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Abstract We study the ℤ2 and U(1) Abelian lattice gauge theories using a bootstrap method, in which the loop equations and positivity conditions are employed for Wilson loops with lengths L ⩽ L max to derive two-sided bounds on the Wilson loop averages. We address a fundamental question that whether the constraints from loop equations and positivity are strong enough to solve lattice gauge theories. We answer this question by bootstrapping the 2D U(1) lattice gauge theory. We show that with sufficiently large L max = 60, the two-sided bounds provide estimates for the plaquette averages with precision near 10 −8 or even higher, suggesting the bootstrap constraints are sufficient to numerically pin down this theory. We compute the bootstrap bounds on the plaquette averages in the 3D ℤ2 and U(1) lattice gauge theories with L max = 16. In the regions with weak or strong coupling, the two-sided bootstrap bounds converge quickly and coincide with the perturbative results to high precision. The bootstrap bounds are well consistent with the Monte Carlo results in the nonperturbative region. We observe interesting connections between the bounds generated by the bootstrap computations and the Griffiths’ inequalities. We present results towards bootstrapping the string tension and glueball mass in Abelian lattice gauge theories.

Abstract We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on the 1D O(N) vector bootstrap. We obtain a remarkably simple bootstrap equation from the O(N) vector crossing equations in the large N limit. The numerical conformal bootstrap bound is saturated by the generalized free field theories, while its extremal functional actions do not converge to any non-vanishing limit. We study the analytical extremal functionals of this crossing equation, for which the total positivity of the SL(2, ℝ) conformal block plays a critical role. We prove the SL(2, ℝ) conformal block is totally positive in the limits with large ∆ or small 1 − z and show that the total positivity is violated below a critical value ∆ TP ∗ $$ {\Delta }_{\textrm{TP}}^{\ast } $$ ≈ 0.32315626. The SL(2, ℝ) conformal block forms a surprisingly sophisticated mathematical structure, which for instance can violate total positivity at the order 10 −5654 for a normal value ∆ = 0.1627! We construct a series of analytical functionals {α M } which satisfy the bootstrap positive conditions up to a range ∆ ⩽ Λ M . The functionals {α M } have a trivial large M limit. However, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high Λ M . Moreover, in the region ∆ ⩽ Λ M , the analytical functional actions are consistent with the numerical bootstrap results, therefore it clarifies the positive structure in the crossing equation analytically. Our result provides a concrete example to illustrate how the analytical properties of the conformal block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N analytical functional bootstrap in higher dimensions.

Abstract We bootstrap the deconfined quantum critical point (DQCP) and 3D Quantum Electrodynamics (QED3) coupled to N f flavors of two-component Dirac fermions. We show the lattice and perturbative results on the SO(5) symmetric DQCP are excluded by the bootstrap bounds with an assumption that the lowest singlet scalar is irrelevant. Remarkably, we discover a new family of kinks in the 3D SO(N) vector bootstrap bounds with N ⩾ 6. We demonstrate coincidences between SU(N f ) adjoint and SO N f 2 − 1 $$ \textrm{SO}\left({N}_f^2-1\right) $$ vector bootstrap bounds due to a novel algebraic relation between the crossing equations. By introducing gap assumptions breaking the SO N f 2 − 1 $$ \textrm{SO}\left({N}_f^2-1\right) $$ symmetry, the SU(N f ) adjoint bootstrap bounds with large N f converge to the 1/N f perturbative results of QED3. Our results provide strong evidence that the SO(5) DQCP is not continuous and the critical flavor number of QED3 is slightly above 2: N f ∗ ∈ 2 4 $$ {N}_f^{\ast}\in \left(2,4\right) $$ . Bootstrap results near N f ∗ $$ {N}_f^{\ast } $$ are well consistent with the merger and annihilation mechanism for the loss of conformality in QED3.

