Holographic Line Defects in $D=4$, $N=2$ Gauged Supergravity

We construct solutions of four-dimensional $N=2$ gauged supergravity coupled to vector multiplets which are holographically dual to superconformal line defects. For the gauged STU and the $SU(1,n)/U(1)\times SU(n)$ coset models, we use the solutions to calculate holographic observables such as the expectation value of the defect and one-point functions in the presence of the defect.


Introduction
Extended objects such as defects, line operators, and interfaces are important tools in the study of quantum field theories. For conformal field theories (CFTs) with holographic duals, in many cases the extended objects have a realization on the gravity side. One example of such a duality is the fundamental string in AdS 5 × S 5 describing a Wilson line in the fundamental representation [1,2], and its generalization to D3 probe branes with an AdS 2 × S 2 worldvolume and D5 probe branes with an AdS 2 × S 4 worldvolume [3,4,5,6,7]. When the number of probe branes becomes large, the backreaction can not be neglected and a fully backreacted type IIB supergravity solution replaces the probe description. For the case of the half-BPS Wilson loop, this solution was found in [8]. 1 In general, the fully backreacted solutions in type II and M-theory are warped product geometries and the solutions are complicated and difficult to obtain. 2 A simpler setting is to consider lower-dimensional gauged supergravities, where the ansatz is simpler and the number of fields is smaller. If the gauged supergravity is a consistent truncation of a higher-dimensional theory, the resulting lower-dimensional solution can be uplifted to produce solutions in the higher-dimensional theory.
In this paper, we consider four-dimensional N = 2 gauged supergravity, which has been used in the past to describe condensed matter systems in three dimensions in order to find holographic models for superfluids and superconductors, see e.g. [12,13,14]. We construct half-BPS supergravity solutions which are dual to line defects in three-dimensional N = 2 superconformal field theories. The metric ansatz is given by AdS 2 × S 1 warped over an interval. We generalize the analysis of [15], which considered pure gauged supergravity, to the case of matter couplings.
The structure of the paper is as follows. In section 2, we review our conventions for four-dimensional N = 2 gauged supergravity coupled to vector multiplets. In section 3, we give a general solution describing a half-BPS line defect, obtained by a double analytic continuation of the black hole solutions first found by Sabra [16]. Since the behavior of the vector multiplet scalars can only be determined implicitly, we consider three examples, namely a single scalar model, the gauged STU model, and the SU(1, n) coset model to obtain explicit solutions. In section 4, we use the machinery of holographic renormalization to calculate holographic observables for the solutions, namely the on-shell action and the expectation values of operators dual to the supergravity fields. In section 5, we explore the conditions for a regular geometry and calculate their consequences. In section 6, we discuss our results and possible directions for future research. Our conventions and some details of the calculations presented in the main body of the paper are relegated to several appendices.

D = 4, N = Gauged Supergravity
In this section, we review four-dimensional N = 2 gauged supergravity coupled to n vector multiplets. We use the conventions and notations of [17,18,19].
The field content of the gauged supergravity theory is as follows. The supergravity contains one graviton e a µ , two gravitinos ψ i µ , and one graviphoton A 0 µ . The gravity multiplet can be coupled to N = 2 matter, and in particular we consider n vector multiplets, which are labeled by an index α = 1, 2, . . . , n. Each vector multiplet contains one vector field A α µ , two gauginos λ α i , and one complex scalar τ α . In this paper we do not consider adding N = 2 hypermultiplets.
It is convenient to introduce a new index I = 0, 1, . . . , n and include the graviphoton with the other vector fields as A I µ . The complex scalars τ α parametrize a special Kähler manifold equipped with a holomorphic symplectic vector where the Kähler potential K(τ,τ ) is determined by In the models we will consider, there exists a holomorphic function F(Z), called the prepotential, that is homogeneous of second order in Z such that The supergravity theory is fully specified by the prepotential F(Z) and the choice of gauging of the SU(2) R-symmetry. We will choose the U(1) Fayet-Iliopoulos (FI) gauging. The only charged fields of the theory are the gravitinos, which couple to the gauge fields through the linear combination ξ I A I , for some real constants ξ I . The two gravitinos have opposite charges ±gξ I for each U(1) gauge factor, where g is the gauge coupling. The bosonic action is 3 where F I µν = ∂ µ A I ν − ∂ ν A I µ are the field strengths and g αβ = ∂ α ∂βK is the Kähler metric of the scalar manifold. We use G µν to denote the four-dimensional metric, so e = √ −det G. The scalar potential is where the kinetic matrix N IJ is given by This is equivalently defined as the matrix which solves the equations where D is the Kähler covariant derivative, The equations of motion are obtained by varying the Lagrangian (2.4) The supersymmetry transformations are given in appendix A.

