Stress Tensor Bootstrap in CFTs

• Stress tensor bootstrap is a framework that applies crossing symmetry, unitarity, and conservation of stress tensor correlators to map out the landscape of conformal field theories.
• It employs numerical optimization, semidefinite programming, and tensor-structure analysis to establish rigorous bounds on central charges and operator dimensions across various dimensions and supersymmetric settings.
• Analytic and mixed-correlator methods within stress tensor bootstrap uncover universal features—such as kinks and intersections—that help discriminate between known theories and emerging CFT scenarios.

Searching arXiv for recent and foundational papers on the stress tensor bootstrap to ground the article in the current literature. Stress tensor bootstrap denotes a family of bootstrap programs centered on the stress tensor $T_{\mu\nu}$ or, in supersymmetric settings, on the universal stress-tensor multiplet. In the standard conformal-bootstrap usage, it studies crossing-symmetric, unitary correlators such as $\langle TTTT\rangle$ or universal four-point functions of stress-tensor-multiplet primaries, using conservation, Ward identities, tensor-structure analysis, and optimization to constrain central charges, OPE coefficients, and low-lying spectra. In adjacent usages, the term also refers to analytic reconstruction of multi-stress-tensor sectors in heavy-light correlators and to symbol or form-factor bootstraps with stress-tensor-multiplet insertions; these are related by bootstrap logic but are kinematically distinct problems (Dymarsky et al., 2017, Beem et al., 2015, Li, 2019, Dixon et al., 2022).

1. Universal role and scope

In parity-preserving three-dimensional CFTs, the stress tensor is a conserved spin-2, parity-even, traceless symmetric operator of scaling dimension $\Delta_T=3$, and every local CFT necessarily contains it. This makes the correlator $\langle TTTT\rangle$ unusually universal: unlike scalar-bootstrap studies, which depend on the existence of a chosen scalar, stress-tensor bootstrap probes essentially all local, unitary 3d CFTs with parity symmetry (Dymarsky et al., 2017, Erramilli et al., 13 Feb 2026).

A closely analogous universality appears in supersymmetric theories. In six-dimensional $(2,0)$ SCFTs, every local interacting theory contains the half-BPS stress-tensor multiplet $D[2,0]$, whose superconformal primary is a scalar $\Phi^{IJ}$ of scaling dimension $\Delta_\Phi=4$ in the $\mathbf{14}$ of $\mathfrak{so}(5)_R$. Bootstrapping the four-point function of $\langle TTTT\rangle$0 is therefore equivalent to bootstrapping the stress-tensor-multiplet four-point function, but in a technically simpler representation (Beem et al., 2015). In 4d $\langle TTTT\rangle$1 SCFTs, the mixed correlator $\langle TTTT\rangle$2, with $\langle TTTT\rangle$3 the superconformal primary of the stress-tensor multiplet and $\langle TTTT\rangle$4 a chiral primary, plays an analogous role as a natural mixed-correlator stress-tensor system (Rakshit et al., 2023).

This universality is the central reason the stress tensor occupies a special place in bootstrap theory. It couples locality, conservation, and energy transport to the algebraic structure of crossing. A plausible implication is that stress-tensor bootstrap bounds can reveal features of the global CFT landscape that scalar-only systems do not directly access.

2. Core structures and numerical formulation

The general bootstrap logic used in stress-tensor studies follows the standard optimization-based pattern: identify observables and consistency equations, supplement them with positivity or unitarity conditions, formulate an optimization problem, and compute rigorous bounds. In the conformal-bootstrap setting, this is the same general machinery described for scalar correlators by the schematic crossing equation

$\langle TTTT\rangle$5

with positivity of OPE-squared data in unitary theories and exclusion or optimization implemented through linear functionals and semidefinite programming (Zheng, 2023).

For 3d stress-tensor bootstrap, the schematic OPE is

$\langle TTTT\rangle$6

Because $\langle TTTT\rangle$7 is parity-even, spin-2, conserved, and a singlet, only specific exchanged representations can appear. In parity-preserving 3d CFTs, the stress-tensor three-point function has two independent parity-even structures, conventionally written as

$\langle TTTT\rangle$8

or equivalently in a normalization with coefficients $\langle TTTT\rangle$9. The average null energy condition implies

$\Delta_T=3$0

and the collider-angle parametrization

$\Delta_T=3$1

packages the bosonic versus fermionic admixture of $\Delta_T=3$2 (Erramilli et al., 13 Feb 2026).

