Very Special Conformal Field Theories and Their Holographic Duals

PHYSICAL REVIEW D 97, 065003 (2018)

Very special conformal field theories and their holographic duals

Yu Nakayama Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

(Received 3 August 2017; published 6 March 2018)

Cohen and Glashow introduced the notion of very special relativity as viable space-time symmetry of elementary particle physics. As a natural generalization of their idea, we study the subgroup of the conformal group, dubbed very special conformal symmetry, which is an extension of the very special relativity. We classify all of them and construct field theory examples as well as holographic realization of the very special conformal field theories.

DOI: 10.1103/PhysRevD.97.065003

I. INTRODUCTION what is a subgroup of the conformal group that is consistent with the very special relativity. Once we determine it, it is To a very good approximation, the Poincaré group (i.e., important to ask the further question what are the actual space-time translation and Lorentz transformation) is sym- field theory realizations. In this paper, we will give metry of particle physics. However, it may not be the examples of what we call very special conformal field fundamental symmetry of the nature. The gravity certainly theories which are field theories with no larger space-time affects the space-time symmetry. Even without gravity, the symmetry than the very special conformal symmetry. Lorentz symmetry may be just accidental low energy Conformal field theories are natural candidates of effective symmetry, or alternatively it may be violated at renormalization group fixed points, and they play signifi- a very large distance. In both cases, the nature may not cant roles in our understanding of critical phenomena. They preserve the Lorentz symmetry but it is violated with a tiny may also play important roles in particle physics. This is amount that has not been observed yet. because most of the scale invariant field theories are Sometime ago Cohen and Glashow introduced the actually conformal (see e.g. [4] for a review), and at the notion of very special relativity as viable space-time high or low energy limit, the physics will be governed by symmetry of elementary particle physics [1,2]. They renormalization group fixed points without any intrinsic proposed that a certain subgroup of the Poincaré symmetry scale. We then expect that the very special conformal field may be the fundamental space-time symmetry. The group theories should play a crucial role in our understanding of spanned by a subgroup of the Poincaré generator Pþ, P−, the very special relativity. This leads us to the next P , Jþ is dubbed very special relativity. A crucial obser- i i immediate question if the scale invariant very special vation there is that if we added CP symmetry, it would be relativity naturally leads to very special conformal field enhanced to the Poincaré group because it exchanges þ and theories. For this purpose, we will study the structure of the − so that we need to add J− and then the commutation i energy-momentum tensor by focusing on the possibility of relations imply the existence of J . Since the CP violation ij the improvement. Another question is if the very special of the standard model is somehow small, the violation of conformal field theories must have secret symmetry the Lorentz symmetry down to the very special relativity enhancement to the full Poincaré conformal symmetry. might be explained to be naturally small as we observe. We will address these issues by offering examples as well Before the advent of the special relativity by Einstein, the as general argument. Poincaré group had been known to be symmetry of the In recent years, it has become more and more common to Maxwell theory of electrodynamics. Actually, the Maxwell study strong dynamics of quantum field theories by using theory in the vacuum (without source) has a larger holography, even without the Poincaré invariance. In this symmetry known as the conformal symmetry (but only paper, we also construct a holographic dual description of in 1 þ 3 dimensions [3]). In this article, we ask the question very special conformal field theories. In particular, in some versions of the very special conformal symmetry it is possible to construct a holographic model not only in the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. effective gravity, but also in the full string theory back- Further distribution of this work must maintain attribution to ground, so we naturally believe that the very special the author(s) and the published article’s title, journal citation, conformal field theories are not theories in the swampland and DOI. Funded by SCOAP3. but are consistent by themselves. In other versions of the

