## Final Report Summary - TESTINGGSD (Testing Gauge String Duality)

Gauge/String duality, also called AdS/CFT correspondence, is a surprising conjecture in theoretical physics. Gauge theory describing the interaction of elementary particles, and string theory as a theory of quantum gravity, can give the same prediction in special situations called the large Nc limit. This conjecture is hard to prove, but it can be checked or demonstrated in simple cases. There are theories which possess a hidden infinite-dimensional symmetry called integrability. It is hoped that, by exploiting integrability, the validity of AdS/CFT correspondence can be tested at any values of a coupling constant.

The objectives of this project are twofold: inventing new methods of integrability and computing new physical observables to test the Gauge/String duality. More specifically, our goals are rephrased as follows.

O1) To formulate Thermodynamic Bethe Ansatz (TBA) equations in the presence of non-trivial boundary conditions.

O2) To find new observables from which a new test of the AdS/CFT correspondence can be performed.

Description of the work performed since the beginning of the project:

Let us split the description of work into three periods in the chronological order.

First period (Sep/2013 - Mar/2014)

We studied the AdS/CFT correspondence in unstable backgrounds. It is believed that TBA equations with certain boundary conditions describe the dimension of double-determinant-like operators in gauge theory at weak coupling. According to AdS/CFT, these operators should correspond to a pair of potentially unstable open strings at strong coupling. Together with six collaborators worldwide, we formulated the desired TBA equations, solved the equations analytically and numerically, and checked the prediction by perturbative methods in gauge theory.

We found unexpectedly that our TBA equations do not have regular solutions beyond certain values of the coupling constant. Thus, for certain operators the integrability methods need to be substantially modified. We suspected that the trouble originates from a subtlety in taking the large Nc limit.

Second period (Apr/2014 – Mar/2015)

Based on our findings in the first period, we decided to revisit the operator mixing problem in gauge theory at finite Nc. Together with a new collaborator, we tackled this problem using two methods, analytical and numerical. In the analytical method, we used perturbative and group-theoretical techniques. In the numerical method, we wrote Mathematica codes and obtained new spectral data.

Our results will be explained shortly. Besides, we found an interesting pattern among the eigenstates of the operator mixing which are degenerate at large Nc.

Third period (Apr/2015 – Sep/2015)

The finite Nc operator mixing problem are related to the four-point functions. The latter subject has attracted a lot of attention recently, owing to remarkable successes of the conformal bootstrap methods. We worked to understand the aforementioned interesting pattern more precisely through this connection.

Description of the main results achieved so far:

We published two research articles (R1) and (R2), shown at the end of this summary. They correspond roughly and respectively to the objectives (O1) and (O2) written above.

R1) We proposed Boundary Thermodynamic Bethe Ansatz (BTBA) equations which describe the dimension of double-determinant-like operators of N=4 supersymmetric Yang-Mills (SYM) theory in four dimensions, which should be dual to the energy of a pair of potentially tachyonic open strings ending on giant-graviton D3-branes in AdS5xS5 spacetime. We showed that the exact energy of TBA equations appearing in the spectral problem of AdS/CFT is generally bounded from below. The numerical solution of our BTBA equations hits this bound, and starts to exhibit pathological behaviours, as shown in Figure 1 below. This finding was quite surprising, because it was commonly assumed that TBA equations always have real and regular solutions.

We interpret this singularity as the implication of the existence of open tachyons at strong coupling. The energy of tachyons is complex, and we should not have the exact energy smoothly interpolating between weak and strong coupling regions. In unstable situations the meaning of the large Nc limit is subtle.

R2) We examined operator mixing problem of N=4 SYM theory, among scalar so(6) sector at one-loop in coupling constant at finite Nc, and obtained three main results. First, by computing 1/Nc corrections to the dimensions of the operators called planar zero modes, we found that they acquire non-positive anomalous dimensions, as shown in Figure 2. It means that a multi-particle state in gravity should form a bound state in absence of repulsive charges. Second, we found that these operators always have a definite number of traces, which can be explained by the large Nc order counting of higher-point correlation functions. Third, we found examples of the operators which are hard to predict in integrability methods. One of our numerical results of the one-loop anomalous dimensions at any Nc is depicted in Figure 3.

(see an attached file for figures)

[Figure 1] Open string energy computed by BTBA, showing large errors around E=0.

[Figure 2] The 1/Nc correction to the dimensions of the planar zero modes is non-positive.

[Figure 3] Operator dimensions as functions of 't Hooft coupling and Nc

Expected final results and their potential impact and use (including the socio-economic impact and the wider societal implications of the project so far):

Expected final results and Potential academic impact

A grand goal of the Gauge/Gravity Duality is to ask how general this duality holds true. It is widely believed that this duality is undisputedly true in special examples in the large Nc limit. A natural question is what happens at finite Nc. However, the latter problem is notoriously difficult.

Our results suggest that part of the finite Nc spectral problems can be solved by studying higher correlation functions in the large Nc limit. By continuing this line of research, we expect to give a precise relation in between. In particular, the leading 1/Nc correction to the anomalous dimensions of planar zero-modes can be thought of a new interpolating function of AdS/CFT, interpolating between week and strong coupling beyond the planar limit.

Recently higher correlation functions of AdS/CFT correspondence receive sharp attention. Based on the success of integrability methods in the spectral problem, people started developing new integrability methods to obtain three-point functions in the large Nc limit. The four-point functions have also been studied in view of conformal bootstrap. How to combine all these developments is an essential question to determine the generality of the Gauge/String Duality in a precise manner.

Potential wider impact:

The main aim of this project is to test a conjecture. We did this, unexpectedly, by finding a counterexample. It illustrates how science develops; it is common to start implementing an idea by the minimal examples and improve the theory.

By fully utilizing computer software, we could express our results in beautiful pictures, which are simple to present. Besides, this project caught attention to a wide range of academic staff and students, both inside and outside EU, partially thanks to the eminence of Mathematical Institute in University of Oxford.

References:

R1) Z. Bajnok, N. Drukker, Á. Hegedüs, R. I. Nepomechie, L. Palla, C. Sieg and R. Suzuki, “The spectrum of tachyons in AdS/CFT,” JHEP 1403 (2014) 055 [arXiv:1312.3900 [hep-th]].

R2) Y. Kimura and R. Suzuki, “Negative anomalous dimensions in N=4 SYM,” Nucl. Phys. B900 (2015) 603-659 [arXiv:1503.06210 [hep-th]].