The spinning crossing equations are matrix-valued because multiple $\Delta_T=3$3 structures can occur. In 3d they are written schematically as

$\Delta_T=3$4

A major technical issue is the choice of a strictly linearly independent set of tensor structures and crossing equations. In the mixed $\Delta_T=3$5 system, conserved three-point structures must remain linearly independent for all allowed $\Delta_T=3$6, including asymptotically as $\Delta_T=3$7, and the conservation equations reduce spinning correlators to “bulk”, “line”, and “point” functional degrees of freedom. For $\Delta_T=3$8, after imposing conformal symmetry, parity, permutation symmetry, and conservation, the independent data consist of $\Delta_T=3$9 bulk, $\langle TTTT\rangle$0 line, and $\langle TTTT\rangle$1 point degrees of freedom; redundant line or point equations can produce near-flat directions and can even fake exclusions of the true Ising region (Chang et al., 2024).

This technical layer is not peripheral. In stress-tensor bootstrap, conservation and basis independence are themselves part of the bootstrap data. They determine which positivity statements and which derivative functionals are mathematically meaningful.

3. Universal three-dimensional stress-tensor bootstrap

The first systematic 3d single-correlator stress-tensor bootstrap analyzed the crossing equations for $\langle TTTT\rangle$2 in the most general parity-preserving, unitary 3d CFT. A central result was that, with no additional assumptions, the numerical bootstrap reproduced the conformal collider bounds from Euclidean crossing: $\langle TTTT\rangle$3 The same analysis produced universal upper bounds on the lightest parity-even and parity-odd scalar operators in the $\langle TTTT\rangle$4 OPE,

$\langle TTTT\rangle$5

and showed that upper bounds on $\langle TTTT\rangle$6 appear only after one excludes large-$\langle TTTT\rangle$7 or mean-field-like spectra by imposing sufficiently strong scalar or spin-2 gaps (Dymarsky et al., 2017).

Later work used the same universal $\langle TTTT\rangle$8 system in a different way: not to isolate one target theory, but to scan the global space of local, unitary, parity-preserving 3d CFTs by optimizing OPE coefficients such as $\langle TTTT\rangle$9 and $(2,0)$0 as functions of the leading parity-even and parity-odd singlet scalar dimensions

$(2,0)$1

The resulting landscape exhibits sharp kinks, ridges, corners, and intersections. Several boundary points correspond to known theories: stress tensor mean field theory at $(2,0)$2, free real scalar at $(2,0)$3, free Majorana fermion at $(2,0)$4, the 3d Ising CFT near $(2,0)$5 and $(2,0)$6, and the large-$(2,0)$7 limit of the $(2,0)$8 family at $(2,0)$9. Interior structures include the free Dirac fermion and large-$D[2,0]$0 QED$D[2,0]$1 at $D[2,0]$2, the Gross–Neveu–Yukawa large-$D[2,0]$3 point at $D[2,0]$4, and the chiral Ising large-$D[2,0]$5 point at $D[2,0]$6 (Erramilli et al., 13 Feb 2026).

The “CFT cartography” program proposed in that work interprets these optimized surfaces as geometry of the CFT landscape itself. The paper is careful not to identify every ridge with a known theory, and it explicitly lists caveats: finite numerical order, single-correlator limitations, restricted visible spectrum, and possible “sharing” effects when extra low-lying spin-2 operators are present. Even with those caveats, a clear implication is that $D[2,0]$7 alone already contains far more theory-discriminating information than one might expect from a universal single-correlator system.

4. Mixed-correlator bootstrap of the 3d Ising stress tensor

A major development was the first high-precision 3d Ising bootstrap that treated the stress tensor as an external operator on the same footing as the leading $D[2,0]$8-odd and $D[2,0]$9-even scalars $\Phi^{IJ}$0 and $\Phi^{IJ}$1. The full set of nonvanishing four-point functions imposed simultaneously is

$\Phi^{IJ}$2

This enlargement accesses sectors invisible in the earlier scalar-only system, including $\Phi^{IJ}$3-even odd-spin operators, parity-odd operators, $\Phi^{IJ}$4, and the two independent structures of $\Phi^{IJ}$5 (Chang et al., 2024).

