Gauge/gravity correspondence [1] is a proposal for concrete realizations of the holographic principle [2]. There have been exciting developments in which quantum gravity on various spacetimes is proposed to be equivalent to gauge theories defined at the spatial boundary. Gauge/gravity correspondence is now being applied to theories beyond the original proposals based on string/M-theory, in such areas as nuclear physics and condensed matter physics (see, e.g., Refs. [3–6] for reviews). It is widely regarded as a powerful tool that allows one to study strongly coupled quantum theory by gravity that is simple.

Gauge/gravity correspondence has not been proven yet, and we do not know how (or why) it works. An unresolved issue that should be closely related to the proof is what kind of correspondence exists, if any, when gauge theory is weakly coupled. There have been very few studies¹ on this limit, compared to the active studies on the other strong-coupling limit. The purpose of this letter is to point out an intriguing fact about this limit, which may suggest a possibility for a new approach towards establishing holography in the weak-coupling limit.

We consider gauge/gravity correspondence associated with two descriptions of D-branes, namely the conjectured equivalence between the D maximally supersymmetric Yang–Mills theory and superstring theory on the near-horizon limit of the D-brane solution [21]. The former is the low-energy effective theory on the D-brane worldvolume based on the open-string lowest mode, and the latter gravity description is based on the closed-string degrees of freedom.

The case is the well studied AdS/CFT correspondence between the (3+1)D supersymmetric Yang–Mills theory and superstring theory on AdS [1]. We will consider the general case with . For , there is no conformal invariance, and exact results are hard to obtain, but there have been quantitative studies (especially for the case) [22–30,32,33,35]. Correlation functions at strong ’t Hooft coupling have been obtained by tree-level supergravity. The results for have been confirmed by Monte Carlo simulations [28,29]², providing strong evidence for the gauge/gravity correspondence without conformal symmetry.

We will consider gauge theory at zero temperature, and study the two-point function of a single-trace operator with large angular momentum on the transverse , which was introduced by Berenstein, Maldacena, and Nastase (BMN) [36] in the study of strings on the plane wave background. We show that the free-field result of gauge theory can be reproduced by string theory, by making an assumption that a string is composed of “bits”, which is natural in view of previous work [16,36–41]. In the following, we will first review gauge/gravity correspondence for general at strong ’t Hooft coupling, so that readers who are not familiar with this topic can follow the line of argument. The main result about weak ’t Hooft coupling is presented near the end.

The near-horizon limit of the metric and the dilaton for the zero-temperature D-brane solution in the string frame are (1) where and with . The integer is the number of D-branes, and is the string length. The Yang–Mills coupling is given by .

For , the near-horizon geometry is AdS; for general , it is related to AdS by Weyl rescaling, (2) where the radial variable in the Poincaré coordinates for AdS is defined by (3)

The boundary is at . The distance in gauge theory roughly corresponds to the region [42,43], as we can see, e.g., in the geodesic approximation.

For , the dilaton and the curvature depend on the radial position, and the background does not have AdS isometries; correspondingly, the gauge theory does not have conformal invariance. Nevertheless, the representation (2) is useful; since null geodesics are not affected by the Weyl factor, supergravity modes, which are massless in 10 dimensions, show similar behavior (obeying a power law) to the conformal case at tree level.

In this work, we always assume that the string coupling is small, , so that we can ignore string loop effects. For , this is satisfied in the near-boundary (UV) region, , and, for 3$]]>, the center (IR) region, . If curvature is small in string units, string higher excitations can be ignored. For , this is satisfied in the IR region, , and for 3$]]>, the UV region, . If we take with the ’t Hooft coupling fixed but large , the above two conditions are satisfied in almost the whole near-horizon region [23]. In that case, the tree-level supergravity is expected to be valid. When we take ’t Hooft coupling to be small, we should consider strongly curved background.

