arXiv subjects

Tobias Hansen

Publications and source records attributed to Tobias Hansen.

At least 19 recordsLinked to original sources

arXiv2026

The type IIA Virasoro-Shapiro amplitude in AdS$_4$ $\times$ CP$^3$ from ABJM theory

We consider tree level scattering of gravitons in type IIA string theory on $AdS_4\times \mathbb{CP}^3$ to all orders in $α'$, which is dual to the stress tensor correlator in $U(N)_k\times U(N)_{-k}$ ABJM theory in the planar large $N$ limit and to all orders in large $λ\sim N/k$. The small curvature expansion of this correlator, defined via a Borel transform, is given by the flat space Virasoro-Shapiro amplitude plus AdS curvature corrections. We fix curvature corrections by demanding that their resonances are consistent with the superconformal block expansion of the correlator and with a worldsheet ansatz in terms of single-valued multiple polylogarithms. The first correction is fully fixed in this way, and matches independent results from integrability, as well as the $R^4$ correction at finite AdS curvature that was previously fixed using supersymmetric localization. We are also able to fix the second curvature correction by using a few additional assumptions, and find that it also satisfies various non-trivial consistency checks. We use our results to fix the tree level $D^4R^4$ correction at finite AdS curvature, and to give many predictions for future integrability studies.

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arXiv2026

Superconformal Weight Shifting Operators

We develop a framework for constructing superconformal blocks for correlators of general supermultiplets in theories with $\mathrm{SU}(m,m|2n)$ symmetry, such as four-dimensional $\mathcal{N}=2$ and $\mathcal{N} = 4$ conformal theories. We use analytic superspace, viewed as the super-Grassmannian $\mathrm{Gr}(m|n,2m|2n)$, which includes 4D Minkowski space ($m=2,n=0$). In this formalism, superblocks for non-half-BPS correlators are analogous to non-supersymmetric conformal blocks for correlators of fields with spin. We construct $\mathrm{SU}(m,m|2n)$-covariant differential operators which generalise the existing conformal weight-shifting operators, and thus allow us to derive all superconformal blocks from the known half-BPS blocks. Our results provide a framework from which to advance the conformal bootstrap in 4D supersymmetric settings, with potential extensions to lower and higher-dimensional SCFTs. The Grassmannian formalism is also seen to offer a natural and often simpler alternative to the embedding space formalism of non-supersymmetric CFTs.

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arXiv2026

The AdS Veneziano amplitude at small curvature

We compute the AdS Veneziano amplitude for type IIB gluon scattering in $AdS_5 \times S^3$ to all orders in $α'$ in a small curvature expansion. This is achieved by combining a dispersion relation in the dual $4d$ $\mathcal{N}=2$ SCFT with an ansatz for the amplitude as a worldsheet integral in terms of multiple polylogarithms. The first curvature correction is fully fixed in this way and satisfies consistency checks in the high energy limit, the low energy expansion as previously fixed using supersymmetric localisation, and for the energy of massive string operators, which we independently compute using a semiclassical expansion. We also combine localisation with this first curvature correction to fix the unprotected $D^4F^4$ correction to the amplitude at finite curvature.

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arXiv2024

On the $AdS_3$ Virasoro-Shapiro Amplitude

We consider tree-level scattering amplitudes for four string tachyons on $AdS_3 \times {\cal N}$ with pure NSNS fluxes. We show that in a small curvature expansion, properly defined, the amplitudes take the form of a genus zero integral given by the Virasoro-Shapiro integrand with the extra insertion of single valued multiple polylogarithms. This is the same structure as the one found for the AdS Virasoro-Shapiro amplitude in higher dimensions.

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arXiv2024

Single-valuedness of the AdS Veneziano amplitude

We consider the Veneziano amplitude for the scattering of gluons in type IIB string theory on $AdS_5 \times S^5/\mathbb{Z}_2$ in the presence of D7 branes. On general grounds curvature corrections around flat space can be thought of as arising from the extra insertion of soft gravitons. This naturally leads to an open string world-sheet representation with the extra insertion of single-valued functions evaluated on the real line. We check that the recently obtained first curvature correction is of this form and use this new constraint to compute the second curvature correction of the AdS Veneziano amplitude.

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arXiv2024

High Energy String Scattering in AdS

We study the AdS Virasoro-Shapiro amplitude in the limit of fixed-angle high energy scattering. A recent representation as a world-sheet integral allows to compute the amplitude in this regime by saddle point techniques, very much as in flat space. This result is then compared to a classical scattering computation in AdS and agreement is found. As a byproduct of this comparison we show that AdS curvature corrections exponentiate in the high energy limit.

