Kangwon National University

Transcription

1 USES of String Field Theory Taejin Lee Kangwon National University Taejin Lee (KNU APCTP July 5th, 08 / 57

2 USES of String Field Theory I will introduce the covariant string field theories and the Polyakov string path integrals in the proper-time gauge. In the proper-time gauge, the string path integral can be written as integrals over the proper-times in a way similar to the Schwinger s proper time representation of Feynman integrals of quantum field theory. For this reason, it becomes feasible in the proper-time gauge to identify the field theoretical expressions of the string path integrals which depict multiple string scatterings. I will discuss the four-massless-particle scattering amplitude in open string theory and the four-graviton scattering amplitude in the closed string theory. The covariant string field theories are useful to study the ultraviolet behavior and the entanglement entropy of strings which are quite different from their counterparts of point particles. I will also discuss applications of the string field theory in the super-gravity, AdS/CFT and related topics. Taejin Lee (KNU APCTP July 5th, 08 / 57

3 Contents Application of Open String Field Theory in Entanglement Entropy Entanglement entropy for open strings on Dp-branes BH Entropy as Entanglement Entropy 3 Entanglement Entropy and AdS/CFT Application of Closed String Field Theory in QM Gravity Quantum Gravity and String Field Theory Quantum Gravity as a Low Energy Limit of SFT 3 Scattering Amplitudes of Closed String Field Theory 4 Scattering Amplitudes of Three Closed Strings/Gravitons 5 Scattering Amplitudes of Four Closed Strings/Gravitons 6 Four-Graviton Scattering 3 Applications of String Field Theories in AdS/CFT and Related Topics Classical Solutions of Super-Gravity UV Behaviors of Scattering Amplitudes 3 UV Completions of QFT in Higher Dimensions 4 Correlators of BMN Operators 5 Higher Loop Corrections in QCD Taejin Lee (KNU APCTP July 5th, 08 3 / 57

4 Entanglement entropy Von Neumann entanglement entropy For a pure state ρ AB = Ψ Ψ AB S(ρ A = tr (ρ A log ρ A = tr (ρ B log ρ B = S(ρ B ρ A = tr B (ρ AB, ρ B = tr A (ρ AB. Renyi entanglement entropies and replica trick S α (ρ A = α log tr(ρα = S a (ρ B, S(ρ A = lim S α (ρ A. α Entanglement entropy is proportional to the area of boundary between the two partitions. L. Bombelli, R. Koul, J. Lee, and R. Sorkin, Phys. Rev. D 34, 373 (986. M. Srednicki, Phys. Rev. Lett. 7, 666 (993. Taejin Lee (KNU APCTP July 5th, 08 4 / 57

5 Taejin Lee (KNU APCTP July 5th, 08 5 / 57 Entanglement entropy for open strings on Dp-branes Locality and Fock space representation of open string field Ψ = = a {Nn B,Ngh n,n=,,3,... } a Ψ a (x µ T a {N B {Nn B,Ngh n } n, Nn gh, n =,, 3,... } ( φ a (x + A a µ(xa µ + ϕa i (xa i +... T a 0.

6 Density matrix for string field String vacuum functional and density matrix Φ[Ψ] = 0 Ψ = N φ{nn} (x,0=ψ {Nn} (x φ {Nn} (x, =0 ρ[ψ, Ψ ] = Ψ 0 0 Ψ = Φ[Ψ] Φ[Ψ ]. D[Φ]e S E (Φ We consider the functionals Ψ = Ψ A Ψ B and Ψ = Ψ A Ψ B which coincide on the half line x < 0. We shall sum over all possible functional Ψ B. Taking two copies of the half planes, we write the reduced density matrix as ( ρ A Ψ, Ψ = D[Φ B ]Φ[Ψ A Ψ B ] Φ[Ψ A Ψ B] = N φ{nn} (x,0 + =ψ {Nn} (x, x A φ {Nn} (x,0 =ψ {Nn} (x, x A D[Φ]e S E (Φ. Taejin Lee (KNU APCTP July 5th, 08 6 / 57