2470-0010=2018=97(6)=065003(7) 065003-1 Published by the American Physical Society YU NAKAYAMA PHYS. REV. D 97, 065003 (2018) very special conformal symmetry, it is an open question if TABLE I. The commutation relation of very special conformal we can construct any holographic dual description. generators. The organization of the paper is as follows. In Sec. II,we Pþ P− P J Jþ Jþ− Kþ ˜ define very special conformal symmetry and study the i ij i D properties of the energy-momentum tensor. In Sec. III we Pþ 00 0 00−Pþ 00 ˜ 2 propose field theoretic descriptions of very special con- P− 00 0 0Pi P− −D P− formal field theories, and in Sec. IV we show some Pi 00 0Pj Pþ 0 Jþi Pi − examples. In Sec. V, we study holographic dual descrip- Jij 00 Pj Jkl Jþi 00 0 − − − − − tions of very special conformal field theories. In Sec. VI, Jþi 0 Pi Pþ Jþi 0 Jþi 0 Jþi Jþ− Pþ −P− 00Jþi 0 Kþ 0 we conclude with discussions. ˜ Kþ 0 D −Jþi 00−Kþ 0 −2Kþ ˜ D 0 −2P− −Pi 0 Jþi 0 2Kþ 0 II. VERY SPECIAL CONFORMAL SYMMETRY Let us first fix our convention. We use the light-cone þ ¼ p1ffiffi ð þ Þ − ¼ p1ffiffi ð − Þ i ¼ conformal symmetry that is consistent with the very special coordinate: x 2 t x , x 2 t x and x (i y, z), 1 1 i relativity. or xþ ¼ − pffiffi ðt − xÞ, x− ¼ − pffiffi ðt þ xÞ and x ¼ x . 2 2 i We list the schematic form of the commutator i½X; Y in þ Similarly, we define the light-cone vectors as A ¼ Table 1 in the most extended case of SIMð2Þ with the very p1ffiffi ð 0 þ xÞ − ¼p1ffiffið 0 − xÞ ¼ − p1ffiffi ð 0 − xÞ¼ 2 A A , A 2 A A or Aþ 2 A A special conformal symmetry. The other special conformal p1ffiffi ð þ Þ ¼−p1ffiffið 0 þ xÞ¼p1ffiffið − Þ symmetry with Tð2Þ, Eð2Þ and HOMð2Þ can be obtained 2 A0 Ax , A− 2 A A 2 A0 Ax . The light- J Jþ− ∂ ¼ ∂ ¼p1ffiffið∂ þ∂ Þ just as a subgroup by neglecting ij, or both in each cone derivatives are defined as þ ∂ þ 2 t x and x case. Since the commutator with Kþ does not give rise to ∂ ¼ ∂ ¼p1ffiffið∂ −∂ Þ μ ¼ þ þ − þ − ∂x− 2 t x . We note that AμB A Bþ A B− them, these are all consistent subalgebras. i þ − − þ i i ð2Þ ð2Þ A Bi ¼−A B −A B þA B ¼−AþB− −AþB− þAiBi. We should note that in the case of T and E , Pþ is The very special relativity is symmetry of the space-time, the