Abstract Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO(N) vector 4-point functions in general dimension D. In the large N limit, upper bounds on the scaling dimensions of the lowest SO(N) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O(N) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1/2 at large N. We show that the bootstrap bounds also admit another infinite family of kinks T D $$ {\mathcal{T}}_D $$ , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks T D $$ {\mathcal{T}}_D $$ appear in general dimensions with a D-dependent critical N * below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO(N) vectors, SU(N) fundamentals, and SU(N) × SU(N) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of T D $$ {\mathcal{T}}_D $$ are subgroups of SO(N), and we speculate that the kinks T D $$ {\mathcal{T}}_D $$ relate to the fixed points of gauge theories coupled to fermions.

Abstract Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the planar four-point functions at arbitrary ’t Hooft coupling λ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the λ expansions of the correlation functions. The λ expansions terminate at order λ 2 and the solutions are exact in λ. Our work generalizes the approach proposed by Maldacena and Zhiboedov to four-point correlators, and it amounts to an on-shell study for the 3D Chern-Simons vector models and their holographic duals in AdS 4. Besides, we show that the same results can also be obtained rather simply from bosonization duality of 3D Chern-Simons vector models. The odd term at order O(λ) in the spinning four-point function relates to the free boson correlator through a Legendre transformation. This provides new evidence on the 3D bosonization duality at the spinning four-point function level. We expect this work can be generalized to a complete classification of general four-point functions of single trace currents.

Abstract We compute the superconformal partial waves of the four-point correlator 〈JJJJ〉, in which the external operator J is the superconformal primary of the 4D N $$ \mathcal{N} $$ = 2 stress-tensor multiplet J $$ \mathcal{J} $$ . We develop the superembedding formalism for the superconformal field theories (SCFTs) with extended supersymmetry. In N $$ \mathcal{N} $$ = 2 SCFTs, the three- point functions JJO $$ \left\langle \mathcal{JJO}\right\rangle $$ with general multiplet O $$ \mathcal{O} $$ contain two independent nilpotent superconformal invariants and new superconformal tensor structures, which can be nicely constructed from variables in superembedding space, and the three-point functions can be written in compact forms. We compute the superconformal partial waves corresponding to the exchange of long multiplets using supershadow approach. The results are consistent with the non-trivial constraints by decomposing the N $$ \mathcal{N} $$ = 2 superconformal blocks into N $$ \mathcal{N} $$ = 1 superconformal blocks. Our results provide the necessary ingredient to study the fascinating 4D N $$ \mathcal{N} $$ = 2 SCFTs using conformal bootstrap.

Abstract We use the conformal bootstrap approach to explore 5D CFTs with O(N) global symmetry, which contain N scalars ϕ i transforming as O(N) vector. Specifically, we study multiple four-point correlators of the leading O(N) vector ϕ i and the O(N) singlet σ. The crossing symmetry of the four-point functions and the unitarity condition provide nontrivial constraints on the scaling dimensions (Δ ϕ , Δ σ ) of ϕ i and σ. With reasonable assumptions on the gaps between scaling dimensions of ϕ i (σ) and the next O(N) vector ϕ i ′ (singlet σ ′ ) scalar, we are able to isolate the scaling dimensions (Δ ϕ , Δ σ ) in small islands. In particular, for large N = 500, the isolated region is highly consistent with the result obtained from large N expansion. We also study the interacting O(N) CFTs for 1≤N ≤100. Isolated regions on (Δ ϕ , Δ σ ) plane are obtained using conformal bootstrap program with lower order of derivatives Λ; however, they disappear after increasing Λ. For N = 100, no solution can be found with Λ = 25 under the assumptions on the scaling dimensions of next O(N) vector Δ ϕ i ′ ≥ 5.0 $$ {\varDelta}_{\phi_i^{\prime }}\ge\ 5.0 $$ (singlet Δ σ ′ ≥ 3.3). These islands are expected to be corresponding to interacting but nonunitary O(N) CFTs. Our results suggest a lower bound on the critical value N c > 100, below which the interacting O(N) CFTs turn into nonunitary.