Eric D Hoker Mani L. Bhaumik Institute for Theoretical Physics Department of Physics and Astronomy, UCLA Ecole Normale Supérieure, Paris, 2018
Introduction String theory naturally generalizes real-analytic Eisenstein series for genus one surfaces, multiple Kronecker-Eisenstein sums, multiple integrations of Green function on the torus, modular graph functions invariant under SL(2, Z). (Green, Russo, Vanhove 2008; ED, Green, Vanhove 2015; ED, Green, Gurdogan, Vanhove 2015) String theory includes contributions from surfaces of all genera = expect modular graph functions for higher genus surfaces. Focus of this talk is on genus two and higher simplest is Kawazumi-Zhang invariant (to be explained below); genus-two string theory predicts an infinite number of higher invariants; genus greater than two string theory offers no predictions, but mathematical constructions produce higher invariants. I will give an account of what we know and do not know to date.
Bibliography Based on ED, Michael Green and Boris Pioline, arxiv:1712.06135,, string invariants, and their exact asymptotics ED, Michael Green and Boris Pioline, (in preparation) Asymptotics of the D 8 R 4 genus-two string invariant and earlier work ED, Michael Green, arxiv:1308.4597, Journal of Number Theory, Vol 144 (2014) 111-150, Zhang-Kawazumi invariants and Superstring Amplitudes ED, Michael Green, Boris Pioline, Rudolfo Russo, arxiv:1405.6226, JHEP 1501 (2015) 031, Matching the D 6 R 4 interaction at two-loops Boris Pioline, arxiv:1504.04182, Journal of Number Theor. 163, 520 (2016), A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces
Expansion in genus and low energy energy genus 0 genus 1 genus 2 g s D 6 R 4 D 4 R 4 R 4 R Supertring Perturbation Theory in g s (2 2h) with h 0 is the genus holds for small string coupling g s 1 but for all energies Supergravity R leading low energy expansion of string theory holds for all couplings g s String induced effective interactions R 4,D 4 R 4,D 6 R 4 Evaluated in superstring perturbation theory Accessible via the four-graviton scattering amplitude
Effective Interactions Four-graviton amplitude in Type II at genus 0, A (0) (s ij ) = R 4 1 stu Γ(1 s)γ(1 t)γ(1 u) Γ(1+s)Γ(1+t)Γ(1+u) R 4 = unique maximally supersymmetric contraction of 4 Weyl tensors External momenta k i for i = 1,2,3,4 with k 2 i = 0 and i k i = 0 Introduce dimensionless Lorentz-invariants s ij = α k i k j /2 s = s 12 = s 34, t = s 13 = s 24, u = s 14 = s 23 with s+t+u = 0 Low energy expansion corresponds to s, t, u 1 1 stu + 2ζ(3)+ζ(5)(s2 +t 2 +u 2 )+2ζ(3) 2 stu+ massless R 4 D 4 R 4 D 6 R 4 Exchange of massive string states produces local effective interactions.
