Lectures on Superstring Amplitudes

Part 1: Bosonic String

Eric D’Hoker Mani L. Bhaumik Institute for Theoretical Physics University of California, Los Angeles

Center for Quantum Mathematics and Physics - 2018 Amplitudes 2018 Summer School Eric D’Hoker Lecture I on Superstring Amplitudes Outline of lectures

Lecture 1 • Bosonic strings and conformal ﬁeld theory

Lecture 2 • Superstring amplitudes

Lecture 3 • Low energy eﬀective interactions Eric D’Hoker Lecture I on Superstring Amplitudes Strings

A string is a 1-dimensional object • – open string = topology of an interval; – closed string = topology of a circle; – physical size Planck length ℓ 10−33cm 10−19 size of the proton. P ≈ ≈ ×

Ultimate goal: uniﬁed theory of particle physics and gravity • – elementary particles correspond to strings and their excited states; – consistently with quantum mechanics and general relativity; – remarkably unique structure.

Immediate goal: relating string amplitudes and ﬁeld theory amplitudes • – at distance scales larger than the Planck length (low energy) a string eﬀectively behaves as a point particle – string theory exhibits powerful structure of amplitudes Eric D’Hoker Lecture I on Superstring Amplitudes String Topology

Consistent interacting string theories • – only closed strings (Type IIA,B and heterotic) – closed and open strings (Type I) – Type II theories have open strings in the presence of D-branes Strings live in a physical space-time M • – M may be a manifold or an orbifold (with mild isolated singularities) – superstring theory predicts 10-dim – but space-time visible to us is 4-dim. requires “compactiﬁcation” ⇒ Under time-evolution strings sweep out a 2-dim. surface •

closed strings

time-evolution (freely propagating) basic interaction (purely topological) Eric D’Hoker Lecture I on Superstring Amplitudes Perturbative String Amplitudes

Quantum probability “scattering” amplitudes • = Feynman functional integral/sum over all surfaces with given boundary components for initial and ﬁnal strings

Closed oriented string perturbation theory • – The only remaining topological characterization is the genus h 0 - probability amplitude includes sum over all genera ≥ 2h−2 - weighed by a factor gs where gs is the “string coupling”

g−2 + g0 + g2 + s s s ···

– genus h = number of “loops” Eric D’Hoker Lecture I on Superstring Amplitudes Structure of string amplitudes

Perturbative part of string amplitude decomposes into a sum over topologies • ∞ = g2h−2 (h) Aperturbative s × A hX=0 – (h) is the amplitude at genus h A

– The perturbative expansion in gs is asymptotic but not convergent (just as in ﬁeld theory)

Non-perturbative part (not considered here) • 2 ⋆ instantons e−1/gs ≈ ⋆ D-branes contribute e−1/gs. ≈ Eric D’Hoker Lecture I on Superstring Amplitudes String Data (closed oriented bosonic strings)

Assume ﬁxed space-time M, with ﬁxed metric G • – Physical space-time has Minkowski signature metric G – Starting point for string theory is often a Riemannian metric (if needed to be analytically continued to Minkowski signature)

The 2-dimensional worldsheet Σ is mapped into space-time M • – The space of all such maps x : Σ M is denoted Map(Σ). → Riemannian metric G induces a Riemannian metric x∗(G) on Σ • – Hence Σ is a Riemann surface (i.e. complex manifold with hol´otransition functions)

Polyakov formulation invokes an independent metric • – Riemannian metric g on Σ – Denote the -dim. Riemannian manifold of such metrics by Met(Σ) – String amplitude∞ at ﬁxed genus h obtained by weighed sum over g, x

(h) = Dg Dx e−IG[x,g] A ZMet(Σ) ZMap(Σ) Eric D’Hoker Lecture I on Superstring Amplitudes

The worldsheet action IG and the measure Dx Basic Criteria • – Intrinsic = invariant under “reparametrizations” Diﬀ(Σ) of Σ – lead to a well-deﬁned QFT (renormalizable)

e.g. Non-linear sigma model action with Riemannian metric G • 1 2 mn µ ν IG[x, g]= ′ d ξ√gg ∂mx ∂nx Gµν(x) α ZΣ

m, n = 1, 2 worldsheet indices µ, ν = 1, ··· ,D space-time Einstein indices

The measure is governed by the L2-norm • 2 2 µ ν δx G = d ξ√g δx δx Gµν(x) k k ZΣ – manifestly intrinsic – renormalizable in a generalized sense (the metric G is renormalized) Eric D’Hoker Lecture I on Superstring Amplitudes Weyl(Σ)-invariance Weyl transformations: g e2σg leaving xµ and G unchanged • mn → mn The classical action IG is Weyl-invariant for any metric G • – but the measure Dx is not Weyl-invariant – which gives rise to a “Weyl-anomaly” = symmetry of classical action not preserved by quantization The action I defines a conformal quantum field theory • G e−WG[g] = Dx e−IG[x,g] ZMap(Σ) – provided WG is Diff(Σ)-invariant – obeys the following Ward identity under Weyl transformations

c 2 δWG[g]= d ξ√gRg δσ 24 ZΣ – where Rg is the scalar curvature of the metric g on the surface Σ The measure Dg is not Weyl-invariant, but the combined amplitude • – is Weyl invariant for central charge c = 26 = dim(M) – later we shall see for the superstring D = 10 = dim(M) Eric D’Hoker Lecture I on Superstring Amplitudes Conformal Field Theory

Stress tensor encodes response of field theory to change in metric • δW [g] T c = G T c = T c mn √gδgmn mn nm m c – Diff(Σ)-invariance requires “a conserved stress tensor” Tmn = 0 – Weyl anomaly requires gmnT c = c R ∇ mn −12 g Traceless stress tensor T obtained by adding a local counter-term • mn – In local complex coordinates (z, z˜) we have Tzz˜ = Tzz˜ = 0 and c c z z 2 z Tzz = Tzz + 2∂zΓzz − (Γzz) Γzz = ∂z ln gzz˜ 6 – Successive derivatives of W in gmn give correlators of Tmn – Their singular part is governed by the OPE and the Ward identities c/2 2T ∂ T T T = + ww + w ww + regular zz ww (z w)4 (z w)2 z w − − − −2−m – The mode expansion Tzz = m z Lm gives the Virasoro algebra P c [L , L ]=(m n)L + m(m2 1)δ m n − m+n 12 − m+n,0 Eric D’Hoker Lecture I on Superstring Amplitudes Negative norm states Consider flat Minkowski M = R26 with metric η = diag( + +) • µ − ··· – Maps x : Σ M satisfy Laplace equation ∂z˜∂zx = 0 for µ = 1, · · · , 26 – Concentrate→ on holomorphic field µ µ −m−1 µ ν µν µ † µ ∂zx = xm z [xm, xn]= mδm+n,0 η (xn) = x−n mX∈Z µ µ – Similarly anti-holomorphic field ∂z˜x produces modes x˜ Single string ground state 0,k labeled by its momentum k satisfies • | i xµ 0,k = kµ 0,k xµ 0,k = 0 for m> 0 0 | i | i m| i – Fock space (holo sector) generated by linear combinations of

µ1 µp x xm 0,k m , , m < 0 m1 ··· p| i 1 ··· p – Lowest excited state ε (k)xµ 0,k has norm µ −1| i ε (k)xµ 0,k 2 = ε (k)ε (k)ηµν 0,k 2 k µ −1| ik µ ν k| ik – component εµ = δµ,0 produces negative norm state (assuming k|0,kik2 > 0) = inconsistent with quantum mechanical probability interpretation Eric D’Hoker Lecture I on Superstring Amplitudes Eliminating negative norm states – conformal symmetry

Conformal symmetry guarantees the existence of Virasoro algebra • c [L , L ]=(m n)L + m(m2 1)δ m n − m+n 12 − m+n,0 – for the bosonic string c = 26 and 1 1 L = x x L = x2 + x x m 2 m−n · n 0 2 0 −n · n nX∈Z nX∈N

