Wilson loop in a $T\bar{T}$ like deformed $\rm{CFT}_2$

In this paper we study string theory in the background $\mathcal{M}_3$ that interpolates between $AdS_3$ in the IR and linear dilaton spacetime $\mathbb{R}^{1,1}\times\mathbb{R}_\phi$ in the UV. Via holographic duality this background corresponds to $\rm{CFT}_2$ deformed by a dimension $(2,2)$ operator. Here we discuss the holographic Wilson loop in such a model and shed more light in support of the non-local structure of the theory (Little String Theory (LST)) in the UV. We also discuss quantum and thermal phase transitions of the boundary theory.


Introduction
Recently there has been a considerable interest [1][2][3][4] in the study of string theory in the background where the background M 3 interpolates between linear dilaton geometry R 1,1 × R φ in the ultraviolet (UV) and AdS 3 in the infrared (IR) and N is a 7-dimensional compact space. From UV perspective this background can be realized as the bulk description of certain 2-dimensional vacua of LST with N 1 fundamental strings [5,6]. While the UV physics is governed by that of the underlying LST, the theory approaches a CFT 2 dual to AdS 3 in the bulk, in the IR. The AdS 3 geometry in the IR corresponds to the near-horizon geometry of the fundamental strings in the linear dilaton background.
However, from the IR perspective, these backgrounds can be realized as an irrelevant deformation of the CFT 2 dual to the above AdS 3 by a certain class of dimenision (2,2) quasi-primary [1][2][3][4]. Such a deformation of the IR CFT 2 involves flowing up the renormalization group (RG) and hence may appear to be ambiguous and ill-defined. However, as has been argued in [1][2][3][4] (both from worldsheet and spacetime point of view), the situation here is much better and one can actually flow up the RG.
In this paper we are going to investigate the behaviour of the holographic Wilson loop operator of the dual boundary theory all along the RG. Since the theory in the UV is non-local, we expect the Wilson loop operator to exhibit non field-theoretic behaviour at short distances. We would investigate the non-locality scale and comment on quantum and thermal phase transitions of the dual spacetime theory.
As we will discuss, in details, in the next section, the large T E behavior of the expectation value of the Wilson loop operator, W , is given by (see e.g. [7]) where T E and L are the lengths of the sides of a rectangular loop (residing on the boundary) with edges parallel to the direction of Euclidean time and space respectively and E is the potential energy of the quark anti-quark pair separated by a distance L. For theories with two fixed points, E(L) interpolates between quark anti-quark potential in the UV CFT to that in the IR CFT. From the behaviour of E as a function of L one can understand the nature of transition from the UV to the IR and determine the different phases of the theory. A natural question to ask is what happens to E(L) if the short distance physics is non-local and not governed by a UV fixed point [1][2][3][4].
As discussed in [1][2][3][4], string theory in M 3 × N is closely related to T T deformed CFT 2 . Following the recent works of [8,9], there has been a considerable progress in understanding different aspects of T T deformed CFT 2 ; see e.g. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] and [27][28][29][30][31] for related works. It is natural to ask if the expectation value of the Wilson loop operator in these models is similar to that of the string background M 3 × N . But unfortunately, there are no simple method to calculate the expectation value of Wilson loop operator in strongly coupled gauge theories. We will therefore assume the duality between the string theory in M 3 × N and T T deformed CFT 2 and investigate the properties of the Wilson loop operator of T T deformed CFT 2 via holography.
The plan of this note is as follows. In section 2 we give a brief review of the necessary aspects of the construction in [1][2][3][4] and previous works on holographic Wilson loop operator [7]. In section 3 we compute the expectation value of the holographic Wilson loop operator and the potential energy of the quark anti-quark system separated by a distance L at zero temperature. We compute its large and small L behaviour and estimate the non-locality scale of the theory in the UV. We find that the theory exhibits second order quantum phase transition.
In section 4 we generalize the holographic results to finite temperature and discuss thermal and quantum phase transitions in the theory. In section 5 we discuss our results and compare ours with those obtained in [4] and comment on few possible avenues for future works.

