Abstract

The genus zero contribution to the four-point correlator leftlangle {mathcal{O}}_{p_1}{mathcal{O}}_{p_2}{mathcal{O}}_{p_3}{mathcal{O}}_{p_4}rightrangle of half-BPS single-particle operators {mathcal{O}}_p in mathcal{N} = 4 super Yang-Mills, at strong coupling, computes the Virasoro-Shapiro amplitude of closed superstrings in AdS5× S5. Combining Mellin space techniques, the large p limit, and data about the spectrum of two-particle operators at tree level in supergravity, we design a bootstrap algorithm which heavily constrains its α′ expansion. We use crossing symmetry, polynomiality in the Mellin variables and the large p limit to stratify the Virasoro-Shapiro amplitude away from the ten-dimensional flat space limit. Then we analyse the spectrum of exchanged two-particle operators at fixed order in the α′ expansion. We impose that the ten-dimensional spin of the spectrum visible at that order is bounded above in the same way as in the flat space amplitude. This constraint determines the Virasoro-Shapiro amplitude in AdS5× S5 up to a small number of ambiguities at each order. We compute it explicitly for (α′)5,6,7,8,9. As the order of α′ grows, the ten dimensional spin grows, and the set of visible two-particle operators opens up. Operators illuminated for the first time receive a string correction to their anomalous dimensions which is uniquely determined and lifts the residual degeneracy of tree level supergravity, due to ten-dimensional conformal symmetry. We encode the lifting of the residual degeneracy in a characteristic polynomial. This object carries information about all orders in α′. It is analytic in the quantum numbers, symmetric under an AdS5 ↔ S5 exchange, and it enjoys intriguing properties, which we explain and detail in various cases.

Highlights

  • In this paper we will consider genus zero string corrections to four-point correlators of single-particle operators Op in AdS5 × S5, following up on previous work [18,19,20] and especially [21], where a formula for the (α )5 amplitude was obtained for arbitrary charges pi=1,2,3,4

  • Operators illuminated for the first time receive a string correction to their anomalous dimensions which is uniquely determined and lifts the residual degeneracy of tree level supergravity, due to ten-dimensional conformal symmetry

  • There it was shown that in the limit in which the charges pi=1,2,3,4 are large, the correlator localises on a classical saddle point, s → scl, s → scl, t → tcl, t → tcl, where the Mellin variables are large, and quite remarkably the combinations scl, tcl and ucl = −scl − tcl become the 10d flat space Mandelstam invariants of a scattering process which is focussed on a bulk point of AdS5 × S5

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Summary

Genus zero amplitudes from the bootstrap

The combination of crossing symmetry and large p stratification, discussed in the previous section, provides us with the initial ansatz for the VS amplitude in AdS5 × S5. The combination of crossing symmetry and large p stratification, discussed, provides us with the initial ansatz for the VS amplitude in AdS5 × S5. Notice that only H1,(0,1) exists, and it is spanned just by Σ. We point out that inside a given Hn,( ,n− ) we can add another level, which is the one given by terms of the form (Σ# × crossing invariants), where usually the latter already appeared at previous orders. The step is to impose constraints on the free parameters in our initial ansatz, at each order in the α expansion. We know from the very beginning that we will not be able to fix the ambiguity of adding previous amplitudes Vk≤n−1 to our result for Vn, within the bootstrap.

Explicit results and remarkable simplifications
The spectrum of two-particle operators at genus zero
SUGRA eigenvalue problem: a review
N2 ηpq
Consequences of the tree-level hidden conformal symmetry
STRINGY eigenvalue problem
Tailoring the bootstrap program
Σ1 Σ2 degree 4 polynomial degree 3 polynomial
Level splitting
General properties of the characteristic polynomial
Conclusions
A On the correlators Op1Op2Op3Op4 at genus zero
Ansatz at all orders: iterative scheme
OPE equations
Superblock decomposition
Spin structures of the flat VS
Full Text
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