center of the algebra. Furthermore, the very special which is a subgroup of the Poincaré group, given by Pþ, P−, conformal algebra with Eð2Þ is isomorphic to the ¼ p1ffiffi ð þ Þ Schrödinger algebra in 1 þ 2 dimensions [7]. Pi, Jþi 2 Jti Jxi , where Pμ is the space-time trans- In order to formulate field theoretic realizations of the lation and Jμν ¼ −Jνμ is the Lorentz transformation. This is very special conformal symmetry, it is convenient to study the minimal version of the very special relativity, and apart the properties of the energy-momentum tensor.2 First of all, from the space-time translation, the Lorentz part of the group the space-time translation requires the existence of the is Tð2Þ. There are a couple of different extensions of the very ν conserved energy-momentum tensor Tμ : special relativity. First, we may supplement Jij, and the Lorentz part becomes Eð2Þ. Second, we may supplement ∂ þ þ ∂ − þ ∂ i ¼ 0 Jþ− instead, and the Lorentz part becomes HOMð2Þ. þTþ −Tþ iTþ Finally, we may supplement both Jij and Jþ− and the þ − i ∂þT− þ ∂−T− þ ∂iT− ¼ 0 Lorentz part becomes SIMð2Þ.1 ∂ þ þ ∂ − þ ∂ i ¼ 0 ð Þ The very special conformal symmetry is defined by þTj −Tj iTj 1 ˜ ¼ þ ¼p1ffiffið þ Þ adding two operators D D Jxt and Kþ 2 Kt Kx , ¼ where D is the generator of the Poincaré dilatation and Kμ is Rso that the chargesR for the space-timeR translations Pþ 0 ¼ 0 ¼ 0 the generator of the Poincaré special conformal transforma- dtTþ , P− dtT− , Pi dtTi are conserved. tion. We first argue that this is the only possibility in any In the very special relativity, the existence of the very special Lorentz transformation Jþi demands the conserva- version of the very special relativity. Suppose we add Ki to μ the very special relativity, then the commutator of Ki and P− tion of the very special Lorentz current Jþi: gives J−i. Similarly the commutator of Ki and Pj gives Jij as well as D. The commutator of J− and Jþ gives Jþ−, so the þ ¼ − − þ − þ i j Jþi x Ti xiTþ entire Poincaré symmetry is recovered. Instead, suppose we − ¼ − − − − − Jþi x Ti xiTþ add K−, then the commutator of K− and Jþi gives Ki and the j − above argument follows. Thus adding Kþ is the only special ¼ − j − j ð Þ Jþi x Ti xiTþ ; 2

