1. 1 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

2. 2 Motivation Type IIB matrix model: [Ishibashi-Kawai-Kitazawa-Tsuchiya `96] A μ : N×N Hermitian matrices ψ: Ten dimensional Majorana-Weyl spinor, N×N matrices Perturbative picture of fundamental strings Vertex operators? Interactions?

3. 3 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.

4. 4 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]

5. 5 Green-Schwarz superstring from IIB matrix model (i) IR limit: the NCYM action becomes commutative N=8 SYM: Perturbative vacua: 8 scalar fields: φ i 8 v representation of SO(8) 16 spinor fields: 8 c, 8 s reps. of SO(8)

6. 6 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.

7. 7 Green-Schwarz superstring from IIB matrix model (iii) Mapping the coordinate system from R 2 into R 1 ×S 1 : We identify the eigenvalue w : a winding number along σ direction Then, the integral is described as

8. 8 Green-Schwarz superstring from IIB matrix model By the rescaling we obtain the action: Multiple strings are obtained in general. n=∑ a w a By identifying, we obtain Green-Schwarz light-cone superstring action.

9. 9 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.

10. 10 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, B μν and C 4 are completely determined.

11. 11 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.

12. 12 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

13. 13 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 …

14. 14 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 (k i ) 2 =0, k + =0. After τ→ - i τ, exponential factor gives to the scattering amplitude.

15. 15 Superstring vertex operators in IIB matrix model Graviton vertex operator (8v×8v) where

16. 16 Superstring vertex operators in IIB matrix model B μν vertex operator (8v×8v) The operator in IIB matrix model which reduces to B ij type (1,1) operator should be either of the following four forms: Thus, we obtain

17. 17 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 : 2-form :

18. 18 Superstring vertex operators in IIB matrix model The correspondence between IIB matrix model / superstring vertex operators h ij, B ij, C ijl, C i and Ψ: completely the same vertex operators

19. 19 Future directions weak coupling ← g YM → strong coupling strong coupling ← g s → 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