A state ψ is “physical” if (L0 1) ψ = Lm ψ = 0 for m N • – Eliminates| i all negative norm states;− | i | i ∈ – Decouples all null states produced by gauge transformations;

– e.g. on states ψ = ε(k) x 0,k | i · −1| i ⋆ L1 constraint imposes k ε(k) = 0 2· ⋆ L0 constraint imposes k = 0 ⋆ L constraints are automatic for m 2 for this particular state m ≥ – the state 0,k itself is a tachyon (to be absent in the superstring) | i Negative norm and null states eliminated by conformal symmetry ⇒ Eric D’Hoker Lecture I on Superstring Amplitudes Conformal symmetry in curved space-times

Condition for Weyl-invariance on the metric G • – Inﬁnitesimal Weyl variation for arbitrary G to one-loop order in α′

2 mn µ ν ′ δWG[g]= d ξ√gg ∂mx ∂nx Rµν(x)δσ + + (α ) ZΣ ··· O

where Rµν is the Ricci tensor of the metric Gµν ′ – Thus, to leading order in α conformal invariance requires Rµν = 0 Eric D’Hoker Lecture I on Superstring Amplitudes Conformal symmetry in curved space-times

Condition for Weyl-invariance on the metric G • – Inﬁnitesimal Weyl variation for arbitrary G to one-loop order in α′

2 mn µ ν ′ δWG[g]= d ξ√gg ∂mx ∂nx Rµν(x)δσ + + (α ) ZΣ ··· O

where Rµν is the Ricci tensor of the metric Gµν ′ – Thus, to leading order in α conformal invariance requires Rµν = 0 Eric D’Hoker Lecture I on Superstring Amplitudes Vertex operators

Small ﬂuctuations in the metric are gravitons • – A string couples to N gravitons in ﬂat space by slightly perturbing the metric N µ G (x)= η + ε (k ) eikiµx + (ε2) µν µν iµν i O Xi=1 – conformal invariance requires G to satisfy the linearized Einstein equations 2 µ ki = 0 ki εiµν(ki) = 0 for i = 1, · · · ,n

Vertex operator formulation is obtained by expanding in powers of ε • i ∞ = g2h−2 Dg Dx [x, g] [x, g] e−Iη[x,g] A s Z Z V1 ···VN hX=0 Met(Σ) Map(Σ) – where the vertex operator for an on-shell physical graviton is given by

µ 2 mn µ ν ikµx i[x, g]= εiµν(ki) d ξ√gg ∂mx ∂nx e V ZΣ – On-shell conditions k2 = k ε = 0 guarantee conformal invariance i i · i Eric D’Hoker Lecture I on Superstring Amplitudes Diﬀ(Σ) ⋉ Weyl(Σ) and Moduli space

Fix topology of Σ • – Diff(Σ) re-parametrizes ξm on Σ by vector field δξm = δvm δg = δv + δv − mn ∇m n ∇n m – Weyl (Σ) δgmn = 2δσgmn with δσ an arbitrary real function of Σ Orbits of Diff(Σ) ⋉ Weyl(Σ) acting on the space Met(Σ) • Met(Σ)

Met(Σ)/Diﬀ(Σ) ⋉ Weyl(Σ)= Mh

Moduli space of compact Riemann surfaces of genus h (no boundaries) • Mh = space of conformal structures (= space of complex structures) 0 h = 0 dimC h =  1 h = 1 M  3h 3 h 2 − ≥  Eric D’Hoker Lecture I on Superstring Amplitudes Some trivial moduli spaces Given an inﬁnitesimal δg can one solve for δσ and δv ? • mn m δg = 2δσg + δv + δv mn mn ∇m n ∇n m mn m – Eliminate the trace part by choosing δσ = g δgmn + mδv – In local complex coordinates (z, z˜), remaining eqs for traceless∇ part δg = v δg = v zz ∇z z z˜z˜ ∇z˜ z˜

– Integrability automatic since z and z˜ act on different functions locally, or in any simply connected∇ set,∇ you can always solve ⇒ The sphere S2 has no moduli (compact) • – Its stereographic projection onto C admits a globally conformally flat metric dz 2 ds2 = | | (1 + z 2)2 | | The Poincaréupper half plane has no moduli (non-compact) • H dz 2 ds2 = | | Im z > 0 (Im z)2 Eric D’Hoker Lecture I on Superstring Amplitudes Moduli deformations of the torus The torus may be viewed as the product of two circles A and B • – The ratio of their lengths and relative angle provide two real moduli – equivalently represented by parallelogram in C with sides pairwise identified

τ τ + 1 / • B •

B = = Σ ≡ A Σ / A 0• • 1

– The complex number τ contains the information of relative lengths and angle Constant metric deformations equivalently provide a complex modulus • – translation invariance on the circles induces translation invariance on the torus – by translation invariance, metric is constant on Σ – constant trace-part of δgmn eliminated by constant σ – but constant δgzz = ∂zvz has no periodic solutions vz constant δg provides the deformation of the complex modulus of the torus. ⇒ zz Eric D’Hoker Lecture I on Superstring Amplitudes Moduli space of the torus

Oriented Riemann surfaces: cycles A and B ordered • – equivalently choose orientation τ = τ C, Im(τ) > 0 ∈ H1 { ∈ }

Space of inequivalent tori = space of inequivalent lattices Λτ = Z τZ • – but diﬀerent values of τ may give the same lattice ⊕ ′ ω ′ ω1 = a ω1 + b ω2 1 ω1 ′ ω2 = c ω1 + d ω2 • •

τ = ω1/ω2 ≡ ′ τ =(aτ + b)/(cτ + d) ′ 0• ω•2 0• •ω2

– identical lattices requires Λτ ′ Λτ and Λτ Λτ ′ – so that a,b,c,d Z and ad ⊂bc = 1 and ⊂a b SL(2, Z) ∈ − c d ∈ – generated by τ τ + 1 and τ τ −1 → →−

Moduli space of tori = space of inequivalent lattices = 1/SL(2, Z) • – standard fundamental domain H 1 1/SL(2, Z) τ 1, τ 1, Re(τ) H ≡ n ∈ H | |≥ | |≤ 2o Eric D’Hoker Lecture I on Superstring Amplitudes Decomposing the measure Dg

At any point g Met(Σ) the measure Dg factors • ∈ Dg = Z Dσ Dv dµ g × × × Mh Jacobian Weyl Diﬀ 0 Mh

– infinitesimal Weyl δgmn = δσgmn – infinitesimal Diff δg = δv + δv 0 mn ∇m n ∇n m – infinitesimal moduli deformations δgmn Goal • – compute Zg – formulate Zg in terms of ghosts – omit volume factors DσDv of the group Diff+(Σ)⋉Weyl(Σ)

