Lectures on Superstring Amplitudes

Transcription

1 Lectures on Superstring Amplitudes Part 2: Superstrings Eric D Hoker Mani L. Bhaumik Institute for Theoretical Physics University of California, Los Angeles Center for Quantum Mathematics and Physics Amplitudes 2018 Summer School

2 Superstring Perturbation Theory Theory of fluctuating random surfaces (closed strings shown) governed by topological expansion in the genus h weighed by g 2h 2 s g 2 s + g 0 s + g 2 s + Bosonic string unstable with closed string tachyon Nature has fermions! Superstrings generalize bosonic string they have fermions no tachyon supersymmetry

3 Approaches to Superstring Perturbation Theory Goal is to obtain superstring amplitudes at all genera Ramond-Neveu-Schwarz formulation of fermionic strings; w/ Gliozzi-Scherk-Olive projection to supersymmetric spectrum; Green-Schwarz space-time supersymmetric formulation; Mandelstam light-cone formulation; String field theory; Topological string theory; Berkovits pure spinor formulation. Different perturbative superstring theories (in 10 dimensions) Type I open & closed, orientable & non-orientable, D-branes Type IIA,B closed orientable, D-branes Heterotic closed orientable E 8 E 8, Spin(32/Z 2 ) Here: RNS formulation, closed orientable superstrings, dimension 10

4 Genus-zero four-graviton superstring amplitude Kinematics of the four-graviton amplitude momenta of gravitons k µ i are conserved i kµ i = 0 choose basis of factorized polarization tensors ε µν i = ε µ i εν i masslessness ki 2 = 0 and transversality kµ i εµ i = kµ i εµ i = 0 for i = 1,2,3,4 kinematic invariants s = s 12 = s 34, t = s 14 = s 23, u = s 13 = s 24 s ij = α (k i +k j ) 2 /4 Tree-level four-graviton amplitude is given by K K 1 stu A (0) (ε i, ε i,k i ) = 1 gs 2 Kinematical factor K given in terms of f µν i Γ(1 s)γ(1 t)γ(1 u) Γ(1+s)Γ(1+t)Γ(1+u) = k µ i εν i kν i εµ i by K = (f 1 f 2 )(f 3 f 4 ) + (f 1 f 3 )(f 2 f 4 ) + (f 1 f 4 )(f 2 f 3 ) 4(f 1 f 2 f 3 f 4 ) 4(f 1 f 2 f 4 f 3 ) 4(f 1 f 3 f 2 f 4 ) for K replace ε i by ε i Equivalently, K K = R 4 with R the linearized Weyl tensor String duality: symmetric in s,t,u Poles in each channel, at s,t,u = 0,1,2,

5 Genus-one four-graviton superstring amplitude Type II four-graviton amplitude to one-loop order (Green, Schwarz 1982) A (1) (ε i, ε i,k i ) = R 4 d 2 τ (Imτ) 2 B(1) (s ij τ) M 1 Partial amplitude B (1) is a modular function in τ M 1 = H 1 /SL(2,Z) B (1) (s ij τ) = Σ 4 4 i=1 d 2 z i ( Imτ exp i<j ) s ij G(z i z j τ) G(z τ) is the scalar Green function on the torus Σ of modulus τ. Analogous formulas for Heterotic strings and more external states. Singularity structure For fixed τ integrations over Σ produce poles in B (1) at positive integers s ij. The integral over τ converges absolutely only for Re(s ij ) = 0. Analytic continuation to s ij C via decomposition of M 1. Branch cuts in s ij starting at integers 0 are produced by τ i region.