An irrelevant deformation of AdS 3 /CF T 2
The authors of [8,9] have shown that a certain irrelevant deformation of a generic CFT 2 by an operator which is bilinear in the stress tensor (to be precise T T ) is, to a large extent, solvable. For example one can calculate the exact spectrum of the theory. At high energy, the theory appears to be well defined in spite of the fact that this involves flowing up the RG. The entropy of the system interpolates between that of a CFT 2 in the IR (Cardy entropy) and one with Hagedorn entropy at very high energies. Thus the short distance behaviour of the theory is not governed by local QFT. In the case of supersymmetric theories, the deforming operator being the top component of the superfield, preserves supersymmetry. Inspired by the results of [8,9], the authors of [1][2][3][4] discussed a string model that shares many properties with the T T deformed CFT 2 .
In those particular cases where the IR CFT 2 has a holographic dual in AdS 3 , the T T deformed CFT 2 studied in [8,9] corresponds to a double trace deformation of the original duality. Generic double trace deformations of AdS/CF T dualities are studies in [32,33]. The authors of [1] made an observation that there exists an irrelevant single trace deformation of the boundary CFT 2 that does the same job as the T T deformation of CFT 2 where µ is the irrelevant coupling of spacetime theory with mass dimension (−1, −1) and D(x) is a certain dimension (2, 2) quasi-primary of the spacetime Virasoro. The construction of the operator D(x) is discussed in [34]. Despite being irrelevant, this single trace deformation D(x) of the spacetime CFT 2 is under control as it induces on the worldsheet a truly marginal deformation of the form 1 where J − and J − are the holomorphic and anti-holomorphic components of the null SL(2, R) currents on the worldsheet whose zero modes gives rise to the boundary Virasoro generators L −1 and L −1 respectively [36]. The coupling λ on the worldsheet is truly marginal as it has worldsheet dimension (0, 0) and is related to the spacetime coupling µ by some dimensionfull constant. The deformation D(x) is in some sense universal because any CFT 2 with AdS 3 dual and NS-NS B-field flux with support in AdS 3 contains such a deformation. The current-current deformation (non Abelian Thirring) (2.2) of the worldsheet sigma model is exactly solvable. Using standard worldsheet techniques, one can read off the deformed metric, dilaton and B-field which we refer to as M 3 . This deformed background is given by [37] where f = 1 + 1 kU 2 , k is the level of the worldsheet SL(2, R) current algebra of the model and g s is the string coupling determined by the asymptotic value (i.e. U → ∞) of the dilaton field.
The background M 3 in (2.3) interpolates between AdS 3 in the IR (i.e. U → 0) to linear dilaton geometry Transition takes place at scales that depends of the coupling λ. Without loss of generality one can set the coupling to a convenient value as discussed in [1,2].
As an example, the interpolating geometry M 3 is realized as follows. We start with a stack of k NS5-branes wrapped around a four dimensional compact manifold (e.g. T 4 or K 3 ). The near horizon geometry of the NS5-branes gives (2.4). The string coupling g s goes to 0 near the boundary (i.e. U → ∞) where the dual field theory lives, but deep in the bulk (i.e. U → 0), g s → ∞. Now if we put N fundamental strings (F1) stretched along R 1,1 , the resulting background is given by (2.3). Upon addition of F1 strings, the IR geometry gets modified and the string coupling stops growing and saturates as g 2 s ∼ 1/N . The smooth interpolation from linear dilaton background in the UV to AdS 3 in the IR corresponds to going from the near horizon geometry of the NS5 system to that of NS5+F1 system. The spacetime CFT 2 in the IR has central charge c = 6kN . The linear dilaton geometry in the UV describes a two dimensional vacua of LST [5] with Hagedorn density of states [38] and diverging entropic c-function [4]. The inverse Hagedorn temperature is given by If the original LST to start with is supersymmetric, the deformation preserves supersymmetry since F1 strings are BPS. As discussed earlier, the single trace deformation (2.1) of the spacetime CFT 2 is not exactly same as the T T deformation as studies in [8,9] but is closely related. To realize the difference let us consider a CFT 2 that has a symmetric product form M N /S N where M is the CFT 2 that forms the building block of the symmetric product orbifold CFT. In such a theory there are two natural choices of T T deformation is the holomorphic component of the stress tensor of the i th block M in the symmetric product. The first is the T T deformation of M N /S N and is double trace, while the second is the T T deformation of the block M and is single trace. The spacetime CFT 2 corresponding to string theory on the bulk AdS 3 is not exactly a symmetric product orbifold CFT but is closely related to it [39,40]. The dual background corresponding to a certain Ramond vacuum (which preserves supersymmetry on a cylinder) of the boundary CFT 2 , is massless BTZ black hole. The strings and five branes that form the background are mutually BPS implying that their potential is flat. This in turn implies that there are continuum of states corresponding to strings moving radially away from the five branes. Such states form a symmetric product CFT, as observed in matrix string theory [41,42].