Line Defect Solution
In this section, we give a general solution describing a half-BPS line defect in four-dimensional N = 2 gauged supergravity, and then construct the solution for three specific choices of the prepotential.

Holographic Line Defects
A conformal line defect in three dimensions is a codimension-two defect which breaks the three-dimensional conformal group SO (3,2) down to an SO(2, 1) × SO(2) subgroup. The subgroup factors represent the unbroken conformal symmetry along the defect and transverse rotations about the defect, respectively. Minkoswski space R 1,2 is related by a Weyl transformation to AdS 2 × S 1 , namely − dt 2 + dr 2 + r 2 dφ 2 = Ω(r) − dt 2 + dr 2 r 2 + dφ 2 . (3.1) Hence in the holographic dual, the SO(2, 1) × SO(2) symmetry can be realized as the isometries of AdS 2 × S 1 , which we choose as the boundary of the four-dimensional asymptotically anti-de Sitter space. Therefore we consider a metric ansatz with AdS 2 × S 1 warped over a radial coordinate. We note that the location of the defect at r = 0 in Minkowski space gets mapped to the boundary of AdS 2 in the AdS 2 × S 1 geometry. Secondly, the absence of a conical singularity on the boundary fixes the periodicity of the angle φ to be 2π. The superconformal algebras in three dimensions are OSp(N |4), where N = 1, 2, . . . , 6, 8. For the CFT dual of four-dimensional N = 2 gauged supergravity, the relevant superalgebra is OSp(2|4) which has four Poincaré and four conformal supercharges. A conformal line defect is called superconformal if it preserves some supersymmetry. In the present paper, we will consider half-BPS defects which preserve an OSp(2|2) superalgebra and hence four of the eight supersymmetries.

General Solution
Four-dimensional N = 2, U(1) FI gauged supergravity admits half-BPS black hole solutions first found in [16]. The line defect solutions with AdS 2 × S 1 geometry are constructed by a double analytic continuation of the black hole solution. The metric and gauge fields are given by for some real constants q I and µ I , where Z I =Z I . Given a prepotential F(Z) and choice of parametrization of the symplectic sections Z I (τ ), the scalars τ α are given implicitly by the equation At the conformal boundary where r → ∞, in order to have asymptotic AdS 4 we need 2 √ 2gθ to be 2π-periodic, i.e. θ ∼ θ + π/ √ 2g. The AdS 4 length scale is then given by We will set 8g 2 = 1 to obtain the usual S 1 periodicity θ ∼ θ + 2π.
The center of the space 4 r = r + corresponds to the largest value of r where f (r) = 0. We consider radii taking values in the range r ∈ [r + , ∞). Demanding a regular geometry also requires r + > 0 and the absence of a conical singularity at the center of the space, both of which can be done by tuning the q I and ξ I parameters. This is explored in further detail in section 5.

Examples
For a general prepotential, the equation (3.3) is very complicated and can only be solved numerically. Consequently, we will explicitly work out the line defect solution for three specific prepotentials, for which we can find explicit expressions for the scalars. An important requirement is the existence of an AdS 4 vacuum, which not all prepotentials admit, see e.g. [19,20].
The cosmological constant at this extremum gives the AdS 4 length scale We choose ξ 1 = 2/ξ 0 to set the AdS 4 length scale to unity. The line defect solution (3.2) has the explicit form, The scalar is given by We have verified that the above fields obey the equations of motion (2.9).

