Recursion relation for general 3d blocks

Abstract

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].

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

“…Moreover, when the momentum-space 3-point functions are known, constructing conformal blocks out of them is simple in the sense that it does not require additional integration [44][45][46]: a recent example where this technology has been put to good use is Ref. [64].…”

A multipoint conformal block chain in d dimensions

“…For instance, explicit expressions are known for four-point scalar conformal blocks in general spacetime dimensions [8][9][10][11][12][13]. A variety of techniques have been developed for computing four-point conformal blocks involving external and internal exchanged operators in arbitrary representations of the Lorentz group in closed-form, integral or efficient series expansions; a partial list includes various recursive methods [11,[13][14][15][16][17][18][19][20][21], shadow formalism [22], use of differential operators [23][24][25][26][27][28][29][30][31][32][33][34][35], Wilson line constructions [36][37][38], integrability methods [39][40][41][42] and holographic geodesic diagram techniques [43][44][45][46][47][48][49][50][51]…”

“…This integral can be worked out by applying Schwinger parametrization and subsequently turning it into a Mellin-Barnes integral. Since these techniques are not new, 21 in the interest of keeping the appendix to a reasonable length, we refrain from presenting all the standard intermediate steps and refer the reader to the references in footnote 21. These computations lead to the following combination of Mellin-Barnes integrals…”

The superconformal equation

“…Conformal blocks for four-point functions of local operators in bosonic conformal field theories are relatively well studied by now, see e.g. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and references therein. On the other hand, while we know of many examples of such theories in d = 3 dimensions, most conformal field theories in d ≥ 4 seem to possess supersymmetry.…”