Kac Moody Eisenstein series in String Theory

Transcription

1 Kac Moody Eisenstein series in String Theory Philipp Fleig Institut des Hautes Études Scientifiques (IHES), Bures-sur-Yvette and Institut Henri Poincaré (IHP), Paris KIAS Seoul, 17. Nov with Axel Kleinschmidt, Daniel Persson and Henrik P. A. Gustafsson arxiv: [hep-th] (FK), [hep-th] (FKP), [math.nt] (FGKP)

2 Goal and Plan Goal: KM Eisenstein series in string theory: why, where and what? I. String theory context General facts about string theory Hidden discrete symmetries (why) String scattering amplitudes (where) Eisenstein series on exceptional KM groups II. KM Eisenstein series by examples (what) Constant term Whittaker Fourier coefficients

3 I. String theory context

4 General facts about string theory Theory of one-dimensional objects ( strings ), generalising the concept of point particles. String sweeps out worldsheet Σ moving in lorentzian target space M String theory: dynamics of embedding map X : Σ M Worldsheet Σ: closed, orientable (super) Riemann manifold Quantum consistency requires 10-dim. traget space, e.g. M = R 1,9

5 General facts about string theory Natural scale α l 2 s with l s m (Planck scale) Low-energy limit: α 0 Infinite tower of string excitations. Bosonic mass spectrum with N = 0,1,2,... m 2 = 2 α N Spin 2 excitation interpreted as the graviton Strings interact by joining and splitting. Interaction strength controlled by g s -coupling constant.

6 String scattering amplitudes Scattering amplitude A: Probability for a particular interaction to take place. Given the in-and out states (momenta k i, polarizations ε i ), the amplitude A is computed by summing over all possible interactions Weighted infinite sum over all loops (genera) A = L=0 g 2(L 1) s A L

7 String scattering amplitudes The amplitude A is a function of: (I.) Asymptotic sates: momenta k i, polarizations ε i, (II.) α (III.) Moduli fields g M Moduli space M depends on the geometry of the target space M. E.g. Kaluza-Klein compactification M = R 1,D 1 T 10 D M class. = E 11 D (R)/K(E 11 D (R)) Quantum effects break the classical moduli space M class. M class. M quant. = E 11 D (Z)\E 11 D (R)/K(E 11 D (R))

8 Hidden symmetries and U-duality in lower dimensions SUGRA hidden symmetry, max. comp. subgroup, U-duality group: D Hidd. sym. E 11 D (R) K(E 11 D ) U-duality E 11 D (Z) 10B SL(2, R) SO(2) SL(2, Z) [Font et al. 92] SO(5,5;R) SO(5) SO(5) SO(5,5,Z) 5 E 6 (R) USp(8) E 6 (Z) 4 E 7 (R) SU(8)/Z 2 E 7 (Z) 3 E 8 (R) Spin(16)/Z 2 E 8 (Z) infinite-dimensional Kac Moody groups: 2 E 9 (R) [Nicolai 87] K(E 9 (R)) E 9 (Z) 1 E 10 (R) [Julia 82, DHN 02] K(E 10 (R)) E 10 (Z) 0 E 11 (R) [West 01] K(E 11 (R)) E 11 (Z)

9 T-and S-duality Mass spectrum of closed string on circle (T 1 ) with radius R: m 2 n2 R 2 + w2 R 2 α 2 n, w: momentum-and winding modes T-duality: R α /R with n w Scalar fields parameterise moduli space M, interpreted as radii R i = exp φ i of torus T 10 D and the string coupling y = g 1 s. S-duality: y y 1

10 Four-graviton scattering in D = 10: tree-level A L=0 Recall A = L=0 g2(l 1) s A L. Tree level (genus 0) contribution: A 0 (s,t,u) = 1 Γ(1 α s)γ(1 α t)γ(1 α u) stu Γ(1 + α s)γ(1 + α t)γ(1 + α u) R4 Mandelstam vars. constaining momenta satisfying k 2 i = 0 s = α 4 (k 1 + k 2 ) 2, t = α 4 (k 1 + k 3 ) 2, s = α 4 (k 1 + k 4 ) 2 Lin. curvature tensor R µνρσ = k µ ε νρ k σ

11 α expansion α -expansion: ( 3 ) A 0 (s,t,u) = + (α ) 3 2ζ (3) + (α ) 3 σ 2 ζ (5) +... R 4 σ 3 }{{} α 5 σ 2 = s 2 + t 2 + u 2 and σ 3 = s 3 + t 3 + u 3 Effective action provides more elegant way of writing α expansion. All loop orders A L can contribute to a given order in α plus additional non-perturbative effects. Focus now on α 3 order.