Genus-one string amplitude Effective R 4 -type interactions in Type II Generated by a multiple integral over a torus Σ 1 = C/(Z+τZ), of modulus τ H, namely τ = τ 1 +iτ 2 with τ 1,τ 2 R and τ 2 > 0, B (1) (s ij τ) = N i=1 Σ 1 d 2 z i τ 2 exp { 1 i<j N } s ij g(z i z j τ) Mathematically, one may consider this integral for arbitrary N g(z τ) is the translation-invariant Green function on Σ 1, τ 2 z z g(z τ) = πδ (2) (z)+π d 2 zg(z τ) = 0 Σ 1 Integrals absolutely convergent for s ij < 1; analytic near s ij = 0; B (1) (s ij τ) is invariant under the modular group SL(2,Z). String amplitude obtained by integral over modulus of the torus, A (1) (s ij ) = H/SL(2,Z) d 2 τ B (1) (s ij τ) requires analytic continuation in s ij (ED, Phong 1994). τ 2 2
Genus-one modular graph functions Taylor series expansion of B (1) (s ij τ) for fixed τ in powers of s ij An integration point z i is represented by a vertex A Green function is represented by an edge = g(z i z j τ) z i z j D 4 R 4 D 6 R 4 D 8 R 4 D 10 R 4 one-loop two-loops three-loops four-loops
Properties of genus-one modular graph functions One-loop graphs with k vertices give real analytic Eisenstein series E s k d 2 z i τ2 k g(z i z i+1 τ) = τ 2 π k mτ +n 2k = E k i=1 Σ (m,n) (0,0) convergent sums for Re(s) > 1; modular SL(2, Z)-invariant Laplace-eigenvalue equation, ( s(s 1))E s = 0 with = 4τ 2 2 τ τ Two-loop graphs evaluate to the series 3 ( ) sr τ 2 C s1,s 2,s 3 (τ) = π m r τ +n r 2 δ( m r )δ( r (m r,n r ) (0,0) r=1 convergent sums for Re(s r ) 1; modular SL(2,Z)-invariant; satisfy inhomogeneous Laplace-eigenvalue equations, e.g. r n r ) C 1,1,1 = 6E 3 ( 2)C 2,1,1 = 9E 4 E 2 2 ( 6)C 3,1,1 = 3C 2,2,1 + 16E 5 4E 2 E 3 (1)
Genus-two surfaces Σ is a compact Riemann surface of genus two Key difference with genus-one: no translation symmetry A 1 A 2 B 1 B 2 Σ Homology and cohomology One-cycles H 1 (Σ,Z) Z 4 with intersection pairing J(, ) Z Canonical basis J(A I,A J ) = 0, J(B I,B J ) = 0 with I,J = 1,2 J(A I,B J ) = δ IJ, J(B I,A J ) = δ IJ Canonical dual basis of holomorphic one-forms ω I in H (1,0) (Σ) A I ω J = δ IJ B I ω J = Ω IJ Period matrix Ω obeys Riemann relations Ω t = Ω, Im(Ω) > 0
Modular transformations and geometry Transformation Sp(4,Z) : H 1 (Σ,Z) H 1 (Σ,Z) leaves J(, ) invariant action on basis cycles given by ( BI A I ) J M IJ ( BJ A J action on 1-forms ω I and periods Ω IJ given by ω ω(cω+d) 1 Ω (AΩ+B)(CΩ+D) 1 M = ) M t JM = J ( ) A B C D Siegel upper half space S 2 { } S 2 = Ω IJ C with Ω t = Ω and Y = Im(Ω) > 0 S 2 = Sp(4,R) SU(2) U(1) = SO(3,2) ds 2 2 = SO(3) SO(2) I,J,K,L=1,2 is Kähler with invariant metric Y 1 IJ d Ω JK Y 1 KL dω LI Moduli space of genus-two surfaces is S 2 /Sp(4,Z) (minus diagonal Ω)
Green function and volume form How to generalize the genus-one formula to a genus-two formula? recall the genus-one formula N B (1) d 2 z { i } (s ij τ) = exp s ij g(z i z j τ) Σ 1 τ 2 i=1 1 i<j N Canonical metric and Kähler form for genus-two Σ modular invariant and smooth κ = i Y 1 IJ 4 ω I ω J κ = 1 I,J Natural Arakelov Green function G(w, z Ω) = G(z, w Ω) Inverse of scalar Laplace operator for canonical metric Σ w w G(w,z Ω) = πδ(w,z)+πκ(w) κ(w)g(w,z Ω) = 0 Σ
