Quantum string test of nonconformal holography

Journal of High Energy Physics, Apr 2017

We compute Lüscher corrections to the effective string tension in the PilchWarner background, holographically dual to \( \mathcal{N} \) = 2∗ supersymmetric Yang-Mills theory. The same quantity can be calculated directly from field theory by solving the localization matrix model at large-N . We find complete agreement between the field-theory predictions and explicit string-theory calculation at strong coupling.

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Quantum string test of nonconformal holography

Received: March Quantum string test of nonconformal holography Xinyi Chen-Lin 0 1 2 3 4 Daniel Medina-Rincon 0 1 2 3 4 Konstantin Zarembo 0 1 2 3 4 Open Access, c The Authors. 0 Department of Physics and Astronomy, Uppsala University 1 Roslagstullsbacken 23 , SE-106 91 Stockholm , Sweden 2 Nordita, Stockholm University and KTH Royal Institute of Technology 3 eld theory by solving the localization 4 SE-751 08 Uppsala , Sweden We compute Luscher corrections to the e ective string tension in the PilchWarner background, holographically dual to N = 2 supersymmetric Yang-Mills theory. The same quantity can be calculated directly from matrix model at large-N . We nd complete agreement between the eld-theory predictions and explicit string-theory calculation at strong coupling. AdS-CFT Correspondence; Gauge-gravity correspondence; Wilson; 't Hooft Contents 1 Introduction 2 The Pilch-Warner background 3 Setup 4 Fradkin-Tseytlin term 5 Bosonic uctuations 6 Fermionic uctuations 7.1 Spectral problem 7.2 Phaseshifts 7.3 Numerics 8 Conclusions A Conventions B The AdS5 S5 limit 7 The semiclassical partition function 7.1.1 Boundary conditions C String partition function in AdS5 C.1 Bosonic uctuations C.2 Fermionic uctuations C.3 The semiclassical partition function D Second order di erential equations WKB expansion of phaseshifts E.1 WKB solutions E.2 Cancellation of divergences E.3 Large momentum expansion for phaseshifts F Numeric error estimate Introduction the large-N limit and then take the 't Hooft coupling 2 = gYMN to be also large. The side of the holographic duality. W (C) = tr P exp ds (iA x_ + jx_ j ) perimeter law: ML= 1 e T ( )ML; T ( ) = probe holography at the the formalism. The Pilch-Warner background W (C) = are summarized in appendix A. The Einstein-frame metric for the PW background is1 [1, 11]: ds2E = (cX1X2) 4 where c 2 [1; 1) and d 2 is the metric of the deformed three-sphere: and are de ned in the SU(2) group-manifold representation of S3, as: i = g 2 SU(2); where i are the Pauli matrices. The function A is given by: while X1;2 are: d 2 = d i = ijk j ^ k; A = c X1 = sin2 X2 = c sin2 + cA cos2 ; + A cos2 : compared to these references. The dilaton-axion is given by: iC(0) = B = e 2i a1 (c; ) = a2 (c; ) = i a3 (c; ) = 1=2sin ; 1=2sin2 cos ; 1=2sin2 cos ; \modi ed" R-R eld strengths are given by: F~(1) = dC(0); F~(3) = dC(2) + C(0) dB(2); and the four-form potential C(4) is given by: where ! = !(c; ) is de ned as: C(4) = 4! dx0 ^ dx1 ^ dx2 ^ dx3; ! (c; ) = where F~(5) satis es F~(5) = F~(5). in nite wall: ularized by a cuto at length L 1. The minimal surface with this boundary is an xc1l = ; ccl = : surface turns around at some c0 1 and goes back to the boundary. As shown in [4], the the large-distance cuto L. We will make the same assumption in calculating quantum corrections to the minimal area law, and will study quantum uctuations of the string around the simple in nite-wall con guration. string on S5 is dictated by the scalar coupling of the Wilson loop (1.1): cl = 0; cl = 0; = 0. e =2jcl = 1=p where now A equals to compared to the Einstein metric in (2.2), is ds2w:s: = Sreg = 3 = where integration over ranges from The area law in the PW from localization. One is quantum without a factor of 1= 0 [12, 13]. The Fradkin-Tseytlin term is usually ignored in dilaton, for instance AdS5 S5, where the Fradkin-Tseytlin term is purely topological. But term has to be taken into account. It has been long recognized that string uctuations play