Non-perturbative Formulation of Superstring Theory
Based on String Geometry
Matsuo Sato (Hirosaki U.)• String geometry and non-perturbative formulation of string theory
arXiv:1709.03587 M.S.
• Topological string geometry
work in progress M.S. , Yuji Sugimoto (Osaka City Univ.)• String geometric phenomenology
work in progress Masaki Honda (Waseda Univ.), M.S.

Contents
0.

Introduction

1.

String geometry2.

Non-perturbative formulation of string theory

3. String geometry solution that represents a perturbative vacuum of string theory
4. Derive all order scattering amplitudes of perturbative string
5. General supersymmetric case that includes open strings
6. String geometry and a new type of supersymmetric matrix models
7. Unification of particles and the space-time
8. Future directions

Motivation
T-duality• IIA string on a background

IIB string on another background

Observed values by strings coincide.
“Space observed by strings are the same.’’
Expected to be “geometric principle of string theory.’’
Q. Is there spaces T-dual to each other?
A. Yes: moduli spaces of Riemann surfaces embedded on-shell in the backgrounds.
Extend off-shell
Extend non-perturbatively. (Consider collections of• Spaces where curves parametrized byreproduce the right moduli space of the Riemann surfaces ina target manifold.
criterion to define string topology
string geometry• Construct string theory by regarding string geometry as geometric principle.

.)

1. String geometry

1. 1 String model spaceGlobal time
• On, there exists an unique Abelian differentialthat has simple poles with residues
at Punctures Pi where, if it is normalized to have purely imaginary periods with respect to all contours.

• Global timeis defined by(Krichever, Novikov 1987)

on Pi with• Determine
0.

︓

conservation law (if we choose the outgoing direction as positive.)

1. Divide Pi s to arbitrary incoming and outgoing sets.
2. Divide -1 to incoming•

are determined uniquely on•

is uniquely determined.

and 1 to outgoing

1/3-1/2

1/3

1/3

-1/2

String model space E・many body states of strings

・: backgrounds (B, dilaton）

︓equivalence class・
atHere,・Collection of string states

1. 2 String toplology

String topology•open neighborhoods.t.

• U is defined to be an open set if there exists
for an arbitrary point PU.

such that anopen neighborhoodPε•The open sets satisfies the axiom of topology.

UU1. 2 String manifold

General coordinate transformation•

does not transform toand vice versa,
becauseis a discrete variable, whereas
are continuous variables
by definition of the neighbourhoods.

•

anddo not transform to each other because the string states aredefined byconstant lines.

•Under these restrictions, the most general coordinate transformation is given byup to diff × Weyl
• String manifoldsare constructed by patching open sets of E by general coordinate transformations.

Example of string manifolds•
where

which has target mertic• D: backgronds including the target metric .

• Open sets of

D is fixed on a string manifold.

Open sets of E

homeomorphic
diffeo:Is induced by the diffeomorphism transformation of the target spaceExample of string manifolds(cont’d)• Trajectories in asymptotic processes on
represents
2-dim. Riemann surfaces in the target manifold.
reproduce the right moduli space.
MExample of string manifolds(cont’d)• By a general trajectory,string states on different two-dimensional Riemann surfaces that have different genus numbers
can be connected continuously.
v.s. the moduli space• String geometry is expected to possess non-perturbative effects.

1. 4 Riemannian string manifold

Riemannian string manifold• cotangent vectors
cotangent space of manifolds are spanned by continuous variables:

as indices.

＝：＝：

Treatsummarize

Take summation by
Invariant underTransformed as scalar under• metric2 Non-perturbative formulation of string theory

Non-perturbative formulation of superstring theory•: the invariant measure ofon
divided by
the volume of the diffeomorphism and the Weyl transformations.

diff × Weyl
: field strength of an u(1) gauge field• The theory is background independent.

diffeomorphism invariance・ Under: symmetric tensor

︓vectorAction is manifestly invariant

Under・Under,,andthe fields that have index

are defined as scalarstransform as scalars.

is invariant.

The action is invariant under* In a supersymmetric case, the action is invariant under3 String geometry solution that represents
a perturbative vacuum of string theory

Perturbative vacuum solution （Extension of Majumdar-Papapetrou solution (1947,1948))•

is a solution to the equations of motion.
where(are all independent. )（is the scalar curvature of)（index volume）• We derive all the perturbative string amplitudes on flat spacetime from the fluctuations around this solution.
• The solution is defined on•where the target metric is fixed to be flat .