A natural genus-two candidate A natural candidate formula for a string amplitude would be N { } C (2) (s ij Ω) = κ(z i ) exp s ij G(z i,z j Ω) i=1 Σ 1 i<j N Integrals absolutely convergent for s ij < 1; analytic near s ij = 0, Expanding in powers of s ij gives genus-two modular graph functions. But... Genus-two string amplitudes are NOT given by C (2) (s ij Ω) For integration over a single copy of Σ κ is the only natural modular invariant volume form. For integration over multiple copies of Σ Sp(4,Z) modular invariants other than i κ(z i) allowed. For example, when N = 2 we can have κ(z 1 )κ(z 2 ) as well as Y 1 IL Y 1 JK ω I(z 1 )ω J (z 1 )ω K (z 2 )ω L (z 2 ) I,J,K,L
Genus-two string amplitude Instead the N = 4 graviton amplitude was calculated (ED, Phong 2005) { B (2) Y Ȳ } (s ij Ω) = Σ 4 (dety) exp s 2 ij G(z i,z j ) 1 i<j 4 The key difference with the candidate C (2) is the structure of Y 3Y = (t u) (z 1,z 2 ) (z 3,z 4 ) s = s 12 = s 34 +(s t) (z 1,z 3 ) (z 4,z 2 ) t = s 13 = s 24 +(u s) (z 1,z 4 ) (z 2,z 3 ) u = s 14 = s 23 where is a holomorphic (1,0) i (1,0) j form on Σ Σ (z i,z j ) = ε IJ ω I (z i ) ω J (z j ) The combination Y Ȳ/(detY)2 is Sp(4,Z)-invariant, and produces a modular invariant B (2) (s ij Ω).
Low energy expansion Contributions to local effective interactions Expand B (2) (s ij Ω) in powers of s ij and integrate over M 2 = S 2 /Sp(4,Z) R 4,D 2 R 4 zero, since Y vanishes for s = t = u = 0 D 4 R 4 Siegel volume form on M 2 D 6 R 4 one factor of G in expansion in powers of s ij B (2) (s ij Ω) = 32(s 2 +t 2 +u 2 )+192stuϕ(Ω)+O(s 4 ij) ϕ(ω) = 1 Y 1 IL 4 Y 1 JK G(x,y)ω I (x)ω J (x)ω K (y)ω L (y) Σ 2 I,J,K,L ϕ(ω) coincides with the Kawazumi-Zhang invariant (ED, Green 2013) introduced as a spectral invariant (Kawazumi 0801.4218 and Zhang 0812.0371) related to the genus-two Faltings invariant (De Jong 2010) formulated in terms of modular tensors (Kawazumi OIST lecture notes 2016) A IJ;KL = G(x,y)ω I (x)ω J (x)ω K (y)ω L (y) Σ 2
Higher string-invariants The KZ-invariant exists for all genera 2 (Kawazumi 2008; Zhang 2008) unknown if correct object for string theory at genus 3. But the Taylor expansion coefficients of B (2) (s ij Ω) are modular graph functions at genus-two; do naturally emerge from string theory at genus-two; provide a string-motivated generalization of KZ-invariants at genus-two. Higher string-invariants (contribute to D 8 R 4 and D 10 R 4 ) (ED, Green 2013) B (2) (2,0) = B (2) (1,1) = = (1,2) (3,4) Σ 4 (dety) 2 (1,2) (3,4) (1,4) (2,3) Σ 4 (dety ) 2 2 ( ) 2 G(1,4) + G(2,3) G(1,3) G(2,4) 2 ( G(1,2) + G(3,4) +G(1,4) + G(2,3) 2G(1,3) 2G(2,4) ) 3
Differential equations and Asymptotics Genus-one modular graph functions satisfy System of inhomogeneous Laplace-eigenvalue equations Laurent polynomial behavior near the cusp of degree bounded by weight Genus-two KZ-invariant ϕ satisfies Characteristic class relations and role in Johnson homomorphism (Kawazumi 2008; Zhang 2008; DeJong 2010) Eigenvalue equations for Sp(4, R)-invariant differential operators (ED, Green, Pioline, Russo 2014; Pioline 2015) Theta-lift analogous to Borcherds for Igusa cusp form (Pioline 2015) Laurent polynomial of degree (1, 1) near non-separating divisor (De Jong 2010; Pioline 2015; ED, Green, Pioline 2017) Genus-two higher string-invariants satisfy Laurent polynomial of bounded degree near non-separating degeneration (ED, Green, Pioline 2017) System of differential equations for invariant differential operators (in progress ED, Green, Pioline 201?)