an important role in gaugeall higher-order around the minimal surface and integrating out the uctuation modes in the one-loop of the string sigma-model in AdS5 S5 was developed in [23] and has been successfully using integrability of the AdS/CFT system [27, 28]. The formalism of [23], originally developed for strings in AdS5 S5, uses the Greenlution: X = Xcl + then yields: W (C) = e Scl det 2 KF ; 1 as the one-loop partition function. determinants that appears in (3.5). Fradkin-Tseytlin term The bosonic part of the sigma-model Lagrangian is LB = hhij @iX @j X G where G denotes the background metric in the string frame and B is the B- eld. We on the classical solution (3.3). ton [12, 13]: LFT = The full bosonic action of the sigma-model is SB = LB + LFT ; evaluating the Fradkin-Tseytlin term. The curvature of the induced world-sheet metric (3.3) is equal to hR(2) = 2 solution, we have: Integration by parts gives Combining the result with (3.4), we get: jcl = SFT = Scl = uctuations For the conformal factor in the string frame we get: (cX1X2) 4 = 1 + the Cartesian four-vector in the tangent space, Up to the requisite accuracy, the string frame metric takes the form: ds2 = i = y = n; in uctuations. B = The Maurer-Cartan forms on S3 can be written as B = i = minnmdnn; B = B- eld becomes L(2) = 2 A 2 A2p 2 1 A 2 where x is the three-dimensional vector of transverse uctuations of the string in the 4d space-time directions. In the derivation we used the identities A0 = 2 A00 = Lagrangian describes the eight transverse modes of the string. elds with appropriate -dependent factors.2 After the requisite eld rede nitions, we get the following uctuation Hamiltonians and multiplicities for the three types of modes, x, , and y: The uctuation operators that enter (3.5) are de ned as of the 't Hooft symbol: Kx = Ky = Ky = Nx = 3 = 1 Ny = 2; S(2) = X Z h = K = world-sheet metric (3.3): unit coe cient: The y- uctuations decomposed into two identical 2 2 systems upon relabelling of indices. Those can be further disentangled by a similarity transformation: U = p U yKyU = X = E a^, where the rescaling a^ ! a^ and partial integration in the action allows us to write rescaling for fermions, thus preserving the measure of the path integral. Ky = K~y tuation operators of the bosonic modes: Kx = = Kx Ky = Kx + 1 1)@2 + A (4 3A ) @ ; The simplest relation is the time reversal symmetry Since the determinants are time-reversal invariant, that maps Ky+ to Ky . + det Ky = det Ky : written in a factorized form by introducing the rst-order operators L = Ap 2 Ly = which are Hermitian conjugate with respect to the scalar product It is easy to check that The operators Kx and K , as a consequence, are intertwined by L and Ly: h 1j 2i = Kx = @2 + LyL; @2 + LLy: KxLy = LyK ; LKx = K L; and their eigenfunctions are related: same spectra and equal determinants:3 / L x. The two operators therefore have the can be written as 3For the intertwined operators Kx and K to have the same spectra it is also necessary that the map is compatible with the choice of boundary conditions. The latter are discussed in section 7.1, and by looking at the ! 1 behaviour of the eigenfunctions, we con rmed that this is indeed = det Kx: L(F2) = JK JK E= j : (6.1) The fermion eld I is a 32-component Majorana-Weyl spinor subject to the constraint 11 I = ^1 ^2:::^n = while Dj and F JK are de ned by: Dj = @j + JK = n=0 (2n + 1)! F(2n+1) ~ ^1 ^2:::^2n+1 ^1 ^2:::^2n+1 (2n+1) : JK matrices de ned by: (1) = (3) = 1 ; (5) = that are in between on the classical solution (3.1), (3.2). To do this, we use the orthonormal frame E ^: Lagrangian:4 L(2) = 2 h +ic(R1R) ic(N3S)NS 4The fermionic operator presented here was calculated using the coordinate of references [1, 2, 11] for which cl = where the coe cients are c(1) = c(!) = c(R3R) = c(R5R) = (2 + A) c(2) = A c(R1R) = c(N3S)NS = 1 = 2 = appendix A. We used the identities (5.10) and the positive chirality condition pendix A, yields the following form of the fermionic Lagrangian: L(2) =2 h + ic(R3R) c(R1R) 1^^4^9 ic(N3S)NS the string in AdS5 S5 from [23, 31]. Dirac