The equations of motion are differential equations with respect toThe functions ofare constants in the solution (determined by the consistency of the fluctuations.)4 Derive all order scattering amplitudes of perturbative stringPropagators around the perturbative vacuum
１．Expand the action around the perturbtive vacuum up to 2nd order:
２．Take.

Then, the fluctuations of the gaguge field are suppressed.

３．Take the harmonic gauge to fix diffeo. Then, the gauge fixing term is added.

４．Take slowly varying field limit:
derivative expansion

５．Normalize to obtain canonical kinetic term:
６. TakeandPropagators around the perturbative vacuum (cont’d)+(terms do not mix with•is one of the modes ofADM decomposition• Differential equation for the propagator)

Schwinger representation of the propagator = path integral of the perturbative strings• In order to compare with perturbative strings,
Take the Schwinger representation of the propagator by using the first quantization formalism.
operators••

conjugate momenta

eigen states

Schwinger representation of the propagator = path integral of the perturbative strings (cont’d)• path integral representation* By introducing•,

constant T → field T(t)move onto Lagrange formalism from the canonical formalism by integrating out.

Schwinger representation of the propagator = path integral of the perturbative strings (cont’d)•• This path integral is obtained
ifgauge is chosen in the next covariant form w.r.t. t diffeo:

Schwinger representation of the propagator = path integral of the perturbative strings (cont’d)• Covariant form w.r.t. t diffeo*•

disappears underis transformed as an einbein.

:* This action is still invariant under the diffeomorphismwith respect to t if
transforms in the same way as t.
• Takegauge.

Schwinger representation of the propagator = path integral of the perturbative strings (cont’d)•• Diff × Weyl transformation gives

︓Euler number• We obtain the all-order perturbative scattering amplitudes that possess the moduli in the string theory,
by inserting asymptotic states.
• The consistency of the fluctuations around the backgrounds

the critical dimension d=26.
(d=10 in the supersymmetric cases)

5 General supersymmetric case that includes open strings

Supersymmetric generalization including open strings
So farGeneral

Riemann surface

super Riemann surface

with or without boundaries
Boundaries have CP factors and map to D-branes: background (B, dilaton）

model space: background(B, dilaton , RR, submanifolds of M that represent D-branes and O-planes
gauge fields on D-branes)(T= IIA, IIB, I)• For T=I,projected• For T=IIA (T=IIB, I), IIA (IIB) GSO projection is attached on
asymptotic states* We can define GSO projection
because functions over the model space are functions ofindex

Non-perturbative formulation of superstring theory•• The theory is background independent.

Supersymmetry is a part of the diffeomorphisms symmetry• These are dimensional reductions indirection of the two-dimensional

local susy trans.

• supercharges• The number of supercharges is the same as of the two-dimensional ones.
• The supersymmetry algebra closes in a field-independent sense as in ordinary supergravities.

Derive the all order perturbative superstring scattering amplitudes• We obtain the all-order scattering amplitudes that possess the supermoduli in the perturbative type IIA, IIB andSO(32) type I superstring,
if we consider the fluctuations after fixing IIA, IIB and SO(32) type I charts, respectively.
•These amplitudes are derived from the single theory.

• The consistency of the fluctuations around the backgrounds

d=10• We obtain amplitudes of the superstrings with Dirichlet and Neumann boundary conditions in the normal
and tangential directions to the D-submanifolds, respectively.
D-submanifolds represent D-brane backgrounds where back reactions from the D-branes are ignored.

6 String geometry and a new type of supersymmetric matrix models

String geometry and a new type of supersymmetric matrix models
Gravity and a matrix moldel (Hanada-Kawai-Kimura 2006)
Equations of motion ofequivalent
where we replace

Equations of motion ofString geometry and a matrix model
Equations of motion ofequivalent

where we replace

Equations of motion ofMore simple（extended）large N reduction ?
（a supersymmetric matrix model that has

is interesting.

indices

Worldsheets can be derived in general by perturbations of matrix models)7 Unification of particles and the space-timeUnification of space-time and particles• space-time and string geometry
asymptotic trajectory on

with target M = string world-sheet in M
M

macro

trajectory of a particle in M

Space-time M is identified by: observing all trajectories of a particle in M.
∴MM

is observed as M macroscopically.

Conversely, we see a string, if we microscopically observe a point of the space-time.
• particle and string geometry
A fluctuation of= string

macroparticle

Conversely, we see a string, if we microscopically observe a particle.
• unification of space-time and particle
Macroscopically,

space-time = string manifold

particle = a fluctuation of string manifold