1. Michael B. Green University of Cambridge
Some modular properties of superstring
scattering amplitudes
Michael B. Green University of Cambridge
Amplitudes 2016, Stockholm
July 05 , 2016
The icon – an apple made of string - indicates that this talk is about string theory and the apple indicates the connection with Isaac Newton and his theory of gravity.
This is a big subject and difficult to describe. I hope to convey its elegance and ambitions.
This talk will range FROM the very VERY BIG to the very VERY SMALL

2. The low energy expansion of string amplitudes
Consider narrowly-focused aspects of the low energy expansion of closed string theory obtained from maximally supersymmetric closed string scattering amplitudes.
Modular invariants of higher-genus Riemann surfaces
Mathematical connections to Multiple-Zeta Values and their elliptic generalisations
Features of String Perturbation theory
With: Eric D’Hoker; Pierre Vanhove; Omer Gurdogan
Recent papers
Non-Perturbative features of string amplitudes
Connects perturbative with non-perturbative effects
Constraints imposed by SUSY, Duality, Unitarity
Modular Forms; Automorphic forms for higher-rank groups; ….
Connections with maximal supergravity.
Implications for ultraviolet divergences of maximal supergravity
Piece of a larger programme investigating

3. The low energy expansion of the effective action
Einstein-Hilbert
Interactions of other supergravity fields
Lowest order term reproduces the results of classical supergravity
- string
length scale
string coupling
constant
scalar field
- dilaton
metric –
compactify space-time to dimensions D < 10
Higher order terms:
Coefficient depends on moduli (scalar fields)
(maximal supersymmetry)
Expansion in powers of
Double expansion – perturbative expansion in powers of

4. + + …. Scalar Fields (Moduli) and S-Duality
groups in series
(real split forms)
Supergravity (low energy limit of string theory):
(Cremmer, Julia)
Scalar fields parameterize a symmetric space
Maximal compact
subgroup
String theory:
Discrete identifications of scalar fields
Only a discrete arithmetic subgroup
of is symmetry of string theory
Duality group
String perturbation theory: Expansion around boundary of moduli space
e.g. in powers of +
+ ….
Vertex operators
acts on complex structure of torus
World-sheet automorphic symmetries

5. Four-Graviton Scattering in Type IIB String Theory
e.g.
Four-Graviton Scattering in Type IIB String Theory
linearized curvature
Symmetric function of Mandelstam invariants (with ).
Has an expansion in power series of and
(non-analytic pieces are essential, but not for this talk)
Coefficients are S-duality ( ) invariant functions of
scalar fields (moduli, or coupling constants).
To what extent can we determine these coefficients?
Boundary data: String Perturbation Theory
( )

6. Four-Graviton Scattering in Type IIB String Theory
e.g.
Four-Graviton Scattering in Type IIB String Theory
linearized curvature
inverse string coupling constant
One complex modulus
Symmetric function of Mandelstam invariants (with ).
Has an expansion in power series of and
(non-analytic pieces are essential, but will be ignored in this talk)
Coefficients are invariant functions of
scalar fields (moduli, or coupling constants).
To what extent can we determine these coefficients?
Boundary data: String Perturbation Theory
( )

7. Tree-level (Virasoro Amplitude)
Tree-level SUPERGRAVITY
Infinite Series of terms. Coefficients are powers of Riemann values
with rational coefficients
Generalisation to N-particle scattering

8. Zeta values and multiple-zeta values
Very brief review
Special values of polylogarithms
Zeta values:
Even zeta values (powers of pi)
Odd zeta values transcendental? Independent?
e.g.
Multi-zeta values (MZV’s)
Special values of multiple polylogarithms
“weight”
“depth”
MZV are numbers with algebraic properties inherited from the algebraic properties
of multiple polylogarithms, e.g.
weight
First non-trivial (irreducible) case is
Arise in dimensional regularisation of renormalisable quantum field theories.

9. N-particle tree amplitudes
Open-string trees: For coefficients of higher derivative interactions of order
(Yang-Mills)
are MZV’s with weight
(Stieberger, Broedel, Mafra, Schlotterer)
coefficients are single-valued MZV’s (svMZV’s)
Closed-string trees: For
(gravity)
(Brown)
(Schlotterer, Stieberger)
Special values of single-valued multiple polylogarithms – have no monodromies
(generalisations of Bloch-Wigner dilogarithm )
No even zeta values
also
First non-trivial case is
weight
How does this generalize to higher genus ??

10. Genus one Genus one amplitude
Integral over complex
structure
Vertex operator
Corr. function
Green function
Low energy expansion - integrate powers of the genus-one Green function over the torus
and over the modulus of the torus – difficult!
(MBG, D’Hoker, Russo, Vanhove)
Expanding in a power series in momenta gives
(with )
Coefficients of higher derivative interactions
Modular invariants for surface
Coefficients of higher derivative interactions:
(genus-one generalisation of the tree-level values)
Feynman diagrams on toroidal world-sheet

11. “Modular Graph Functions”
(D’Hoker, MBG, Vanhove)
“Modular Graph Functions”
is sum of world-sheet Feynman diagrams.
Each of these is a modular function - invariant under
The Green function on a torus of complex structure :
doubly periodic function
Momentum-space Propagator:
integer world-sheet momenta l1 l5 l2 l3 l6 l4 1 2 3 4 lS
labels number of propagators on line S
General contribution to 4-particle amplitude:
Modular function
“Weight”
contributes to

12. World-sheet Feynman diagrams
Multiple sums:
Non-holomorphic SL(2) Eisenstein series
e.g.
sequence
e.g.
( ) vertices
…..