matrices. We take the following representation for the 4 4 Dirac matrices a^ and a^0 described in appendix A ^0 = i 2 ^50 = operator in terms of 2 2 operators, instead of more complicated 4 4 operators that one As in the case of bosons, we rescale the elds in order to normalize the coe cient in front of @ to one. The requisite rescaling is the help of eqs. (5.10): D0 = = @ The operators D are related by time reversal: det KF = det4 D0 det2 D+ det2 D = det4 D0 det4 D : The semiclassical partition function in the analysis of the bosonic modes: Squaring the Dirac operator, we nd: D0 = ( 3D0)2 = Kx and K . Since Kx and K = det4 Kx, and W (C) = e Scl det2 D L = Ap 2 Ly = ; (7.4) tor (6.11) then takes the form: The operators Ky can also be neatly expressed through L, Ly: Ky = Using the formula for the determinant of a block matrix: = det AD BD 1CD if [C;D]=0 = det its original Dirac form. det Ky = det will be explained in section 7.1. By introducing two Dirac-type Hamiltonians: we can bring (7.3) to the form: HB = HF = W (C) = e Scl det2 (@ det2 (@ the one-loop correction to the Wilson loop expectation value. Spectral problem ln det (@ H) = L tr ln (i! tors (7.10): can absorb the measure into the wavefunction: = E : = ( 2 = E The resulting eigenvalue problem, HB = HF = L^ = Ap 2 L^y = = 1 and ! 1. Near the boundary, A = 1 + O (( The potential terms in the intertwiners vanish at in nity, and (7.16) become free, massive Dirac operators. bouncing o an in nite wall at of which the eigenvalue is given by At large , The Dirichlet boundary conditions for the string uctuations require the growing, nonnormalizable solution to be absent: for the (normalizable) eigenfunction is: and the Dirac operators (minus the eigenvalue) asymptote to ansatz (proportional to a constant vector) two solutions are found: E = @ A + O(1) = 0: A = E = to holes. The asymptotic wavefunctions are plane waves: B ' CB1 @ B;F (p) are the phaseshifts experienced by particles/holes as they re ects from the wall at Phaseshifts we can impose ducial boundary conditions at some large = R. For instance, (1 + 3) (R) = 0: pnR + (pn) = (p) = 1 d (p) from which we nd the density of states: on the O(1) momentum-dependent distortion due to the phaseshift. Taking into account (7.26), we rewrite (7.12) as ln det (@ H) = L ln i! + Integration by parts gives H) = L 4 Z +1 d! Z 1 dp The integral over ! is a half-residue at in nity and we nally obtain: ln det (@ H) = 4 Z 1 dp We can write: T ( ) = to express the determinants in (7.11) through phaseshifts we get: The large-N localization predicts and the complete result is UV nite, at least in the one-loop approximation. Cancellation cancel out as expected. Another check on our formalism is to see that in AdS5 S5 the quantum string asymptotes to AdS5 S5 near the boundary, and the AdS result can be viewed as a limiting rst, prior to computing the phaseshifts (see appendix B). The AdS5 in AdS5 in order to evaluate conditions (7.23) at ! 1, and then numerically evolve the wavefunctions far away from functions to plane waves and nd their phaseshifts. This procedure is done for a range in appendix F. are plotted in shifts (E.8) is also shown, displaying nice agreement for large p. making the area under the curve, and with it , a nite quantity. = 1:01 Conclusions area under the curve, , is given in equation (7.36). agreement with the eld-theory predictions. has been observed [37], and to backgrounds with N theory predictions are available yet. Acknowledgments Conventions In this article we chose Minkowskian signature ( + + : : : +) for the background metric for indices used here is given by: a^, ^b, c^ = 0, 1,. . . , 4 a^0, b^0, c^0 = 5, 6,. . . , 9 ^, ^, ^, ^. . . = 0, 1,. . . , 9 , , , . . . = 0, 1,. . . , 9 i, j = 0, 1 I, J, K = 1, 2 AdS5 tangent space indices S5 tangent space indices S5 tangent space indices S5 coordinate indices World-sheet indices Spinor indices ric ^^ = ( 1; 1; : : : ; 1), for the indices we will use the background metric tensor G , rally, coordinate indices and tangent space indices ^ are related using the standard vierbein