Superstring vertex operators in type
IIB matrix model
Satoshi Nagaoka (KEK)
with Yoshihisa Kitazawa (KEK & Sokendai)
String Theory and Quantum Field Theory
Kinki university/August 7 2007
1
Introduction
Type IIB matrix model: [Ishibashi-Kawai-Kitazawa-Tsuchiya `96]
Aμ: N×N Hermitian matrices
ψ: Ten dimensional Majorana-Weyl spinor, N×N matrices
• Dimensional reduction of 10-dim SYM theory to 0-dim
• Matrix regularization of type IIB Green-Schwarz
superstring
2
Introduction
(Green-Schwarz light-cone closed) superstring:
• Supersymmetry transformation determines the interaction
vertex for GS light-cone superstring.
• T-duality transformation
radius R
winding number w
α’/R
momentum p+
3
Introduction
• In this talk, Green-Schwarz light-cone closed superstring
theory is derived from type IIB matrix model.
• Low energy excitations of closed string are described by
the matrices.
4
Plan of Talk
IIB matrix model
Light-cone GS string action
1.
2.
3.
4.
5.
SUSY trf.
SUSY trf.
vertex operators
vertex op.
Introduction
GS light-cone superstring action from IIB matrix model
Supersymmetry transformation
Superstring vertex operators in IIB matrix model
Summary
5
GS superstring from IIB matrix model
Background is determined by the classical solutions.
Expanding this action around 2D background :
2D N=8 U(n) noncommutative Yang-Mills theory
8 scalar fields: φi
8v representation of SO(8)
16 spinor fields:
8c, 8s rep. of SO(8)
6
GS superstring from IIB matrix model
1. Mapping the coordinate system from R2 into R1×S1 as
The origin of x coordinate is the point where vertex
operators are inserted.
2. In the low energy limit,
• Noncommutative product → commutative product :
• Diagonal components are favored rather than off-diagonal
components.
• Gauge fields on 2-dim decouple to other fields.
7
GS superstring from IIB matrix model
We obtain the action:
• w : a winding number along σ direction
• Multiple strings are obtained in general.
n=∑i wi
3. By identifying
superstring action.
, we obtain GS light-cone
8
GS superstring from IIB matrix model
Duality relation:
IIA D0
T-duality
IIB D1 (2D-YM)
S-duality
IIB F1
T-duality
IIA F1
DVV’s matrix string
F1: fundamental string
D1: D-string
D0: D-particle
D1
S-duality
F1
T-duality
F1
string in IKKT model
[Dijkgraaf-Verlinde-Verlinde `97]
9
Supersymmetry transformation
N=2 SUSY transformation in IIB matrix model :
In 2D background and low energy limit, this transformation
reduces to
This transformation leaves the GS light-cone string action
invariant.
10
Construction of vertex operator in GS superstring
16 supercahrges :
left mover
right mover
For the vertex operators,
the coefficients B and F are determined by the SUSY trf.
11
Construction of vertex operator in GS superstring
Light-cone open superstring vertex operators are
composed of bosonic (vector) and fermionic (spinor)
operators:
where
k: external momentum, (ki)2=0 , k+=0
ζi, ua : polarization vectors (spinors) which represent the
wave functions for vector (spinor) states
12
Construction of vertex operator in GS superstring
There are 256 massless states in type IIA superstring
theory. Field contents are composed of
NS-NS sector: 8v×8v=[0]+[2]+(2)=1+28+35v
R-R sector: 8c×8s=[1]+[3]=8v+56v
NS-R sector: 8v×8s=[1]+[3]=8c+56c
R-NS sector: 8c×8v=[1]+[3]=8s+56s
Vertex operators for closed string modes are
constructed by the product of left-mover and right-mover
(~open string):
13
Vertex operator in type IIB matrix model
Vertex operators in type IIB SUGRA multiplet are
constructed for type IIB matrix model. [Kitazawa (2002),
Iso-Terachi-Umetsu (2004), Kitazawa-Mizoguchi-Saito (2006)]
(1) Linear couplings to the background fields
Sint= ∑i Vi (A,ψ) fi
(2) Related with each other by the SUSY transf.
∑i Vi (δA, δψ) fi = ∑i Vi (A,ψ) δfi
14
Vertex operator in type IIB matrix model
Type IIB supergravity multiplet
Dilaton vertex operator:
Dilatino vertex operator:
:
Graviton vertex operator:
:
where
15
Superstring vertex operator from type IIB matrix model
1) 2D background
8 scalar fields, 16 spinor fields
2) We construct corresponding states for IIB matrix model
as closed string states (~(left-mover) × (right-mover)).
3) Since left and right mover of the fermion have an
opposite chirality in 2D, we factorize operators by their
chiralities.
4) Type IIA supergravity modes are derived by the
compactification along S1 in matrix model. (~T-duality
transformation).
5) We consider the kinematics (ki)2=0, k+=0
16
Superstring vertex operator from type IIB matrix model
Example: graviton
Graviton vertex operators include the contribution such as
In the second term, although there are four right-handed
spinors, there is no left-handed spinor.
Since we have defined the Fock space of this theory as
that of closed string states, level matching condition
forbids the contribution from these terms.
17
Superstring vertex operator from type IIB matrix model
Many terms vanish by the level matching condition.
The remaining terms are
This operator is equivalent with the vertex operator for
graviton in GS light-cone superstring.
The interaction of multi-graviton gives the same amplitude.
Gravitino, Cμνρ
the same vertex operators
18
Summary
IIB matrix model
Light-cone GS string action
SUSY trf.
SUSY trf.
vertex operators
vertex op.
We have derived Green-Schwarz light-cone superstring
theory from type IIB matrix model.
In the low energy limit, 2-dim NC background in type IIB
matrix model reduces to GS superstring action.
We have derived supersymmetry transformation for GS
light-cone string from IIB matrix model.
19
Summary
IIB matrix model
Light-cone GS string action
SUSY trf.
SUSY trf.
vertex operators
vertex op.
We have identified superstring vertex operators with those
in type IIB matrix model.
Multi-string interaction
• Recombination of (D-)string?
• Closed superstring field theory?
20