The critical O(N) σ-model at dimension 2 &lt; d &lt; 4 and order 1/N2: Operator product expansions and renormalization

Lang, K.; Rühl, W.

doi:10.1016/0550-3213(92)90028-a

Cited by 69 publications

(90 citation statements)

References 9 publications

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“…We will then match our results with the 1/N expansions derived for the [4][5][6][7][8][9][10][11][12], evaluating them in d = 6− . The perfect match of the coefficients in these two 1/N expansions provides convincing evidence that the IR fixed point of the cubic O(N ) theory (1.2) indeed describes the same physics as the UV fixed point of the (φ i φ i ) 2 theory.…”

Section: This Leads To the Well-knownmentioning

confidence: 99%

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CriticalO(N)models in6−εdimensions

2014

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We revisit the classic O(N ) symmetric scalar field theories in d dimensions with interaction (φ i φ i ) 2 . For 2 < d < 4 these theories flow to the Wilson-Fisher fixed points for any N . A standard large N Hubbard-Stratonovich approach also indicates that, for 4 < d < 6, these theories possess unitary UV fixed points. We propose their alternate description in terms of a theory of N + 1 massless scalars with the cubic interactions σφ i φ i and σ 3 . Our one-loop calculation in 6 − dimensions shows that this theory has an IR stable fixed point at real values of the coupling constants for N > 1038. We show that the 1/N expansions of various operator scaling dimensions match the known results for the critical O(N ) theory continued to d = 6 − . These results suggest that, for sufficiently large N , there are 5-dimensional unitary O(N ) symmetric interacting CFT's; they should be dual to the Vasiliev higher-spin theory in AdS 6 with alternate boundary conditions for the bulk scalar. Using these CFT's we provide a new test of the 5-dimensional F -theorem, and also find a new counterexample for the C T theorem.

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Section: This Leads To the Well-knownmentioning

confidence: 99%

“…We first note that the coefficients in the 1/N expansions derived for various quantities in [4][5][6][7][8][9][10][11][12] may be continued to the range (1.1) without any obvious difficulty. Thus, the UV fixed points make sense at least in the 1/N expansion.…”

Section: Introductionmentioning

confidence: 99%

CriticalO(N)models in6−εdimensions

2014

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“…The large N analysis of the interacting O(N ) vector model has been carried out for arbitrary dimension d in [39][40][41][42][43][44][45][46][47][48]. For the two cases studied in this work, d = 5 and d = 5.95, we record that the dimensions for the O(N ) fundamental φ i and the two lowest singlet operators σ and…”

Section: < D <mentioning

confidence: 99%

“…3 We emphasize that the lower bound on ∆ σ 2 (where this lower bound is chosen to be strictly greater than ∆ σ ) must be chosen judiciously if one's goal is to study the interacting O(N ) theories proposed in [55,56]. Indeed, in these theories, one has ∆ σ 2 = 4+O(1/N ), where the leading 1/N correction is negative [39][40][41][42][43][44][45][46][47][48]. At least at large N , one therefore expects the scaling dimension of σ 2 to be slightly less than four.…”

Section: Introductionmentioning

confidence: 99%

BootstrappingO(N)vector models in4<d<6

2015

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We use the conformal bootstrap to study conformal field theories with O(N ) global symmetry in d = 5 and d = 5.95 spacetime dimensions that have a scalar operator φ i transforming as an O(N ) vector. The crossing symmetry of the four-point function of this O(N ) vector operator, along with unitarity assumptions, determine constraints on the scaling dimensions of conformal primary operators in the φ i × φ j OPE. Imposing a lower bound on the second smallest scaling dimension of such an O(N )-singlet conformal primary, and varying the scaling dimension of the lowest one, we obtain an allowed region that exhibits a kink located very close to the interacting O(N )-symmetric CFT conjectured to exist recently by Fei, Giombi, and Klebanov. Under reasonable assumptions on the dimension of the second lowest O(N ) singlet in the φ i × φ j OPE, we observe that this kink disappears in d = 5 for small enough N , suggesting that in this case an interacting O(N ) CFT may cease to exist for N below a certain critical value. *

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“…We believe this prescription is physically well motivated, because we are interested in the anomalous dimensions of the CFT 2 which are related to the UV properties of the theory. Furthermore, within the large-N approach anomalous dimensions are often computed in a similar manner for generic d [49,50] and, when finite, the d → 2 limit can be taken.…”

Section: Jhep06(2015)103mentioning

confidence: 99%

Integrability and the conformal field theory of the Higgs branch

Sax

Sfondrini

Stefański

2015

J. High Energ. Phys.

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In the context of the AdS 3 /CFT 2 correspondence, we investigate the Higgs branch CFT 2 . Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weakcoupling limit, in which anomalous dimensions of single-trace composite operators can be calculated. At one loop, the calculation reduces to finding the spectrum of a spin-chain with nearest-neighbour interactions. This CFT 2 spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS 3 side of the duality. We compute the one-loop dilatation operator in a non-trivial compact subsector and show that it corresponds to an integrable spin-chain Hamiltonian. This provides the first direct evidence of integrability on the CFT 2 side of the correspondence.

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The critical O(N) σ-model at dimension 2 < d < 4 and order 1/N2: Operator product expansions and renormalization

Cited by 69 publications

References 9 publications

CriticalO(N)models in6−εdimensions

CriticalO(N)models in6−εdimensions

BootstrappingO(N)vector models in4<d<6

Integrability and the conformal field theory of the Higgs branch

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