7 Entanglement entropy for massive scalar field: Review The partition function Z(n for a massive scalar field in (p + dimensions on the n-sheeted Riemann surface ln Z(n = ln Det [ + m ], m ln Z(n = d p+ x G n (x, x. R n where G n (x, x is the Green s for a massive field in the conical space and x = (x 0, x,..., x p = (x 0, x, x. Making use of the expicit expression of the Green s function in the coincident limit, x x, we get m ln tr ρn = Z(n ln m Z( n = n 4n A d p p (π p m + p where A is the area of boundary hypersurface, perpendicular to (x 0, x -plane, A = d p x. Taejin Lee (KNU APCTP July 5th, 08 7 / 57

8 Entanglement entropy for open string Entanglement entropy of a massive field S A = A ds s d p p (π p exp { s ( p + m}. A direct extension to string field yields the entanglement entropy of open string in the boson sector (N gh n = 0, n =,,... : S Open A = A ds s N B = n= na I n a I n, d p p (π p Tr exp { s ( p + N B } I = 0,,..., d where Tr denotes the trace over the Fock space as well as over U(N group space. Taejin Lee (KNU APCTP July 5th, 08 8 / 57

9 From the bosonic harmonic oscillator algebra Tr e sn B = N { exp s } nnn B {Nn B} n= (8π p = N e d+ 0 t p 4 s d+ η ( is π where η(τ is the Dedekind eta-function, we find the entanglement entropy for open string in boson sector S Open A = A N {( dt exp d + } πt t 4 where t = s/(π. η(it d+ Taejin Lee (KNU APCTP July 5th, 08 9 / 57

10 Taking into account contributions of ghost sector, S Open A = A ds s N gh = n= n d p p (π p Tr exp { s ( p + N B + N gh } ( F gh, F gh = ( a gh n agh n + a gh n agh n. n= ( a gh n agh n + a gh n agh n Using {N gh } e sngh ( Fgh = n= ( e sn ( s is = e η, π we find that the entanglement entropy may be written as S Open A = A N {( dt 5 d exp πt (8π p 0 t t p 4 } η(it d. Taejin Lee (KNU APCTP July 5th, 08 0 / 57

11 IR behavor of the entanglement entropy for open string IR Behavor: Apart from the leading divergence, the entanglement entropy for open string is at most logarithmically divergent. Its IR behavor is finite for p and logarithmically divergent for p =. In the asypmtotic region t, the integrand becomes t p+ η(it 4 = { } e πt O(e πt. t p+ Taejin Lee (KNU APCTP July 5th, 08 / 57

12 UV behavors of the entanglement entropy for open string UV Behavor: Except for the tachyon contribution, the entanglement entropy is UV finite for p 4 and logarithmically divergent for p = 5. Using the modular transformation formula η( /τ = ( iτ / η(τ, and rewriting the integral for S Open A S Open A = A N (8π p in terms of s = /t, we get 0 ds s (p 7 η(is 4. The integrand in UV region where s, may be expanded asymptotically as { } s (p 7 η(is 4 = s (p 7 e πs O(e πs. Taejin Lee (KNU APCTP July 5th, 08 / 57

13 BH entropy as Entanglement entropy Entanglement entropy in QFT S E = A(Σ, φ(xφ(y ɛd ɛ d Entanglement entropy measures short-distance correlations between the two partitions (interior and exteria regions of black hole G. thooft, Nucl. Phys. B 56, 77 (985. D. Kabat, Nucl. Phys. B 453 8, (995. S. N. Solodukhin, Phys. Rev. Lett. 97, 060 (006. Taejin Lee (KNU APCTP July 5th, 08 3 / 57