1The total very special relativity algebra has various names in 2We work on the Noether or canonical construction of the the literature. The combination of Eð2Þ and Pμ is sometimes energy-momentum tensor, but an equivalent way is to couple the called the Bargmann algebra or massive Galilean algebra (see, theory to nonrelativistic Newton-Cartan gravity and study e.g., [5] and references therein). The combination of SIMð2Þ and the gravitational source as in [8,9]. This has been mostly studied Pμ is called ISIMð2Þ algebra in [6]. in the Eð2Þ case.

065003-2 VERY SPECIAL CONFORMAL FIELD THEORIES AND … PHYS. REV. D 97, 065003 (2018) Z − − − i ¼ 0 from which we require Ti Tþ .IfJij is conserved, δ ¼ λ 3 ð Þ j i S d xdtJþ; 5 we further require Ti ¼ Tj . Similarly, if Jþ− is conserved, þ − Tþ ¼ T− we require . where Jþ is a vector primary operator with Poincaré scaling With the very special conformal symmetry, we further Δ ¼ 5 “ ” ˜ μ dimension J . Here the irrelevance is with respect to demand the conservation of the dilatation current D the Poincaré dilatation D and the deformation is actually 3 ˜ þ − þ i þ ˜ D ¼ð2x ÞT− þ x T marginal with respect to D. Indeed, one may easily check i ˜ that the added operator is invariant under D and Kþ at the ˜ − ¼ð2 −Þ − þ i − D x T− x Ti classical level, but breaks the Lorentz symmetry down to ˜ j − j i j ð2Þ D ¼ð2x ÞT− þ x Ti ð3Þ E . One may generalize the construction with rank n R tensor Jþþ with the Poincaré scaling dimension ˜ ˜ 0 with D ¼ dtD . Furthermore the very special conformal Δ ¼ 4 þ n. current must be conserved: In the Tð2Þ case, we instead consider adding the “irrelevant deformation” ðx Þ2 Z Kþ ¼ðx−Þ2T þ þ x−xiT þ þ i T þ þ − i 2 þ 3 δS ¼ λ d xdtAþx; ð6Þ ð Þ2 − − 2 − − i − xi − Kþ ¼ðx Þ T− þ x x Ti þ Tþ 2 where Aþx ¼ −Axþ is an antisymmetric tensor with the Δ ¼ 5 ðx Þ2 Poincaré scaling dimension A . At the classical level, j ¼ð −Þ2 j þ − i j þ i j ð Þ ˜ Kþ x T− x x Ti 2 Tþ 4 the added operator is invariant under D and Kþ, but breaks ð2Þ R the Lorentz symmetry down to T . In particular, it is not 3 0 For the conservation of Kþ ¼ d xKþ, it is sufficient that invariant under Jij. One may consider the higher rank − i ½þ the energy-momentum tensor is “traceless”: 2T− þTi ¼0 tensor deformations that have more antisymmetric x and − i þ and symmetric −Ti − Tþ ¼ 0. indices. The traceless condition is not necessary because the The case with SIMð2Þ and HOMð2Þ is much more energy-momentum tensor has an ambiguity in its defini- nontrivial. In [1], it was argued that there is no local field tion. For instance, it allows the improvement of the form theory constructions because of the lack of the nontrivial ˜ ν ν 1 ν ν ρ spurion field. They instead considered the nonlocal filed Tμ ¼ Tμ þ 3 ð∂μ∂ − δμ∂ ∂ρÞL with a certain local oper- 2 − þ i ¼ð−2∂ ∂ þ ∂2Þ theory examples. We will discuss the conformal extension ator L. Thus, when T− Ti þ − i L, one can always introduce the traceless energy-momentum in Sec. IV D. Here we would like to propose local but ˜ ν singular constructions that at least make sense classically. tensor Tμ so that the very special conformal symmetry In the SIMð2Þ case, we consider the deformation is realized. Finally, we note that while the very special conformal Z δ ¼ λ 3 Jþ ð Þ symmetry is the only nontrivial extension of the very S d xdt ˜ ; 7 special relativity that contains the “conformal transforma- Jþ tion,” there are many other possibilities with only extra ˜ ˜ where Jμ and Jμ are primary vector operators, and the “dilatation” symmetry with Dλ ¼ D þ λJxt. The very Poincaré scaling dimensions must obey Δ − Δ˜ ¼ 4. special conformal symmetry Kþ is compatible only with J J λ ¼ 1 (or λ ¼ 0 in which case we recover the full Poincaré Compared with (5), this deformation preserves the addi- conformal symmetry). tional Jþ− symmetry. There is an obvious generalization with higher rank tensors, which we will not dwell on. Similarly, in the HOMð2Þ case, we consider the defor- III. FIELD THEORY DEFORMATIONS TO VERY mation SPECIAL CONFORMAL FIELD THEORIES Z In this section we discuss general recipes to construct δ ¼ λ 3 Aþx ð Þ S d xdt ˜ ; 8 very special conformal field theories from the Poincaré Aþx invariant conformal field theories by adding local oper- ˜ ators. The construction in particular applies to Lagrangian where Aμν and Aμν are primary antisymmetric tensor field theories because they are based on the free conformal operators, and the Poincaré scaling dimensions must obey field theories. In the next section we will show some Δ − Δ ¼ 4 A A˜ . examples. Let us first consider easier cases with Eð2Þ and Tð2Þ 3The different notion of relevance between D and D˜ is also symmetry, which we can construct via local nonsingular de- emphasized in [10], where they consider the similar deformations formations. In the Eð2Þ case, we may start with a conformal of Poincaré conformal field theories to obtain Schrödinger field theory by adding the “irrelevant” deformation invariant field theories.