We will focus on the two-point functions of a “BMN operator” [36], (4) where is a complex combination of two of the scalar fields (that arise from dimensional reduction of the (9+1)D gauge field) in the D maximally supersymmetric Yang–Mills theory. The integer corresponds to a quantum number for the symmetry (angular momentum along a great circle in the – plane).

For , this operator is one of the BPS operators, which belong to a short representation of the superconformal algebra. Its scaling dimension is given by the free-field value , and is protected against quantum corrections. The BPS operators correspond to supergravity modes.

For , the analogs of the BPS operators, which are related to the ones for by “T-duality” (or dimensional reduction), couple to supergravity modes [44–46]. The full spectrum of the supergravity modes has been obtained for the D0-brane background [23,24], and the corresponding operators in the D SYM have been identified [23,24] with the help of “generalized conformal symmetry” [47]³. The operator (4) belongs to defined in Ref. [44], and corresponds to the supergravity mode called in Ref. [23].

We assume that the general relation due to Gubser, Klebanov, Polyakov [48], and Witten [49] (GKPW) between the bulk partition function and the generating functional for correlation functions in gauge theory holds: (5)

Here, is the boundary condition for a bulk field , imposed at the D boundary of the AdS-like space, and is the operator that couples to . Calculations are performed in the Euclidean signature. In the limit of weak string coupling, the bulk partition function is given by the classical action, and can be calculated by tree-level supergravity using bulk-to-boundary propagators. By applying the GKPW prescription to the near-horizon D-brane background, the two-point function of the operator (4) has been found to be [23] (6) where we have ignored an overall constant factor. The constant in the exponent takes the value⁴

The correlator (6) obeys a power law⁵, even though the coupling constant has a dimension for . The power is different from the free-field value for . Strong-coupling dynamics together with supersymmetric cancellations should be responsible for this behavior in gauge theory, but the mechanism is not understood analytically yet. For , this power law has been confirmed by Monte Carlo simulation [28,29].

Supergravity modes are massless in 10D spacetime, but when we regard them as Kaluza–Klein modes, angular momentum on corresponds to mass⁶ in AdS. In the large- limit, the geodesic approximation can be used to obtain the correlator.

We will consider Euclidean AdS. Then, a geodesic of a massive particle connects and on the boundary (separated, e.g., in the Euclidean time direction). We will regulate the infinite volume of the AdS, and put the boundary at the radial position . The geodesic is a half sphere in the Poincaré coordinates. In terms of the proper time , it is given by (8) where is a parameter corresponding to the distance of the two points along the boundary . The geodesic reaches the boundary at an infinite proper time; thus we introduce a cutoff . The relation between the cutoffs and is given from Eq. (8) as .

The two-point function is obtained by evaluating the geodesic length: (9)

The exponent indeed agrees with the leading (-dependent) exponent in Eq. (6) obtained by the GKPW prescription.

The null geodesic in 10D spacetime, given by Eq. (8) together with the part, is a classical solution of the closed-string worldsheet theory, in which the string is in a point-like configuration. Fluctuations around the null geodesic can be studied by expanding the string action to the quadratic order around the classical solution⁷. After eliminating unphysical modes by gauge fixing and the use of constraints, we get eight bosonic and eight fermionic fields, which are massive on the worldsheet [25,26]⁸. The origin of their mass is the curvature of the background spacetime.

The bosonic part of the quadratic action is [25,26]⁹ (10) where , and . The radius of the direction is proportional to the angular momentum, [25,26,29]. The terms involving the spatial derivative have a factor , which depends on the position on the geodesic, so the frequencies of string excited states are time dependent. The mass for the bosonic fields depends on whether the field describes fluctuations along AdS or [25]: (11)

The quadratic action for fermions has been obtained similarly, starting from the Green–Schwarz action [26]. The mass for the fermionic fields is¹⁰ (12)

The ground state of the closed string corresponds to the operator (4) [36]. The classical amplitude (9) is corrected by contributions from the fluctuations¹¹. The frequencies of the lowest (-independent) modes are equal to the masses (11) and (12). Their contribution to the zero-point energy is (13)

By including this correction¹² in the classical result (9), we recover the GKPW result [26], (14) with the value of given in Eq. (7).