6 Loop momenta Loop momenta may be exposed Choose a canonical basis of homology cycles A,B. Choose loop momentum p flowing through the cycle A, M 1 d 2 τ (Imτ) 2 B(1) (s ij τ) = d 10 p R 10 M 1 Σ 4 F(z i,k i,p τ) 2 Chiral amplitude F is locally holomorphic in τ and z i F(z i,k i,p τ) = e iπτp2 +2πip i k iz i ϑ 1 (z i z j τ) s ij dτ at the cost of non-trivial monodromy i<j 4 dz i i=1 F(z i +δ i,l A,k i,p τ) = e 2πik l p F(z i,k i,p τ) F(z i +δ i,l B,k i,p τ) = F(z i,k i,p+k l τ) Modular invariance of A (1) guarantees independence of choices. Hermitian pairing of F and F is familiar from 2-d CFT where loop momentum p labels conformal blocks of 10 copies of c = 1.

7 UV-finiteness Thanks to modular invariance, all string amplitudes are UV-finite shown for the closed bosonic string at genus one (Shapiro 1972) holds for all modular invariant superstrings to all loops (i.e. all genera) For genus-one: All chiral amplitudes have a universal factor F(z i,ε i,k i,p I τ) = e iπpµ τp µ Modular invariance allows one to choose a fundamental domain where Im(τ) bounded from below H 1 /SL(2,Z) = { τ C,Im(τ) > 0, τ 1, Re(τ) 1 2 Analogous, more complicated, choices to higher genus Uniform Gaussian suppression at large loop momenta UV finiteness }

8 RNS formulation of superstrings M = R 10 flat Minkowski space-time with Lorentz group SO(1,9) x µ scalars on worldsheet Σ, map Σ into M ψ µ spinors on Σ but Lorentz vector under SO(1,9) Worldsheet supersymmetry = Σ is a super Riemann surface Two sectors : NS bosons SO(1,9)-tensors R fermions SO(1, 9)-spinors With Minkowski signature Σ ψ µ and ψ µ are independent Majorana-Weyl spinors of opposite chirality With Euclidean signature Σ ψ µ and ψ µ must be independent complex Weyl spinors Globally, on a compact Riemann surface of genus h, All ψ µ are sections of a the same spin bundle S (and ψ µ of S) 2 2h distinct spin structures for S (and 2 2h independently for S) GSO projection requires independent summation over spin structures

9 Quantization of worldsheet spinor fields Illustrate Ramond and Neveu-Schwarz sectors independence of chiralities Dirac action and equation for flat M = R 10 with metric η All components of ψ µ + are sections of the same spin bundle S Complex structure J with local complex coordinates (z, z) Dirac action, I ψ [ψ,j] = 1 d zdzψ + z µ ψ ν 2π +η µν Dirac equation z ψ µ + = 0 has locally holomorphic solutions, but products of operators produce singularities ψ µ +(z)ψ ν +(w) = ηµν z w +regular each component ψ µ generates a CFT with central charge c = 1 2. Σ

10 Quantization of worldsheet spinor fields (cont d) Quantization on flat cylinder or conformal equivalent flat annulus cylinder w = τ +iσ with identification σ σ +2π annulus centered at z = 0, conformally mapped by z = e w one-forms related by dz = e w dw, spinors by (dz) 1 2 = e w/2 (dw) 1 2 fields related by conformal transformation ψ cyl (z) = e w/2 ψ ann (w) Two possible spin structures NS R ψ µ cyl (τ,σ +2π) = ψµ cyl (τ,σ) or ψµ ann(e 2πi z) = +ψ µ ann(z) ψ µ cyl (τ,σ +2π) = +ψµ cyl (τ,σ) or ψµ ann(e 2πi z) = ψ µ ann(z) Free field quantization in annulus representation NS ψ µ (z) = r 1 2 +Zbµ r z 1 2 r {b µ r,b ν s} = η µν δ r+s,0 R ψ µ (z) = n Z dµ nz 1 2 n {d µ m,d ν n} = η µν δ m+n,0