12 Amplitude at order α 3 By direct calculation [Green, Gutperle 97] E 10 3/2 (z) = constant term - perturb. {}}{ 2ζ (3)y 3/2 + 4ζ (2)y 1/2 + 4 y m µ 2 (m)k 1 (2π m y)e 2πimx m 0 }{{} Fourier coefficients - non-perturb. Non pert. = 2 m µ 2 (m)e S [ inst 1 + O(y 1 ) +... ] m 0 D(-1) instanton action S inst = 2π( m y + imx) = 2π( m /g s + imχ) and instanton measure µ 2 = d m d 2

13 Amplitude as α 3 series In D = 10 dimensions, with z = x + iy ( A 10 = 3σ3 1 + α 3 E 10 3/2 (z) + (α ) 3 σ }{{} 2 α 5 ) 10 E5/2 (z) +... R 4 In D dimensions (after compactification) ( ) A D = ld 6 3σ3 1 + E3/2 D (g) + ED 5/2 (g)σ R 4 (1) E D s (g) are functions of the moduli fields g M class.

14 Automorphic properties Recall ( ) A 10 = 3σ3 1 + α 3 E3/ (α ) 3 σ 2 E5/ R 4 E3/2 10 satisfies props.: [Green, Sethi 98] U-duality SL(2, Z) invariant satisfies diff. equation 4τ2 2 τ τ E3/2 10 = 3 4 E10 3/2 well-behaved growth for Im(z) = y = g 1 s and similarly for E 10 5/2 Automorphic function...study these automorphic functions now...

15 E 10 3/2 and E10 5/2 as SL(2) Eisenstein series E3/2 10 SL(2) (z) = 2ζ (3)E3/2 (z) and E5/2 10 SL(2) (z) = ζ (5)E5/2 (z) SL(2) Eisenstein series written as coset sum Es SL(2) (z) = Im(γ z) s γ B(Z)\SL(2,Z) Borel subgroup B(Z) = Fourier expansion (constant term): 1 0 E SL(2) s (x + iy)dx = y s + ξ (s) ξ (1 + s) y1 s {( ) } 1 k, k Z 0 1 ξ (s) = π s/2 Γ(s/2)ζ (s) (completed Riemann ζ -function).

16 Langlands Eisenstein series Eisenstein series for finite-dimensional groups: [Langlands 76] Function of g G/K(G) Defined as G(Z)-invariant coset sum: E G λ (g) = e λ+ρ H(γg) γ B(Z)\G(Z) Map H : G(R) h(r) (Cartan subalgebra) λ h C like a parameter Technical notation: Weyl vector ρ Pairing : h h C

17 Langlands Eisenstein series for E 11 D In our context G = E 11 D : simple root α i, fund. weight Λ i D 11 D Function of g M class. String theory requires: λ = 2sΛ 1 ρ with s = 3/2,5/2 E D 3/2 = 2ζ (3)E E 11 D λ=3λ 1 ρ and ED 5/2 = ζ (5)E E 11 D λ=5λ 1 ρ Defn. in the Kac Moody cases, D 2: Affine: theory established in [Garland 01] Hyperbolic: small rank examples [Carbone, Lee, Liu 13]

18 II. KM Eisenstein series by examples

19 Fourier expansion of Eisenstein series Expand with respect to parabolic subgroup P Mostly take P = B, with unipotent radical N Iwasawa decomp. G = NAK, B = NA. General (abelian) Fourier expansion Eλ G (g) = C λ (a) + W ψ (λ,g) +... ψ 1 with W ψ (λ,g) = Eλ G (ng)ψ(n)dn N(Z)\N(R)