To decompose Dg we study tensor spaces (alias line bundles) on Σ • Eric D’Hoker Lecture I on Superstring Amplitudes Tensor Spaces - Line Bundles on Σ A one-form φ = φ dz + φ dz˜ on Σ decomposes into K K¯ • z z˜ ⊕ K = φ dz is the (space of sections of the) canonical bundle on Σ { z } for m Z define Km = φ dzm and K¯ m = φ dz˜m K−m ∈ { z···z } { z˜···z˜ }≈ L2 inner product for φ ,φ Km • 1 2 ∈ −m ∗ (φ1,φ2)= dzdz˜ √g (gzz˜) φ1φ2 ZΣ The spaces Km and Kn with m = n are mutually orthogonal 6 m (m) (m) Covariant derivative on φ K decomposes φ = z φ + φ • ∈ ∇ ∇ ∇z¯ (m) m m+1 (m) † z z : K K mutual adjoint operators ( z ) = ∇ → ∇ −∇(m+1) z : Km Km−1 with (m) = g z ∇(m) → ∇z˜ zz˜∇(m) Riemann-Roch and Vanishing Theorems • (m) (1−m) dimC Ker dimC Ker = (2m 1)(h 1) ∇z˜ − ∇z˜ − − (m) Ker = 0 for h 2 and m 1 (no holóvector fields for h ≥ 2) ∇z˜ ≥ ≤− (m) Ker = 0 for h = 0 and m 1 (no holóforms on the sphere) ∇z˜ ≥ Eric D’Hoker Lecture I on Superstring Amplitudes Decomposing the tangent space to Met(Σ) Orthogonal decomposition of T (Met(Σ)) • g z˜ z Tg(Met(Σ)) = δσgzz˜ δgzz = gzz˜ δηz δgz˜z˜ = gzz˜ δηz˜ { } ⊕ ⊕ { } δσ K0 δη z˜ K K¯ −1 δη z K¯ K−1 ∈ z ∈ ⊗ z˜ ∈ ⊗ z˜ (1) z˜ Diff acts by δη = z δv • 0 z ∇ (1) −1 – For h 1, the range of the operator z is NOT all of K K¯ ≥ ∇ (1) ⊗ – The orthogonal complement of the range of z is given by ∇ (1) (1) † ¯ −1 2 Range z Ker z = K K K ∇ ⊕ ∇ ⊗ ≈ j (2) (1) † Holomorphic quadratic differentials φ Ker z˜ Ker z • ∈ ∇ ≈ ∇ – Hence we may identify Ker (2) = T ∗ ( ) ∇z¯ (1,0) Mh – One-forms δmj T ∗ ( ) given by linear forms on K¯ K−1 ∈ (1,0) Mh ⊗ j j z j δm =(δη,φ )= dzdz˜ δηz˜ φzz ZΣ (1) – Weyl-invariant pairing and vanishes on δη Range z ∈ ∇ – Riemann-Roch and Vanishing give dimC = 3h 3 for h 2 Mh − ≥ Eric D’Hoker Lecture I on Superstring Amplitudes Decomposing the measure Dg (cont’d) Parametrize by a slice in Met(Σ) transverse to Weyl ⋉ Diff • Mh 0 Met(Σ)

orbits of Diﬀ(Σ) ⋉ Weyl(Σ) mj, m˜ j local coordinates on g(mj, m˜ j) Mh

Carry out a change of integration variables • z¯ z Tg(Met(Σ)) = δσgzz¯ δηz δηz¯ { }⊕ ⊕ { } – Orthogonality implies that the measure factorizes Dg = DσDηDη¯ – The change of variables is given by (repeated indices j are summed) ∂g δη z = (−1)δvz +(µ ) z˜δmj (µ ) z = gzz¯ z˜z˜ z˜ ∇z˜ j z j z˜ ∂mj ∂g δη z˜ = (1)δvz˜ + (˜µ ) z˜δm˜ j (˜µ ) z˜ = gzz¯ zz z ∇z j z j z ∂m˜ j Eric D’Hoker Lecture I on Superstring Amplitudes Ghosts

Use standard rules to introduce ghosts for the determinant • – gauge transformations (δvz, δvz˜) (cz, c˜z˜) Grassmann-odd ghosts z˜ z → – conjugate (δηz , δηz˜ ) (bzz, ˜bz˜z˜) Grassmann-odd anti-ghosts – extended ghost action →

d2z b ∂ cz + µ δmj + ˜b ∂ c˜z˜ +µ ˜ δm˜ j Z zz z˜ j z˜z˜ z j Σ h i – Here δmj,δm˜ j are diﬀerential one-forms which are Grassmann odd

Integrating out δmj,δm˜ j gives the standard ghost representation • D(xµ,b, ˜b,c, c˜) e−IG−Igh δ( b,µ )δ( ˜b, µ˜ )dmjdm˜ j Z V1 ···VN h ji h ji Yj

– where Igh is the standard ghost action

2 z z˜ Igh = d z bzz∂z˜c + ˜bz˜z˜∂zc˜ ZΣ h i – gauge ﬁxed formulation has BRST invariance – for the sphere and the torus, quotient out by conformal automorphisms Eric D’Hoker Lecture I on Superstring Amplitudes Bosonic string has tachyon and no fermions: unphysical Warm-up : tree-level tachyon scattering amplitude • – tachyon vertex operator (k )= d2z g(z ): eiki·x(zi): V i Σ i i – scalar Green function on the sphereR withp metric |dz|2/(1 + |z|2)2 z w 2 xµ(z)xν(w) = ηµνG(z, w) G(z, w)= ln | − | h i − (1 + z 2)(1 + w 2) | | | | Sphere has no moduli, ghost and scalar partition functions are constant • N N N ′ d2z g(z ): eiki·x(zi): = d2z z z α ki·kj h i i i i | i − j| iY=1 p iY=1 i

Tree-level closed string amplitudes are bilinears in open string amplitudes • – Closed string amplitudes on the sphere, vertex operators in interior – Open string amplitude on upper half plane, vertex operators on boundary – Consider open and closed string 4-tachyon amplitudes

1 (0) k1·k2 k2·k3 (0) 2 2k1·k2 2k2·k3 Aopen(s, t) = dξ|ξ| |1 − ξ| Aclosed(s, t, u) = d z|z| |1 − z| Z0 ZS2 – Parametrize z = α + iβ then z-integrand is analytic function of β with branch points at β = iα and β = i(1 α) – Deform β-contour from real to± imaginary axis,± but− pick up phases 1 ∞ 2 2k1·k2 2k2·k3 k1·k2 k2·k3 k1·k2 k2·k3 d z|z| |1 − z| = sin(πk2 · k3) dξ|ξ| |1 − ξ| dη|η| |1 − η| ZS2 Z0 Z1 – Converting the second integral back to , we obtain the KLT relation Aopen (0) (s,t,u) = sin(πk k ) (0) (s, t) (0) (t, u) Aclosed 2 · 3 Aopen Aopen – Does the worldsheet secretly have a Minkowski signature structure ? – No generalization known to loop level 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 - 2018 Amplitudes 2018 Summer School Eric D’Hoker Lecture II on Superstring Amplitudes Superstring Perturbation Theory

Theory of ﬂuctuating random surfaces (closed strings shown) • 2h−2 – governed by topological expansion in the genus h weighed by gs

g−2 + g0 + g2 + s s s ···

Bosonic string • – unstable with closed string tachyon – Nature has fermions !

Superstrings generalize bosonic string • – they have fermions – no tachyon – supersymmetry Eric D’Hoker Lecture II on Superstring Amplitudes 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 E , Spin(32/Z ) 8 × 8 2 Here: RNS formulation, closed orientable superstrings, dimension 10 • Eric D’Hoker Lecture II on Superstring Amplitudes Genus-zero four-graviton superstring amplitude

Kinematics of the four-graviton amplitude • µ µ – momenta of gravitons ki are conserved i ki = 0 µν µ ν – choose basis of factorized polarization tensorsP εi = εi ε˜i 2 µ µ µ µ – masslessness ki = 0 and transversality ki εi = ki ε˜i = 0 for i = 1, 2, 3, 4 – kinematic invariants s = s12 = s34, t = s14 = s23, u = s13 = s24 s = α′(k + k )2/4 ij − i j Tree-level four-graviton amplitude is given by • (0) 1 ˜ 1 Γ(1 s)Γ(1 t)Γ(1 u) (εi, ε˜i,ki)= 2 − − − A gs × KK× stu Γ(1 + s)Γ(1 + t)Γ(1 + u) – Kinematical factor given in terms of f µν = kµεν kνεµ by K i i i − i i K = (f1f2)(f3f4) + (f1f3)(f2f4) + (f1f4)(f2f3)

−4(f1f2f3f4) − 4(f1f2f4f3) − 4(f1f3f2f4)

– for ˜ replace εi by ε˜i – Equivalently,K ˜ = 4 with the linearized Weyl tensor – String duality:K× symmetricK R in s,t,uR – Poles in each channel, at s,t,u = 0, 1, 2, ··· Eric D’Hoker Lecture II on Superstring Amplitudes Genus-one four-graviton superstring amplitude