14 Puzzle of non-minimal coupling Renormalization of Newton constant = 4G ren 4G + s (4π (d N s D s (d. (d ɛ d 6 In the presence of the non-minimal coupling ξr for scalar field theory, = ( 4G ren 4G + (4π (d (d ɛ d 6 ξ. However, the entanglement entropy in a Ricci flat background (R = 0 is not affected by the non-minimal coupling. S. Solodukhin, Living Rev. Relativity 4, 8 (0. Taejin Lee (KNU APCTP July 5th, 08 4 / 57

15 BH entropy for extremal and near extremal black holes in string theory Microstates of Black Hole In string theory, extremal black hole is described by BPS states: N BPS = exp (S BH. BH entropy is obtained by counting the microstates of the BPS solitons. A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (996. C. G. Callan and J. M. Maldacena, Nucl. Phys. B 47, 59 (996. G. Horowitz and A. Strominger, Phys. Rev. Lett. 77, 368 (996. Taejin Lee (KNU APCTP July 5th, 08 5 / 57

16 Entanglement entropy and AdS/CFT Ryu-Takayanagi Formula If we choose a closed subspace Σ on the spatial boundary of the AdS space to define the entanglement entropy with respect to Σ, the entropy may be determined by A(Γ, the area of minimal surface Γ, in the bulk AdS space: S Σ = A(Γ/4G N, Σ = Γ. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 860 (006. S. Ryu and T. Takayanagi, JHEP 0608, 045 (006. Taejin Lee (KNU APCTP July 5th, 08 6 / 57

17 Quantum Gravity and String Field Theory Classical General Relativity Derived from Quantum Gravity Boulware and Deser, Ann. Phys. 89 (975: A quantum particle description of local (noncosmological gravitational phenomena necessarily leads to a classical limit which is just a metric theory of gravity. As long as only helicity ± gravitons are included, the theory is precisely Einsteins general relativity. Closed String Field Theory Closed string theory contains massless spin particles in its spectrum. The low energy limit of the covariant interacting closed string field theory must be the Einstein s general relativity. The closed string field theory may provide a consistent framework to describe a finite quantum theory of the spin particles, the gravitons. We need to examine the graviton scattering amplitudes of the covariant string field theory and compare them with those of the perturbation theory of the gravity in the low energy region. Taejin Lee (KNU APCTP July 5th, 08 7 / 57

18 Gravity as a low energy limit of a closed string field theory Quantum gravity and closed string field theory. Closed string field theory is finite: It may resolve the problem associated with non-renormalizability of the perturbative quantum gravity. Closed string field theory is unitary: It may provide a resolution to the problems related to the information loss and the Bekenstein-Hawking entropy of black holes 3. Closed string field theory has well-defined UV and IR behaviors: It may help us to study the dark matter physics and related cosmological problems. 4. Open/Closed string field theory as a framework to study various dualities in string theory including AdS/CFT correspondence. Taejin Lee (KNU APCTP July 5th, 08 8 / 57

19 UV behavior of String and SFT String Scattering Amplitudes with Vertex Operators String scattering amplitudes with vertex operators may not capture correct UV behavior of string theory Taejin Lee (KNU APCTP July 5th, 08 9 / 57

20 String Field Theory in the Proper-Time Gauge II. Closed string field theory in the proper-time gauge We will construct a covariant string field theory and calculate three-string scattering amplitude and the four-string scattering amplitude in the low energy limit by extending the previous works on the open string field theory. ( ( (3 (3 ( ( (4 Taejin Lee (KNU APCTP July 5th, 08 0 / 57

21 Fock Space Representation of the Closed String Field Theory in the Proper-Time Gauge Closed String Field Theory in the Proper-Time Gauge S = Φ KΦ + g 3 ( Φ Φ Φ + Φ Φ Φ. The closed string field theory in the proper-time gauge generates the string scattering diagrams, which can be represented by the Polyakov string path integrals: S P = 4πα M dτdσ αβ X µ X ν hh σ α σ β η µν, µ, ν = 0,..., d. String Scattering Amplitudes [ ] A M = D[X ]D[h] exp i dτdσl. M Taejin Lee (KNU APCTP July 5th, 08 / 57

