General bootstrap equations in 4D CFTs

We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers (k,k) specifying the transformations under chiral su(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin (k,k) and scaling dimension ∆ satisfy ∆ ≥ max{k,k}. If |k −k| > 4, this is stronger than the unitarity bound.

“…by conformal symmetry, and can be produced by a variety of techniques [14][15][16][17]. For the specific case of chiral operators of interest to us we follow [18].…”

Section: Jhep02(2018)131mentioning

confidence: 99%

“…Using a translation, we write the general formula (2.15) for the three-point function in the coordinates 17) where in the above > 0 enforces the operator ordering. When we perform the relevant integrations, the i terms will tell us how to pick the appropriate contour.…”

Section: Example Calculation: the (3 0) Operatormentioning

confidence: 99%

Universal bounds on operator dimensions from the average null energy condition

Córdova

Diab

2018

“…Here, the δ 3 (p − q) comes from translation in invariance, δ m,m ′ comes from invariance under little group of p, 16 the power of p 2 is fixed by scaling invariance, and θ(p) restricts p to be in the forward null cone as required by positivity of energy. The only undetermined part here are the coefficients B m (∆, j).…”

Section: Two-point Functionmentioning

confidence: 99%

“…(3.38)As in the scalar example in the beginning of this section, we can now use this relation to analyze the poles of the conformal block, which come from the factor B m (∆, j) −1 . We start with the classification of such poles 16. Recall that the delta function δ 3 (p − q) sets q = p.…”

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confidence: 99%

Recursion relation for general 3d blocks

Erramilli

Iliesiu

Kravchuk

2019

Self Cite

We derive closed-form expressions for all ingredients of the Zamolodchikovlike recursion relation for general spinning conformal blocks in 3-dimensional conformal field theory. This result opens a path to efficient automatic generation of conformal block tables, which has immediate applications in numerical conformal bootstrap program. Our derivation is based on an understanding of null states and conformally-invariant differential operators in momentum space, combined with a careful choice of the relevant tensor structures bases. This derivation generalizes straightforwardly to higher spacetime dimensions d, provided the relevant Clebsch-Gordan coefficients of Spin(d) are known. Contents-i -A.5 Reality of OPE coefficients and conformal blocks 48 B Details about the two-point function in Fourier space 49 C Details about differential operators for null states 53 C.1 Operator parity and action on past-directed momentum 56 D Deriving f a,b j,I (θ) 58which are labeled by a pair of three-point tensor structures a and b, a four-point tensor 1 This is not to say that the results of [6,[25][26][27][28][29][30][31][32] are in any way incomplete or impractical. On the contrary, it was shown in these works how any spinning conformal block in any number d of spacetime dimensions can be computed, and these results were successfully used in d = 3 and d = 4 in [6][7][8][9][10]. We are merely pointing out that these methods require calculations that, even though can be carried out in each particular case, are difficult to implement in full generality. 2 The idea of writing such a recursion relation was initially developed by Zamolodchikov for 2d Virasoro blocks in [35] and was later generalized to any dimension for conformal blocks of external scalar operators by Kos, Poland, and Simmons-Duffin in [21,36], and finally generalized to spinning blocks by Penedones, Trevisani, and Yamazaki in [29].

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confidence: 99%

Superfluids, vortices and spinning charged operators in 4d CFT

Cuomo

2020

We include vortices in the superfluid EFT for four dimensional CFTs at large global charge. Using the state-operator correspondence, vortices are mapped to charged operators with large spin and we compute their scaling dimensions. Different regimes are identified: phonons, vortex rings, Kelvin waves, and vortex crystals. We also compute correlators with a Noether current insertion in between vortex states. Results for the scaling dimensions of traceless symmetric operators are given in arbitrary spacetime dimensions. * gabriel.cuomo@epfl.ch arXiv:1906.07283v2 [hep-th] 22 Jun 2019 1 IntroductionConformal field theories (CFTs) play a key role in particle and condensed matter physics. As fixed points of the renormalization group flow, they act as landmarks in the space of quantum field theories (QFTs). Through the AdS/CFT correspondence [1,2], they promise to shed light on quantum gravity. They also describe critical points for second order phase transitions. Finally, CFTs are also among the few examples of interacting QFTs where exact results are available without supersymmetry. Recently, the bootstrap program [3,4] achieved much progress in the study of CFTs, both through numerical [5,6] and analytical [7,8,9] techniques.Basic observables in CFTs are correlation functions of local operators in the vacuum. Despite this, sometimes one can make predictions for the CFT data defining the theory studying the dynamics of finite density states [10]. This is a consequence of the state/operator correspondence [11,12], which relates states in radial quantization to local operators with the same quantum numbers. So far, this idea has been mainly applied in the investigation of the superfluid phase in conformal field theories [10,13,14,15,16,17,18,19,20]. Indeed superfluids are the most natural candidates to describe states at large internal quantum numbers in CFTs. They admit a simple and universal effective field theory (EFT) description [21,22] which allows the computation of correlators in a perturbative expansion controlled by the charge density. The same strategy was recently applied also in the context of non-relativistic CFTs [23,24,25].As the angular momentum is increased, the superfluid starts rotating and vortices develop [26]. These can be included in the EFT as heavy topological defects [27,28,29]. In [30], this EFT was used to describe operators with large spin and large charge in three dimensional CFTs. In this work, we study the predictions of the vortex EFT for four dimensional CFTs.