Bounds on 4D conformal and superconformal field theories

By David Poland, David Simmons-Duffin. 2010

293 citations recorded by the source corpus.

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and $ \mathcal{N} = 1 $ superconformal field theories. In any CFT containing a scalar primary ϕ of dimension d we show that crossing symmetry of $ \left\langle {\phi \phi \phi \phi } \right\rangle $ implies a completely general lower bound on the central charge c ≥ fc(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients τIJ and flavor charges. We extend these bounds to $ \mathcal{N} = 1 $ superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Φ and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Φ × Φ† OPE, and show that there is an upper bound on the dimension of Φ†Φ when dim Φ is close to 1. We also present even more stringent bounds on c and τIJ. In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/683c58ddfb6f133f992074b0b43c4f6177999465

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Nuclear and High Energy Physics

Citations in the corpus

Infinite Chiral Symmetry in Four Dimensions (2013, background): “Generally speaking, these techniques are equally applicable to theories with and without supersymmetry, and despite promising early results [5–8], it has not been entirely clear the extent to which supersymmetry improves the situation.”
Bootstrapping the O(N ) vector models (2013, background): “A promising approach to extracting more of the spectrum in these theories is to consider the locations of the zeros of α(VO) along the boundary [6, 14, 16].”
Radial Coordinates for Conformal Blocks (2013, methodology): “Such a strategy was used in [2, 4, 6–8], and a similar one in [5].”
Central charge bounds in 4D conformal field theory (2010, background): “The morning of the day this paper was submitted to arxiv, a nice paper [19] appeared which, among other things, also derives lower bounds on the central charge.”
The ϵ ?> -expansion from conformal field theory (2015, methodology): “The conformal bootstrap—using conformal symmetry to constrain the dynamics of various physically interesting critical theories—is an old idea [1–3] which in the last few years has been shown to be much more powerful than previously thought [4–11].”
Projectors, shadows, and conformal blocks (2012, background): “In particular, it would be interesting to apply it to superconformal theories, perhaps providing a way to bypass the complicated superconformal block calculations in [14, 42, 43].”
Solving the 3d Ising Model with the Conformal Bootstrap II. $$c$$c-Minimization and Precise Critical Exponents (2012, methodology): “This establishes the equivalence of the primal method and the dual (or “extremal functional”) method discussed in [2,10].”
Boundary and interface CFTs from the conformal bootstrap (2015, methodology): “The method proposed in [12, 16] which we refer to as the linear functional method, relies on unitarity to find forbidden regions in the space of the CFT data, by considering particular channels in the conformal block decomposition of a four-point function.”
Bootstrapping the O(N) archipelago (2015, unknown): “Previously such bounds were explored in d = 4 in [6, 10] and in d = 3, 5 in [22].”
Bootstrapping 3D fermions (2015, background): “The bound has similar features to analogous bounds on CT coming from scalar 4-point functions in four dimensional CFTs [55–57].”
Universality of long-distance AdS physics from the CFT bootstrap (2014, methodology): “The bootstrap was originally developed in the case of CFT2 [1, 2], and has recently seen extensive analytic [7, 9, 14, 40–43] and numerical [3, 7, 44–61] application in CFT≥3.”
Critical exponents of the 3d Ising and related models from conformal bootstrap (2014, background): “In both cases the zeros of the scalar product with |L,∆,∆φ〉 provide the operator spectrum [14, 18], hence a solution to the crossing symmetry; on the other hand 〈Λ| must fulfil the much stronger constraint”
Convexity and liberation at large spin (2012, background): “Here we discuss d = 4 N = 1 SCFTs (see also [69] and references therein).”
The analytic bootstrap and AdS superhorizon locality (2012, methodology): “This provides a non-trivial consistency check of our results and those of [12].”
The S-matrix bootstrap. Part I: QFT in AdS (2016, methodology): “Precisely when the numerical OPE coefficient bound is saturated we can extract an approximate solution to the crossing symmetry equations as a side result from the numerical analysis [31, 32].”
A universal inequality for CFT and quantum gravity (2009, unknown): “(Also see [16, 17, 18, 19, 20, 21, 22].)”
Emergent spacetime and holographic CFTs (2011, methodology): “See [68], [69], [70], [71], [72], [73] for recent work were the unitarity of the CFT in d > 2 was used to derive certain constraints for the data ∆i, Cijk, in special cases.”