22 String Scattering Amplitudes of Closed String Field Theory and Polyakov String Path Integral Strategy of Calculation of String Scattering Amplitudes Evaluate the scattering amplitudes by using the Polyakov string path integral Re-express the Polakov string path integrals in terms of the oscillator operators 3 Identify the Fock space (operator representations of the string field theory vertices 4 Choose appropriate external string states, corresponding to the various particle states and evaluate the scattering amplitudes. 5 Compare the resultant scattering amplitudes with those of YM theory for open string and GR for closed string in the zero-slope limit Taejin Lee (KNU APCTP July 5th, 08 / 57

23 Closed String Theory: Review Free String Theory S = 4πα dτdσ X X. Decomposition of X in terms of left-movers and right-movers Mode expansions X L (τ, σ = x L + X R (τ, σ = x R + where x = x L + x R. X (τ, σ = X L (τ + σ + X R (τ σ. α α p L(τ + σ + i p R(τ σ + i α α n 0 n 0 n α ne in(τ+σ, n α ne in(τ σ, Taejin Lee (KNU APCTP July 5th, 08 3 / 57

24 Closed String Theory: Review Canonical commutation relations α [x L, p L ] = [x R, p R ] = i, [α m, α n ] = [ α m, α n ] = mδ(m + n. Momentum eigentate with eigenvalue P n, n 0: {( P n = πn exp Pn α n + P n α n α n α n n n n Mapping from cylindrical surface onto the complex plane z = e ρ = e ξ+iη, π η π. Green s function on complex plane (ξ > ξ, = ξ ξ, G C (z, z = ln z z = max(ξ, ξ n= e n n P } n P n 0. 4n ( e in(η η + e in(η η. Taejin Lee (KNU APCTP July 5th, 08 4 / 57

25 Closed String Interaction in the Proper Time Gauge CS mapping from the world sheet of three closed string scattering onto the complex plane. For the three-string vertex in the proper-time gauge, ρ = ln(z + ln z. The local coordinates ζ r = ξ r + iη r, r =,, 3 defined on invidual string world sheet patches are related to z as follows: e ζ = e τ 0 z(z, e ζ = e τ 0 z(z, e ζ 3 = e τ 0 z(z. For convenience we choose, by using SL(, C invariance Z = 0, Z =, Z 3 =. Taejin Lee (KNU APCTP July 5th, 08 5 / 57

26 Local Coordinates ρ = τ + iσ ρ = τ + iσ σ π 0 ζ ζ η π η 0 π 0 τ 0 τ 0 π η 3 ζ 3 σ π 0 ζ ζ η π η 0 π 0 τ 0 τ 0 π η 3 ζ 3 Taejin Lee (KNU APCTP July 5th, 08 6 / 57

27 Neumann Functions of the Closed String Vertices Fourier components of the Green s function on complex plane G C (ρ r, ρ s = ln z r z s = δ rs { n= e n n } ( e in(η s ηr + e in(η s ηr max(ξ, ξ + n,m C rs nme n ξr + m ξ s e inηr e imη s. Taejin Lee (KNU APCTP July 5th, 08 7 / 57

28 Integral Formulas for C rs nm C rs 00 = ln Z r Z s, r s, α i C 00 rr = ln Z r Z i + τ (r 0 α r α r i r C n0 rs = C 0n sr = dz e nζr (z, n, n πi z Z s C rs nm = nm Z r Z r dz πi Zs dz πi Reality conditions of the Green s function (z z e nζr (z mζ s (z, n, m. C nm rs rs = C n m, C rs nm = 0, n, m. Taejin Lee (KNU APCTP July 5th, 08 8 / 57

29 Scattering Amplitude of Three Strings Scattering amplitude W = DX exp ( i M r= P r (σ X (τ r, σdσ { { = [det ] d/ exp ξ r P0 4 r r } + n,m = P exp C rs nme n ξr + m ξ s P (r n P(s m ( r ξ r L (r 0 V [N]. n= dτdσl n P(r n P (r n Taejin Lee (KNU APCTP July 5th, 08 9 / 57