Type II four-graviton amplitude to one-loop order (Green, Schwarz 1982) • d2τ (1)(ε , ε˜ ,k )= 4 (1)(s τ) i i i Z 2 ij A R M1 (Im τ) B | – Partial amplitude (1) is a modular function in τ = /SL(2, Z) B ∈M1 H1 4 2 (1) d zi (sij τ)= exp sij G(zi zj τ) B | Z 4 Im τ − | Σ iY=1 Xi

Singularity structure • – For ﬁxed τ integrations over Σ produce poles in (1) at positive integers s . B ij – The integral over τ converges absolutely only for Re(sij) = 0. – Analytic continuation to sij C via decomposition of 1. – Branch cuts in s starting at∈ integers 0 are producedM by τ i region. ij ≥ → ∞ Eric D’Hoker Lecture II on Superstring Amplitudes 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,

2 2 d τ (1) 10 2 (sij τ)= d p (zi,ki,p τ) Z (Im τ) B | ZR10 Z Z 4 F | M1 M1 Σ

Chiral amplitude is locally holomorphic in τ and z • F i 4 2 iπτp +2πip kizi −sij (zi,ki,p τ)= e i ϑ1(zi zj τ) dτ dzi F | P − | i

(z + δ A,k ,p τ) = e2πikℓ·p (z ,k ,p τ) F i i,ℓ i | F i i | (z + δ B,k ,p τ) = (z ,k ,p + k τ) F i i,ℓ i | F i i ℓ| – Modular invariance of (1) guarantees independence of choices. – Hermitian pairing of Aand ¯ is familiar from 2-d CFT where loop momentum pFlabelsF conformal blocks of 10 copies of c = 1. Eric D’Hoker Lecture II on Superstring Amplitudes RNS formulation of superstrings

M = R10 ﬂat Minkowski space-time with Lorentz group SO(1, 9) • – xµ scalars on worldsheet Σ, map Σ into M – ψµ 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 • – ψµ and ψ˜µ are independent Majorana-Weyl spinors of opposite chirality

With Euclidean signature Σ • – ψµ and ψ˜µ must be independent complex Weyl spinors – Globally, on a compact Riemann surface of genus h, ⋆ All ψµ are sections of a the same spin bundle S (and ψ˜µ of S˜) ⋆ 22h distinct spin structures for S (and 22h independently for S˜)

GSO projection requires independent summation over spin structures • Eric D’Hoker Lecture II on Superstring Amplitudes Quantization of worldsheet spinor ﬁelds

Illustrate • – Ramond and Neveu-Schwarz sectors – independence of chiralities

Dirac action and equation for ﬂat M = R10 with metric η • µ – All components of ψ+ are sections of the same spin bundle S – Complex structure J with local complex coordinates (z, z˜) – Dirac action,

1 µ ν Iψ[ψ, J]= dzdz˜ ψ+∂z˜ψ+ηµν 2π ZΣ µ – Dirac equation ∂z˜ψ+ = 0 has locally holomorphic solutions, – but products of operators produce singularities ηµν ψµ (z)ψν (w)= + regular + + z w µ − 1 – each component ψ generates a CFT with central charge c = 2. Eric D’Hoker Lecture II on Superstring Amplitudes 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 = ew 1 1 – one-forms related by dz = ew dw, spinors by (dz)2 = ew/2 (dw)2 w/2 – fields related by conformal transformation ψcyl(z)= e ψann(w) Two possible spin structures • NS ψµ (τ, σ + 2π)= ψµ (τ, σ) or ψµ (e2πi z) =+ ψµ (z) cyl − cyl ann ann R ψµ (τ, σ + 2π)=+ ψµ (τ, σ) or ψµ (e2πi z) = ψµ (z) cyl cyl ann − ann Free field quantization in annulus representation • µ µ −1−r µ ν µν NS ψ (z)= 1 Z br z 2 br ,bs = η δr+s,0 r∈2+ { } P 1 R ψµ(z)= dµ z−2−n dµ ,dν = ηµνδ n∈Z n { m n} m+n,0 P Eric D’Hoker Lecture II on Superstring Amplitudes Quantization of worldsheet spinor fields (cont’d)

Lorentz generators of SO(1, 9) : [J µν, ψκ(z)] = ηνκψµ(z) ηµκψν(z) • − J µν = bµ bν bν bµ NS −r r − −r r XN 1 r∈ −2 1 J µν = [dµ,dν]+ dµ dν dν dµ R 2 0 0 −n n − −n n nX∈N Fock space construction produces two sectors • µ ⋆ NS ground state deﬁned by br 0; NS = 0 for all r> 0 – 0; NS is unique and in trivial| representationi of SO(1, 9) –| Fock spacei = linear combinations of bµ1 bµp 0; NS , r > 0 −r1 ··· −rp| i i – All states in tensor reps of SO(1, 9) are space-time bosons.

µ ⋆ R ground state deﬁned by dn 0, α;R = 0 for all n> 0 – 0, α;R is degenerate and| in spinori rep. of SO(1, 9), states labelled by α –| Fock spacei = linear combinations of dµ1 dµp 0, α;R , n > 0 −n1 ··· −np| i i – All states in spinor reps of SO(1, 9) are space-time fermions. Eric D’Hoker Lecture II on Superstring Amplitudes 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

µ Type II spin structures of ψ± 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 ⇒ Eric D’Hoker Lecture II on Superstring Amplitudes Summation over spin structures (cont’d)

Fix a canonical homology basis of cycles AI, BI of H1(Σ, Z) I = 1, ··· ,h • – with canonical intersection pairing #(AI, AJ)=#(BI, BJ ) = 0 and #(AI, AJ)= δIJ

A2 A1 Σ B1 B2

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, Z2) Eric D’Hoker Lecture II on Superstring Amplitudes 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 Mh = {J}/Diff(Σ) of genus h compact Riemann surfaces Complex super manifold (locally C1|1 with coordinates z θ) • | – holótransition functions z|θ → z′(z,θ)|θ′(z,θ) generate N = 2 super conformal Super Riemann surface (locally C1|1 with coordinates z θ) • ′ ′ | – holótransition functions z θ z θ rescale Dθ = ∂θ + θ∂z – Transition functions define| →= 1| superconformal structure – Globally: T Σ has a completelyN non-integrable subbundle of rankJ 0 1 | Moduli space of compact super Riemann surfaces: Mh = /Diff(Σ) • = equivalence classes of superconformal structures {J } J 0|0 h = 0 dimC Mh =  1|0 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. Eric D’Hoker Lecture II on Superstring Amplitudes Superstring worldsheets and moduli spaces

Heterotic • i –Left : RS ΣL, moduli space L coord resp. z˜ and m˜ M i α – Right : SRS ΣR, moduli space MR coord resp. (z,θ) and (m , ζ ) – Worldsheet is a cycle Σ ΣL ΣR of dim 1 1 ⊂ × ∗ | subject to Σred = diag(ΣL red × ΣR red) : z˜ = z + nilpotent – Moduli space is a cycle Γ L MR of dim 3h 3 2h 2 for h 2 ⊂M × i ∗ −i | − ≥ subject to Γred = diag(ML red × MR red) : (m ˜ ) = m + nilpotent (reduced space obtained by setting all nilpotent variables to zero)

