Supersymmetric Warped Conformal Field Theory

In this work, we study the supersymmetric warped conformal field theory in two dimensions. We show that the Hofman-Strominger theorem on symmetry enhancement could be generalized to the supersymmetric case. More precisely, we find that within a chiral superspace $(x^+,\th)$, a two-dimensional field theory with two translational invariance and a chiral scaling symmetry can have enhanced local symmetry, under the assumption that the dilation spectrum is discrete and non-negative. Similar to the pure bosonic case, there are two kinds of minimal models, one being $N=(1,0)$ supersymmetric conformal field theories, while the other being $N=1$ supersymmetric warped conformal field theories (SWCFT). We study the properties of SWCFT, including the representations of the algebra, the space of states and the correlation functions of the superprimaries.


Introduction
Symmetry plays an essential role in quantum field theories. The theories with more symmetries could be better constrained such that their dynamics might be investigated even nonperturbatively. For example, the supersymmetric field theories have better UV behaviors, and the conformal invariant theories are expected to be solvable in the framework of conformal bootstrap.
In two dimensions (2D), the global symmetries in a quantum field theory with scaling symmetry could be enhanced. As shown by J. Polchinski in late 1980s [1], a 2D Poincaré invariant QFT with scale invariance could become conformal invariant, provided that the theory is unitary and the dilation spectrum is discrete and non-negative. In 2011, D. Hofman and A. Strominger [2] relaxed the requirement of Lorentz invariance and studied the enhanced symmetries of the theory with chiral scaling. They obtained two kinds of minimal theories, one being the two-dimensional conformal field theory (CFT 2 ) [3] and the other being the so-called the warped conformal field theory (WCFT 2 ) [4]. In a warped CFT 2 , the global symmetry group is SL(2, R) × U (1), and it is enhanced to an infinite-dimensional group generated by an Virasoro-Kac-Moody algebra. Very recently, the symmetry enhancement in 2D QFT was generalized to the cases with global translations and anisotropic scaling symmetries [5]. In such 2D Galilean field theories with anisotropic scaling, the enhanced local symmetries are generated by the infinite dimensional spin-ℓ Galilean algebra with possible central extensions, under the assumption that the dilation operator is diagonalizable and has a discrete and non-negative spectrum. WCFT 2 has rich structures similar to CFT 2 . Though they are not Lorentzian invariant, WCFT 2 shares the modular covariance like CFT 2 . For finite temperature WCFT 2 defined on a torus, the modular property can be used to evaluate the density of states at high temperature, which gives a Cardy-like formula for the thermal entropy of WCFT 2 [4]. Due to the infinite symmetries, WCFT 2 is highly constrained. The form of the two-and three-point functions are determined by the global warped conformal symmetry while the four-point functions can be determined up to an arbitrary function of the cross ratio [6]. Specific models of WCFT 2 include chiral Liouville gravity [7], free Weyl fermion [8,9], free scalars [10],and also the Sachdev-Ye-Kitaev models with complex fermions [11,12]. For the study on other aspects of WCFT 2 , see [11,[13][14][15][16][17][18].
On the other hand, WCFT 2 plays an important role in the study of holography beyond the usual AdS/CFT correspondence. In [19], it has been shown that under the Compère-Song-Strominger (CSS) boundary conditions, the asymptotic symmetry group of the AdS 3 gravity is generated by an Virasoro-Kac-Moody algebra. This leads to the conjecture that under the CSS boundary conditions, the AdS 3 gravity could be dual to a holographic warped conformal field theory. This AdS 3 /WCFT correspondence has been studied in [14,16,[20][21][22][23][24]. Moreover WCFT 2 could also appears in the WAdS 3 /WCFT 2 correspondence [25][26][27][28], in which the bulk gravity is a three-dimensional topological massive gravity.
In this paper, we would like to generalize the study on WCFT 2 to the supersymmetric case. We first study the supersymmetries on the warped flat geometry [9] in two dimensions, which is essentially equivalent to a Newton-Cartan geometry [29][30][31][32][33][34] with an additional scaling structure. The supersymmetrization could be done by including Grassmannian coordinates into the bosonic directions to make warped "superspace". However, it turns out that the minimal supersymmetry could be realized in a chiral N = (1, 0) superspace. We then study the enhanced local symmetries, following the approach developed in [1] and [2]. Just as in bosonic case, we find two classes of minimal enhanced algebra. One generates the local symmetries of N = (1, 0) SCFT 2 , while the other one generates the symmetries of the supersymmetric warped conformal field theory (SWCFT 2 ). Furthermore, we discuss the radial quantization and the state-operator correspondence in SWCFT 2 , analogous to the usual WCFT 2 case. We study the correlation functions of superprimaries in SWCFT 2 as well. We notice that the correlation functions share the similar structure as the ones in the holomorphic sector of N = (1, 0) SCFT 2 , with additional modifications from U (1) symmetry.
The remaining parts are organized as follows. In Section 2, we discuss the supersymmetries on the warped geometry and set our notations. In Section 3, we generalize the Hofman-Strominger theorem to the supersymmetric case and show that the global symmetries are enhanced to the local ones. In Section 4, we consider the Hilbert space and the representation of the NS sector of the SWCFT 2 . After establishing the state-operator correspondence, we discuss the transformations of the super-primaries. Then we calculate the two-point functions and three-point functions of the superprimary operators in the NS sector of the SWCFT, and discuss the higher-point functions. We conclude and give some discussions in Section 5. In Appendix, we discuss the conserved currents in the superspace and show that we can consistently work in the N = (1, 0) superspace.