30 Neumann Functions of Three-Closed-String Vertex Neumann functions of three-closed string vertex and those of three-open string vertex: C 00 rs = N 00 rs = ln Z r Z s, r s, C rr 00 = N rr 00 = i r α i C n0 rs = C n0 rs = rs N n0 = n C rs nm = C rs = n m = dz πi nm Z r C rs n m = C rs nm = 0. ln Z r Z i + τ α r α 0, r dz πi N rs mn Zs dz πi Z r z Z s e nζr (z, n, (z z e nζr (z mζ s(z, n, m, Taejin Lee (KNU APCTP July 5th, / 57

31 Factorization of Three-Closed-String Scattering Amplitude A[,, 3] = g {k (r } { ( exp r,s n,m { ( exp τ 0 r α r N rs nm { ( exp r,s n,m { ( exp τ 0 r α r N rs nm α (r n p (r α(r m α n (r α(r m p (r + n } + n } 0. N rs n0 N rs n0 α (r n α (r n p(s } p(s } Taejin Lee (KNU APCTP July 5th, 08 3 / 57

32 Factorization of Three-Closed-String Scattering Amplitude Factorization of three-closed-string scattering amplitude A closed [,, 3] = A open [,, 3] A open [,, 3]. Scattering amplitude of three closed strings can be completely factorized into those of three open strings except for the zero modes. Question: Can we factorize general closed string scattering amplitudes into those of open string theory? Realization of the Kawai-Lewellen-Tye (KLT relations in the framework of the second quantized string theory. Taejin Lee (KNU APCTP July 5th, 08 3 / 57

33 Three-Graviton Scattering Amplitude Decomposition of the spin- field into graviton, anti-symmetric tensor, and scalar field { h µν = (h } { } { } µν + h νµ η µν d hσ σ + (h µν h νµ + η µν d hσ σ. We choose the covariant gauge condition µ h µν = 0, which becomes de Donder gauge condition for the graviton µ h µν d νh σ σ = 0. For three-graviton scattering, we choose the external string state as Ψ 3G = 3 { r= h µν (p r α (rµ α(rν } 0. Taejin Lee (KNU APCTP July 5th, / 57

34 Three-Graviton Scattering Amplitude Three-graviton scattering amplitude 3 ( 3 A [3 graviton] = dp (r δ p (r = r= ( g 3 { 3 0 ( 3 t= ( 3 n= r= e τ 3 0 r= αr i= g 3 Ψ 3G E[3] Closed [,, 3] 0 ( 3 3 dp (i δ i= h µν (p (i a (iµ ã (iν N t a (t p N n ã (n p l,m= } 5 i= 3 r,s= p (i N rs a (r N lm ã (l ã (m a (s Taejin Lee (KNU APCTP July 5th, / 57

35 Three-Graviton Scattering Amplitude We note that A [3 graviton] can be written also as A [3 graviton] = ( 3 ( 3 g 3 8 dp (i δ p (i i= i= { 3 } 0 h µν (p (i a (iµ ã (iν E Open [3 Gauge] Ẽ Open [3 Gauge] 0. i= Making use of the Neumann functions of the open string N = 4, N = 4, N 33 =, N = N = 4, N 3 = N 3 =, N 3 = N 3 =, N = N = 4, N3 =, Taejin Lee (KNU APCTP July 5th, / 57

36 Three-Graviton Scattering Amplitude We are able evaluate the three-graviton interaction term as follows ( ( g 3 ( 3 A [3 graviton] = dp (i δ p (i i= i= h µ ν (p ( h µ ν (p ( h µ3 ν 3 (p (3 { 4 ηµ µ p µ ηµ µ 3 p µ + 3 ηµ µ 3 p µ 4 ηµ µ p µ ηµ 3µ p µ + } 3 ηµ 3µ p µ { 4 ην ν p ν ην ν 3 p ν + 3 ην ν 3 p ν 4 ην ν p ν ην 3ν p ν + 3 ην 3ν p ν }. Taejin Lee (KNU APCTP July 5th, / 57