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arXiv2023

The AdS Virasoro-Shapiro Amplitude

We present a constructive method to compute the AdS Virasoro-Shapiro amplitude, order by order in AdS curvature corrections. At kth order the answer takes the form of a genus zero world-sheet integral involving weight 3k single-valued multiple polylogarithms. The coefficients in our ansatz are fixed, order by order, by requiring: crossing symmetry; the correct supergravity limit; the correct structure of poles, determined by dispersive sum rules; and the dimensions of the first few Konishi-like operators, available from integrability. We explicitly construct the first two curvature corrections. Our final answer then reproduces all localisation results and all CFT data available from integrability, to this order, and produces a wealth of new CFT data for planar N=4 SYM at strong coupling.

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arXiv2023

On the spectrum and structure constants of short operators in N=4 SYM at strong coupling

We study short operators in planar $\mathcal{N}=4$ SYM at strong coupling, for general spin and $SO(6)$ symmetric traceless representations. At strong coupling their dimension grows like $Δ\sim 2\sqrtδ λ^{1/4}$ and their spectrum of degeneracies can be analysed by considering the massive spectrum of type II strings in flat space-time. We furthermore compute their structure constants with two arbitrary chiral primary operators. This is done by considering the four-point correlator of arbitrary chiral primary operators at strong coupling in planar $\mathcal{N}=4$ SYM, including the supergravity approximation plus the infinite tower of stringy corrections that contributes in the flat space limit. Our results are valid for generic rank $n$ symmetric traceless representations of $SO(6)$ and in particular for $n \gg 1$, as long as $n \ll λ^{1/4}$.

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arXiv2023

AdS Virasoro-Shapiro amplitude with KK modes

We determine the first curvature correction for the string amplitude of two supergravity states and two Kaluza-Klein modes on $\text{AdS}_5 \times \text{S}^5$, which is dual to the correlator $\langle \mathcal{O}_2 \mathcal{O}_2 \mathcal{O}_p \mathcal{O}_p \rangle$ of half-BPS operators in $\mathcal{N}=4$ SYM theory. The result has the form of an integral over the Riemann sphere as for the usual Virasoro-Shapiro amplitude, with the insertion of single-valued multiple polylogarithms of weight three. The result fixes OPE data of single-trace operators in $\mathcal{N}=4$ SYM theory at strong coupling, including operators with non-zero $R$-charge and odd spin. We successfully check our results by comparing to data available from integrability, localisation and consistency with a $10d$ effective action.

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arXiv2023

Emergent world-sheet for the AdS Virasoro-Shapiro amplitude

We construct a representation for the first AdS curvature correction to the Virasoro-Shapiro amplitude, as an integral over the Riemann sphere. The integrand is that of the Virasoro-Shapiro amplitude in flat space, with the extra insertion of a linear combination of single-valued multiple polylogarithms of weight three. The integral representation implies an elegant, manifestly single-valued representation for the Wilson coefficients of the low energy expansion.

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arXiv2023

AdS Virasoro-Shapiro from single-valued periods

We determine the full $1/\sqrtλ$ correction to the flat-space Wilson coefficients which enter the AdS Virasoro-Shapiro amplitude in $\mathcal{N}=4$ SYM theory at strong coupling. The assumption that the Wilson coefficients are in the ring of single-valued multiple zeta values, as expected for closed string amplitudes, is surprisingly powerful and leads to a unique solution to the dispersive sum rules relating Wilson coefficients and OPE data obtained in [1]. The corresponding OPE data fully agrees with and extends the results from integrability. The Wilson coefficients to order $1/\sqrtλ$ can be summed into an expression whose structure of poles and residues generalises that of the Virasoro-Shapiro amplitude in flat space.

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arXiv2022

AdS Virasoro-Shapiro from dispersive sum rules

We consider the four-point correlator of the stress-energy tensor in ${\cal N}=4$ SYM, to leading order in inverse powers of the central charge, but including all order corrections in $1/λ$. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small $α'$/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