Gauged STU Model
The STU model is given by considering n = 3 vector multiplets with the prepotential This theory has three complex scalars τ 1 , τ 2 , τ 3 and the scalar manifold is three copies of SU(1,1) U (1) . When all ξ I = ξ > 0 are equal, this theory is a consistent truncation of N = 8 gauged supergravity [21,22]. For reference on this model, see [23]. Using the parametrization . (3.10) The expressions for the kinetic matrix and scalar potential are complicated, but simplify for real scalars τ α =τᾱ, which will be the case for the line defect solution.
The potential has extrema at Positivity of e K requires us to choose the positive root. The cosmological constant at this extremum gives the AdS 4 length scale We pick the non-zero constants ξ I in a way that sets the AdS 4 length scale to unity. The line defect solution (3.2) has the explicit form, 14) The scalars are This solution is also the double analytic continuation of the hyperbolic black hole solution in [24]. As consistency checks, we have verified that the above solution obeys the equations of motion (2.9) and is half-BPS. The latter was done by a direct calculation, independent of [16], which can be found in appendix A.

SU(1, n) Coset Model
Another model which admits an AdS 4 vacuum has the prepotential F(Z) = i 4 Z I η IJ Z J , and can be formulated with any number of vector multiplets. η IJ is a Minkowski metric, which we will take to be η = diag(−1, +1, . . . , +1). The scalar manifold of this theory is SU(1,n) U(1)×SU(n) . Using the parametrization (Z 0 , Z α ) = (1, τ α ), the Kähler potential is Once again, the kinetic matrix and scalar potential have simpler forms for real scalars τ α = τᾱ. The matrix η IJ is used to lower indices, e.g.
This potential has an extremum at τ α = −ξ α /ξ 0 . 5 The cosmological constant at this extremum gives us the AdS 4 length scale where ξ 2 = ξ I η IJ ξ J . We pick a time-like ξ I with ξ 2 = −2 that will set the AdS 4 length scale to unity. The line defect solution (3.2) has the explicit form, The scalars are We have verified that the above fields obey the equations of motion (2.9).

Holographic Calculations
In this section, we use the machinery of holographic renormalization [25,26] to calculate the on-shell action and the one-point functions of dual operators of the boundary CFT in the presence of the defect, namely the stress tensor, scalar, and currents. This is done explicitly for the three examples in section 3.3.

General Procedure
First, we put the metric into the Fefferman-Graham (FG) form, where i, j = 1, 2, 3 run over the AdS 2 and S 1 indices and z → 0 is the conformal boundary. This is done by taking z = z(r) so that the appropriate coordinate change is obtained by the solution to the ordinary differential equation which can be integrated perturbatively in 1/r. This coordinate change gives the FG expansions of the fields, which we assume will take the form where A I 0 and A I 1 are 1-forms on the x 1 , x 2 , x 3 coordinates. The constants τ α 0 are the AdS 4 vacuum values of the scalars, which depend on the model. There is no gravitational conformal anomaly (i.e. a term proportional to z 3 log z in the expansion of g ij ) since d = 3 is odd.
In the three-dimensional boundary CFT, the conformal dimensions of the dual operators corresponding to the scalars τ α and vector fields A I are determined by the linearized bulk equations of motion near the AdS boundary. For instance, using the expansion τ α ∼ τ α 0 +z ∆τ in the linearized equation of motion for the scalar, we find that the scaling dimension of the dual operator is related to the mass-squared of the field by the equation (4.4) The mass-squared is −2 for all scalars of the three examples considered in this paper. This mass-squared is within the window where both standard and alternative quantization are possible [27], which implies that the scaling dimension of the dual operator can be either ∆ τ = 1 or ∆ τ = 2. Similarly, using the expansion A I ∼ z ∆ A −1 dθ in the linearized equation of motion for the vector field gives us We must have ∆ A = 2 as the vector field sources a conserved current of the boundary CFT. These scaling dimensions naturally fit into the flavor current multiplet 1 of the d = 3, N = 2 boundary CFT, using the notation of [28]. This short multiplet contains, in addition to the spin-1 operator [2] (0) 2 with scaling dimension ∆ = 2, two scalar operators [0] 2 is also present, as usual.
In the four-dimensional gauged supergravity, for a well-defined variational principle of the metric we need to add to the bulk action given by the Lagrangian (2.4) the Gibbons-Hawking boundary term, where h ij is the induced metric on the boundary and K ij is the extrinsic curvature. In FG coordinates, these take the form, The action I bulk + I GH diverges due to the infinite volume of integration. To regulate the theory, we restrict the bulk integral to the region z ≥ and evaluate the boundary term at z = . Divergences in the action then appear as 1/ k poles. 6 Counterterms I ct are added on the boundary which subtract these divergent terms. The counterterms have been constructed in [23] and are compatible with supersymmetry. They are where R[h] is the Ricci scalar of the boundary metric and W is the superpotential. In all, the renormalized action, is finite. We can then take functional derivatives to obtain finite expectation values of the dual CFT operators. Let T ij be the boundary stress tensor, O α be the operators dual to τ α , and J I i be the current operators dual to A I µ .