Supersymmetries on Warped Geometry
Let us start from a two-dimensional unitary local field theory with translational invariance and a chiral scaling symmetry. The transformation of coordinates under these symmetries are where λ a b is a scaling matrix: As shown in [2], the theory would have enhanced local symmetries. There are two kinds of minimal theories. One kind is the two-dimensional conformal field theory (CFT 2 ), while the other kind is the two-dimensional warped conformal field theory (WCFT 2 ). For WCFT 2 , in addition to the symmetries (2.1), there is a generalized boost symmetry where Λ a b is the boost matrix The WCFT 2 can be defined consistently in a warped geometry, which is a variant of the Newton-Cartan geometry with an additional scaling structure [4]. In the warped geometry, there are one vector and one one-form h ab can be used to lower the indices, but one should keep in mind that h ab is not the metric of the warped geometry.
In the warped geometry, one may define the fermionic representations. The first step is to consider the gamma matrix algebra. The gamma matrix algebra is given by the warped Clifford algebra where the gamma matrices are: The lower-index gamma matrices are defined by This definition proves quite useful as it allows us to define the boost generator as (2.10) One can check that it acts on the gamma matrices as they are in a vector representation The operators generating the translations will be donated by The two-dimensional spinor space is spanned by Ψ 0 , Ψ 1 as follows From this it is easy to see that 14) The definition for the dual representation is where the ǫ AB is given by One can easily show that the quantity ΨΨ is a scalar under the boost. Now, let us introduce the supercharge operator Q = (Q 0 , Q 1 ) T . The commutators of supercharges are They can be written in terms of the component operators From these commutators, one can easily find that H and Q 1 are superpartners under the action of Q 1 , so are P and Q 0 .
For simplicity, we will donate Q 1 by Q + and Q 0 by Q − in the following discussion. Moreover, we will donate x a = (x 0 , x 1 ) by x a = (x + , x − ), and the Grassmannian coordinates by A general superfield is defined on the superspace, and can be expanded as a power series (2.20) The transformation of any field Φ under the generator G is given by In the appendix, we discuss the conserved charges of the theory in the superspace and their corresponding supercurrents. We find that there exists a minimal superspace in which the right-moving supersymmetry can be turned off consistently. As we show in the next section, even only with the supersymmetry in the left-moving sector, the right-moving global symmetry gets enhanced and supersymmetrized as well.

Enhanced Symmetries
In this section, we study the enhanced symmetry of two dimensional quantum field theory, whose global symmetry is generated by the left-moving translation H, the dilation D, the right-moving translation P and the supersymmetries Q + . We will work in the chiral superspace (x + , x − , θ + ). For simplicity, we denote θ + = θ. Now that for the global charges their related supercurrents depend only on one Grassmannian coordinate θ, we will discuss the enhanced symmetry in this N = 1 chiral superspace. As in [2], we assume that the eigenvalue spectrum of D is discrete and non-negative and there exists a complete basis of N = 1 local superfields A general superfield can be expanded as It satisfy: where λ i is the superweight of Φ i and C dΦ i = 0 for any closed contour C. The translational plus the dilational invariance and the supersymmetry restrict the form of the vacuum two-point where f ij are some unknown functions.