37 Three-Graviton Scattering Amplitude A [3 graviton] is precisely the three-graviton interaction term which may be obtained from the Einstein s gravity action. where κ = A [3 graviton] = κ g 7 3 = 3πG 0. 3 ( 3 dp (i δ p (i i= i= h µ ν (p ( h µ ν (p ( h µ3 ν 3 (p (3 { } η µ µ p (µ 3 + η µ µ 3 p (µ + η µ 3µ p (3µ { } η ν ν p (ν 3 + η ν ν 3 p (ν + η ν 3ν p (3ν Taejin Lee (KNU APCTP July 5th, / 57

38 Scattering Amplitude of Four Strings Using the Cremmer-Gervais identity, we may write the scattering amplitude of four closed strings as follows A[,, 3, 4] = g r r<s Z r Z s p(r p(s r<s dz r Z a Z b Z b Z c Z c Z a Z r Z s { d Z a d Z b d Z c αs αr { ( {k (r } exp 4 r,s n,m + ( ( C n0 rs α n (r p(s + C n0 rs α n (r r,s n +τ ( } (r p (r 0. α r r ( ( p (r + αr αs ( C nm rs α n (r α m (s + C nm rs α n (r α m (r p(s ( } p (s Taejin Lee (KNU APCTP July 5th, / 57

39 Shapiro-Virasoro Amplitude Scattering of four closed string tachyons A Tachyon [,, 3, 4] = g d Z Z p( p( Z p( p(3 = g d Z Z ( s 8 Z ( u 8 = πg Γ ( s ( ( 8 Γ t 8 Γ u 8 Γ ( + 8 s ( ( Γ + t 8 Γ + u. 8 Mandelstam variables: s = (p + p, t = (p + p 3, u = (p + p 4. The Koba-Nielsen variables: Z = 0, Z = Z, Z 3 =, Z 4 =. Taejin Lee (KNU APCTP July 5th, / 57

40 Local coordinates for four-closed-string scattering Schwarz-Christoffel transformation 4 ρ = α r ln(z Z r = ln z + ln(z Z ln(z + iπ. r= Equivalently, e ρ = z(z Z z. Interaction points on the complex plane are determined by which has two solutions: ρ z = 4 r= α r z Z r = 0 z ± = ± Z. These two solutions define two interaction points ρ(z ± = ln z ±(z ± Z z ± + iπ Taejin Lee (KNU APCTP July 5th, / 57

41 Local coordinate patches Two interaction points on the world sheet: ( τ = Re ln ( Z, τ = Re ln + Z ( σ = Im ln Z + π, σ = Im ln Global coordinates on the local patches: ζ + τ + iσ iπ, on I ζ + τ + iσ, on II ρ = ζ 3 + τ + iσ + iπ, on III ζ 4 + τ + iσ, on IV., ( + Z + π. Taejin Lee (KNU APCTP July 5th, 08 4 / 57

42 Local coordinate patches ( II III (3 (4 z Z ( I IV (4 ( i ( (3 Taejin Lee (KNU APCTP July 5th, 08 4 / 57

43 Conformal mapping from the string world-sheet to the complex plane The conformal mapping from the local string patches to the complex plane may be written as follows : e ζ = e τ +iσ (z z(z Z, e ζ = e τ +iσ (z z(z Z, on the st string patch, on the nd string patch, e ζ 3 = e τ iσ z(z Z, z on the 3rd string patch, e ζ 4 = e τ iσ z(z Z, z on the 4th string patch. Taejin Lee (KNU APCTP July 5th, / 57