20 Character ψ Integrate against character ψ : N(Z)\N(R) U(1). Non-trivial only on abelian part of N: only depends on [N,N]\N rk(g) variables χ i Fourier mode with charges m i : ψ = e 2πi j m j χ j Terminology for ψ: all m j = 0: trivial, ψ = 1 all m j 0: generic some but not all m j = 0: degenerate

21 Constant term (trivial ψ) Constant term given by Langlands formula [Langlands 76] C λ (a) = N(Z)\N(R) E E 11 D λ (ng)dn = M(w,λ)e wλ+ρ H(g) w W Weyl group W. Elements w = w ik...w i2 w i1

22 Properties of M(w,λ) factor M(w,λ) = c(k) = α>0 wα<0 ξ (k) ξ (k + 1) c( λ α ) and α > 0 are positive roots ξ (k) ξ (k+1) k 5 Multiplicative property M(w w,λ) = M( w,λ)m(w, wλ).

23 Langlands formula for E 11 D For G = E 11 D we note: M(w,λ)e wλ+ρ H(g) = w W(E 11 D ) Obvious complications in the Kac Moody case: However Weyl group W is infinite. Infinite set of (positive) roots α. { finite sum, D 3 infinite sum, D 2! Find systematic vanishing of M(w, λ)-coeffs. for special λ Collapse to finite sum for λ = 2sΛ 1 ρ with s = 3/2,5/2,... perturbative terms even computable for D 2! [FK 12]

24 λ = 2sΛ 1 ρ: reduction step I First note: c( λ α i ) = c( 1) = 0 for all α i α 1. Easy to show that corresponding M(w, λ) vanishes. Can restrict the sum over W in Langlands formula to S = {w W wα i > 0 for all α i α 1 } S: set of minimal reflections needed to construct Weyl orbit of Λ 1 weight [FK 12] Note: S still infinite! M(w,λ) = α>0 wα<0 c( λ α )

25 E E 10 λ=3λ 1 : reduction step II ρ Look for positive roots α, such that c( λ α = 1) = 0 w 1 id. 1 π 2 3ζ (3) 1 Construct tree of Weyl words S using orbit method [FK 12] 2 Sift through nodes along each branch, until M(w,λ) = 0 3 Reduce to set C λ = {w W M(w,λ) 0} S C λ is finite for the string theory cases. 1 3ζ (5) 2πζ (3) 15ζ (7) 4π 2 ζ (3) π 2 3ζ (3) w 3 w 1 w 4 w 3 w 1... π 3 45ζ (3) 4π 4 945ζ (3) 0 π 2 3ζ (3) 0...

26 Constant term expression for E E 10 λ=3λ 1 ρ Variables r i parameterise the Cartan subalgebra E E 10 3Λ 1 ρ (ng)dn N(Z)\N(R) = r1 3 + r3 6 r π4 945ζ (3) r9 6 r π3 45ζ (3) r7 4 r8 3 + π2 r πγ E 3ζ (3) r 6 ζ (3) r 4 + π2 r3 2 2πr 4 log(4πr 5 ) + 4πr 4 log(r 4 ) 2πr 4 log(r 3 ) 3ζ (3) r 1 ζ (3) ζ (3) ζ (3) 2πr 4 log(r 2 ) ζ (3) + π2 3ζ (3) r ζ (5) 2πζ (3) r8 5 r ζ (7) 4π 2 ζ (3) r7 10 r 1,...,r 10 are related to the radii of T 9 torus and the string coupling.

27 Collapse for range of s values across dimensions: E E 11 D 2sΛ 1 ρ s E E E E E E Number of Weyl words with non-vanishing coefficients M(w, λ) Ellipsis: row to be continued with last number explicitly written out (for D 2 this is conjectural) ŜL(2) provides instructive example for collapse mechanism [FK 12]

28 Fourier coefficients Recall, C λ (a) + ψ W ψ (λ,g) Whittaker Fourier coefficients W ψ Explicit expression for W ψ when ψ generic (all m j 0) (c.f. Casselman-Shalika) But For λ s appearing in string theory, generic W ψ (λ) = 0 (so far). For Kac Moody groups all generic W ψ (λ) = 0. [Liu 11] Need to study W ψ for degenerate ψ