From left global symmetries to local symmetries
The global charges H, D, P are associated to the supercurrents H, D and P, respectively.
All of these supercurrents have shift freedom [2], which can be used to "gauge" the currents to satisfy the canonical commutation relations This implies that H ± , P ± are local operators, but D ± must have explicit dependence on the The D ± 's have explicit coordinate dependence. Let us write the current in terms of local operators as in [2]. Defining S ± by One can easily find that and So we conclude (S + , S − ) are local operators of weight (1, 0).

The conservations of the dilation current and left-translation current yield
which leads to (3.14) Then we use the shift freedom in the currents to shift away S + One can check that the commutators and the conservations of the currents remain consistent.

Now, the equation (3.10) becomes
and Because of S − is a local operator of weight zero, then from the general form of the two-point function, we have which implies Assuming the conservation law and the equation (3.19) yield This fact immediately leads to the existence of two sets of conserved charges. Defining where ξ = 1 2 α(x + ) + θa(x + ) with α and a being the function of x + , we have There is another set of conserved charges which may lead to other symmetries [2,9]. As we are going to discuss the minimal algebra, we The algebra spanned by the conserved bosonic charges T 1α has been done in [2]: where the prime denotes the derivative with respect to x + , α ′ ≡ ∂ + α. This is the same as the algebra of the left-moving conformal generators on the Minkowski plane with the central charge c 0 .
Let us now work out the algebra spanned by adding the fermionic charges T 0a . The global charges are The actions of H and D on H + imply This in turn implies that the action of T 0a on h 1 is The scaling symmetry plus the locality imply that O a must be of the form O a = c 1 a with c 1 being a local operator of weight 1 2 . But the Jacobi identity with the third operator T 1β implies that c 1 = 0. So we arrive at Next, the action of and hence where g a is to be determined. Integrating both sides with − 1 The scaling symmetry and the exchange symmetry under a ↔ b imply g a = c 2 a ′ , where c 2 is a constant number. The Jacobi identity with the third operator T 1α implies that c 2 = c 0 3 . Then

From right global symmetries to local symmetries
In general, P ± can be written in form of The fact that P − is a local superfield of weight − 1 2 implies that p 2 is a weight-zero local field. From the two-point function of p 2 , we get ∂ + p 2 = 0. The current conservation then implies The supersymmetry requires p 0 = p 0 (x + ) and p 3 = 0, hence p − is a singlet under the supersymmetry. Now we have In the case P + = 0, we have infinitely many charges given by The algebra spanned by T 1α gives the right-moving Virasoro algebra on Minkowski plane [2]. In this case, the enhanced local symmetry is generated by the left-moving super-Virasoro algebra and the right-moving Virasoro algebra. It gives the local symmetry of N = (1, 0) SCFT 2 .
In the case P − = 0, we have infinitely many left-moving charges The bosonic sector of the algebra are simply [2] i To find the fermionic sector of the enhanced symmetry, we need to consider other commutators. Firstly we study the commutator [T 0a , J 1η ]. Note that the action This in turn implies that the action of J 1η on h 0 is The scaling symmetry plus the locality imply that O 0η must be of the form O 0η = c 3 η, where c 3 is a local operator of weight 1 2 . Consider the zero mode of J 1η (η = 1) ≡ J 11 , which act as This leads to the fact that c 3 must be independent of x + .
On the other hand, c 3 is an operator of weight 1 2 under the chiral scaling, we conclude that c 3 must be zero. Now integrating both sides of the equation Let us now work out the algebra spanned by J 0c . Due to the fact that the zero mode J 11 acts as ∂ − , we have i[J 11 , p 0 ] = 0. This implies i[J 11 , J 0c ] = 0 and hence Again, the scaling symmetry plus the locality imply X c = 0, then We also need the commutator The scaling symmetry plus the locality implies Z α must be of the form Z α = c 4 α with c 4 being a local operator of weight 1 2 . The action of D 1 on p 0 implies that Y = − p 0 2 and c 4 = p 0 , hence Then we have

Mode expansion
The supersymmetric Virasoro-Kac-Moody algebra consists of a super-Virasoro algebra and the semidirect product of the super-Virasoro and super-Kac-Moody algebras