prescription: = E ^V ^ ; V ^ = E ^ = E ^E ^ ^^ : the properties: Just as in [40], we will choose matrices a^ and a^0 such that: (a^ ^b) = a^^b = ( + + + +) ; (a^0 ^b0) = a^0^b0 = (+ + + + +) ; a^1a^2a^3a^4a^5 = i a^1a^2a^3a^4a^5 ; a^0 a^0 a^0 a^04a^05 = a^0 a^0 a^0 a^04a^0 : 1 2 3 1 2 3 The 32 following way: a^ = a^0 = 1 C = C where 1 is the 4 4 identity matrix, i are the Pauli matrices, while C and C0 are the y 0^ = T C. In 10 dimensions, a positive chirality 32-component spinor can be decomposed in the following way: here on the left hand-side ^ corresponds to a 32 tion (A.4), while positive chirality. On the right hand-side, K = 0K (K = 1; 2) is a 16-component spinor, while the 16 16 matrices a^ and a^0 represent a^ 1 and 1 a^0 , respectively. In the main text, a^ and a^0 denote these 16 16 matrices unless otherwise speci ed. Equawith 32 32 matrices into lower dimensional 16 16 matrices. The AdS5 S5 limit The Pilch-Warner background asymptotes to AdS5 S5 near the boundary. To see this, ds2E = dx2 + dz2 + d 2 + cos2 d 2 + sin2 d 2 which is the usual metric of AdS5 S5 with d 2 describing the three-sphere d 2 = z = solution c = , while the AdS5 S5 computation in [23, 31] employs as classical solution . This means that the spatial world-sheet coordinates of the Pilch-Warner and S5 computations are related by PW ! 1 + A2dS . In order not to overload our relation between the two. Having AdS5 S5 a trivial dilaton, we see from (B.1) that the operators presented in sections 5 and 6. To obtain the appropriate AdS5 is necessary to simultaneously make the substitutions expand to rst order in ! 0. For the bosonic operators in (5.19){(5.21), this results in S5 operators it 1 @ , and then Ky ! Nx = 3 = 1 Ny = 2 expected as AdS5 S5 has no B- eld. As we will see in appendix C.1, the above bosonic operators are related to the ones found in [23, 31]. To take the AdS5 S5 limit of the fermionic operator (6.6), we will perform the same ux in AdS5 S5 is the ve-form and there is no NS-NS three-form. After the required substitutions and expansion, the nal result is is a 16-component spinor and a^ are the 16 16 matrices described in apve-form, respectively. in order to have a worldMinkowskian world-sheet signature. String partition function in AdS5 S5 background was rst done in [23]. The straight Wilson line in AdS5 S5 has trivial expectation value, not renormalized by quantum corrections. A particularly simstraight line in AdS5 S5 is given in the appendix B of [41]. Here we illustrate how the we use in the main text for the Pilch-Warner background. resulting operators have a structure similar to the Kx, K and D0 Pilch-Warner operators. S5 case. Due to the much simpler eld content of AdS5 S5, there is no need for consistency check of the formalism that we use. tion function will consist exclusively of the quadratic transverse uctuations. As shown remaining AdS5 transverse modes). The contribution of these uctuations is given by the action [23, 31] S2B = a^;^b 2f0;2;3g Comparing with the standard normalization for bosons R p d d , we see that the ogous to the one done for bosons in section 5 be written in the following way S(2) = a^;^b 2f0;2;3g given by the determinant of 2 di erential operators: ZBosons / det which are naturally the AdS5 equations (B.2)). We will take as starting point the AdS5 S5 limit of the fermionic Pilch-Warner oper S5 limits of the bosonic Pilch-Warner operators (recall L(2) = 2 h KF , can be seen as a 16 16 block matrix composed of 8 identical 2 2 blocks. In order to cancel the front of the @ derivatives, we perform the following eld rede nition a slightly di erent choice of representation for the Dirac matrices. L(2) = 2ph X written in terms of the determinant of a 2 2 operator ZFermions / det Hi i = pi2 i with i 2 f0; 1; 2g: is a consequence of the choice of boundary conditions. in (C.8), we consider the square of this operator ZFermions / det to the one observed for the Pilch-Warner operators of equation (7.2). The semiclassical partition function and (C.9), we have that the semiclassical partition function is given by8 tr