Differential equations Theorem 1 Laplace eigenvalue equation (ED, Green, Pioline, Russo 2014) ( 5)ϕ = 2πδ SN δ SN has support on separating node (into two genus-one surfaces) is the Laplace-Beltrami operator on S 2 with Siegel metric = 4 Y IK Y JL IJ KL IJ = 1 2 (1+δIJ ) I,J,K,L Ω IJ proven by theory and methods of deformations of complex structures Theorem 2 Quartic differential operator eigenvalue equation (Pioline 2015) 0, 2 are Maass-Siegel operators (32 2 0 15)ϕ = 0 0 = ε IJ ε KL IK JL IJ = Y IK Y JL KL System of differential equations satisfied by higher string invariants?
Asymptotics For genus one, there is only one type of degeneration, as modulus τ i Behavior of holomorphic Eisenstein series G 2k (τ) = B 2k 2k + n=1 σ 2k 1 (n)q n q = e 2πiτ Behavior of non-holomorphic Eisenstein series, y = πim(τ) E k (τ) = 2ζ(2k) ( ) 2k 3 ζ(2k 1) π 2k yk +4 +O( q ) k 1 (4y) k 1 Behavior of general genus-one modular graph functions. Finite degree Laurent polynomial in y plus exponentials, eg two-loop modular graph functions (ED, Bill Duke 2017) C a,b,c (τ) = c w y w + c 2 w y w 2 + c 2 w = w 2 m=1 w 1 k=1 c w 2k 1 ζ(2k + 1) y 2k+1 w γ m ζ(2m + 1)ζ(2w 2m 3) with w = a+b+c and c w,γ m,c w 2k 1 Q. + O( q )
Degenerations of genus-two Riemann surfaces Ω = Separating degeneration ( τ v v σ ) v 0 Non-separating degeneration σ i Maximal degeneration (or tropical limit ) τ,v,σ i
Non-separating degeneration A genus-two surface Σ degenerates to a torus Σ 1 with two punctures p a,p b keep the cycles A 1,B 1,A 2 fixed, and let B 2 ( ) τ v Ω = Im(σ) v σ τ is the modulus of Σ 1 and v = p b p a ω 1 = p b p a The genus-two Sp(4,Z) restricts to SL(2,Z) Z 3 (Fourier-Jacobi group) SL(2,Z) subgroup of M Sp(4,Z) such that MB 2 = B 2 is a 0 b 0 M = 0 1 0 0 τ = (aτ +b)/(cτ +d) c 0 d 0 v = v/(cτ +d) σ 0 0 0 1 = σ cv 2 /(cτ +d) Z 3 subgroup shifts v by Z+τZ and Re(σ) by Z; The degeneration parameter σ is not invariant under SL(2, Z)
Degeneration of Siegel modular forms A genus-two ( rank two ) Siegel modular form S of modular weight k is holomorphic and transforms under Sp(4, Z) by, S(Ω ) = det(cω+d) k S(Ω) Ω = (AΩ+B)(CΩ+D) 1 Fourier expansion in powers of e 2πiσ near non-separating node S(Ω) = e 2πimσ φ m (v τ) m=0 φ m (v τ) is a Jacobi form of modular weight k and index m transforms under the residual modular group SL(2, Z) by ( ) v φ m aτ +b cτ +d = (cτ +d) k e 2πimcv2 /(cτ+d) φ m (v τ) cτ +d (Eichler and Zagier, The theory of Jacobi forms, 1985)
Degeneration of the KZ-invariant Non-separating degeneration of non-holomorphic modular functions is governed by a real SL(2,Z)-invariant parameter t > 0 t det(imω) ( ) τ v Ω = Imτ v σ with the non-separating node characterized by t Theorem 3 Non-separating degeneration of KZ-invariant (Pioline 2015) ϕ(ω) = πt 6 + 1 2 g(v τ)+ 1 ( ) E 2 (τ) g 2 (v τ) +O(e 2πt ) 4πt g(v τ) is the torus Green function; E 2 is the non-holo Eisenstein series; g 2 (v τ) = Σ 1 d 2 z/τ 2 g(v z τ)g(z τ); Derived using the Laplace-eigenvalue equation for ϕ. The Laurent polynomial is of finite degree in the variable t but it is not of finite degree in, say, t+1