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These deformations are singular in the sense that the the conformal transformation, one can see that it is a ˜ inverse of the operator such as 1/Jþ may not be well- primary vector operator. defined quantum mechanically. We, however, point out that While we may check the invariance of the action directly, in quantum mechanics, the inverse of the position operator we may also compute the energy-momentum tensor: is frequently used in conformal quantum mechanics (as well as in the Coulomb potential problem). It is however ν ν ν Tμ ¼ −∂μϕ ∂ ϕ − ∂μϕ∂ ϕ possible that with these singular deformations, the ground ν ρ ρ 2 state may break the very special conformal symmetry − δμð−∂ρϕ ∂ ϕ − iλ jϕj ðϕ ∂ρϕ − ϕ∂ρϕ Þ spontaneously. ν 2 − iλ jϕj ðϕ ∂μϕ − ϕ∂μϕ Þ: ð12Þ It is not entirely impossible to construct SIMð2Þ invariant very special conformal field theories by using nonlocal interactions as proposed in [1]. As we have already It has the desired properties that ensure the very special mentioned, since there is no obvious general argument, conformal symmetry discussed in the previous section. In − ¼ − i 2 − þ i ¼ we will separately show examples in the next section. particular, note that Ti Tþ and T− Ti ð−2∂ ∂ þ ∂2Þjϕj2 þ − i up to the use of the equations of IV. SOME EXAMPLES motion so that one can improve the energy-momentum tensor to make it traceless. A. An example: Eð2Þ case Let us note that the interaction is irrelevant in the Let us consider a field theory model with a complex Poincaré sense. If we take the infrared limit with the scalar ϕ having the action Poincaré scaling [i.e., ðxþ;x−;xiÞ → λðxþ;x−;xiÞ and Z λ → ∞], the interaction vanishes. However, in the special þ − i þ 2 − i 3 μ μ 2 conformal scaling limit [i.e., ðx ;x ;xÞ → ðx ; λ x ; λx Þ S ¼ d xdtð−∂μϕ ∂ ϕ − iλ jϕj ðϕ ∂μϕ − ϕ∂μϕ ÞÞ and λ → ∞], it survives. In other word, the violation of the Z Poincaré invariance may be regarded as a UV effect. To see ¼ 3 ð∂ ϕ∂ ϕ þ ∂ ϕ∂ ϕ − ∂ ϕ∂ ϕ d xdt þ − þ − i i this point more clearly, let us note that in the conventional λjϕj4 interaction gives the ϕðk1Þϕðk2Þ → ϕðk3Þϕðk4Þ − λjϕj2ðϕ∂ ϕ − ϕ∂ ϕÞÞ ð Þ i þ þ ; 9 scattering amplitude of δ4ðk1 þ k2 þ k3 þ k4Þ, while our 4 1 2 3 4 1 2 interaction in (9) gives δ ðk þ k þ k þ k Þðkþ − kþÞ, λμ ¼ðλþ λ− λy λzÞ¼ðλ 0 0 0Þ where ; ; ; ; ; ; is a lightlike vec- which vanishes in the Poincaré IR limit kþ → 0, but it does tor invariant under the special relativity. not vanish in the very special conformal limit where kþ is The model is invariant under the space-time translation fixed. Beyond the tree level, we have to study the Pþ, P−, Pi and very special Lorentz transformation that renormalization group equation with respect to our dilata- λμ∂ ¼ λ∂ makes μ þ invariant, i.e., Jþi and Jij. It is also tion D˜ rather than the Poincaré one D. In some situations ˜ invariant under the dilatation D under which one can argue that the deformations are marginal as studied in [10]. In our holographic example in Sec. V, the coupling δ ϕ ¼ð2 −∂ þ i∂ þ ΔÞϕ ð Þ D˜ x − x i ; 10 constant is again marginal, so in these cases, the very special conformal invariance is preserved even quantum where Δ ¼ 1. In addition, it is invariant under the very mechanically. special conformal transformation

δ ϕ ¼ð2 −Δ þ 2ð −Þ2∂ þ 2 − i∂ þ 2∂ Þϕ ð Þ B. An example: Tð2Þ case Kþ x x − x x i xi þ : 11 In order to obtain very special conformal field theories This model can be regarded as an example of the with the Tð2Þ symmetry, we need to add an antisymmetric deformation (5). Here, we take the original theory as a tensor with the Poincar´e dimension 5. Since we have no theory of free massless complex scalar ϕ, and take the such primary operator out of one complex scalar, we 2 deformation Jþ ¼jϕj ðϕ ∂þϕ − ϕ∂þϕ Þ. Jþ has the consider introducing an additional real scalar field φ and Poincar´e conformal dimension ΔJ ¼ 5, and by checking consider the action