Type II • i α –Left : SRS ΣL, moduli space ML coord resp. (˜z, θ˜) and (m ˜ , ζ˜ ) i α – Right : SRS ΣR, moduli space MR coord resp. (z,θ) and (m , ζ ) – Worldsheet is a cycle Σ ΣL ΣR of dim 1 2 – Moduli space is cycle Γ ⊂ M × M of dim |3h 3 4h 4 for h 2 ⊂ L × R − | − ≥ subject to z˜∗ = z + nilpotent and (m ˜ i)∗ = mi + 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) Eric D’Hoker Lecture II on Superstring Amplitudes Worldsheet action for Type II superstrings Worldsheet is Σ Σ Σ • ⊂ L × R – ΣL has superconformal structure ˜ with local coordinates z˜ θ˜ – Σ has superconformal structure J with local coordinates z|θ R J | Superconformal invariant matter action • – worldsheet matter ﬁeld Xµ(˜z, z θ,θ˜ )= xµ(˜z, z)+ θψµ(˜z, z)+ θ˜ψ˜µ(˜z, z)+ θθF˜ µ(˜z, z) |

– Worldsheet action in local coordinates (Dθ = ∂θ + θ∂z) µ ˜ ˜ ˜ µ Im[X , , ]= [dzdz˜ dθdθ]Dθ˜X DθXµ J J ZΣ | – Superconformal algebra on ﬁelds generated by 1 1 1 = S + θT S = ψµ∂ x T = ∂ xµ∂ x + ψµ∂ ψ Szθ zθ zz zθ 2 z µ zz −2 z z µ 2 z µ 1 µ 1 µ 1 µ ˜ ˜ = S˜ ˜ + θ˜T˜ S˜ ˜ = ψ˜ ∂ x T˜ = ∂ x ∂ x + ψ˜ ∂ ψ˜ Sz˜θ z˜θ z˜z˜ z˜θ 2 z˜ µ z˜z˜ −2 z˜ z˜ µ 2 z˜ µ Eric D’Hoker Lecture II on Superstring Amplitudes Deformations of superconformal structures

Under deformation of ˜ for Σ and for Σ • J L J R ˜ z ˜ z˜ ˜ δI = [dzdz˜ dθdθ] Hθ˜ zθ + Hθ z˜θ˜ ZΣ | S S – in components by integrating out θ,θ˜ ,

˜ δI = dzdz˜ µ z T + χ θ S +µ ˜ z˜ T +χ ˜ θ S˜ Z z˜ zz z˜ zθ z z˜z˜ z z˜θ˜ Σred – recover Beltrami diﬀerentials µ, µ˜ and worldsheet gravitino ﬁelds χ, χ˜

z ˜ z θ ˜ z˜ z˜ ˜ θ˜ Hθ˜ = θ(µz¯ + θχz˜ ) Hθ = θ(˜µz + θχ˜z ) – Finite deformations of the metric with µ˜ =µ ¯ and χ˜ =χ ¯ integrate to the standard 2-dim = 1 supergravity action N (Brink, Di Vecchia, Howe; Deser, Zumino 1976)

Type II superstring perturbation theory requires µ˜ =µ ¯ and χ˜ =χ ¯ • 6 6 Eric D’Hoker Lecture II on Superstring Amplitudes Type II string amplitude

z˜ z Parametrize deformations H˜ , H˜ by slice ˜(m˜ ), (m) in M M • θ θ {J J } L × R z ˜ z mA z mA mA i α Hθ˜ = Dθ˜V + HAδ HA = ∂Jθ˜ /∂ = (m ,ζ ) ˜ z˜ ˜ z˜ ˜ mA˜ ˜ z˜ mA˜ mA i ˜α Hθ = DθV + HA˜δ ˜ HA˜ = ∂Jθ /∂ ˜ ˜ = (m ˜ , ζ )

z z z θ z ghost ﬁelds V → C = c + θγ Hθ˜ → Bzθ = βzθ + θbzz z˜ ˜z˜ z˜ ˜ θ˜ z˜ ˜ ˜ ˜˜ V → C =c ˜ + θγ˜ Hθ → Bz˜θ˜ = βz˜θ˜ + θbz˜z˜ – Super conformal invariant ghost action ˜ ˜ z ˜ z˜ mA ˜ ˜ mA˜ Igh = [dzdz˜ |dθdθ] BzθDθ˜C + Bz˜θ˜DθC + BzθHAδ + Bz˜θ˜HA˜δ ˜ ZΣ The integrand for the full amplitude is given by • ˜ A A −Im−Igh D(XBBC˜ C˜) V1 ···Vn [dm˜ dm ] δ(hB,˜ H˜ ˜i)δ(hB,HAi) e Z A YA,A˜ – are BRST-invariant vertex operators. V1 ···Vn – 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) Eric D’Hoker Lecture II on Superstring Amplitudes Loop momenta and Chiral amplitudes

h independent loop momenta pµ defined to flow across A cycles • I I µ µ pI = dz∂zx A I I Chiral Amplitudes (ED, Phong 1988) • µµ˜ µ µ˜ – Massless NS bosons with factorized polarization tensor εi = εi ε˜i – Chiral amplitude at fixed loop momenta is given by

µ µ p B dz∂zx H zS A F (J ,ε ,k ,p ) = V ···V e I I e Σ θ˜ zθ δ(hB,H i)dm R i i I 1 N H R A YA – Correlation functions computed with chiral Green functions h···i Full Superstring Amplitudes • – obtained by pairing left and right and integrating over Γ M M ∈ L × R (h) µ ˜ µ µ (εi, ε˜i,ki)= dpI L( , ε˜i,ki,pI ) R( , εi,ki,pI ) A ZR10 ZΓ F J F J

– integration over vertex operator insertion points included in integration over Γ – cfr “double copy construction” in supergravity calculations Eric D’Hoker Lecture II on Superstring Amplitudes Parametrization of super moduli

Superconformal structure Mh specified by transition functions • J ∈ θ – Concrete calculations use parametrization by gravitino field χz˜ 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 + µdz˜|2 z – realized in CFT by inserting Σ dzdz˜ µz˜ Tzz to all orders in µ R Local parametrization of supermoduli (in superconformal-invariant theory) • – Start with Σred with complex structure given by J Mred – Deform super conformal structure by inserting T and∈ S

z θ dzdz˜ µz˜ Tzz + χz˜ Szθ ZΣred

– χ and µ parametrized by local odd coordinates on Mh For h = 2, even spin structures, hol´oprojection M exists • 2 →M2 – via the super period matrix (ED, Phong 2001) For h 5 no hol´oprojection M exists (Donagi, Witten 2013) • ≥ h →Mh Eric D’Hoker Lecture II on Superstring Amplitudes

The super period matrix (even spin structures)

Start from conformal structure J for Σ with hol´o1-forms ω • red I ωJ = δIJ ωJ = ΩIJ I, J = 1, 2 A B I I I I Deform to superconformal structure on Σ with superhol´oforms ωˆ • J I ωˆJ = δIJ ωˆJ = Ωˆ IJ I, J = 1, 2 A B I I I I – Explicit formula for the super period matrix Ωˆ for even spin structure δ i Ωˆ IJ = ΩIJ ωI(z)χ(z)Sδ(z, w Ω)χ(w)ωJ(w)+ µ ωIωJ − 8π Z 2 | Z Σred Σred

– Ωˆ IJ is locally supersymmetric; Ωˆ IJ = Ωˆ JI; and Im Ωˆ > 0 – Every Ωˆ corresponds to an ordinary Riemann surface – Szeg¨okernel S (z, w Ω) is non-singular in the interior of δ | M2 Projection using Ωˆ is holomorphic and natural for genus 2 ⇒ Eric D’Hoker Lecture II on Superstring Amplitudes Projecting and pairing Chiral Amplitudes

Chiral Amplitudes on M2 • α – Natural parametrization of M2 by (Ωˆ IJ , ζ ) (even spin structure δ) – involves measure dκ[δ](Ωˆ, ζ) and correlation functions [δ](ε ,k ,p Ωˆ, ζ) C i i I|

Projection to chiral amplitudes on 2 • – by integrating over ζ and summingM over δ at ﬁxed Ωˆ