44 Neuumann functions for four-closed-string scattering C n0 rs = C 0n sr = n Z r dz πi z Z s e nζr (z, n, C 0 = eτ +iσ Z C 0 4 = 0, C 0 = eτ +iσ C 3 0 = eτ +iσ C 0 3 = e τ iσ C 0 4 = e τ iσ Z, C 0 = eτ +iσ Z Z, C 0 = eτ +iσ Z, C 4 0 = 0, C 3, C 33 0 = e τ iσ ( Z, C 4 Z, C 0 3 = eτ +iσ 0 = e τ iσ Z, ( Z, ( Z, C 34 0 = C 44 0 = 0, 0 = e τ iσ, C 43 0 = e τ iσ Z, ( Z. Taejin Lee (KNU APCTP July 5th, / 57

45 Neuumann functions for four-closed-string scattering C rs nm = nm Z r dz Zs dz πi πi (z z e nζr (z mζs(z, n, m. C = eτ +iσ ( Z Z 4, C 3 C 4 = eτ +iσ τ iσ C 4 = eτ +iσ τ iσ = eτ +iσ τ iσ = eτ +iσ τ iσ Z, C 3 ( Z 34. C Z = e τ iσ ( Z, Z Z, ( Z. Taejin Lee (KNU APCTP July 5th, / 57

46 Scattering amplitude for four gravitons A [4] = g 0 + { 4 Z 4 4 d Z r<s i= ( r,s {(! 4 r,s ( r,s h µν (p (i a (iµ ã (iν Z r Z s p(r p(s e 4 [4]rr r= C 00 } ( C rs a (r a (s C rs 0ã (r p (s {(! 4 r,s C rs ã (r ã (s + } 0. r,s C rs 0 a (r p (s ( r,s C rs a (r a (s } C rs ã (r ã (s Taejin Lee (KNU APCTP July 5th, / 57

47 A [4] = g 0 d Z Z s 4 Z t 4 hµ ν h µ ν h µ3 ν 3 h µ4 ν 4 { 4 Z ηµ µ η µ 3µ 4 + 4η µ µ 3 η µ µ ( Z ηµ µ 4 η µ µ 3 η µ µ ( Z Z ( Z p (µ 3 + p (4µ 3 +η µ µ 3 ( Z Z ( p (µ + Z p (4µ η µ µ 4 ( Z Z ( ( Z p (4µ + p (3µ η µ µ 3 ( Z Z ( ( Z p (3µ + p (4µ +η µ µ 4 ( Z Z ( Z p (3µ + p (µ η µ 3µ 4 ( Z Z ( Z p (3µ + p (µ ( Z p (µ 4 + p (3µ 4 ( Z p (µ 4 + p (3µ 4 ( ( Z p (µ 3 + p (µ 3 ( p (µ 4 + ( Z p (µ 4 ( Z p (µ 3 + p (4µ 3 ( p (µ + Z p (4µ }{ Z Z Taejin Lee (KNU APCTP July 5th, / 57

48 Kawai-Lewellen-Tye (KLT Relations I = = d Z r<s Z r Z s p(r p(s (Z n Z m ( Z p ( Z q d Z Z m s 8 ( Z q t 8 (Z n s 8 ( Z p t 8. If we write Z = x + iy, then we treat x and y as independent two complex variables. The integrand is an analytic function of y with four branch points, ±x, ±i( x. We may deform the contour of y which is along the real line to the contour along the imaginary axis. ( π I = sin 8 t dξ ξ s 8 ξ t 8 ξ m ( ξ q 0 dη η s 8 η t 8 η n ( η p. Taejin Lee (KNU APCTP July 5th, / 57

49 Kawai-Lewellen-Tye (KLT Relations Factorization ( πt I = sin I (n, pi (m, q, 8 I (n, p = ( p Γ ( u 8 + n + p Γ ( t 8 p + Γ ( s 8 + n, I (m, q = Γ ( s 8 m + Γ ( t 8 q + Γ ( u 8 m q + where n, p, m, q are integers or half-integers. Taejin Lee (KNU APCTP July 5th, / 57