Z 1 ¼ 3 −∂ ϕ∂μϕ − ∂ φ∂μφ − λμνðφð∂ ϕ∂ ϕ − ∂ ϕ∂ ϕÞÞ S d xdt μ 2 μ i μ ν ν μ μν þ iλ ð∂μφðϕ ∂νϕ − ϕ∂νϕ Þ − ∂νφðϕ ∂μϕ − ϕ∂μϕ ÞÞ ; ð13Þ

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μν where λ is an antisymmetric tensor that has a nonzero dimension ΔA ¼ 5, and by checking the conformal trans- component only in λþx ¼ −λxþ ¼ λ. Actually, the two formation, one can see that it is an antisymmetric tensor interaction terms are identical upon integration by part, primary operator. but the combination is judiciously chosen so that it is a Again we may check the invariance of the action conformal primary operator. This is an example of under Kþ directly, but we can also compute the energy- deformations of (6): the introduced interaction momentum tensor: μν μν λAþx ¼−iλ ðφð∂μϕ ∂νϕ−∂νϕ ∂μϕÞÞþiλ ð∂μφðϕ ∂νϕ− ϕ∂νϕ Þ−∂νφðϕ ∂μϕ−ϕ∂μϕ ÞÞ has the Poincar´e

μ μ μ μ Tν ¼ −∂ ϕ ∂νϕ − ∂νϕ ∂ ϕ − ∂ φ∂νφ μα αμ − iλ φð∂νϕ ∂αϕ − ∂αϕ ∂νϕÞ − iλ φð∂αϕ∂νϕ − ∂νϕ ∂αϕÞ αμ þ iλ ð∂αφðϕ ∂νϕ − ϕ∂νϕ Þ − ∂νφðϕ ∂αϕ − ϕ∂αϕ ÞÞ μα − iλ ð∂αφðϕ ∂νϕ − ϕ∂νϕ Þ − ∂νφðϕ ∂αϕ − ϕ∂αϕ ÞÞ: ð14Þ

It has the desired properties that ensure the very special and the identity conformal symmetry discussed in the previous section. In − ¼ − i 2 − þ i ¼ particular, note that Ti Tþ and T− Ti δμν ð−2∂ ∂ þ ∂2Þjðϕj2 þ 1 φ2Þ ϵμαρσ ν ¼ ϵ ρσ αβ ð Þ þ − i 2 up to the use of the equations FρσF α 4 αβρσF F : 18 of motion so that one can improve the energy-momentum tensor to make it traceless. Having understood the conservation of the energy- momentum tensor, let us go back to the space-time C. Not an example: Maxwell-Chern-Simons theory symmetry of this theory. The point is that the energy In the Introduction, we mentioned the symmetry of the momentum has the properties free Maxwell theory. While the free Maxwell theory admits no relevant deformations with the Poincar´e invariance, it − þ i ¼ 0 admits a nontrivial relevant deformation only with the very Ti Tþ þ i special relativity. It takes the form of the Chern-Simons 2Tþ þ Ti ¼ 0 interaction with the action given by 1 þ þ i ¼ − ϵiαρσ ≠ 0 Z Ti T− 2 k− FρσAα 3 1 μν 1 μνρσ S ¼ d xdt − FμνF þ ϵ kμFρσAν ; ð15Þ 2 − þ i ¼ − ϵ−αρσ ≠ 0 ð Þ 4 2 T− Ti k− FρσAα : 19 where kμ ¼ðkþ;k−;ky;kzÞ¼ð0;k;0; 0Þ is a lightlike The first equation guarantees that Jþi is conserved, and the vector invariant under the special relativity. ˜ second equation guarantee that D−1 is conserved. However, The action is obviously invariant under Pþ, P−, P , Jþ , i i the fourth equation says that it is not invariant under D˜ , nor and Jij. In addition it is invariant under the dilatation ˜ Kþ. In particular, one cannot improve the energy-momen- D−1 ¼ D − Jxy [11]. Note that this dilatation is different ˜ ˜ tum tensor. We also note that although D−1 is conserved from D ¼ D þ J that is compatible with the very special þ i xy with the traceless condition 2Tþ þ Ti ¼ 0, K− is not a conformal symmetry. Thus, the theory is invariant under the symmetry because the third equation does not vanish. dilatation, but is not invariant under the very special A different look at this model is the scaling behavior. conformal transformation. Unlike the ones studied in our very special conformal field To see it, let us study the energy-momentum tensor [11] theories, the deformation here is relevant in the Poincar´e scaling (i.e. ΔJ ¼ 3). The Chern-Simons interaction makes ν 1 ν ρσ να 1 ναρσ Tμ ¼ δμFρσF − F Fμα − kμϵ FρσAα: ð16Þ the photon massive at long distance and therefore it is an IR 4 2 effect rather than the UV effect. On the other hand, we recall that to preserve the very special conformal symmetry, It is a little bit nontrivial to check the conservation ν we needed ΔJ ¼ 5 and the deformation must have been a ∂νTμ ¼ 0 : We need to use the equation of motion UV effect. This is an intuitive reason why this model does not exhibit the very special conformal symmetry even ∂ μν þ ϵμνρσ ¼ 0 ð Þ μF kμ Fρσ 17 though it is scale invariant.