R(εi,ki,pI|Ω)ˆ = dκ[δ](Ωˆ,ζ) C[δ](εi,ki,pI|Ωˆ,ζ) Z Xδ ζ

L(˜εi,ki,pI|Ω)ˆ = dκ[δ˜](Ωˆ, ζ˜) C[δ˜](˜εi,ki,pI|Ωˆ, ζ˜) Zζ˜ Xδ˜ – for heterotic, is chiral half of bosonic string, has no integral in ζ˜ – phase factorsL determined by Sp(4, Z) modular invariance

Pairing left and right chiral amplitudes, integrating over p and Ωˆ • I (2)(ε , ε˜ ,k )= dΩˆ dpµ (ε ,k ,p Ω)ˆ (˜ε ,k ,p Ω)ˆ i i i Z Z I i i I i i I A M2 R | L |

– Integral over pI is Gaussian and can be carried out explicitly. Eric D’Hoker Lecture II on Superstring Amplitudes Genus two

Siegel Upper half space • S2 = Ω = Ω C with I, J = 1, 2 and Y = ImΩ > 0 S2 { IJ JI ∈ } – Sp(4, R) acts by Ω (AΩ+ B)(CΩ+ D)−1 → AB 0 −I M tJM = J M = J = C D I 0 – has Sp(4, R)-invariant metric ds2 and volume form dµ S2 2 2 2 −1 ¯ −1 ds2 = YIJ dΩJK YKL dΩLI I,J,K,LX=1,2 Compact Riemann surfaces Σ • – Choose canonical homology basis of AI, BI cycles for H1(Σ, Z). – ωI dual holomorphic (1,0) forms,

ωJ = δIJ ωJ = ΩIJ A B I I I I – Riemann relations imply Ω 2; – Modular group Sp(4, Z); moduli∈ S space = /Sp(4, Z). M2 S2 Eric D’Hoker Lecture II on Superstring Amplitudes Genus-two Type II four-graviton amplitude

Type II four-graviton amplitude (ED, Phong 2001 – 2005) • (2)(ε , ε˜ ,k ) = g2 ˜ dµ (2)(s Ω) i i i s Z 2 ij A KK M2 B | ¯ (2) (sij Ω) = Y∧ Y 2 exp sij G(zi, zj Ω) B | Z 4 (detImΩ) | Σ Xi

– G(zi, zj) is the genus-two scalar Green function; – ∆(zi, zj) is a bi-holomorphic form independent of s,t,u.

∆(z, w) = ω1(z) ∧ ω2(w) − ω2(z) ∧ ω1(w)

Y = (t − u)∆(z1,z2) ∧ ∆(z3,z4) + (s − t)∆(z1,z3) ∧ ∆(z4,z2)

+(u − s)∆(z1,z4) ∧ ∆(z2,z3) – reproduced (with fermions) in pure spinor formulation (Berkovits, Mafra 2005)

Singularity structure • – For ﬁxed Ω integrations over Σ produce poles in at positive integers s . B ij – The integral over Ω requires analytic continuation beyond Re(sij) = 0. – Branch cuts in s starting at integers produced from Ω , Ω i ij 11 22 → ∞ Eric D’Hoker Lecture II on Superstring Amplitudes Genus-two Heterotic four-graviton amplitude

Heterotic four NS boson amplitude at genus 2 (ED, Phong 2005) • (2)(ε , ε˜ ,k ) = g2 dµ (2)(˜ε ,k Ω) O i i i s Z 2 O i i A K M2 B |

(2) O(˜εi,ki) O (˜εi,ki Ω) = Y∧ W exp sij G(zi, zj) B | Z 4 2 Σ (det ImΩ) Ψ10(Ω) Xi

– Ψ10(Ω) is the Igusa cusp form. Dependence of the operator on the channel: • ⋆ 4 gravitons 4 O ⋆ 2 gravitonsR + 2 gauge bosons 2tr( 2); ⋆ 4 gauge bosons (tr 2)2 R F ⋆ 4 gauge bosons tr(F 4) – For example, F 4 ¯ iki·x˜(zi) h i=1 ε˜i · ∂x˜(zi) e i WR4(˜εi,ki) = 4 Q iki·x˜(zi) h i=1 e i – Gauge parts are obtained by the correlatorsQ of the current (0, 1)-forms. Eric D’Hoker Lecture II on Superstring Amplitudes UV-ﬁniteness and one-loop amplitudes

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)

All chiral amplitudes have a universal loop momentum factor • iπpµΩ pµ (z , ε ,k ,p Ω) = e I IJ J FR i i i I| ×··· – Modular invariance allows one to choose a fundamental domain where Im(Ω) bounded from below – For genus one, choose the standard fundamental domain 1 1/SL(2, Z)= τ C, Im(τ) > 0, τ 1, Re(τ) H ∈ | |≥ | |≤ 2 – Analogous, more complicated, choices to higher genus

Uniform Gaussian suppression at large loop momenta ⇒ UV finiteness to all genera ⇒ Eric D’Hoker Lecture II on Superstring Amplitudes Singularities in the projection M 2 → M2 Projection M2 2 is holó, but integration extends to boundary • – are there singularities→M in the projection M ? 2 → M2 τ u u → 0 separating node Ω = u σ σ → i∞ non-separating node – Key ingredient in Ωˆ is the Szegökernel ϑ[δ](z − w|Ω) Sδ(z, w|Ω) = ϑ[δ](0|Ω)E(z, w)

– As u → 0 we have ϑ[δ](0|Ω) → ϑ[δ1](0|τ) ϑ[δ2](0|τ)

– Even δ = [δ1,δ2] with δ1,δ2 odd produces a singularity in Sδ and Ωˆ

Physical eﬀects • – singularity killed by ψ-zero modes in R10 (space-time susy) – contribution when susy is broken by radiative corrections (Witten 2013) – Two-loop vacuum energy in Heterotic strings on CY orbifold C3/Z Z 2 × 2 ⋆ is zero for E8 E8 E6 E8 with unbroken susy ⋆ non-zero for Spin(32)×/Z → SO(26)× U(1) with broken susy 2 → × (Atick, Sen 1988; ···; ED, Phong 2013; Berkovits, Witten 2014) Eric D’Hoker Lecture II on Superstring Amplitudes Singularities in the projection M 3 →M3 Some basic structure theorems • – A hyper-elliptic surface is a branched double cover of the sphere S2; – 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 M3 (referred to as the hyper-elliptic divisor)

The genus-three period matrix (for even spin structure) • i Ωˆ = Ω ω (z)χ(z)S (z, w Ω)χ(w)ω (w)+ (χ4) IJ IJ − 8π ZZ I δ | J O

– For Ω on the hyper-elliptic divisor of M3 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 Ωˆ (Witten 2015) Lectures on Superstring Amplitudes

Part 3: Low energy eﬀective interactions

Eric D’Hoker Mani L. Bhaumik Institute for Theoretical Physics University of California, Los Angeles

Center for Quantum Mathematics and Physics - 2018 Amplitudes 2018 Summer School Eric D’Hoker Lecture III on Superstring Amplitudes Superstring Perturbation Theory and Supergravity

genus 0 genus 1 √α′ energy genus 2

D6R4 D4R4 R4 R gs

Superstring perturbation theory in powers of the string coupling g • s – holds for weak coupling gs – and for all energies Classical supergravity “ ” • – leading low energy expansionR of string theory – holds for all couplings gs String induced eﬀective interactions 4,D4 4,D6 4 • – Evaluated in perturbation theory forRg 1R R s ≪ Eric D’Hoker Lecture III on Superstring Amplitudes Low energy expansion of tree-level amplitudes

Closed superstring tree-level four-graviton amplitude • 1 R4 Γ(1 − s)Γ(1 − t)Γ(1 − u) α′ A(0)(ε , ε˜ ,k ) = s = − (k + k )2 i i i 2 ij i j gs stu Γ(1 + s)Γ(1 + t)Γ(1 + u) 4 – symbolically stands for the Weyl tensor – R4 symbolically stands for a scalar contraction dictated by supersymmetry R