Stress Tensor Expectation Value
The expectation value of the boundary stress tensor is defined to be [29] T ij ≡ −2 √ −g 0 δI ren δg ij 0 . (4.10) The variation decomposes into two contributions: one coming from the regularized action and one coming from the counterterms. As usual [30], the former is given by The latter is straightforward to compute, and is given by Therefore, (4.13) By construction of the counterterms, this limit exists. 6 In even boundary dimensions, a term proportional to log may also appear.

Scalar Expectation Values
The expectation value of the operator O α is similarly defined by (4.14) The variation has contributions from the bulk action and the counterterms, and is For real scalars, supersymmetry implies O α = 0. A proof of this statement can be found in appendix B.

Current Expectation Values
The expectation value of the current operator J I is defined by The only contribution to the variation comes from the bulk action, and is (4.17)

On-Shell Action
We can evaluate the on-shell action for the line defect solution by further simplifying the bulk action to a total derivative [31], where Vol(S 1 ) = 2π and Vol(AdS 2 ) = −2π is the regularized volume of AdS 2 .

Examples
In this section, we use the general expressions derived in section 4.1 to compute observables for the three examples considered in this paper.

Single Scalar Model
Let us consider the defect solution (3.7, 3.8) for the single scalar model. The FG expansion of the radial coordinate r from solving the ordinary differential equation (4.2) is Using this coordinate change, the metric, gauge fields, and scalar can be expanded in FG coordinates. The one-point functions in the presence of the line defect can then be evaluated by computing the limits (4.13, 4.14, 4.16) directly. For the renormalized on-shell action (4.9), the finite terms at the conformal boundary cancel, leaving just the term obtained by evaluating (4.18) at r = r + . In the end, we obtain the following expectation values: (4.20)

Gauged STU Model
Let us consider the defect solution (3.14, 3.15) for the gauged STU model. Some of the calculations for this model are identical to those found in [24]. The FG expansion of the radial coordinate r from solving the ODE (4.2) is where we have defined the constants Using this coordinate change, the fields of the defect solution can be expanded in FG coordinates. We obtain the following on-shell action and one-point functions, Note that the expression for I ren is identical to that of the single scalar model, but the radius r + = r + (ξ I , q I ) will be different.