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D. Another example: Nonlocal SIMð2Þ case The configuration of the metric and the Proca field is manifestly invariant under Pþ, P−, Pi, Jþi and Jij. It is also The nonlocal model studied by Cohen and Glashow can ˜ be made very special conformal invariant by replacing the invariant under the dilatation D explicit mass term by the vacuum expectation value of a x− → λ2x−;xi → λxi;z→ λz; xþ → xþ ð25Þ scalar field. Consider a Weyl fermion ψ coupled with a real scalar φ with the nonlocal action as well as the very special conformal transformation Kþ Z μ − i 1 n γμ − x x ¼ 4 − ∂μφ∂ φþ ψγ¯ ∂μψ þφ2ψ¯ ψ ð Þ x → ;xi → ; S d x μ i μ μ 20 1 þ − 1 þ − 2 n ∂μ ax ax z a x2 þ z2 with nμ ¼ðnþ;n−;nx;nyÞ¼ðn; 0; 0; 0Þ, which is SIMð2Þ → þ → þ − i ð Þ z 1 þ − ;xx 2 1 þ − : 26 invariant. The idea, being of phenomenological interest, is ax ax that upon giving a vacuum expectation value to φ, this The background is same as the holographic dual description describes a massive charged fermion without breaking of nonrelativistic (Schrödinger) conformal field theories the Uð1Þ symmetry, which is impossible with Lorentz [12,13] except that we do not compactify the xþ direction. invariance. We should note that the background does not solve the Introducing nonlocality, however, makes the use of the vacuum Einstein equation without the Proca field. There is very special conformal symmetry less trivial because most a good reason for this: as we discussed in Sec. II, very of the representation theory based on the state operator special conformal field theories possess the energy- correspondence does not apply. Nor does the operator momentum tensor that has more nontrivial components production expansion, which is at the heart of the bootstrap than the one in the Poincaré conformal field theory. We approach to conformal field theories. On the other hand, the therefore need more dynamical degrees of freedom in the violation of the non-locality is governed by the very bulk setup, which is naturally achieved by the Proca field. special symmetry or CP violation, so it may be tamed This local argument is stronger than the global one based perturbatively. on the symmetry alone that gives us the form of the metric but not the matter field [14]. V. HOLOGRAPHIC MODEL We point out that the above effective gravity construction may be uplifted to the full string theory compactification. In Let us consider a holographic model of very special particular, one may embed the solution into the type IIB conformal field theories. In this paper, we only work with supergravity by using the so-called TST transformation the case with Eð2Þ symmetry and leave the other cases for [15–17]. This technique was used in the context of the future works. We begin with the five dimensional Einstein- holographic dual of Schrödinger field theories, and the Proca theory with the action holographic dual for Eð2Þ invariant very special conformal Z pffiffiffiffiffiffi 1 1 2 field theories can be simply constructed by decompactify- 5 MN m M 4 S ¼ d x −g R − Λ − FMNF − AMA : ing the light-cone direction. The supergravity solution may 2 4 2 be unstable once we compactify the light-cone direction, ð21Þ but our solution is stable since we do not compactify it. This seems consistent with the stability of the dual gauge ¼ ∂ − ∂ with FMN MAN NAM. This theory has a solution theories under the deformations we studied (see, e.g., [20]). with the metric