Wilson loop
In this subsection we give a brief review of the holographic Wilson loop operator; see e.g. [7] for details. The Wilson loop operators in any gauge theory are highly non-local operators in the theory and are defined by where C is a closed loop in spacetime where the gauge theory lives, A is the gauge field and P denotes path ordering of the gauge connection along the contour C.
The physical meaning that one can associate with the Wilson loop is that of the phase factor that shows up in transporting a given external charged particle in some representation along some closed path C. The trace in (2.6), in principle, can be taken over any representation, but we will restrict ourselves only to the fundamental representation. The potential energy between a quark and an anti-quark pair in the gauge theory can be calculated from the expectation value of the Wilson loop operator W (C) . To calculate W (C) , let us first Wick rotate to Euclidean signature and then consider a rectangular loop with sides of length T E and L. Treating T E as the Euclidean time, one would expect large can be interpreted as the quark anti-quark potential energy. For large N gauge theories at large 'tHooft coupling, the complicated problem of computation of W is mapped to a classical problem of finding the minimal surface Σ in the holographic bulk dual such that ∂Σ = C.
The partition function in any quark system in any gauge theory is W . At large N and large 'tHooft limit, one can equate this to the classical string partition function Z string [∂Σ = C] = e −S associated with a single F1 string where Σ is a genus zero compact worldsheet Riemann surface that minimizes the string action S (Nambu-Goto or Polyakov) and satisfies ∂Σ = C. In superstring theory, under S-duality a F1 string gets mapped to a D1-brane, in that case one needs to replace the Nambu-Goto or Polyakov action in the string partition function by the Dirac-Born-Infeld (DBI) action .