70)
i{T 0a , J 0c } = 2J 1(ac) . (3.71) Let us put the theory on a cylinder and find the mode expansion of the above algebra. The coordinate transformation is Using the new coordinate φ, we choose test functions α n = (x + ) n+1 = e i(n+1)φ , a r = e i(r+ 1 2 )φ , η n = e inφ and c r = e i(r− 1 2 )φ , where n ∈ Z and r ∈ Z + 1 2 for the Neveu-Schwarz(NS) sector or r ∈ Z for the Ramond(R) sector. Letting L n = iT 1αn , G r = iT 0ar , P n = J 1ηn and S r = J 0cr , then the commutation relations in terms of the charges {L n , P m , G r , S s } are as follows 1 . The super-Virasoro algebra is generated by The super-Kac-Moody algebra is generated by The semi-direct product part of the super-Virasoro and super-Kac-Moody algebras is generated Another remarkable point is on the supersymmetry in the right-moving sector. Our construction starts from the left-moving superspace, but the right-moving sector gets supersymmetrized as well. This could be understood from the diffeomorphism of WCFT 2 . Recall that the diffeomorphism of WCFT 2 is generated by and The diffeomorphism could be generalize to chiral superspace such that the supersymmetry in the left-moving sector is transferred to the right-moving sector. Actually, the superconformal transformation in the left-moving sector is [39] x Here f (x + ), p(x + ) are holomorphic functions and F (x + ), φ(x + ) are anticommuting holomorphic functions, satisfying the following relations The transformation in the right-moving sector is where G(x + ) is an anticommuting holomorphic function. Considering the infinitesimal version of the above transformations, we find that the generators of the super-Virasoro-Kac-Moody algebra could be realized by

Properties of SWCFT 2
Now we have found two kinds of minimal theories in N = (1, 0) superspace, starting from a 2D QFT with chiral scaling and translation symmetry. One is the N = (1, 0) supersymmetric conformal field theory, whose local symmetries consist of a left-moving super-Virasoro algebra (SVA) and a right-moving Virasoro algebra. The other is the supersymmetric warped conformal field theory, whose local symmetries are generated by supersymmetric Virasoro-Kac-Moody algebra (SVCMA). In this section, we discuss the representations of this algebra, the stateoperator correspondence and then the correlation functions in SWCFT 2 .

Primary states and descendants
In all our subsequent discussions, we consider the NS sector of SWCFT 2 and hence r, s ∈ Z + 1 2 . We want to define the states in this theory at t = 0 by doing radial quantization. For this purpose, we consider the following complex coordinates  Using the algebra obtained previously, we have  At level 1 2 , we have (4.11) At level 1, we have (4.12) At level 3 2 , we have which is in the base We can derive some simple unitarity bounds on the plane charges by requiring the norm of the states to be non-negative. First, we have ||L n |∆, Q|| ≥ 0 =⇒ ∆ ≥ 0, c ≥ 0, (4.14)

Transformation laws of superprimary fields
We now consider the transformation laws of the primary superfields. The local operator at position (x + , x − , θ) is related to the one at the origin by the transformation Next we would like to find the explicit form of the commutator [L n , Φ(z)](n ≥ 0) for a primary field Φ(z). First, we have Using the Baker-Campbell-Hausdorff (BCH) formula, we get Then we obtain (4.20) In particular, we have Using the above relations, we finally obtain Consider an n-point function of super-primary fields where T stands for time ordering, and z i ≡ {x + i , x − i , θ i }. Throughout this paper, we will always assume x + i > x + j for i < j. Since the vacuum state |0 is OSP (1|2) × U (1) invariant, the n-point function is invariant under the action of L 0 , L ±1 , G ± 1 2 . This leads to the following differential equations corresponding to the generators L −1 , P 0 , L 0 , G −1/2 , G 1/2 respectively The first equation implies that G (n) should be a function of x + ij ≡ x + i − x + j . While the fourth equation implies that G (n) should be a function of For the x − part, the second equation implies G (n) should be a function of Consequently, the correlation function is of form We stress that because of the OSP (1|2) × U (1) symmetry of the vacuum, the correlation functions must have the OSP (1|2) × U (1) structure.
Let us first consider the two-point function G (2) which must be of form where {C i } are constants.
Next consider the differential equation arising from dilations L 0 , we find the conditions Moreover, the special transformation L +1 leads to the condition which gives As a result, we have the two-point function where we have set the normalization to unit. In the component fields, the nonvanishing twopoint functions are super-conformal theory is given in [40], and the U (1) part is given by [6], thus the three-point function in the NS sector of the SWCFT primaries is where we have defined The C 123 and C 123 are two structure constants of the three-point function. The quantity Ξ ijk is given by For the n-point function (n > 3) there are 3n coordinates {x + i , x − i , θ i }, and 5 constraints from OSP (1|2) invariance and one constraint from U (1) invariance. The x − i dependence can be totally determined by the U (1) invariance, thus the n-point function is essentially a function of (2n − 5) OSP (1|2) invariants, which are given by [37] Ξ ijk , Θ ijkl ≡ s ij s kl s li s jk . (4.42) The general form of n-point function can be written as Here F (Ξ ijk , Θ ijkl ) is an undetermined function, and ∆ ij are real constants which satisfy (4.44)