11 Quantization of worldsheet spinor fields (cont d) Lorentz generators of SO(1,9) : [J µν,ψ κ (z)] = η νκ ψ µ (z) η µκ ψ ν (z) J µν NS = ( b µ ) rb ν r b ν rb µ r J µν R r N 1 2 = 1 2 [dµ 0,dν 0]+ n N ( d µ nd ν n d ν nd µ ) n Fock space construction produces two sectors NS ground state defined by b µ r 0;NS = 0 for all r > 0 0;NS is unique and in trivial representation of SO(1,9) Fock space = linear combinations of b µ 1 r 1 b µ p r p 0;NS, r i > 0 All states in tensor reps of SO(1,9) are space-time bosons. R ground state defined by d µ n 0,α;R = 0 for all n > 0 0,α;R is degenerate and in spinor rep. of SO(1,9), states labelled by α Fock space = linear combinations of d µ 1 n 1 d µ p n p 0,α;R, n i > 0 All states in spinor reps of SO(1,9) are space-time fermions.

12 Summation over spin structures Theory with bosons and fermions requires both NS and R sectors to include both, one must sum over two spin structures of the annulus Type II spin structures of ψ µ ± are independent of one another space-time fermions are in the R NS and NS R sectors which could never arise if spin structures for opposite chiralities coincided On the torus, viewed as cylinder + identification spin structures along cycle of cylinder produce R and NS sectors sum over spin structures along conjugate cycle produces GSO-projection reduces to half the states in both R and NS sectors R-sector: space-time spinor of definite chirality NS-sector: eliminates the tachyon sum over all spin structures

13 Summation over spin structures (cont d) Fix a canonical homology basis of cycles A I,B I of H 1 (Σ,Z) I = 1,,h with canonical intersection pairing #(A I,A J ) = #(B I,B J ) = 0 and #(A I,A J ) = δ IJ A 1 A 2 B 1 B 2 Σ Transformations which maps one canonical basis into another linear with integer coefficients preserve the intersection matrix: Sp(2h, Z) On Riemann surface of higher genus h sum over all spin structures along A-cycles produces R and NS sectors along B-cycles produces GSO-projection mapped into one another by Sp(2h,Z 2 )

14 Super Riemann surfaces Ordinary Riemann surface (locally C with coordinate z) complex manifold: holomorphic transition functions z z (z); complex structure = conformal structure J Moduli space M h = {J}/Diff(Σ) of genus h compact Riemann surfaces Complex super manifold (locally C 1 1 with coordinates z θ) holó transition functions z θ z (z,θ) θ (z,θ) generate N = 2 super conformal Super Riemann surface (locally C 1 1 with coordinates z θ) holó transition functions z θ z θ rescale D θ = θ +θ z Transition functions define N = 1 superconformal structure J Globally: T Σ has a completely non-integrable subbundle of rank 0 1 Moduli space of compact super Riemann surfaces: M h = {J}/Diff(Σ) = equivalence classes of superconformal structures J dim C M h = 0 0 h = or 1 1 h = 1 even or odd spin structure 3h 3 2h 2 h 2 odd modulus at h = 1 odd spin structure is a book keeping device; odd moduli really first appear at genus 2, as curved super spaces.

15 Superstring worldsheets and moduli spaces Heterotic Left : RS Σ L, moduli space M L coord resp. z and m i Right : SRS Σ R, moduli space M R coord resp. (z,θ) and (m i,ζ α ) Worldsheet is a cycle Σ Σ L Σ R of dim 1 1 subject to Σ red = diag(σ Lred Σ Rred ) : z = z + nilpotent Moduli space is a cycle Γ M L M R of dim 3h 3 2h 2 for h 2 subject to Γ red = diag(m Lred M Rred ) : ( m i ) = m i + nilpotent (reduced space obtained by setting all nilpotent variables to zero) Type II Left : SRS Σ L, moduli space M L coord resp. ( z, θ) and ( m i, ζ α ) Right : SRS Σ R, moduli space M R coord resp. (z,θ) and (m i,ζ α ) Worldsheet is a cycle Σ Σ L Σ R of dim 1 2 Moduli space is cycle Γ M L M R of dim 3h 3 4h 4 for h 2 subject to z = z + nilpotent and ( m i ) = m i + nilpotent Super-Stokes theorem ensures independence of the choice of cycles in amplitudes with BRST invariant vertex operators consistent definition of superstring amplitudes to all genera (Witten 2012)