At low energy sij 1 • – massless string| | exchanges ≪ produce non-local contributions; – massive string exchanges produce local eﬀective interactions – string-induced corrections to supergravity; eg. in Type II 1 1 + 2ζ(3) + ζ(5)(s2 + t2 + u2) + 2ζ(3)2stu + ζ(7)(s2 + t2 + u2)2 + stu 2 ··· massless 4 D4 4 D6 4 D8 4 R R R R – D2k 4 contraction of covariant derivatives D and 4 R R Eric D’Hoker Lecture III on Superstring Amplitudes Eﬀective interactions from Type IIB superstrings

SL(2, Z)-duality symmetry of Type IIB superstrings • – requires effective interactions to be SL(2, Z)-invariant; 4 Z – Einstein frame metric GE and E invariant under SL(2, ) – combine axion χ dilaton Φ in ρR= χ + ie−Φ – transforms by Möbius transformations under SL(2, Z) aρ + b ρ a,b,c,d, Z, ad bc = 1 → cρ + d ∈ − R NS – Flux fields F3 ,F3 transform linearly; F5 is invariant Effective interactions from four-graviton amplitude in Type IIB • G (ρ) 4 + (ρ)D4 4 + (ρ)D6 4 + (ρ)D8 4 + Z E 0 E 4 E E 6 E E 8 E E p E R E R E R E R ··· – For each p the real-valued function (ρ) is SL(2, Z)-invariant Ep aρ + b = (ρ) Ep cρ + d Ep – namely it is a real-analytic modular function (not to be confused with meromorphic modular functions) Eric D’Hoker Lecture III on Superstring Amplitudes Real-analytic Eisenstein series

A famous family of real-analytic modular functions •

– For Re(s) > 1 one deﬁnes Es by Kronecker-Eisenstein sums ′ ρs E (ρ)= 2 ρ = ρ + iρ , ρ , ρ R s πs m + ρn 2s 1 2 1 2 ∈ m,nX∈Z | | – They are SL(2, Z)-invariant and eigenfunctions of the Laplacian ∆E (ρ)= s(1 s)E ∆ = 4ρ2∂ ∂ s − s 2 ρ ρ¯ – Their asymptotic expansion for ρ = weak string coupling 2 → ∞ ρs 2Γ(s 1)ζ(2s 1) E (ρ) = 2ζ(2s) 2 + − 2 − + (e−2πρ2) s s 1 s−1 π s−2 O Γ(s)π ρ2 Eric D’Hoker Lecture III on Superstring Amplitudes Eﬀective interactions and Eisenstein series

String perturbation theory calculations in string frame • Φ/2 – Convert Einstein metric GEµν to string metric Gµν = e GEµν √ 2k 4 e(k−1)Φ/2 2k 4 GE E2k(ρ)DE RE = GE2k(ρ)D R p – Consider combinations involving Eisenstein series 2 4 −2Φ 4 π 4 GE E3 (ρ)RE ≈ e ζ(3)R + R p 2 3 4 4 4 −2Φ 4 4 2π 2Φ 4 4 GE E5 (ρ)DERE ≈ e ζ(5)D R + e D R p 2 135 2 4 2 6 4 −2Φ 2 6 4 2π 6 4 π −2Φ 6 4 GE E3 (ρ) DERE ≈ e ζ(3) D R + ζ(3)D R + e D R p 2 3 9 6 8 4 −2Φ 8 4 16π −4Φ 8 4 GE E7 (ρ)DERE ≈ e ζ(7)D R + e D R p 2 14175 – Compare with low energy expansion of tree-level 1 1 + 2ζ(3) + ζ(5)(s2 + t2 + u2) + 2ζ(3)2stu − ζ(7)(s2 + t2 + u2)2 + ··· stu 2 R4 D4R4 D6R4 D8R4 Eric D’Hoker Lecture III on Superstring Amplitudes D-instantons, S-duality and supersymmetry

Space-time supersymmetry and S-duality • – D-instantons (Green, Gutperle, Vanhove 1996), space-time susy (Green, Sethi 1997)

0(ρ)= E3 (ρ) E 2 – matches tree-level and genus-one results from string perturbation theory – Vanishing contribution from genus-two (ED, Gutperle, Phong 2005)

M-theory perturbation theory on torus (Green, Kwon, Vanhove 1999; GV 2005) • 4(ρ) = E5 (ρ) E 2 2 (∆ 12) 6(ρ) = E3 (ρ) − E 2 – matches genus two (ED, Gutperle, Phong 2005) E4 – matches genus-two (ED, Green, Pioline, R. Russo 2014) E6 genus three (Gomez, Mafra 2015) Non-renormalization theorems: no perturbative corrections • – for for h 2 E0 ≥ – for 4 for h 3 – for E for h ≥ 4 E6 ≥ Eric D’Hoker Lecture III on Superstring Amplitudes Low energy expansion at genus one

Recall genus-one Type II four-graviton amplitude (M = H /SL(2, Z)) • 1 1 d2τ (1)(ε , ε˜ ,k )= 4 (1)(s τ) i i i Z 2 ij A R M1 (Im τ) B | Expand the partial amplitude (1) for s 1 for ﬁxed τ • B | ij| ≪ 4 2 (1) d zi (sij τ)= exp sij G(zi zj τ) B | Z 4 Im τ − | Σ iY=1 Xi

Graph in the expansion of D2w 4 = Modular Function • R ⇒ D4 4 R • •

D6 4 • R • • • • • • D8 4 • R • • • • • • • • • D10 4 R • • • • • •

• • •

• • • • Eric D’Hoker Lecture III on Superstring Amplitudes Modular graph functions

D4 4 R • •

D6 4 • R • • • • • • D8 4 • R • • • • • • • • • D10 4 R • • • • • •

• • •

• • • • one-loop two-loops three-loops Eric D’Hoker Lecture III on Superstring Amplitudes One-loop : Eisenstein series

One-loop worldsheet Feynman diagram with k bivalent vertices • k 2 ′ k d zi τ2 G(zi zi+1 τ)= k 2s = Ek(τ) Z τ2 − | π m + nτ iY=1 Σ m,nX∈Z | | – Our old friend: non-holomorphic Eisenstein series for integer index k

Recall properties of E (τ) • s – absolutely convergent for Re(s) > 1; analytically continue to s C ∈ – reﬂection relation Γ(s)Es(τ) = Γ(1 s)E1−s(τ) – satisﬁes a Laplace-eigenvalue equation− on H1 2 ∆ s(s 1) Es(τ)=0 ∆=4τ2 ∂τ ∂τ¯ − − a b – modular invariant E (aτ+b)= E (τ) under ∈ SL(2, Z) s cτ+d s c d Eric D’Hoker Lecture III on Superstring Amplitudes Two-loops : modular graph functions

Feynman diagrams evaluate to the modular functions • ′ 3 3 3 ar τ2 Ca1,a2,a3(τ)= δ mr! nr! 2 π mr + nrτ mrX,nr∈Z, Xr=1 Xr=1 rY=1 r=1,2,3 | | 2w 4 – contribute to D with the weight given by w = a1 + a2 + a3 – satisfy (inhomogeneous)R Laplace-eigenvalue equations

w = 3 C1,1,1 = (∆ 0)C1,1,1 = 6E3 • • −

2 w = 4 C2,1,1 = • (∆ 2)C2,1,1 = 9E4 E2 • • − −

w = 5 C3,1,1 = • • (∆ 6)C3,1,1 = 3C2,2,1 + 16E5 4E2E3 • • − −

w = 5 C2,2,1 = • (∆ 0)C2,2,1 = 8E5 •• • − – Note that eigenvalues are of the form s(s 1) for s = 1, 2, 3; − Eric D’Hoker Lecture III on Superstring Amplitudes Structure Theorem