Conclusion and Discussion
In the present work we studied supersymmetric extension of the warped conformal field theory. Under the assumption that the dilation operator is diagonalizable, and has a discrete, non-negative spectrum, we generalized the Hofman-Strominger theorem to the supersymmetric case. Specifically, we showed that a two-dimensional quantum field theory with two transla- symmetry determined the dependence on x − completely.
One possible future direction is to generalize the minimal supersymmetry to the extended one. Our construction is based on the chiral superspace (x + , θ). It is worthy of generalizing the study to the full superspace, including the Grassmannian partner of the x − coordinate.
In the minimal CFT 2 case, this may lead to the N = (1, 1) SCFT 2 . But it is not clear of its consequence in the WCFT 2 case. The study can be pushed to the case of N ≥ 2 extended supersymmetry as well. Besides, it is interesting to study the supersymmetrization of the other 2D models with scaling symmetry. The supersymmetric GCA has been studied in [41], but for more general AGFT [5] its supersymmetric version has not been worked out.
It would be interesting to study the other properties of SWCFT 2 : the modular properties of the torus partition function, the warped conformal bootstrap [6,16], the entanglement entropy, etc.. It is also interesting to construct explicitly simple examples realizing the SWCFT 2 . This may help us to understand the theory better.
It could be expected that for the holographic SWCFT 2 , it is dual to a supersymmetric AdS 3 gravity under appropriate asymptotic boundary conditions. It would be nice to find the explicit boundary conditions and see how they break half of the supersymmetries.

Appendix: conserved charges in the superspace
We start from the global symmetries of the theory. It is generated by the left-moving translation H, the dilation D, the right-moving translation P and the supersymmetries Q + and Q − .
By assumption these charges annihilate the vacuum. Their non-vanishing (anti)commutation relations are The superspace is a coset space G/I, where G is the whole symmetry group and I is the dilation symmetry. A group element in G may be written in the form where δ, δ, ǫ + , ǫ − and λ are some infinitesimal constants. The coset element can be written as The transformations on the superspace are the natural action of the group G on the coset space from which we read the induced transformations in the superspace Then we can obtain the differential representations of the global charges For each of the charges H, D, P , Q + and Q − , there is a conserved Noether current. In particular, with the supersymmetries, there exist corresponding supercurrents. In general the supercurrents may have the form where a i (i = 1, 2, 3, 4) are constant numbers. The charges associated to the components of supercurrents can be read by The supersymmetric transformations of the supercurrents are In the following, we will donate the currents associated to the charges H, P , Q + and Q − by h ± , p ± , q + ± and q − ± , respectively. As it is not clear at this moment how these currents are related to each other by the supersymmetries, we first assume each of them belongs to a supercurrent donated by H ± , P ± , Q + ± and Q − ± , then we will find out the relationship between the currents by their transformations under the supersymmetries. For Q + ± , in order to be consistent with (5.18)  We have similar relations for H ± , P ± , and Q − ± . From these relations, we find that the form of Q + can only be Q + ± = a 1 q + ± + a 2 θ + h ± + a 3 θ − p ± , (5.22) and the form of Q − must be Q − ± = a 1 q − ± + a 2 θ + p ± , (5.23) with all none-zero coefficients a i , for i = 1, 2, 3. We see that the P belongs to two different supermultiplets. On the other hand, the fact that the operator Q − is nilpotent indicates that we may consider a smaller superspace. In fact we can regard Q − as the superpartner of P , and consider only one global supercharge Q + in the theory. It turns out that the smaller superspace {x + , x − , θ + } is enough to describe our theories consistently.