16 Worldsheet action for Type II superstrings Worldsheet is Σ Σ L Σ R Σ L has superconformal structure J with local coordinates z θ Σ R has superconformal structure J with local coordinates z θ Superconformal invariant matter action worldsheet matter field X µ ( z,z θ,θ) = x µ ( z,z)+θψ µ ( z,z)+ θ ψ µ ( z,z)+ θθf µ ( z,z) Worldsheet action in local coordinates (D θ = θ +θ z ) I m [X µ, J,J] = [d zdz d θdθ] D θx µ D θ X µ Superconformal algebra on fields generated by Σ S zθ = S zθ +θt zz S zθ = 1 2 ψµ z x µ T zz = 1 2 zx µ z x µ ψµ z ψ µ S z θ = S z θ + θ T z z S z θ = 1 2 ψ µ z x µ T z z = 1 2 zx µ z x µ ψ µ z ψµ

17 Deformations of superconformal structures Under deformation of J for Σ L and J for Σ R ( δi = [d zdz d θdθ] H θ z S zθ + H ) θ z S z θ Σ in components by integrating out θ,θ, ( ) δi = d zdz µ z z T zz +χ z θ S zθ + µ z z T z z + χ z θ S z θ Σ red recover Beltrami differentials µ, µ and worldsheet gravitino fields χ, χ H θz = θ(µ z z +θχ z θ ) H θ z = θ( µ z z + θ χ z θ) Finite deformations of the metric with µ = µ and χ = χ integrate to the standard 2-dim N = 1 supergravity action (Brink, Di Vecchia, Howe; Deser, Zumino 1976) Type II superstring perturbation theory requires µ µ and χ χ

18 Type II string amplitude Parametrize deformations H θ z,h θz by slice { J( m),j(m)} in M L M R H θ z = D θv z + H A δm A H A = J θ z / m A m A = (m i,ζ α ) H θ z = D θ Ṽ z + HÃδ mã HÃ = J θ z / mã m A = ( m i, ζ α ) ghost fields V z C z = c z + θγ θ H θz B zθ = β zθ + θb zz V z C z = c z + θ γ θ Super conformal invariant ghost action I gh = Σ H θ z B z θ = β z θ + θ b z z ( ) [d zdz d θdθ] B zθ D θc z + B z θd C z θ + B zθ H A δm A + B z θ HÃδ mã The integrand for the full amplitude is given by D(XB BC C)V 1 V n [d mãdm A ]δ( B, HÃ )δ( B,H A )e I m I gh Ã,A V 1 V n are BRST-invariant vertex operators. Picture Changing Operator formalism (Friedan, Martinec, Shenker 1986) may be obtained as singular limit for χ supported at points globally regular reformulation via vertical integration (Sen, Witten 2016)

19 Loop momenta and Chiral amplitudes h independent loop momenta p µ I defined to flow across A I cycles p µ I = dz z x µ A I Chiral Amplitudes (ED, Phong 1988) Massless NS bosons with factorized polarization tensor ε µ µ i = ε µ ε µ i i Chiral amplitude at fixed loop momenta is given by F R (J,ε i,k i,p I ) = V 1 V N e pµ I B dz zx µ I e Σ H θ z Szθ A δ( B,H A )dm A Correlation functions computed with chiral Green functions Full Superstring Amplitudes obtained by pairing left and right and integrating over Γ M L M R A (h) (ε i, ε i,k i ) = dp µ R 10 I F L ( J, ε i,k i,p µ I )F R(J,ε i,k i,p µ I ) Γ integration over vertex operator insertion points included in integration over Γ cfr double copy construction in supergravity calculations

