Planar diagrams in light-cone gauge

Transcription

1 Planar diagrams in light-cone gauge M. Kruczenski Purdue University Based on: hep-th/

2 Summary Introduction Motivation: large-n, D-branes, AdS/CFT, results D-brane interactions: lowest order, light-cone gauge D-brane interactions in planar approximation Dual closed string Hamiltonian: H = H 0 - P P: hole insertion Supergravity result: H = H 0 - P

3 Calculation of P in the bosonic string (Neumann coeff., scattering from D-branes) Comparison of P and P Notes on superstring and field theory cases Conclusions

4 Large N limit mesons,,... q ( t Hooft) String picture Quark model q q q Fund. strings ( Susy, 10d, Q.G. ) QCD [ SU(3) ] Large N-limit [SU(N)] Strong coupling Effective strings Lowest order: sum of planar diagrams (infinite number) Suggested using light-cone gauge / frame.

5 D-branes (Polchinski) Region of space where strings world-sheets can end. Open strings. Low energy: SU(N) gauge th. Emission of graviton D-branes have mass If N, # of D-branes is large mass large deforms space e.g. D3- brane: [2] [6] Suggests an interesting rep. of the large-n limit

6 AdS/CFT (Maldacena) We can extract the gauge theory, namely the low energy limit and obtain a duality (no direct derivation) Open strings Sugra background N = 4 SYM II B on AdS 5 xs 5 g s = g 2 YM R / l = ( g N) / s ; YM S 5 : X 12 +X 22 + X 6 2 = R 2 AdS 5 : Y 12 +Y Y 52 -Y 6 2 =-R 2 2 YM N, λ = g N fixed large small string th. field th.

7 Planar approximation ( g s 0, N, g s N fixed ) X (g s N) n x g s 2 We expect that summing these diagrams gives the propagation of closed strings in the supergravity background. In the bosonic case we get: H = H 0 - P (open strings) (supergravity)

8 D-brane interactions Open string: zero point energy N [schematic: divergences have to be regulated] σ: 0 p +

9 Length = p + Open string Agree Closed string Z=

10 Boundary states Conditions for boundary state Solutions

11 Higher orders (Include open string interactions) Open strings can split and join. g s p +, the length is conserved, (g s N) 4 g s 4 N 2 = (g s N) 4 / N 2 n (g s N) n τ σ L σ R τ = τ τ = τ τ = τ = propagation of a single closed string σ n slits

12 This gives: τ = τ 3 τ σ L σ R τ = τ 2 σ τ = τ 1 We define the operator P( L, R) that maps the string from 1 - to 1 +

13 To sum we use the closed string point of view Where. Define We can define: Hclosed = H λ 0 P λ = g N s P essentially inserts a hole

14 Possible problems We need This may need corrections if the path integral is not well defined. For example if two slits collide there can be divergences that need to be subtracted. This can modify P and include higher order corrections in. In fact, at first sight this seems even necessary since the propagation of closed strings in the supergravity bakg. Depends on the metric that has non-trivial functions of. We analyze this problem now. Even if there are extra corrections, P as defined contains important information as we will see.

15 Closed strings in the D3-brane background Take: Since &E = 0 we set E=1 and get: and Not good

16 Which is indeed of the form Hclosed = H λ 0 P with The near horizon (field th.) limit is: which describes closed strings in AdS 5 xs 5

17 What is P in the open string side? π 0 π σ 0 σ 0 τ =0 Identified (cylinder) Scattering of a closed string by a D-brane Vertex representation of P: Dirichlet: Neumann

18 Solution: (Neumann) Compute coefficients., π σ 0 i iy z 0 π σ 0 τ =0 Conf. transf. y r,s=1,2

19 Result: with

20 All together we get: Small holes ( 0 0) Gives: Tachyon pole

21 We reproduced the operator P in a certain (small hole) limit. There are extra terms due to the fact that we consider the bosonic string. (tachyon). Should be absent in the superstring. There are also extra terms which do not correspond to the q 2 0 pole. However we should take into account that the hamiltonian form the background is classical and we should have expected further corrections. In pple. the Hamiltonian we proposed should reproduce order by order the planar diagrams (by definition).

22 Notes on superstrings We need to add fermionic degrees of freedom. A ~, A, right moving and A ~, A left moving. There is an SO(6)=SU(4) symmetry. The index A is in the fundamental or anti-fundamental (upper or lower). Conditions: (preserving half the supersymmetry)

23 Can be solved again in terms of a vertex state: However extra operator insertions are required: π 0 σ 0 σ 0 x x π τ =0 These insertions complicate the calculations. Further work needed. In this case we should get, at low energy a gauge theory

24 Notes on field theory t Hooft: propagator in light cone frame: τ Local in and non local in. We want to flip. But we should get a local evolution in the new. Not clear if it is possible.

25 Simple example ( 3 theory): τ X( σ ) ( σ) c X b X ( σ) b X ( σ) a τ 0 Y = 0 Y = 0 x 0 X ( σ) c x 1 X ( σ) a x 0 Y = L = m σ σ 1 2

26 The diagrams equals Z 1 Z 2 : In field theory: t t 0 k p p k Z

27 Conclusions The sum of planar diagrams is determined by an operator P acting on closed strings. It inserts a hole in the world-sheet. The ( ) dual closed string Hamiltonian is: Hclosed = H λ 0 P For bosonic D-branes we obtained P explicitely. From it, after taking a limit we obtained a Hamiltonian similar to the one for closed strings in a modified background. There can be corrections to H but, nevertheless, the operator P contains important information (e.g. bkgnd.)

28 In field theory we can use a ( ) duality if we get a representation local in the new. In that case we can define a dual H = H 0 - P that contains the information on the planar diagrams. Less ambitious than obtaining a dual string theory.