50 Four-graviton scattering Taejin Lee (KNU APCTP July 5th, / 57

51 Using the momentum conservation, on-shell consition s + t + u = 0, and the gauge condition, in the zero-slope limit A [4] = g π 0 h µ ν h µ ν h µ3 ν 3 h µ4 ν 4 { stu ut ηµ µ η µ 3µ 4 + st ηµ µ 3 η µ µ 4 + su ηµ µ 4 η µ µ 3 η µ µ (tp (µ 3 p (µ 4 + up (µ 3 p (µ 4 η µ µ 3 (tp (µ p (3µ 4 + sp (3µ p (µ 4 η µ µ 4 (up (µ p (4µ 3 + sp (4µ p (µ 3 η µ µ 3 (up (µ p (3µ 4 + sp (3µ p (µ 4 η µ µ 4 (sp (4µ p (µ 3 + tp (µ p (4µ 3 η µ 3µ 4 (tp (3µ p (4µ + up (4µ p (3µ }{ µ ν }. Taejin Lee (KNU APCTP July 5th, 08 5 / 57

52 Four-Graviton Scattering Amplitudes B. S. DeWitt, PRD 6, 39 (967: Calculated the four-graviton scattering amplitude by perturbatively expanding the Einstein GR. J. Schwarz, Phys. Rep. 89, 3 (98: Four-graviton scattering amplitude in the type-ii super-string theory (by using graviton vertex operator. 3 S. Sannan, PRD 34, 749 (986: Equivalence between the DeWitt s four-graviton scattering amplitude and that of string theory calculation. Taejin Lee (KNU APCTP July 5th, 08 5 / 57

53 Four-Graviton Scattering Amplitude in the perturbative Einstein GR B. S. DeWitt, PRD 6, 39 (967 Taejin Lee (KNU APCTP July 5th, / 57

54 Four-Graviton Scattering Amp by using vertex operators Γ ( s 8 Γ ( t 8 ( Γ u ( 8 Γ + u 8 A [4] = κ 8 Γ ( + 8 s ( Γ + t 8 { ut ηµ µ η µ 3µ 4 + st ηµ µ 3 η µ µ 4 + su ηµ µ 4 η µ µ 3 h µ ν h µ ν h µ3 ν 3 h µ4 ν 4 ( η µ µ (tp (µ 3 p (µ 4 + up (µ 3 p (µ 4 η µ µ 3 tp (µ p (3µ 4 ( +sp (3µ p (µ 4 η µ µ 4 up (µ p (4µ 3 + sp (4µ p (µ 3 η µ µ 3 (up (µ p (3µ 4 + sp (3µ p (µ 4 η µ µ 4 (sp (4µ p (µ 3 + tp (µ p (4µ 3 } η µ 3µ 4 (tp (3µ p (4µ + up (4µ p (3µ }{ µ ν. Taejin Lee (KNU APCTP July 5th, / 57

55 Classical Solutions with D-Brane Sources and Black p-branes Closed string field theory action { S = Φ KΦ + g ( Φ Φ Φ + Φ Φ Φ + Φ Dp }. 3 Classical equation of motion with a D-brane source, J D Perturbative solutions: Φ = K iɛ J D g K iɛ KΦ + gφ Φ = J D. { K iɛ J D } K iɛ J D The zero-th order by P. Di Vecchia, M. Frau, I. Pesando, S. Sciuto, A. Lerda, and R. Russo (997. Taejin Lee (KNU APCTP July 5th, / 57

56 Higher Loop Corrections in QCD M-Loop Diagrams of YM M-Loop Open String Diagrams with Vertex Operators Tree Diagrams of Closed String with M external states Higher Loop Corrections to Correlators of BMN Operators Taejin Lee (KNU APCTP July 5th, / 57

57 Collaborators String scattering amplitudes: S. H. Lai, J. C. Lee and Y. Yang (NCTU, Taiwan Unitarity of graviton scattering amplitudes: T. Inami (SKKU, RIKEN 3 Classical solutions: Kanghoon Lee (IBS 4 Numerical study and DMFT : Hoonpyo Lee, H. Y. Park (KNU Taejin Lee (KNU APCTP July 5th, / 57