Up to now, we have reviewed the framework for gauge/gravity correspondence for the D-brane and the analyses are expected to be valid at strong ’t Hooft coupling. Assuming the validity of this framework, let us now consider what happens at zero ’t Hooft coupling.

We make one additional assumption: we assume that the operator (4) corresponds to a superstring whose worldsheet spatial direction is discretized into bits¹³. This is natural if we recall the proposal by BMN [36] (see also subsequent work [16,37–41]) that string excitations on the ground state (4) are represented in gauge theory by inserting “impurities” (fields with , or derivatives with ) into the sequence of .

Weak ’t Hooft coupling corresponds to strongly curved backgrounds. In this limit, the string tension is much smaller compared to the scale of the curvature of the background. If string tension is strictly zero (corresponding to zero ’t Hooft coupling), no binding force exists between bits, and we can think of the string as a collection of non-interacting bits (particles). In this case, the zero-point energy would be the sum of contributions from bits, i.e., times , which we obtained in Eq. (13). By including this correction, the correlator now becomes (15)

This is the free-field result: the mass dimension of a scalar field in dimensions is , and the operator consists of scalar fields.

We have shown that the free-field result of the D maximally supersymmetric Yang–Mills theory can be reproduced from the bulk string theory for the two-point function of a particular operator . Our result is based on a natural assumption that the string corresponding to this operator is made of bits. We regard this result as strong evidence that gauge/gravity correspondence works for general and also at weak ’t Hooft coupling.

This result suggests a very concrete picture of strings at strong curvature, i.e., at substringy distances: The mode that we usually consider to be a graviton is really a collection of bits. It is important to fully elucidate its consequences. (This is not particular to ; it is equally important for .)

It remains to be seen whether our approach can be promoted to a calculational framework applicable for finite ’t Hooft coupling. Having reproduced a zero ’t Hooft coupling result from the bulk, it is hoped that perturbative expansions of gauge theory in terms of ’t Hooft coupling can be reproduced as well¹⁴. Discretized string action is not unique; thus one should identify the interactions between bits appropriately for this purpose.

Let us make some comments to clarify the meaning of our proposal. First, in our proposal, we are assuming . In this limit in which we can ignore string loops, we should be able to treat the background as fixed and non-fluctuating, even though it is strongly curved¹⁵. (In that sense, we are not really in the quantum gravity regime.) Second, in the present work, we considered the large- limit, which allowed us to perform a semi-classical study of string theory, with its gauge symmetry completely fixed. It is important to formulate the correspondence at weak ’t Hooft coupling for finite , without the semi-classical approximation and gauge fixing, whether or not it is calculationally feasible.

A crucial question is whether one can construct a consistent string theory with a discrete worldsheet. By discretization we may lose characteristic features of string theory¹⁶ such as the following ones. First, open–closed string duality is due to the presence of an infinite tower of modes on the worldsheet. By discretization, the closed-string level number is cut off at . This is not necessarily a problem, since the SYM theory consists of only the lowest modes of the open string. It is very important to examine whether is just the right number of degrees of freedom necessary for the duality in this context to hold. Second, the elimination of the UV divergence in string theory is due to modular invariance, which is a remnant of conformal symmetry. It is not at all clear what its counterpart is in a discretized theory. For our proposal at , the problem of string loop divergence is not relevant, but if we want to extend it to finite , this problem would be very important.

To gain insight into the above issues, it would be helpful to extend our analysis to more general cases. One direction would be to a class of operators with spins along the AdS direction. These correspond to macroscopic strings rotating in AdS [59]¹⁷. Another direction of extension would be to finite temperature. Finding the correct type of discretized string theory in this case may shed light on the Hagedorn behavior [60].