superstring vertex operators in type iib matrix model arxiv:0708.1077[hep-th], 0710.0709[hep-th]
DESCRIPTION
Superstring vertex operators in type IIB matrix model arXiv:0708.1077[hep-th], 0710.0709[hep-th]. Satoshi Nagaoka (KEK) with Yoshihisa Kitazawa (KEK & Sokendai) KEK workshop/March 5 2008. Motivation. Type IIB matrix model: [Ishibashi-Kawai-Kitazawa-Tsuchiya `96] A μ : N×N Hermitian matrices - PowerPoint PPT PresentationTRANSCRIPT
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Superstring vertex operators in type IIB matrix model
arXiv:0708.1077[hep-th], 0710.0709[hep-th]
Satoshi Nagaoka (KEK)
with Yoshihisa Kitazawa (KEK & Sokendai)
KEK workshop/March 5 2008
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Motivation
Type IIB matrix model: [Ishibashi-Kawai-Kitazawa-Tsuchiya `96]
Aμ: N×N Hermitian matrices
ψ: Ten dimensional Majorana-Weyl spinor, N×N matrices
Perturbative picture of fundamental strings
Vertex operators?
Interactions?
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Approach
Noncommutative (NC) gauge theory
• Worldsheets are obtained from two dimensional NC background.
• Supersymmetry transformation for IIB matrix model (superstring) completely determines the vertex operators for IIB matrix model (superstring).
IIB matrix model SUSY trf. vertex op.
Light-cone string action SUSY trf. vertex op.
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Green-Schwarz superstring from IIB matrix model
Expanding this action around 2-dim background:
2-dim N=8 U(n) noncommutative Yang-Mills theory i , j= 2,…,9 [Aoki-Ishibashi-Iso-Kawai-Kitazawa-Tada `99]
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Green-Schwarz superstring from IIB matrix model
(i) IR limit: the NCYM action becomes commutative N=8 SYM:
Perturbative vacua:
8 scalar fields: φi 8v representation of SO(8) 16 spinor fields: 8c, 8s reps. of SO(8)
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Green-Schwarz superstring from IIB matrix model
• We assume that all the eigenvalues of matrices do not coincide with each other at any points on the worldsheet.
(ii) all the excitations of off-diagonal modes become massive. In the low energy limit,
Gauge fields on 2-dim decouple to other fields.
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Green-Schwarz superstring from IIB matrix model
(iii) Mapping the coordinate system from R2 into R1×S1:
We identify the eigenvalue
• w : a winding number along σ direction
Then, the integral is described as
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Green-Schwarz superstring from IIB matrix model
By the rescaling
we obtain the action:
Multiple strings are obtained in general.
n=∑a wa
By identifying , we obtain Green-Schwarz lig
ht-cone superstring action.
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Supersymmetry transformation
N=2 supersymmetry transformation in IIB matrix model :
On the 2-dim NC background, it reduces in the low energy limit to
This transformation leaves the Green-Schwarz light-cone superstring action invariant.
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Vertex operators in IIB matrix model
Vertex operators (type IIB supergravity multiplet) [Kitazawa (2002), Iso-Terachi-Umetsu (2004), Kitazawa-Mizoguchi-Saito (2006)]
Dilaton vertex operator:
Dilatino vertex operator:
:
Graviton vertex operator:
:
where
Graviton, gravitino, Bμν and C4 are completely determined.
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Superstring vertex operators in IIB matrix model
1) Trace
2-dim background 8 scalar fields, 16 spinor fields
By the scaling,
and the field redefinition,
we obtain
where dimension (1,1) operators in (…) are proportional to 1.
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Superstring vertex operators in IIB matrix model
2) Physical states
In IIB matrix model, we construct closed string states in a radial quantization as the product of separate states corresponding to the left and right movers.
Open string physical states |Λ, k > are obtained by inserting the vertex operators in the far past
There are (8+8) massless ground states in each sector:
| i > , | à >
They are mapped to each other
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Superstring vertex operators in IIB matrix model
3) Chirality
Since left and right mover of the fermion have an opposite chirality in 2-dimension, we factorize operators by their chiralities.
4) Type IIB → type IIA
Type IIA supergravity modes are derived by singling out the light-cone direction. For R-R 4-form tensor,
: 3-form
: 2-form
…
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Superstring vertex operators in IIB matrix model
5) Consistency with the light-cone formalism
In our formalism, correlation functions between gauge fields in the exponential factor are calculated as
where
On the other hand, in the light-cone gauge formalism,
with (ki)2=0, k+=0 .
After τ→ - i τ, exponential factor gives
to the scattering amplitude.
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Superstring vertex operators in IIB matrix model
Graviton vertex operator (8v×8v)
where
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Superstring vertex operators in IIB matrix model
Bμν vertex operator (8v×8v)
The operator in IIB matrix model which reduces to Bij type (1,1) operator should be either of the following four forms:
Thus, we obtain
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Superstring vertex operators in IIB matrix model
R-R 3-form (8c×8s)In the type IIB supergravity multiplet, RR 4-form vertex op.
From this vertex operator, we obtain 4,3 and 2-form fields:
4-form :
3-form :
3-form :
2-form :
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Superstring vertex operators in IIB matrix model
The correspondence between IIB matrix model / superstring vertex operators
hij, Bij, Cijl , Ci and Ψ: completely the same vertex operators
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Future directions
weak coupling ← gYM→ strong coupling
strong coupling ← gs → weak coupling
UV IR → 0
Perturbative (NC)YM/ holographic description →Free string
(Green-Schwarz string)
Multi-string interaction
• Light-cone superstring field theory
• Twist field formalism