Holographic Wilson loop
Let us consider a quark anti-quark pair at x = L/2 and x = −L/2 on the boundary of the background manifold M 3 . Here "quark" means an infinitely massive W-boson connecting the stack of k NS5-branes with the one 5-brane (on the boundary) which is far away from the stack. The DBI action of the D1-brane hanging in the bulk with two of its ends fixed at the boundary, is given by where G M N is the 10 dimensional target space metric and X M s are the embeddings of the D1-brane in the target space and (τ, σ) are Euclidean coordinates on the worldvolume of the D1-brane. Here we choose to work in the static gauge, namely τ = t and σ = x. The equation of motion is given by where we have used the boundary condition that U (x = 0) = U 0 and ∂ x U | x=0 = 0. This allows us to express the length of the separation of the two ends of the D1-brane, L, on the boundary as where y = U U 0 , and K(m) and E(m) are complete elliptic integrals of the first kind and of the second kind respectively, given by In the deep AdS 3 regime (i.e. U 0 → 0), L behaves as (3.5) The 1/U 0 dependence of L is also observed in AdS 5 × S 5 as discussed in [7]. In the linear dilaton regime (i.e. U 0 → ∞), L reaches a constant proportional to the string length √ α . 2 Large U 0 expansion of L is given by the L vs U 0 plot: (i) As L → ∞, U 0 → 0 implying that the bottom of the minimal surface goes deep inside the AdS 3 regime. In this regime the interaction between the quark antiquark pair is governed by physics of the AdS 3 region of the full bulk geometry as depicted by RG.
Here the spacetime physics is that of LST in UV. The separation of the two ends of the D1-brane is precisely same as that observed in the case of hairpin (or paperclip) branes in linear dilaton geometry (see e.g. [43,44]). Fixed separation of the two ends of the D1-brane on the boundary signals non-locality of the spacetime theory (LST) in the UV.
(iii) In the linear dilaton regime (i.e. U 0 → ∞), one can check that dL This implies that there is a minimum L min satisfying 0 < L min < β H /2. This behaviour is puzzling as it implies that ∀L ∈ (L min , β H /2), there are two possible configurations of the D1-brane in the bulk (corresponding to two possible values of U 0 ) that minimizes the action. Numerically one can calculate L min = 2.945 2π β H . We will see from the discussion that follows, that one of the solution is energetically more favourable than the other. Such double valued behaviour of L also appears in AdS black holes (see e.g. [44,45]).
The potential energy of the quark anti-quark system is obtained by plugging (3.2) into (3.1). This integral (3.1), as expected, would give infinite result because it includes the mass of the D1-brane stretching all the way to infinity. One can regularize the expression by subtracting the disconnected solutions of a pair of D1branes stretching from U = 0 to U = ∞. So the energy of the system (i.e. the energy of the connected solution above the pair of disconnected ones), thus obtained, is finite and is given by In the deep AdS 3 regime (i.e. U 0 → 0), E behaves as whereas in the linear dilaton regime (i.e. U 0 → ∞), E behaves as  (i) As stated earlier, the energy E along the y-axis in figures (2) and (3) is the difference between the energies of the connected and disconnected solutions. That E ≤ 0 for all allowed values of U 0 (or equivalently L) shows that the disconnected solution is never energetically favourable over the connected one except when E = 0 where both the solutions are equally favourable. The large and small L configurations are respectively obtained when: (i) the connected D1-brane is hanging deep in the AdS 3 regime (i.e. U 0 → 0) and (ii) when the connected D1-brane is hanging in the linear dilaton regime (i.e. U 0 → ∞). Both these limits agree with the previously known results.
(ii) The double valuedness of L in the E vs L plot in figure (3) together with the kink signals phase transition, in particular second order quantum phase transition which we will analyse, in details, in the discussion that follows. Figure (3) shows that there are two branches namely branch I which we call the AdS 3 branch and branch II which we call the linear dilaton (LD) branch. It is obvious from the plot that the branch I is energetically more favourable than branch II.
(iii) As pointed out earlier, the system undergoes second order phase transition at L = L C = L min . This phase transition takes place at zero temperature, hence it is a pure quantum phenomenon (see e.g. [46] for a review on quantum phase transitions). The force F = − dE dL between the two end points of the D1-brane or equivalently between the quark anti-quark pair is continuous and always attractive (i.e. F ≤ 0), but the derivative of the force (i.e. dF dL ) is discontinuous at L = L C as shown in figure (4). The energy function has a power series expansion around L = L C along both branches: AdS 3 and linear dilaton, given by where C I s and C II s are coefficients that can be determined numerically. 3 This shows that the first derivative of the energy function is continuous at L = L C , but the second derivative is discontinuous at L = L C with both branches diverging as ±|L − L C | − 1 2 . One can read off the critical exponents of the force F from the first derivative of the energy function E I and E II , which turn out to be 1 2 along both branches. The point to be noted is that this critical exponent is independent of k.

Bulk calculation: finite temperature
We now generalize the analysis discussed in the previous section to finite temperature. The background (2.3) at finite temperature takes the form T U 2 and U T is the radius of the horizon of the black hole in M 3 . The temperature of the black hole is related to the horizon radius U T via: .

(4.2)
For √ kU T 1 the horizon is deep inside the AdS 3 regime in M 3 , and to good approximation is describes by BTZ black hole. On the other hand when √ kU T 1 the horizon is deep inside the linear dilaton regime and the solution is well described by the coset SL(2,R)×U (1)