SU(1, n) Coset Model
For the defect solution (3.19, 3.20) of the SU(1, n) coset model, the FG expansion of the radial coordinate r is where η IJ is used to raise the indices of ξ I and q I . Using this coordinate change and expanding the fields in FG coordinates, the on-shell action and one-point functions are

Regularity
In this section, we impose two regularity conditions on the solutions. First, we demand that the geometry smoothly closes off at the largest positive zero of f (r) without a conical singularity in the bulk spacetime. This condition is analogous to the regularity condition imposed on Euclidean black hole solutions. Second, we fix the periodicity of the S 1 at the conformal boundary such that when the AdS 2 × S 1 boundary is conformally mapped to R 1,2 there is no conical deficit on the boundary. This condition is different from the one imposed in the holographic calculation of supersymmetric Rényi entropies [32,33,34,35], which use solutions that are related by double analytic continuation. For these solutions, the periodicity is related to the Rényi index n.
The regularity conditions will impose constraints on the parameters of the solutions. Since the general solution is only implicit, a detailed analysis is performed for the examples presented in this paper. We will show that for the single scalar and coset models, these conditions imply a bound on the expectation value of the boundary stress tensor.

General Statements
Given the metric the center of the space r = r + is defined to be the largest zero of f (r) = −1 + r 2 H(r). We can identify four criteria a regular geometry should satisfy: (a) positivity of the zero, r + > 0, This can be simplified to As the second factor is the sum of two positive quantities, a conical singularity can be avoided if we satisfy the condition H (r + ) = 0. As r + is determined implicitly in terms of the q I , ξ I constants through the equation f (r + ) = 0, this condition can be viewed as a constraint on the possible values q I , ξ I can take. Additionally, we will see that criterion (a) manifests as an inequality on q I , ξ I that we must satisfy.

Single Scalar Model
The single scalar model is simple enough that the conditions for a regular geometry can be solved exactly. Let us define x I ≡ q I /ξ I , but still pick the AdS length scale to be unity, i.e. keep ξ 0 ξ 1 = 2. The metric functions become Let us first satisfy the criterion r + > 0. Solving f (r) = 0, When the first factor is zero, we have a solution where we took the + sign to get the largest root. This solution exists when (x 0 + x 1 − 1) 2 − 4x 0 x 1 ≥ 0, which is a region on the x 0 x 1 -plane bounded by a parabola, shown in figure 1a.
The red shaded region indicates where r 1 does not exist and the blue shaded region indicates where r 1 > 0. When the second factor of (5.5) is zero, we have another solution where we also took the + sign. We have also marked regions where this solution exists and is positive in figure 1b. In regions where r 1 and r 2 both exist and r 1 > 0, we have r 1 > r 2 . Therefore, we can take r + = r 1 and restrict the (x 0 , x 1 ) parameter space to the blue shaded region of figure 1a.
(a) r 1 (b) r 2 Figure 1: Candidate r + for the single scalar model.
Let us now avoid the conical singularity by satisfying H (r + ) = 0. Calculating the derivative of H(r) in (5.4) and plugging in r + = r 1 from (5.6), we get the condition This is a parabola, marked by the black curve in figure 1a in the region where r + > 0. For the single scalar model to admit a regular geometry, the parameters x I = q I /ξ I must satisfy this condition. As a corollary, we can note that This implies that the components of the boundary stress tensor (4.20) have bounded expectation value. Additionally, the pure AdS 4 vacuum (x 0 = x 1 = 0) is the only solution with regular geometry and T ij = 0.

SU(1, n) Coset Model
The coset model is also simple enough that the conditions for a regular geometry can be solved exactly. We can note that actually has the same form as (5.4), where This map is always well-defined as (q I ξ I ) 2 + 2q I q I ≥ 0, which can be checked by rotating to the frame where ξ I = ( √ 2, 0, 0, . . . ). Thus all our results for the single scalar model can be carried over. The bound (5.9) for the single scalar model translates to the same bound on T ij for the coset model: (5.12) The condition (5.8) for a regular geometry translates to 0 = (q I ξ I ) 2 + 2q I q I + 2q I ξ I . (5.13) We can show that the only regular geometry with vanishing T ij is the AdS 4 vacuum. If we rotate to the frame where ξ I = ( √ 2, 0, 0, . . . ), the only q which satisfies q I ξ I = 0 and q I q I = 0 is q I = 0. A general ξ then has a q in the orbit of q I = 0, which is still the zero vector.