20 Parametrization of super moduli Superconformal structure J M h specified by transition functions Concrete calculations use parametrization by gravitino field χ z θ Local parametrization of moduli (in conformal-invariant theory) Conformal structure J with metric g = dz 2 in local coordinates (z, z) deform conformal structure by Beltrami differential to g = dz + µd z 2 realized in CFT by inserting Σ d zdzµ z z T zz to all orders in µ Local parametrization of supermoduli (in superconformal-invariant theory) Start with Σ red with complex structure given by J M red Deform super conformal structure by inserting T and S Σ red d zdz (µ z z T zz + χ z θ S zθ ) χ and µ parametrized by local odd coordinates on M h For h = 2, even spin structures, holó projection M 2 M 2 exists via the super period matrix (ED, Phong 2001) For h 5 no holó projection M h M h exists (Donagi, Witten 2013)

21 The super period matrix (even spin structures) Start from conformal structure J for Σ red with holó 1-forms ω I A I ω J = δ IJ B I ω J = Ω IJ I,J = 1,2 Deform to superconformal structure J on Σ with superholó forms ˆω I A I ˆω J = δ IJ B I ˆω J = ˆΩ IJ I,J = 1,2 Explicit formula for the super period matrix ˆΩ for even spin structure δ ˆΩ IJ = Ω IJ i ω I (z)χ(z)s δ (z,w Ω)χ(w)ω J (w)+ µω I ω J 8π Σ red Σ 2 red ˆΩ IJ is locally supersymmetric; ˆΩ IJ = ˆΩ JI ; and Im ˆΩ > 0 Every ˆΩ corresponds to an ordinary Riemann surface Szegö kernel S δ (z,w Ω) is non-singular in the interior of M 2 Projection using ˆΩ is holomorphic and natural for genus 2

22 Projecting and pairing Chiral Amplitudes Chiral Amplitudes on M 2 Natural parametrization of M 2 by (ˆΩ IJ,ζ α ) (even spin structure δ) involves measure dκ[δ](ˆω,ζ) and correlation functions C[δ](ε i,k i,p I ˆΩ,ζ) Projection to chiral amplitudes on M 2 by integrating over ζ and summing over δ at fixed ˆΩ R(ε i,k i,p I ˆΩ) = δ L( ε i,k i,p I ˆΩ) = δ ζ ζ dκ[δ](ˆω,ζ)c[δ](ε i,k i,p I ˆΩ,ζ) dκ[ δ](ˆω, ζ)c[ δ]( ε i,k i,p I ˆΩ, ζ) for heterotic, L is chiral half of bosonic string, has no integral in ζ phase factors determined by Sp(4, Z) modular invariance Pairing left and right chiral amplitudes, integrating over p I and ˆΩ A (2) (ε i, ε i,k i ) = dˆω dp µ I R(ε i,k i,p I ˆΩ)L( ε i,k i,p I ˆΩ) M 2 Integral over p I is Gaussian and can be carried out explicitly.

23 Genus two Siegel Upper half space S 2 S 2 = {Ω IJ = Ω JI C with I,J = 1,2 and Y = ImΩ > 0} Sp(4,R) acts by Ω (AΩ+B)(CΩ+D) 1 M t JM = J M = ( A B C D ) J = ( 0 I I 0 ) S 2 has Sp(4,R)-invariant metric ds 2 2 and volume form dµ 2 ds 2 2 = Y 1 IJ d Ω JK Y 1 KL dω LI I,J,K,L=1,2 Compact Riemann surfaces Σ Choose canonical homology basis of A I,B I cycles for H 1 (Σ,Z). ω I dual holomorphic (1,0) forms, A I ω J = δ IJ B I ω J = Ω IJ Riemann relations imply Ω S 2 ; Modular group Sp(4,Z); moduli space M 2 = S 2 /Sp(4,Z).