2 M N VI. DISCUSSIONS ds ¼ gMNdx dx 2 2ð −Þ2 −2 þ − þ i þ 2 Cohen and Glashow proposed very special relativity as a ¼ − c dx þ dx dx dx dxi dz ð Þ 4 2 22 small but natural deviation from the Poincar´e invariance. z z Let us revisit a possible classification of such deformations and the Proca field from the renormalization group viewpoint. Within the renormalization group paradigm, the deformations can cdx− be classified by the operators in the undeformed A ¼ A dxM ¼ − ð23Þ M z2 Poincar´e invariant field theory. Cohen and Glashow argued that such deformations must be a vector operator in the case provided of Eð2Þ case and an antisymmetric tensor in the Tð2Þ case, and none exist in the other cases without violating locality. Λ ¼ −6;m2 ¼ 8: ð24Þ We, instead proposed local but singular deformations. It is

The constant c is regarded as an exactly marginal parameter 4 þ The decompactified theory in the type IIB supergravity is dual that corresponds to the coupling constant λ in the previous to dipole deformations of N ¼ 4 super Yang-Mills theory, which field theory construction. has been studied, for example, in [18–20].

065003-6 VERY SPECIAL CONFORMAL FIELD THEORIES AND … PHYS. REV. D 97, 065003 (2018) an interesting question if such deformations make sense invariance in d ¼ 2. On the other hand, the analogue of quantum mechanically. Tð2Þ (or Eð2Þ since there is no distinction) invariance in ˜ For this purpose, it would be quite important to ask if d ¼ 2 (i.e., Pþ, P−, Kþ, D) implies warped conformal SIMð2Þ or HOMð2Þ invariant very special conformal field invariance [24,25] and the theory does not have to be fully theories can be defined in the abstract operator language conformal invariant, example of which may be found in the without relying on the Lagrangian descriptions. In particu- literature [26]. lar, we might suspect that SIMð2Þ invariant very special The construction of the holographic model for SIMð2Þ conformal field theories must secretly possess the full or HOMð2Þ invariant very special conformal symmetry is Poincaré conformal invariance. Then the situation is much challenging. The fact that the spurion cannot be introduced similar to the case with the emergence of Poincaré in the field theory analysis may be of great hindrance in conformal invariance from the mere scale symmetry. In d ¼ constructing the holographic dual, but it is important to 2 and d ¼ 4, unitary scale invariant field theories with understand why this is the case from the operator algebra. Poincaré invariance actually implies conformal invariance This should give us deeper understanding of the origin of (under various technical assumptions) [21–23]. holography. We suspect that we need a drastic change of the Indeed, in d ¼ 2, the analogue of SIMð2Þ (or HOMð2Þ gravity sector. since there is no distinction) invariance means we have Pþ, ˜ P−, Kþ, D and Jþ−. Then with the unitarity, the condition ACKNOWLEDGMENTS is sufficient to guarantee that we have K− although the algebra itself does not need it. This is just a corollary This work is in part supported by JSPS KAKENHI Grant of the claim that scale invariance implies conformal No. 17K14301.

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