Gauged STU Model
For the gauged STU model, it is not practical to solve f (r) = 0 to find r + as f is a quartic polynomial. However, we still expect the criterion r + > 0 to impose an inequality on the four-dimensional parameter space (x 0 , x 1 , x 2 , x 3 ) and the condition of avoiding a conical singularity to reduce this to a three-dimensional hypersurface. However, note that unlike the single scalar and coset models, the expectation value T ij is not bounded. In appendix C we give special cases of the STU model with regular geometry which can have arbitrarily large x 0 + x 1 + x 2 + x 3 .

Discussion
In this paper, we constructed solutions of four-dimensional N = 2 gauged supergravity by a double analytic continuation of the half-BPS black hole solutions first found by Sabra [16]. While the black hole solutions exist for arbitrary prepotentials, explicit expressions for the scalars fields involve algebraic equations which in general can only be solved numerically. We considered three explicit examples of matter-coupled gauged supergravities, namely the single scalar model, the gauged STU model, and the SU(1, n)/U(1) × SU(n) coset model to find solutions and calculate holographic observables.
The solutions we find are holographic duals to line defects in three-dimensional SCFTs. The defect is characterized by a non-trivial expectation value of the R-symmetry and flavor currents along the S 1 factor in the AdS 2 × S 1 description of the defect. After conformally mapping to Minkowski space, this corresponds to a holonomy when encircling the line defect.
The expectation values of the real scalar operators vanish for general models as a consequence of supersymmetry.
For a conformal defect on AdS 2 × S 1 , the expectation value of the stress tensor can be parameterized by a single coefficient h,  [47,48,49,50] and the arguments should carry over to the defects dual to the solutions studied in the paper. 7 The solutions we find are related to supergravity solutions [24,33,34,35] which are holographic duals for a supersymmetric version of Rényi entropy first formulated in [32]. We note two differences. First, the solutions we find in Minkowski time signature have real gauge fields, unlike the duals cited above. 8 Second, we impose the condition that the periodicity of the circle in AdS 2 × S 1 boundary is such that after a conformal map we obtain flat space without a conical singularity. On the other hand, in the holographic duals to the Super-Rényi entropy, the conical singularity is related to the Rényi index n. We note that in [24,33,34,35] the holographic calculation of the Rényi entropy was compared to a localization calculation and agreement was found, and it would be interesting to see whether such a calculation can be performed for the holonomy defects described in this paper.
Another interesting question is whether more general solutions going beyond the examples discussed in this paper can be found. First, it would be interesting to study (numerical) solutions for more complicated superpotentials. Second, it would be interesting to see whether one can go beyond the gauged supergravity approximation and find solutions dual to holonomy defects in ten-or eleven-dimensional duals of N = 2 SCFTs. Uplifting the solutions found in this paper might prove to be a useful guide in this direction [22].

C STU Model Special Cases
Here we give a construction for STU models with regular geometry and arbitrarily large x 0 + x 1 + x 2 + x 3 . The approach we took to find these models was different than that of section 5.2. Instead of solving the condition f = 0 and then H = 0, we first solved H = 0 and then f = 0. The benefit is that H is a lower-degree polynomial and is technically simpler to solve. The downside is that this generates spurious solutions: it is possible that the r we obtain is not the largest root r + , and r + does not satisfy the equation H = 0. These spurious solutions then need to be removed by hand. To summarize our findings, consider the following construction: 1. Let x 0 be any positive number. 3. Consider an STU model with unit AdS 4 length scale where Numerically solve the equation f (r) = 0 for r, There exist exactly two solutions: a positive solution greater than −x 1 , and a negative solution less than −x 0 . Let r + be the positive solution.
The steps above give a STU model with regular geometry. To prove that x 0 +3x 1 is arbitrarily large, we need a better bound than −x 0 /3 < x 1 < 0. To satisfy (C.1) for large x 0 , we have Therefore x 0 + 3x 1 ≈ x 0 for large x 0 , and can be arbitrarily large.