Holographic Wilson loop
The DBI action of the D1-brane hanging in M 3 at finite temperature is given by With the given initial conditions U (x = 0) = U 0 and ∂ x U | x=0 = 0, the equation of motion is given by (4.4) Thus, the separation between the two ends of the D1-brane or equivalently the quark anti-quark pair is given by where y = U U 0 and y T = U T U 0 . In the linear dilaton regime (i.e. U 0 → ∞) L behaves as As expected L approaches β H /2 as U 0 → ∞ (as in the zero temperature case). The other extreme limit where U 0 → U T with U T held fixed in the AdS 3 regime, the separation L goes to 0. Figure (5) shows a schematic variation of L as a function of U 0 for the case where the horizon is deep inside AdS 3 regime ( figure (5(a))) and in linear dilaton regime ( figure (5(b))). Unlike the zero temperature case, L here is bounded from above. Let the maximum possible value that L can assume be L max . Figure (5(a)) shows that when the horizon is sitting deep inside the AdS 3 regime, the curve has a peak where dL dU 0 vanishes (and AdS 3 regime, and L max = β H /2 when the horizon is in the linear dilaton regime. L 0 is a function of U T that smoothly goes to ∞ as U T → 0. For U T in the AdS 3 regime and U 0 greater than the value where figure (5(a)) develops a peak, the overall behaviour of L as a function of U 0 is similar to that of M 3 at zero temperature.
The energy of the quark anti-quark pair above the disconnected solution is given by (4.7) In the limit U 0 → U T , the energy E goes to zero. In the linear dilaton regime (i.e. U 0 → ∞), the energy goes to a positive constant (proportional to U T ); Note that the above equation is valid irrespective of the location of the horizon. Equality in (4.8) holds when the temperature goes to 0. At finite temperature, the asymptotic value of energy (i.e. E(U 0 → ∞)) is positive, implying that the disconnected solution is energetically more favourable than the connected solution in linear dilaton background. Figure (6) shows a schematic variation of E as a function of U 0 for the case where the horizon is deep inside AdS 3 regime (figure (6(a))) and in linear dilaton regime ( figure (6(b))). Figure (7) shows the E vs L plot as one takes U T from zero to the linear dilaton regime. The black dotted arrows indicate how E vs L plot changes as U T increases from 0 to √ kU T 1. Based on figure (6) and figure (7), the following comments are in order: (i) In the zero temperature case, we have seen that L is bounded from below by L min and unbounded from above. For all L > L min there is a connected solution which is energetically favourable over the disconnected solution. As we turn on temperature, we see that this is nolonger the scenario. Finite temperature automatically puts an upper bound on L which goes to infinity as the temperature goes to 0 and to β H /2 for very high temperature. Not all allowed values of L produce a negative energy solution. For temperature T less than certain critical value, T C , there is a continuous interval in L where there exist connected solutions (see figure (7(b)&(c))). But for T > T C , the size of this continuous interval goes to 0 and there only exist disconnected solutions (see figure (7(e)&(f))). Figure (7(d)) corresponds to temperature T = T C . For T < T C , there exists a neighbourhood around T C in which the minimum of the energy (which is the value of the energy at the kink in figure (7)) behaves as E min ∼ |T − T C | α , where the critical exponent α can be computed numerically. This can be realized as a first order thermal phase transition from connected to disconnected solutions. This is the same thermal phase transition encountered in the entanglement entropy analysis in [4]. Similar phase transitions also appeared in [47]. (ii) For T < T C , there is the usual second order phase transition (same as the zero temperature case) coming from the discontinuity of d 2 E dL 2 at the kink. As in the case of zero temperature, one can calculate the critical exponents along both branches, AdS and linear dilaton. The sudden disappearance of this second order phase transition for T > T C can be realized as follows. As we turn on the temperature, thermal fluctuations start competing with quantum fluctuations and as the temperature is increased beyond certain critical value T C , the thermal fluctuations dominate completely over quantum fluctuations and we do not have connected solutions at all. The disconnected solution becomes energetically more favourable.

Discussion
The main goal of this paper is to study the models discussed in [1][2][3][4] through the lens of holographic Wilson loop. We investigated the consequences of non-locality on the quark anti-quark potential energy and the quantum and thermal phase transitions of the theory. The followings are some of the important observations that we have.
(i) In the UV the separation between the quark anti-quark pair (or equivalently the the two ends of a D1-brane hanging in the bulk) cannot be decreased below a certain minimum distance (β H /2) both at zero and finite temperature with connected solution in the bulk. This signals the non-local nature of LST. The non-locality scale observed in the entanglement entropy analysis in M 3 [4] is β H /4. This difference is probably due to the fact that different probes see different non-locality scales, though each is proportional to β H .
(ii) At zero temperature the theory exhibits second order quantum phase transition at L = L C . The force between the quark anti-quark pair exhibits a critical behaviour with exponent 1/2 along both branches, AdS 3 and linear dilaton.
This quantum phase transition is not captured by the analysis of entanglement entropy in M 3 [4] implying that this phenomenon is sensitive to how the theory has been is probed. This in particular, is not a robust property of the theory itself, rather it is an artefact of the interaction between the external quark anti-quark pair introduced to probe the theory.
(iii) At finite temperature the theory exhibits first order thermal phase transition at T = T C from connected to disconnected solution. The same thermal phase transition was observed in [4]. The second order quantum phase transition, encountered in the zero temperature case, persists for T < T C .
It would be interesting to perform perturbative calculation of the expectation of the Wilson loop operator in the T T deformed CF T 2 and compare with the exact results of section 3 and 4. String theory has predictions of different phase transitions in the spacetime theory. It would be nice to understand these transitions from direct field theory analysis.