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arXiv2021

Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group

We study the stress tensor four-point function for $\mathcal{N}=4$ SYM with gauge group $G=SU(N)$, $SO(2N+1)$, $SO(2N)$ or $USp(2N)$ at large $N$. When $G=SU(N)$, the theory is dual to type IIB string theory on $AdS_5\times S^5$ with complexified string coupling $τ_s$, while for the other cases it is dual to the orbifold theory on $AdS_5\times S^5/\mathbb{Z}_2$. In all cases we use the analytic bootstrap and constraints from localization to compute 1-loop and higher derivative tree level corrections to the leading supergravity approximation of the correlator. We give perturbative evidence that the localization constraint in the large $N$ and finite complexified coupling $τ$ limit can be written for each $G$ in terms of Eisenstein series that are modular invariant in terms of $τ_s\proptoτ$, which allows us to fix protected terms in the correlator in that limit. In all cases, we find that the flat space limit of the correlator precisely matches the type IIB S-matrix. We also find a closed form expression for the $SU(N)$ 1-loop Mellin amplitude with supergravity vertices. Finally, we compare our analytic predictions at large $N$ and finite $τ$ to bounds from the numerical bootstrap in the large $N$ regime, and find that they are not saturated for any $G$ and any $τ$, which suggests that no physical theory saturates these bootstrap bounds.

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arXiv2021

The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop

We derive an optical theorem for perturbative CFTs which computes the double discontinuity of conformal correlators from the single discontinuities of lower order correlators, in analogy with the optical theorem for flat space scattering amplitudes. The theorem takes a purely multiplicative form in the CFT impact parameter representation used to describe high-energy scattering in the dual AdS theory. We use this result to study four-point correlation functions that are dominated in the Regge limit by the exchange of the graviton Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering is dominated by double Pomeron exchange and receives contributions from tidal excitations of the scattering states which are efficiently described by an AdS vertex function, in close analogy with the known Regge limit result for one-loop string scattering in flat space at finite string tension. We compare the flat space limit of the conformal correlator to the flat space results and thus derive constraints on the one-loop vertex function for type IIB strings in AdS and also on general spinning tree level type IIB amplitudes in AdS.

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arXiv2020

Operator expansions, layer susceptibility and two-point functions in BCFT

We show that in boundary CFTs, there exists a one-to-one correspondence between the boundary operator expansion of the two-point correlation function and a power series expansion of the layer susceptibility. This general property allows the direct identification of the boundary spectrum and expansion coefficients from the layer susceptibility and opens a new way for efficient calculations of two-point correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function $\langleϕ_i ϕ^i\rangle$ of the O(N) model at the extraordinary transition in 4-$ε$ dimensional semi-infinite space to order $O(ε)$. The bulk operator product expansion of the two-point function gives access to the spectrum of the bulk CFT. In our example, we obtain the averaged anomalous dimensions of scalar composite operators of the O(N) model to order $O(ε^2)$. These agree with the known results both in $ε$ and large-N expansions.

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arXiv2019

Analytic Bootstrap for Boundary CFT

We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator $\langle ϕϕ\rangle$ in $ϕ^4$ theory at the Wilson-Fisher fixed point is computed to order $ε^2$ in the $ε$ expansion.

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arXiv2019

Dispersion Relation for CFT Four-Point Functions

We present a dispersion relation in conformal field theory which expresses the four point function as an integral over its single discontinuity. Exploiting the analytic properties of the OPE and crossing symmetry of the correlator, we show that in perturbative settings the correlator depends only on the spectrum of the theory, as well as the OPE coefficients of certain low twist operators, and can be reconstructed unambiguously. In contrast to the Lorentzian inversion formula, the validity of the dispersion relation does not assume Regge behavior and is not restricted to the exchange of spinning operators. As an application, the correlator $\langle ϕϕϕϕ\rangle$ in $ϕ^4$ theory at the Wilson-Fisher fixed point is computed in closed form to order $ε^2$ in the $ε$ expansion.

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arXiv2018

AdS Weight Shifting Operators

We construct a new class of differential operators that naturally act on AdS harmonic functions. These are weight shifting operators that change the spin and dimension of AdS representations. Together with CFT weight shifting operators, the new operators obey crossing equations that relate distinct representations of the conformal group. We apply our findings to the computation of Witten diagrams, focusing on the particular case of cubic interactions and on massive, symmetric and traceless fields. In particular we show that tree level 4-point Witten diagrams with arbitrary spins, both in the external fields and in the exchanged field, can be reduced to the action of weight shifting operators on similar 4-point Witten diagrams where all fields are scalars. We also show how to obtain the conformal partial wave expansion of these diagrams using the new set of operators. In the case of 1-loop diagrams with cubic couplings we show how to reduce them to similar 1-loop diagrams with scalar fields except for a single external spinning field (which must be a scalar in the case of a two-point diagram). As a bonus, we provide new CFT and AdS weight shifting operators for mixed-symmetry tensors.

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