπ log(φ (x, y)/(xy)) (2πi) 2. x = y =1 This is related to the holomorphic π 0 (s) period near s = 1/p 2 = 0 π 0 = s dr 0(s) ds Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

39 The sunset Gromov-Witten invariants The numbers N l1,l 2,l 3 are local Gromov-Witten expressed in terms of the virtual integer number of degree l rational curves by N l1,l 2,l 3 = d l 1,l 2,l 3 1 d 3 n l 1 d, l 2 d, l 3 d. l (100) k>0 (k00) (110) (210) (111) (310) (220) (211) (221) N l 2 2/k / n l l (410) (320) (311) (510) (420) (411) (330) (321) (222) N l / /4 n l Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

40 The sunset Gromov-Witten invariants For the all equal masses case m 1 = m 2 = m 3 = 1, the mirror map is Q = e R 0 = q ( ) 3 (1 q n ) nδ(n) ; δ(n) := ( 1) n 1, n n 1 where ( ) 3 n = 0, 1, 1 for n 0, 1, 2 mod 3. The local Gromov-Witten numbers N l 6 = 1, 7 8, 28 27, , , , , , , , , , , , , , , , , 4913, , Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

41 The sunset mirror symmetry The sunset elliptic curve is embedded into a singular compactification X 0 of the local Hori-Vafa 3-fold Y := {1 s(m 2 1 x +m2 2 y +m2 3 )(1+x 1 +y 1 )+uv = 0} (C ) 2 C 2, limit of a family of elliptically-fibered CY 3-folds X z The base given by Φ is a toric del Pezzo surface of degree 6 We have an isomorphism of A- and B-model Z-variation of Hodge structure H 3 (X z0 ) H even (X Q 0 ), and taking (the invariant part of) limiting mixed Hodge structure on both sides yields the sunset Feynman integral given by the second regulator period of the motivic cohomology class is identified to the local Gromov-Witten prepotential for the 3-fold X Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

42 Mirror symmetry for elliptically fibered CY 3-fold In the degeneration limit the Yukawa coupling CY 3-fold X leads to the local Yukawa of the sunset elliptic curve Y ijk = Ω δi δ j δ k Ω = Y0ij loc Y = Ω d Ω ds X The holomorphic prepotential of [Huang, Klemm, Poretschkin] F(Q 1, Q 2, Q 3, Q 4 ) = c ijkt i t j t k 3! + c ij 2! ti t j + c i t i + c + β H 2 (M,Z) n β 0 Li 3(Q β ) is mapped to the sunset integral with the identification of the Kähler parameter Q r = exp(2πit r ) = m 2 r Q for r = 1, 2, 3 [Klemm private communication] m1 2 = (Q 1Q 2 Q 4 ) 1 3 ; m Q = (Q 1 1Q 2 Q 4 ) 3 ; m 3 1 Q = (Q 1 1Q 3 Q 4 ) 3 ; Q = (Q 3 2 Q 2 1 Q 2 Q 3 Q 4 ) Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41

43 Mirror symmetry for higher sunset integrals The same construction applies to the 3-loop sunset graph where Φ 4 = 0 defines a family of K 3 The same is conjectured to be true for the 4-loop sunset graph where Φ 5 = 0 defines a family of CY 3-fold. [Doran, Kerr] Not modular in general [Hulek, Verill]. Therefore (elliptic) polylogarithm not enough from 4-loop At higher-loop loop the geometry is more intricate Need to go beyond the smoothness hypothesis for K P used in [Lian, Todorov, Yau] # Need to extend the construction of the motivic cohomology classes and the regulator period of [Doran, Kerr] Pierre Vanhove (IPhT & HSE) Feynman integrals & Mirror symmetry 1/03/ / 41