Spiky strings, light-like Wilson loops and pp-wave anomaly
We consider rigid rotating closed strings with spikes in AdS dual to certain higher twist operators in SYM theory. In the limit of large spin when the spikes reach the boundary of AdS, the solutions with different numbers of spikes are related by conformal transformations, implying that their energy is determined by the same function of the ‘t Hooft coupling that appears in the anomalous dimension of twist 2 operators or in the cusp anomaly. In the limit when the number of spikes goes to infinity, we find an equivalent description in terms of a string moving in an AdS pp-wave background. From the boundary theory point of view, the corresponding description is based on the gauge theory living in a 4d pp-wave space. Then, considering a charge moving at the speed of light, or a null Wilson line, we find that the integrated energy momentum tensor has a logarithmic UV divergence that we call the “pp-wave anomaly”. The AdS/CFT correspondence implies that, for SYM, this pp-wave anomaly should have the same value as the cusp anomaly. We verify this at lowest order in SYM perturbation theory. As a side result of our string theory analysis, we find new open string solutions in the Poincare patch of the standard AdS space which end on a light-like Wilson line and also in two parallel light-like Wilson lines at the boundary.
The AdS/CFT correspondence provides a concrete example of the relation between gauge theories in the large-N limit and string theory malp (). In particular, SYM theory is seen to be dual to type IIB string theory on . In establishing this relation an important role is played by classical string solutions that can be mapped to “long” gauge-theory operators with large effective quantum numbers. An example is provided by strings rotating in the part of ; it improved our understanding of the AdS/CFT and produced numerous interesting checks of the correspondence (see rev () for reviews).
Another interesting case, that will concern us here, is the relation between strings rotating in and twist two operators GKP () as well as its generalization to the relation between spiky strings and higher twist operators bgk (); k (). In field theory at weak coupling kor (); make () and also, via AdS/CFT, at strong coupling kru (); Makeenko (); krtt (); am1 (); am2 (), it can be seen that the anomalouos dimension of twist two operators grw () is related, for large spin, to the cusp anomaly of a light-like Wilson loop. The cusp anomaly defines a function of the ‘t Hooft coupling ,
which has been studied at small coupling using SYM perturbation theory lip () and at strong coupling using string sigma model perturbation theory ft2 (); ftt (); rt (). Recently, the development of the integrability approach culminated in the proposal of an integral equation es (); bes () that describes the function to any order in both weak bes () and strong kle (); bas () coupling expansion and which passed all known perturbative tests.
In the present paper we consider the higher twist operators of the general form which have spin and twist and which where argued in k () to be dual to certain rotating spiky string configurations in . In the limit of large spin, keeping the number of spikes fixed, the corresponding string solutions have their spikes reaching the boundary of and are dual to Wilson loops with parallel light-like lines.
We study these solutions concentrating on the “arcs” between the two spikes. The shape of these arcs is determined solely by the angle between the two spikes. Moreover, we show that the solutions corresponding to different values of this angle are related to one another222This was independently observed by M. Abbott and I. Aniceto by embedding the spiky solutions in the sinh-Gordon model Abbott (). by isometries of . Since the folded rotating string of GKP () is a particular case when the angle between the spikes is , this implies that the anomalous dimensions of all the dual operators should be determined by the same function .
A case of particular interest is when the angle between the adjacent spikes becomes small (). This corresponds to the number of spikes going to infinity and one can then concentrate just on a single arc. It turns out that an appropriate way to take this limit is by a certain rescaling of the coordinates in the metric. Then the metric in global coordinates reduces to a background that can be interpreted as a pp-wave on top of in Poincare coordinates. At the same time, the boundary metric transforms from to a 4d pp-wave, one which is also conformal to the Minkowski space .
The string solution in this limit ends on two parallel light-like lines at the boundary. Computing the conserved momenta associated to such solution it follows that the function is determined just by a divergence near the boundary and can be found by considering a surface ending on just a single light-like line.
From the point of view of the boundary theory, i.e. gauge theory in the 4d pp-wave background, the light-like Wilson line corresponds to a point-like charge moving at the speed of light in the direction in the pp-wave metric
We should then compute
where is a UV cutoff and is the expectation value of the null component of the momentum operator in the presence of the Wilson line
Here is the gauge theory energy-momentum tensor. The function which controls the divergence of the energy-momentum near the charge may be called a “pp-wave anomaly”. The analysis on the string-theory side suggests that it should be related to the twist 2 anomalous dimension, or the cusp anomaly, by
This leads to a novel interpretation of the cusp anomaly in the case of a conformal field theory. If the gauge theory is not conformal is not necessarily related to the cusp anomaly, and defines a new quantity which may be interesting to study.