29 Whittaker Fourier coefficients for non-generic ψ Degenerate ψ, non-trivial along set of simple roots Π Π. E.g. m = (m 1,0,...0,m 11 D ) G 2 = A 1 A 1 Formula for W ψ : D 11 D W ψ (λ,g) = w c w 0 C ψ M(w 1 c,λ)e (w cw 0 ) 1 λ+ρ H(g) W G ψ (w 1 c ψ is generic w.r.t. G explicit expression for W G ψ w c w 0 C ψ special parameterisation for C ψ = {w W wπ < 0}. Obtain collapse from combined condition [FKP 13] [c.f. C λ,ψ = {w 1 c C λ w c w 0 C ψ } λ,1) Hashizume]

30 The case of λ = 3Λ 1 ρ Abelian Fourier coefficients can be expressed entirely as a sum of Whittaker Fourier coefficients on A 1 subgroups W ψ (λ,na) = ψ 0 α Π ψ α c α (a)w ψ α (λ α,1)ψ α (n), ψ α assoc. with simple roots α; m α only non-zero charge. W ψ α is a generic Whittaker Fourier coefficient on the A 1 associated with the simple roots α. c α (a): function of variables parametrizing the Cartan torus. λ α: λ projected onto A 1 subgroup. [FKP 13]

31 Fourier coefficients of E E 10 λ=3λ 1 ρ Fourier coefficients in terms of (modified) Bessel-type functions a = v h j j j ψ (m,0,0,0,0,0,0,0,0,0) (0,m,0,0,0,0,0,0,0,0) (0,0,m,0,0,0,0,0,0,0) (0,0,0,m,0,0,0,0,0,0) (0,0,0,0,m,0,0,0,0,0) (0,0,0,0,0,m,0,0,0,0) (0,0,0,0,0,0,m,0,0,0) (0,0,0,0,0,0,0,m,0,0) (0,0,0,0,0,0,0,0,m,0) (0,0,0,0,0,0,0,0,0,m) c α (a)w ψ α a (χ α,1) v 2 3 v 1 1 B ( 3/2,m v 2 1 v 1 ) 3 v 2 2 B 0,m(v 2 2 v 1 4 ) ξ (3) ξ (2)v 4B 1,m(v 2 3 v 1 1 v 1 4 ) ξ (3) v 4 B 1/2,m (v 2 4 v 1 2 v 1 3 v 1 5 ) ξ (3) v 2 5 B 0,m(v 2 5 v 1 4 v 1 6 ) ξ (3)v 6 ξ (2)v 3 6 B 1/2,m(v 2 6 v 1 5 v 1 7 ) ξ (3)v 2 7 v 4 7 v 3 8 B 1,m ( v 2 7 v 1 6 v 1 8 ξ (4)v 5 8 v 4 9 B 3/2,m(v 2 8 v 1 7 v 1 9 ) ξ (3) ξ (5)v 6 9 v 5 10 B 2,m(v2 9 v 1 8 v 1 10 ) ξ (3) ξ (6)v 7 10 B 5/2,m(v 2 10 v 1 9 ) ξ (3) ) B s,m j (a α j ) := 2 ξ (2s ) aα j s 1/2 m j 1/2 s µ 2s 1(m j )K s 1/2(2π m j a α j )

32 Summary Manifestation of discrete symmetries in string theory Obtained (complete) information about perturbative and non-perturbative effects in string theory from Eisenstein series Validity of results confirmed by taking various physical limits Hints towards distinguished role of E 10 symmetry in fully non-perturbative definition of string theory Use a physically intuitive approach to mathematics. More to come in Axel s and Daniel s talks... Thank you for your attention!

33 Summary Manifestation of discrete symmetries in string theory Obtained (complete) information about perturbative and non-perturbative effects in string theory from Eisenstein series Validity of results confirmed by taking various physical limits Hints towards distinguished role of E 10 symmetry in fully non-perturbative definition of string theory Use a physically intuitive approach to mathematics. More to come in Axel s and Daniel s talks... Thank you for your attention!