ln(!2 + H2)d! ; 1 L Z H1 = de nitions for the operators H1;2 H2 = in rewriting the spectral problems (C.12) in at space. Moreover, having no linear time 1 form without recurring to a 2 2 representation, as was done for the PW case. general, we pick the solution with the highest power of as this will provide an adequate normalization condition. As in the Pilch-Warner case, string uctuations will be thought in nite walls ( = 0) = 0 ; ( = R) = 0 ; 8i 2 f0; 1; 2g tization condition. For the asymptotic operator H0, we see that the eigenfunction 0 has an oscillatory 0 = A0 sin (p0 ) ; the eigenfunctions are of the form Ai sin (pi + i (pi)) : asymptotic operators Hi (i 2 f1; 2g) will be given by p0R = piR + i (pi) = the partition function as Z / Exp 1 L Z tr ln(!2 + H0) d! ; 2 = A2 sin (p2 ) : By comparing the large behaviour of these solutions with equation (C.14), we see that 1 = for the semiclassical partition function operators H1;2 H1 = LLy ; H2 = LyL ; where L = @ 2 and Ly = to the scalar product h 1j 2i = RR d d 2 @ , with the latter being conjugate operators with respect 2 / Ly 1 is compatible with the choice of boundary conditions. Second order di erential equations Here we will write down the explicit equations for our operators H^B;F : OB1;F 2 = OB2;F 1 = 1)A2@2 + 2A(2 3 A)@ + 1 E2 + (1 + E)V ( ) + UB1;F 2( ); 1)A2@2 + 3 A) + E2 + (1 + E)V ( ) + UB2;F 1( ); 1 A2V 0( ) E + V ( ) 9Note that the zero modes of L and Ly, behaviour at large and are therefore excluded in the phaseshift computation. UB1( ) = UF 1( ) = UB2( ) = UF 2( ) = E + V ( )) AV 0( ) E + V ( )) 24 4 + 6 2 + 3 A + 16 3 ; 24 4 + 6 2 + 3 A + 16 3 1 A + 16 3 + 8 1 A + 16 3 + 8 where the di erent U( ) are: equations for Si0( ) that can be solved recursively. Then, Si( ) = Si0(x) dx: WKB expansion of phaseshifts 1=p to be small). Our WKB ansatz is written as ( ) = eiS( ); S( ) = p X p iSi( ) (p ! 1): determinant, and the real part is related to the phaseshift by: Re(S( )) = p + (p) WKB solutions , respectively) is given by: S00;B;F; = S10;B;F; = S20;B; = S20;F; = S30;B; = S30;F; = 2 A2 + 4 (A 2 A2 + 4 (A 1)3=2 1)3=2 A + 9 4 + 2 + 2 A2 3 2 + 2 A + 6 4 + 2 + 1 A2 + 4 2 1)3=2 1)3=2 2 + 4 A + 6 4 + 5 2 + 5 A2 tions above. book.10 Cancellation of divergences which comes from the S40 terms. Using (E.3), the phaseshift di erence at large p is: B = 2048 p3 81 3 860160 3880800 2 + 1245825 3 18350080 p5 = 17:2856 p 3 + 139:805 p 5 + : : : 10See the online repository https://github.com/yixinyi/PhaseShiftMethod. Large momentum expansion for phaseshifts The large-p behaviour for is (p) = p 0 + 1 + (p ! 1): For the leading order, we absorb the linear term in (E.3) into the integration by using the identity compute the next-to-leading order term in the limit domain as shown below: ! 1, by splitting the integration 0 = where we used the expansion as explained in appendix F. i = Re Si0 (x) dx; i = 1; 2: modes, is: Numeric error estimate Our numeric algorithm11 consists of three main parts: at " = 3 ; max], where = 2p denotes the wavelength. 1, for a region of order max that we chose to be the interval [ max PhaseShiftMethod. 3. Numerical integration of (7.35) over a nite range [pmin; pmax]. the result, and larger a sum, namely for the standard error propagation formula gives error( ) = p tuX error(fi)2; fi = pi = i p: error( ) = integrand that multiplies the phaseshifts. the integrand error is Putting all the numbers together, we have that error(fi) = error( ) = error( )pmax = which is much smaller, hence the total error is the numeric error: 12Higher order Sn0( ) O( 2) as well, but they are subleading in large p. error( )total = Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. B 594 (2001) 209 [hep-th/0004063] [INSPIRE]. Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE]. (2014) 001 [arXiv:1311.1508] [INSPIRE]. coupling, JHEP 11 (2014) 057 [arXiv:1408.6040] [INSPIRE]. Math. Phys. 181 (2014) 1522 [arXiv:1410.6114] [INSPIRE]. 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Quantum string test of nonconformal holography, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)095