The paper is organized as follows. In section 2 we shall consider the infinite spin limit of the spiky string solution of k () and show the equivalence of its single-arc portion to the straight rotating string by performing a conformal boost in global coordinates of .
In section 3 we shall focus on a special case when the arc between the two spikes gets small and approaches the boundary (i.e. when the number of spikes in the original solution goes to infinity). It can be studied by first taking a certain limit of the metric of the global space (analogous to the Penrose limit) that produces the metric in Poincare coordinates with a special pp-wave propagating on top of it.
In section 4 we shall find the string solution in this pp-wave background that corresponds to the original small-arc configuration, i.e. a world surface of an open string that ends on two parallel null lines at the boundary of . We shall compute the null components of the corresponding momentum showing that has a logarithmic UV divergence and that . We shall also consider a “half-arc” solution that ends on a single null line and which is already sufficient to compute the pp-wave anomaly coefficient at strong coupling and find that it is equal to the strong-coupling value of the cusp anomaly (1.5).
In section 5 we shall perform the corresponding computation in the boundary gauge theory and confirm the relation (1.5) also at leading order in weak-coupling expansion. Section 6 will contain some concluding remarks.
In Appendix A we will show that the pp-wave background found in section 3 is still locally equivalent to Poincare patch of and also demonstrate how to construct open string solutions in that end on two or one null lines at the boundary. Appendix B will give details of the solution of the Maxwell equations in 4d pp-wave background with a light-like source which is used in section 5. Appendix C will contain a discussion of a generalization of the infinite spin spiky string solution from section 2 to the case of a non-zero angular momentum in .
Ii Infinite spin limit of the spiky string
Below we shall consider a limit of the rotating spiky string of k (), namely, the large spin limit, for fixed number of spikes, in which the end-points of the spikes reach the boundary of the . Since the spiky string is a generalization of the rotating folded string of GKP (), the limit is similar to the one in ft2 (); ftt (). In particular, the limiting solution is sufficient in order to reproduce the large spin limit behavior of the energy (). Also, since this limiting string touches the boundary, the corresponding world-surface has, as in krtt (), an open-string, i.e. Wilson loop, interpretation.
ii.1 Rotating spiky string solution
Consider the part of metric
and a rigidly rotating string configuration described by the ansatz
The Nambu string action and conserved quantities are given by
where is the string tension and we defined the momenta as , . Here is the energy and is the spin.
As follows from the above action, the first integral of the equation for is k ()
where the constant of integration will be the minimal value of . The maximal value corresponds to , i.e. . The resulting solution describes a string with spikes. For large spin we obtain the relation
where is the number of spikes and the terms we ignore are constant or vanishing in the limit. This energy-spin relation is determined entirely by the infinite spin limit. This is also the limit when , i.e. it corresponds to the case when the ends of the spikes approach the boundary of . In that limit the shape of the string simplifies as we discuss in the next subsection.
ii.2 Infinite spin limit
Let us consider the solution of the previous subsection in the limit when the spikes touch the boundary. Such solution corresponds to the value and gives the dominant contribution to the energy at large spin. Interestingly, it happens to have a very simple analytic form. As follows from (2.6) for
so that integrating this equation we get
We can choose so that and . Then
Since when we have where is the number of spikes. In general, the spiky string is a function of two parameters, the angular momentum and the number of spikes or, equivalently, , and – the radii at the valleys and the spikes. Since we took , keeping fixed, only remains as a parameter, or equivalently the number of spikes given here by .
Since in this limit the spikes touch the boundary, we can now take only one arc between the two spikes and ignore the rest of the string. In the open-string picture krtt (), this arc corresponds to a Wilson loop surface ending on two parallel light-like lines at the boundary (the spikes move at the speed of light).
Let us rewrite the above solution in the global embedding coordinates. First, note that
Now, if we use the parameterization (with )
We see that the world surface as a 2d surface in is determined by the following system of 4 equations
Using the symmetry of we can put the second quadratic equation into a simple form, thus finding
Here are defined by
We used (2.11), i.e. that .
We see therefore that all solutions that we found in this limit, parameterized by , are actually related to one another by the two boosts in the 1-5 and 2-6 hyperbolic planes. Namely, a generic arc of the spiky string connecting two spikes reaching the boundary is related by these boosts to the infinite spin limit ft2 (); ftt () of the straight folded rotating string of GKP () which in our present notation corresponds to or (the center of the string is at the center of ).
As follows from (2.17), this surface can be parametrized simply by333 This corresponds to the choice of conformal gauge krtt (). For simplicity, we use the same notation and for the conformal-gauge world-sheet coordinates. These are rescaled coordinates: the original ones in conformal gauge should contain the scale factor (related to ), i.e. while the original varies between , the rescaled one varies between . The same remark will apply to in below.