Non-separating node from a punctured surface Standard construction of a surface near a non-separating node (Fay 1973) start from a genus-one surface with two punctures p a,p b local coordinates z a,z b which vanish respectively at p a,p b identify points z a z b = t between two annuli C a = C b and C a = C b A 1 C a p a C a C a p b C b B 1 C b C b in practice not easy to implement, unless one can define cycles C naturally Σ
Non-separating node from a punctured surface (cont d) Key is the existence of a real single-valued harmonic function f(z) on Σ such that in the degeneration limit f(z) as z p a and f(z) + as z p b for large t cycles prescribed by f(c a ) 2πt and f(c b ) +2πt z ( ) f(z) = 2πt+4πIm ω 2 Im(v)ω 1 /Im(τ) Funnel construction of the non-separating degeneration C a f = t C a C z a f = +t z a p a B 1 a C b C b The cycle A 2 is homologous to the cycles C a,c a,c a and C b,c b,c b ; The cycles are pairwise identified by C a C b, C a C b and C a C b ; The points are pairwise identified by z a z b ; The cycle B 2 may be chosen to be a simple curve connecting z a to z b. A 1 p b z b C b
Degeneration of higher string invariants Consider the full genus-two string amplitude B(s ij Ω) = Y Y Σ 4 (dety ) exp 2 { i<j } s ij G(z i,z j Ω) Expanding in powers of s ij and collecting all terms homogeneous of degree w gives modular graph functions of weight w for genus-two surfaces Σ B w (s ij Ω) = Y Y Σ 4 (dety ) 2 ( s ij G(z i,z j Ω) Theorem 4 Under non-separating degeneration of Σ (ED, Green, Pioline 2017) B w (s ij Ω) has a Laurent polynomial of degree (w,w) in t B (k) B w (s ij Ω) = w k= w i<j ) w B (k) w (s ij v,τ)t k + O(e 2πt ) w (s ij v,τ) are invariant under SL(2,Z) Sp(4,Z) B (k) w ( v s ij cτ + d, aτ + b ) cτ + d = B (k) w (s ij v,ρ) Note similarity with the Laurent polynomial expansion for genus one; B w (k) (s ij v,τ) generalize Jacobi forms to a non-holomorphic setting; combine genus-one modular graph functions and elliptic polylogarithms.
Ingredients in proof of Theorem 4 Uniform asymptotics of the Green function G(x,y Ω) = πt 12 + g(x y τ) f(x)f(y) 4πt + + O(e 2πt ) see DGP 2017 for the complete expression (improving on Wentworth 1991) Measure 4π 2 Y = dz 1 dz 2 dz 3 dz 4 s ij zi f(z i ) zj f(z j ) The cycles C a,c b have exponentially vanishing coordinate radius z C a satisfies z p a 2 e 2πt Power dependence in t arise from poles at p a,p b in the integrand Extract variation in t-dependence (cfr RG with cut-off t) i<j δf = δt f = +t p a B 1 p b C a δc a C a C b C b A 1 C b
Summary and outlook Low energy expansion of string theory has revealed a rich structure of Modular graph functions for genus-one Riemann surfaces; Kawazumi-Zhang and higher string invariants for genus-two surfaces. Asymptotics for higher string invariants Remarkable structure in Laurent polynomial for non-sep degeneration Tested versus maximal degeneration limit (ED, Green, Pioline 2018) Similar result for separating degeneration (ED, Green, Pioline 2018) Crucial for the discovery of relations between modular graph functions Differential equations for higher string invariants In progress (ED, Green, Pioline) Asymptotics is expected to be a key guide