C (τ) are linear combinations of C p(τ) satisfying (ED, Green, Vanhove 2015) • a,b,c w;s; ∆ s(s 1) Cw;s;p = Fw;s;p Es′ − − C F ′ – w;s;p and w;s;p of weight w = a + b + c (with Es′ assigned weight s ); ′ – F p is a polynomial of total degree 2 in E ′ with 2 s w; w;s; s ≤ ≤ w − 1 s − 1 s = w − 2m m = 1, ··· , p = 0, ··· , 2 3 Examples at low weight • w = 3 s = 1 0(1) w = 4 s = 2 2(1) w = 5 s = 1, 3 0(1) ⊕ 6(1) w = 6 s = 2, 4 2(1) ⊕ 12(2) w = 7 s = 1, 3, 5 0(1) ⊕ 6(1) ⊕ 20(2) System of diﬀerential relations to all loop orders (ED, Green, Kaidi, Vanhove 2016) • Relation with polylogarithms & multiple zeta values • (ED, Green, Vanhove 2015; Francis Brown 2017) Eric D’Hoker Lecture III on Superstring Amplitudes Type IIB eﬀective interactions at genus-two

Recall Type II four-graviton amplitude at genus 2, •

(2)(ε ,k ) = 4 dµ (2)(s Ω) i i Z 2 ij A R M2 B |

(2) (sij Ω) = ¯ exp sij G(zi, zj) B | Z 4 Y∧ Y Σ Xi

The ZK-invariant is given as follows • −1 −1 −1 −1 8ϕ(Ω) = YIJ YKL − 2YIL YJK G(x, y)ωI(x)ωJ (x)ωK(y)ωL(y) Z 2 I,J,K,LX Σ – equivalent to deﬁnition via Arakelov geometry (Zhang 2007, Kawazumi 2008)

6 4 Coeﬃcient of genus-two D interaction involves dµ2 ϕ(Ω) M2 • R R – Direct evaluation appeared completely out of reach ... until ... Eric D’Hoker Lecture III on Superstring Amplitudes The Zhang-Kawazumi invariant for genus-two

6 4 Coeﬃcient of genus-two D interaction involves dµ2 ϕ(Ω) M2 • R R – Direct evaluation appeared completely out of reach ... until ...

Theorem (ED, Green, Pioline, R. Russo 2014) • (∆ 5)ϕ = 2πδ − − SN – ∆ is the Laplace-Beltrami operator on with Siegel metric ds2; M2 2 – δSN has support on separating node (into two genus-one surfaces) – The integral over reduces to an integral over ∂ M2 M2 1 2π3 dµ2 ϕ = dµ2 ∆ϕ + 2πδSN = ZM2 5 ZM2 45 – Exact agreement with predictions from S-duality and supersymmetry Eric D’Hoker Lecture III on Superstring Amplitudes Non-analytic contributions at low energy

Non-analytic parts of the amplitudes at low energy • – arise from boundary of moduli space contribution to the integral over – dominant contribution at low energy is from supergravity B – plus string corrections

Look at two-particle unitarity cut in the s-channel •

1 4 1 4

Discs † A = A A 2 3 2 3

i Discs Aε1,ε2,ε3,ε4(p1,p2,p3,p4) = 10 d k 2 2 δ(k )δ((q − k) )Aε ,ε ,ε ,ε (p1,p2, −k,k − q)Aε ,ε ,ε ,ε (k,q − k,p3,p4) Z (2π)10 1 2 r s r s 3 4 Eric D’Hoker Lecture III on Superstring Amplitudes Non-analytic part of the genus-one amplitude

Obtain the genus-one discontinuity from tree-level • – Use the fact that the kinematic factor is the same at all genera h A(h) (p ,p ,p ,p ) = R4 (p ,p ,p ,p )A(h)(s, t, u) ε1,ε2,ε3,ε4 1 2 3 4 ε1,ε2,ε3,ε4 1 2 3 4 red – and satisﬁes the recursive formula (Bern, Dixon, Dunbar, Perelstein, Rozowsky 1998)

4 4 4 4 R (p1,p2, −k,k−q)R (k,q−k,p3,p4) = s R (p1,p2,p3,p4) ε1,ε2,εr,εs εr,εs,ε3,ε4 ε1,ε2,ε3,ε4 εXr,εs – to obtain an eﬀective discontinuity formula 10 (1) d k 2 2 (0) ′ ′ (0) ′′ ′′ i Discs A (s, t, u) = δ(k )δ((q − k) )A (s, t , u )A (s, t , u ) red Z (2π)10 red red ′ 2 ′ 4 ′′ 2 ′′ 2 – where t = −(p1 − k) , u = −(p2 − k) , t = −(p4 − k) , u = −(p3 − k) 1 A(0) (s, t, u) = + 2ζ(3) + ζ(5)(s2 + t2 + u2)+ ··· red stu Substitution into s-channel unitarity relation gives (by power-counting) • (1) 4 6 Discs Ared(s, t, u) = #s + #ζ(3)s + #ζ(5)s + ··· (1) 4 6 Ared(s, t, u) = #s ln(−s) + #ζ(3)s ln(−s) + #ζ(5)s ln(−s)+ ··· Eric D’Hoker Lecture III on Superstring Amplitudes Absence of non-analytic contributions

Discontinuity relation gives non-analytic contributions • – At genus-one use previously obtained result

(1) (s,t,u) = #s ln( s) + #ζ(3)s4 ln( s) + #ζ(5)s6 ln( s)+ Ared − − − ··· – Eﬀective interaction D2 4 vanishes by s + t + u = 0 R – Genus-one 4,D4 4,D6 4,D10 4 eﬀective interactions are completelyR determinedR R by theR analytic part of the amplitude

– Local eﬀective interaction D8 4 can be ﬁxed only after non-analytic partR has been properly normalized Eric D’Hoker Lecture III on Superstring Amplitudes Non-analytic plus analytic parts from genus-one amplitude

Derivation of full genus-one D8 4 from string theory amplitude • R – non-analytic part arises from τ i : partition moduli space → ∞ = = τ , Im(τ) < L M1 M1L ∪M1R M1L { ∈M1 } = τ , Im(τ) > L M1R { ∈M1 } – Full amplitude is a sum (1) = (1) + (1) A AL AR d2τ (1) (ε , ε˜ ,k )= 4 (1)(s τ) L,R i i i Z 2 ij A R M1L,R (Im τ) B | – Both (1), (1) depend on L, but sum is independent of L AL AR – (1) is analytic in s but (1) exhibits non-analyticity at s = 0 AL ij AR ij Eric D’Hoker Lecture III on Superstring Amplitudes Explicit calculation for D8 4 R Since (1) is analytic in s , evaluate using modular graph functions • AL ij

2πζ(3) 1 ζ′(4) ζ′(3) A(1) = R4 ln L − +ln2+ − + power-behaved in L L 45 4 ζ(4) ζ(3) For L 1, approximate integrand of (1) by supergravity + corrections • ≫ AR 4πζ(3) 17 1 ζ′(4) ζ′(3) (1) = +ln2+ (s4 + t4 + u4) 4 A D8R4 45 5 − 4 ζ(4) − ζ(3) R

4ζ(3) s4 ln( 2πs)+ t4 ln( 2πt)+ u4 ln( 2πu) 4 − 45 − − − R – Note: no ambiguities, no inﬁnities, no renormalization required ! – Transcendentality ... (ED, Green, in progress)

Genus-two story ... • (ED, Green, Pioline 2017, 2018, and in progress) Eric D’Hoker Lecture III on Superstring Amplitudes Outlook

Some additional developments • – Clariﬁcation of super Riemann surfaces with R-punctures (Witten 2012) – There exists a super-period matrix for R-punctures (Witten; ED, Phong 2015) – New relations between open and closed string amplitudes (Schlotterer et al.)

Some outstanding issues • – Systematic structure of low energy eﬀective interactions w/ Green, Pioline ⋆ in terms of properties of modular graph functions ⋆ calculation without requiring subtleties of supermoduli space ⋆ UV divergences in supergravity and eﬀective interactions

– Ambi-twistor strings

– string perturbation theory on curved spaces with RR ﬂux, e.g. AdS S5 5 ×