Bounds on 4D conformal and superconformal field theories

Abstract

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N = 1 superconformal field theories. In any CFT containing a scalar primary φ of dimension d we show that crossing symmetry of φφφφ implies a completely general lower bound on the central charge c ≥ f c (d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients τ IJ and flavor charges. We extend these bounds to 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 τ IJ . In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.

The (2, 0) superconformal bootstrap

“…6 This result rules out the existence of exotic (2, 0) theories with a central charge smaller than that of the A 1 theory. A much stronger conclusion follows if one accepts the standard bootstrap wisdom [59,60] that the crossing equation has a unique unitarity solution whenever a bound is saturated. We are then making the precise mathematical conjecture that for c ¼ 25 the CFT data contained in (1.3) are completely determined by the bootstrap equation.…”

“…Indeed, this will be the reason that we can make progress in studying the conformal phase of (2, 0) SCFTs despite the absence of a conventional definition. Thus in broad terms this work will mirror many recent bootstrap studies [37][38][39]59,60,. We will not review the basic philosophy in any detail here.…”

BootstrappingO(N)vector models in4<d<6

“…9 Recall that in 6 − dimensions, the Lagrangian (1.1) has three fixed points that were refer to as the critical theory, Theory 2, and Theory 3. They correspond, respectively, to the red dot, the black dot close to the red dot, and the other black dot in Figure 2 of [55].…”

No unitary bootstrap for the fractal Ising model

“…The seminal work [8] proposed an efficient numerical procedure for extracting information about the space of all conformal field theories, and has been followed by many other works in a variety of contexts [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Similarly, the studies [24][25][26][27] have shown that it is even possible to analytically derive completely generic constraints on the spectrum of CFTs.…”

“…Similarly, the studies [24][25][26][27] have shown that it is even possible to analytically derive completely generic constraints on the spectrum of CFTs. Numerical works have been possible due to increased computer power in the last decades, and rely crucially on an increased understanding of conformal blocks -see [15,[28][29][30][31][32][33][34][35].…”