Then we get from (2.18)
so that the induced metric is . This determines the expression for our solution (i.e. and given by (2.12)) when it is transformed to conformal gauge.
As was argued in k (), the spiky string state corresponds to higher twist operators with maximal anomalous dimension. The above discussion shows that, for large spin, the anomalous dimension of all such operators is determined by the same universal scaling function that appears for twist two operators. This is simply because the corresponding surfaces are related by conformal boosts and thus the associated string partition functions should be the same to all loop orders (see krtt ()).
Iii “Near-boundary string” limit
The solution (2.10) found in the limit admits two special cases. One is when the variable which determines the angular distance between the two spikes () approaches , i.e . Then and the arc becomes the straight string () passing through the center of with its ends reaching the boundary. In this limit the solution () looks singular, implying that some rescaling is to be made.
Another special case corresponds to , i.e. , when the arc between the two adjacent spikes becomes small and represents an open fast-rotating string located close to the boundary with its ends moving along null lines at the boundary. According to (2.18), this case corresponds to an infinite boost of the straight string passing through the center.444In this limit so that the total number of spikes of the original closed string goes to infinity. While each arc still contributes to the energy, the total energy (2.7) of the closed string then becomes infinite.
i.e. this corresponds to an infinite boost in the two (1,5),(2,6) planes.
In this limit the ends of the string follow two light-like lines at the boundary which are very close to each other.
Since for the string is located close to the boundary we may expect that we can ignore the periodicity in and get the corresponding solution in the Poincare patch by identifying
The solution (3.2) then reads as
ending at the boundary on the two parallel null lines .
However, (3.4) is not an exact solution in the Poincare patch. Still, as we shall now show, one can take a particular (infinite-boost) limit of the global metric and obtain a new metric for which eq.(3.4) will be an exact solution.
Let us start by writing the global metric as
where and parameterize the 3-sphere. We can now make a change of coordinates 555In what follows we shall use a formal notation in which .
and a rescaling by a parameter
Here we also introduced a (spurious) mass scale . In the limit when we get and while , i.e. this limit focuses on a small fast-rotating string located near the boundary and near the origin of the transverse space. The end-points of the string (which are then close to its center of mass) follow the massless geodesic .
Taking the limit in (3.5),(3.7) and keeping only the leading -independent terms we get the metric which such small string “sees”:666This limit is similar to the Penrose limit used in the case of the geodesic pen () but notice that here we do not rescale the overall coefficient of the metric, i.e. the string tension. In fact it appears to be a special case of the “conformal Penrose limit” considered in guven ().
This metric may be interpreted as a pp-wave in the space.777For a discussion of various “pp-wave on top of ” solutions see cv (); cha (). Note that the mass parameter can be set to 1 by a boost in the plane.
According to the AdS/CFT duality, the corresponding string theory should be dual to the SYM gauge theory defined on the 4d pp-wave background888For the structure of the action of this gauge theory see mee (); see also qqj () for some studies of the quantum gauge theory in pp-wave backgrounds.
This does not contradict the fact that (3.8) was obtained as a limit of the “empty” space – the metric (3.9) is conformally-flat (the 5d metric (3.8) is, in fact, locally equivalent to the metric cha (), see Appendix A).
Iv Cusp anomaly from null Wilson line in a pp-wave: strong coupling
Suppose we consider the planar SYM theory in the pp-wave metric (3.9) and compute the expectation value of the Wilson loop bounded by two infinite parallel light-like lines in the direction : , -const. Then according to AdS/CFT, the strong ‘t Hooft coupling limit of the expectation value of such Wilson loop should be determined maldrey () by the string action for the minimal-area surface in the pp-wave metric (3.8) ending on these two null lines.
Let us find the corresponding string solution by assuming that it has the form
Then the string action corresponding to (3.8) becomes999One can check directly that the above ansatz is indeed consistent with all the relevant equations.
Minimizing with respect to we obtain (we set in what follows)
For changing from to we have tracing both halves of the “arc” – from 0 to and then from to 0.
This solution agrees with eq.(3.4) after a rescaling of (as follows from comparing the part of the solution, cf. (3.4) and (3.6)), and this is again just a limit of the solution we have found in section 2.
Let us now compute the conserved quantities101010Here , where is the induced metric corresponding to (3.8). In general, if then .
Using eq.(4.3) we can convert the integrals over into the integrals over . The latter will be divergent at so we will insert a cut-off, . Thus
where the factor of 2 reflects the fact that covers twice. Thus solving for we get
Then, at leading order in ,
and since, to leading order, , we get
This agrees with (2.7) since here we are considering one arc () of the full -spike closed string.