24 Genus-two Type II four-graviton amplitude Type II four-graviton amplitude (ED, Phong ) A (2) (ε i, ε i,k i ) = gs 2 K K dµ 2 B (2) (s ij Ω) M 2 B (2) Y Ȳ ( ) (s ij Ω) = (detimω) 2exp s ij G(z i,z j Ω) Σ 4 i<j G(z i,z j ) is the genus-two scalar Green function; (z i,z j ) is a bi-holomorphic form independent of s,t,u. (z,w) = ω 1 (z) ω 2 (w) ω 2 (z) ω 1 (w) Y = (t u) (z 1,z 2 ) (z 3,z 4 ) + (s t) (z 1,z 3 ) (z 4,z 2 ) +(u s) (z 1,z 4 ) (z 2,z 3 ) reproduced (with fermions) in pure spinor formulation (Berkovits, Mafra 2005) Singularity structure For fixed Ω integrations over Σ produce poles in B at positive integers s ij. The integral over Ω requires analytic continuation beyond Re(s ij ) = 0. Branch cuts in s ij starting at integers produced from Ω 11,Ω 22 i

25 Genus-two Heterotic four-graviton amplitude Heterotic four NS boson amplitude at genus 2 (ED, Phong 2005) A (2) O (ε i, ε i,k i ) = gsk 2 dµ 2 B (2) O ( ε i,k i Ω) M 2 B (2) O ( ε i,k i Ω) = Ψ 10 (Ω) is the Igusa cusp form. Y W O ( ε i,k i ) ( Σ 4 (detimω) 2 Ψ 10 (Ω) exp Dependence of the operator O on the channel: 4 gravitons R 4 2 gravitons + 2 gauge bosons R 2 tr(f 2 ); 4 gauge bosons (trf 2 ) 2 4 gauge bosons tr(f 4 ) For example, i<j ) s ij G(z i,z j ) W R 4( ε i,k i ) = 4 i=1 ε i x(z i )e ik i x(z i ) 4 i=1 eik i x(z i ) Gauge parts are obtained by the correlators of the current (0, 1)-forms.

26 Singularities in the projection M 2 M 2 Projection M 2 M 2 is holó, but integration extends to boundary are there singularities in the projection M 2 M 2? Ω = ( τ u u σ Key ingredient in ˆΩ is the Szegö kernel ) S δ (z,w Ω) = u 0 separating node σ i non-separating node ϑ[δ](z w Ω) ϑ[δ](0 Ω)E(z, w) As u 0 we have ϑ[δ](0 Ω) ϑ[δ 1 ](0 τ)ϑ[δ 2 ](0 τ) Even δ = [δ 1,δ 2 ] with δ 1,δ 2 odd produces a singularity in S δ and ˆΩ Physical effects singularity killed by ψ-zero modes in R 10 (space-time susy) contribution when susy is broken by radiative corrections (Witten 2013) Two-loop vacuum energy in Heterotic strings on CY orbifold C 3 /Z 2 Z 2 is zero for E 8 E 8 E 6 E 8 with unbroken susy non-zero for Spin(32)/Z 2 SO(26) U(1) with broken susy (Atick, Dixon, Sen 1988; Dine, Seiberg, Witten; ED, Phong 2013; Berkovits, Witten 2014)

27 Singularities in the projection M 3 M 3 Some basic structure theorems A hyper-elliptic surface is a branched double cover of the sphere S 2 ; All genus 1 and all genus 2 surfaces are hyper-elliptic; Hyper-elliptic surfaces form a co-dim 1 sub-variety in the interior of M 3 (referred to as the hyper-elliptic divisor) The genus-three period matrix (for even spin structure) ˆΩ IJ = Ω IJ i ω I (z)χ(z)s δ (z,w Ω)χ(w)ω J (w)+o(χ 4 ) 8π For Ω on the hyper-elliptic divisor of M 3 there always exists an even spin structure δ such that ϑ[δ](0 Ω) = 0 the presence of the extra Dirac zero modes kills effects of this singularity Beautiful proposal for the genus 3 superstring measure (Cacciatori, Dalla Piazza, van Geemen 2008) Another even δ does produce a subtle singularity in ˆΩ (Witten 2015)