13. Inhomogeneous Laplace
Direct analysis looks forbidding. But these functions satisfy simple Laplace
equations with Laplacian
Simple examples of Laplace equations :
Eisenstein series
Solution:
(also Zagier)
Inhomogeneous Laplace
eigenvalue equations
Degeneracy – simultaneous inhomogeneous Laplace eigenvalue equations.

14. Coefficients of (weight-4)1 2 3 4 `
Coefficients of (weight-5) 1 2 3 4 1 2 3 4

15. Relation to Single-valued Elliptic Multiple Polylogarithms
(D’Hoker, MBG, Gurdogan, Vanhove)
A Modular Graph Function is a Single-Valued Elliptic Multiple
Polylogarithm evaluated at a special value of its argument z1 z3 z2 z4
A typical Modular Graph Function:
i.e. Split one vertex of a modular graph
function and leave it unintegrated z1 z3 z2 z4 z5
Now Consider
with
It is easy to see that
is single valued (in ) elliptic multiple polylogarithm
Generalisation of single-valued elliptic polylogarithm of Zagier (1990)

16. Modular graph functions of arbitrary weight
Single-Valued Elliptic Multiple Polylogarithms
(D’Hoker, MBG, Gurdogan, Vanhove)
Special values of
As with MZV’s, these elliptic functions satisfy a fascinating set of polynomial relationships
– we have found a few of these (with great difficulty!)
e.g.
weight 5
polynomial of weight 5 in functions of different depth (different no. of loops).
weight 6
polynomial of weight 6 in functions of different depth.
e.g.
Proof uses experimental fact:
Consider: sum of weight- modular graph functions
such that
i.e. Laurent series vanishes
Then:
where

17. Modular graph functions of a given weight satisfy
polynomial relations with rational coefficients
Conjecture:
Elliptic generalisation of the rational polynomial relations between multiple polylogarithms
and single-valued MZV’s
What is the basis of modular graph functions?
Question:
Some (presumably) related issues in open string loop amplitudes (Broedel, Mafra, Matthes, Schlotterer),
which involve “holomorphic” elliptic multiple polylogarithms (Brown,Levin).
Important generalisation to modular graph forms

18. Integration over fundamental domain
Integrating over (and dealing with non-analytic contributions) gives the one-loop expansion:
Genus-one expansion coefficients (after much work):
These coefficients are analogous to the tree-level coefficients:
What is the connection between them ?

19. Genus Two Genus Three Higher Orders
Amplitude is explicit but difficult to study Low energy expansion:
(D’Hoker, Gutperle, Phong)
Result:
(D’Hoker, MBG, Pioline, R. Russo)
Genus Three
Technical difficulties analysing 3-loops. Gomez and Mafra evaluated
the leading low energy behaviour using pure spinor formalism, giving
Higher Orders
New problems - No explicit expression

20. Non-perturbative extension
How powerful are the constraints imposed by (maximal) SUSY and Duality ??
Duality relates different regions of moduli space –
Connects perturbative and non-perturbative features in a highly nontrivial manner.
Consider simple example:
inverse string coupling constant
10-dimensional Type IIB - maximal supersymmetry
One complex modulus
Focus on the simplest nontrivial duality group
Duality group
Relates strong and weak coupling.

21. Non-perturbative extension
invariant functions
Using:
Nonlinear supersymmetry
Duality between M-theory (quantum 11-dimensional supergravity on two-torus)
and and string theory compactified on a circle
Solutions: Non-holomorphic Eisenstein series
With b.c: Power behaviour as

22. Non-holomorphic Eisenstein series
Parabolic subgroup
Poincare series –
manifest
invariant (generalises to higher rank duality groups)
Solution of Laplace eigenvalue eqn.
Fourier series
Zero mode TWO POWER-BEHAVED TERMS (perturbative) :
divisor sum
Non-zero modes D-INSTANTON SUM

23. Low order interaction coefficients
MBG, Gutperle, Vanhove
Non-renormalisation beyond 1-loop for
Perturbative terms: tree-level genus-one D-instantons
Perturbative terms: tree-level genus-two D-instantons
NO genus-one term (no ¼-BPS states)
Non-renormalisation beyond 2 loops for

24. NOT an Eisenstein series but satisfies Inhomogeneous Laplace equation
Next order
The square of the
coefficient of
NOT an Eisenstein series but satisfies Inhomogeneous Laplace equation
MBG, Vanhove
This equation was conjectured by consideration of duality between two-loop eleven-dimensional
supergravity compactified on a two-torus and type IIB compactified on a circle.
Structure also motivated by (but not directly derived from) supersymmetry.
(See also, Yifan Wang, Xi Yin, 2015)
The solution of this equation has some weird and wonderful (puzzling) features.
Zero mode of solution (zero net D-instanton number):
Sum of D-instantons
genus zero
genus one
genus two
genus three
Precise agreement with explicit string theory loop calculations

25. A fantasy concerning UV properties of supergravity
The coefficients of the UV divergences in maximal supergravity up to 3 loops in
various dimensions > 4 are precisely reproduced by log terms in modular coefficients.
More Generally:
To what extent do string theory dualities constrain the structure of
perturbative supergravity? – ultraviolet divergences??
Fantasy:
Superstring perturbation theory is free of UV divergences. Can we understand
the UV properties of supergravity by careful consideration of the low energy
limit of string theory?