We conclude that, if we are interested in the strong-coupling limit of anomalous the dimension of operators with large spin, it suffices to consider this particular string solution in the pp-wave background. This is analogous to the “open-string” computation of this anomalous dimension from the null cusp surface in the Poincare patch in kru (); the two world-sheet surfaces are indeed related by an analytic continuation krtt () and a global-coordinate boost (3.1).
It is useful to notice that the above expressions (4.7) and (4.8) were essentially determined by the contribution near , i.e. the result (4.9) follows from the properties of the solution (4.4) near the boundary. For that reason we may repeat the above discussion for a surface ending not on two but just on one null line; the role of will then be played by an explicit IR cutoff .
Indeed, let us consider a world-line of a single massless “quark” at . The corresponding “straight-string” world surface ending on this world line is then (both in the standard and in the pp-wave background (3.8))
Here we restored the dependence on the pp-wave scale to indicate that a non-zero value of is found only in the pp-wave case, i.e. if .
or, using (4.10),
Again, this is one half smaller compared to (4.12) due to the fact that here we had only one null line instead of two.
The conclusion is that in order to compute the scaling function that multiplies in the anomalous dimension it is sufficient to find the UV () divergence of the momenta corresponding to the surface in the pp-wave metric (3.8) ending on the null Wilson line, i.e. on a single line in the direction.
This “elementary” or “half-arc” solution thus reproduces 1/2 of the anomaly coefficient of the two null line surface or the the one-spike solution, and thus 1/4 of the anomaly captured by straight folded rotating string (or two-spike solution).
As we shall discuss below, one can do also a similar computation for a null Wilson line in the weakly coupled gauge theory defined in the corresponding 4d pp-wave background (3.9). In the conformal gauge theory, the single null Wilson line computation will give the same result as the cusp anomaly in the fundamental representation, which is 1/4 smaller than the twist 2 anomalous dimension, i.e. the dimension of the operator like dual to the closed folded rotating string.
V Cusp anomaly from null Wilson line in a pp-wave: weak coupling
As we have found in the previous section, an alternative way to compute the strong-coupling limit of the twist 2 anomalous dimension (or, equivalently, the cusp anomalous dimension) is to consider an open-string world surface in the “ plus pp-wave” background (3.8) that ends on a null line at the boundary of .
This suggests that one should be able to find the weak-coupling limit of the twist 2 anomalous dimension by considering a similar set-up in the boundary theory – the SYM gauge theory in the pp-wave background (3.9). Namely, we should study a field produced by a single charge moving at the speed of light along the direction in the 4d pp-wave background. More explicitly, we would like to reproduce the relation (4.16) at weak coupling, i.e.
where (that we may call a “pp-wave anomaly” coefficient) in the conformal gauge theory case111111To the leading order in that we will consider below the distinction between the conformal and non-conformal cases will not be visible. will turn out to be proportional to the twist 2 anomalous dimension
The latter has the well-known perturbative expansion grw ()
The scaling function is proportional to the cusp anomalous dimension, i.e. it can be found also as a singular part of the expectation value of the Wilson loop with a cusp formed by two null lines in flat (Minkowski) space kor (). At the lowest order in gauge coupling and in the planar limit (, ) the cusp anomaly (in the fundamental representation) is
The null Wilson line along is BPS (both in flat space and in the pp-wave case), so the corresponding expectation value is trivial, . Instead, we are to find the logarithmic UV anomaly in the light-cone energy in the presence of a null line or the relation (5.1) between and . Let us define
Here are the components of the gauge theory energy-momentum tensor and the expectation value is computed in the gauge theory defined in the pp-wave background (3.9). are the generators in the fundamental representation normalized as ().
As we shall see, will be logarithmically UV divergent and thus we may define the “pp-wave anomaly” as
where is a UV cutoff and the factor of two is due to the fact that is quadratically divergent in the cut-off. Then (5.1) will follow from a scaling argument described below.
At lowest order in the gauge coupling the expectation value (5.5) is given simply by the one-gluon exchange, i.e. by the gaussian path integral saturated by the classical gauge field configuration with a source provided by the null Wilson line. We can then simply replace the gauge field by an abelian one including the factor from the trace in the final expression. The corresponding abelian action is then121212Recall that in our notation .
Notice that for the pp-wave in (3.9) we have so we will ignore this factor in what follows.
We are then to solve the equations of motion for (5.8) ()
and evaluate (5.5) on the solution. As shown in Appendix B, the relevant solution is ()
Computing the energy-momentum tensor
The relevant components of the momentum (including the non-abelian group-theory factor which in the planar limit is , see (5.5)) are
will be logarithmically divergent both in the UV and in the IR, while will be quadratically UV divergent. To extract this divergence it is sufficient to introduce an UV regularization in (5.13) by
Then doing the integrals in (5.14) we get