The numerical consequences are substantial. At $\Phi^{IJ}$6, the allowed Ising island in $\Phi^{IJ}$7 is roughly $\Phi^{IJ}$8–$\Phi^{IJ}$9 smaller than in the previous $\Delta_\Phi=4$0-$\Delta_\Phi=4$1 mixed-scalar bootstrap. The final values reported are

$\Delta_\Phi=4$2

$\Delta_\Phi=4$3

$\Delta_\Phi=4$4

The same analysis gives the Ising bound

$\Delta_\Phi=4$5

for the lightest parity-odd scalar in the $\Delta_\Phi=4$6-even sector visible to $\Delta_\Phi=4$7 (Chang et al., 2024).

Methodologically, the paper codifies a concrete 3d spinning-bootstrap toolkit. It solves conservation before bootstrapping, works in explicit $\Delta_\Phi=4$8-basis and $\Delta_\Phi=4$9 bases, uses the coordinates

$\mathbf{14}$0

and at $\mathbf{14}$1 reaches $\mathbf{14}$2 functional components. The implementation relied on blocks_3d, improved interpolation for blocks, optimized memory use by more than $\mathbf{14}$3, and large SDPs run on $\mathbf{14}$4 nodes with $\mathbf{14}$5 cores each. These are not merely engineering details: the paper presents them as necessary conditions for stable dual feasible jumps and for the reliability of the mixed stress-tensor system.

The conceptual lesson is that conserved operators are bootstrap-efficient. They come with protected dimensions, Ward identities, shared OPE parameters across several correlators, and access to otherwise hidden sectors.

5. Supersymmetric stress-tensor multiplets

In 6d $\mathbf{14}$6 SCFTs, the stress-tensor bootstrap is formulated as the crossing problem for the universal four-point function of the scalar superconformal primary $\mathbf{14}$7 in the stress-tensor multiplet $\mathbf{14}$8. Superconformal Ward identities reduce the correlator to one two-variable function $\mathbf{14}$9 and one meromorphic one-variable function $\mathfrak{so}(5)_R$0. Defining

$\mathfrak{so}(5)_R$1

the meromorphic crossing equation becomes

$\mathfrak{so}(5)_R$2

which is exactly the crossing equation for a 2d chiral operator of dimension two. The 2d chiral-algebra identification fixes $\mathfrak{so}(5)_R$3 completely in terms of the central charge $\mathfrak{so}(5)_R$4, with $\mathfrak{so}(5)_R$5; the remaining bootstrap problem is the crossing equation for the unknown dynamical function $\mathfrak{so}(5)_R$6 (Beem et al., 2015).

Numerically, this leads to a strong lower bound on the central charge. The strongest rigorous bound reported is

$\mathfrak{so}(5)_R$7

and the sequence of bounds extrapolates convincingly to

$\mathfrak{so}(5)_R$8

which is exactly the $\mathfrak{so}(5)_R$9 value. The paper therefore argues that every interacting unitary $\langle TTTT\rangle$00 SCFT without higher-spin currents satisfies $\langle TTTT\rangle$01, with saturation by the $\langle TTTT\rangle$02 theory. At $\langle TTTT\rangle$03, the stress-tensor-multiplet four-point function is argued to be the unique unitary solution of the crossing equation. The paper also estimates the lightest unprotected scalar in the $\langle TTTT\rangle$04 theory to lie in the interval

$\langle TTTT\rangle$05

(Beem et al., 2015).

A distinct but related 4d $\langle TTTT\rangle$06 development computes the $\langle TTTT\rangle$07-channel superconformal partial waves for the mixed correlator

$\langle TTTT\rangle$08

where $\langle TTTT\rangle$09 is the superconformal primary of the stress-tensor multiplet and $\langle TTTT\rangle$10 is a chiral primary. The exchanged generic long multiplet is

$\langle TTTT\rangle$11

and the resulting superblocks are expressed as finite sums of ordinary bosonic conformal blocks. For odd $\langle TTTT\rangle$12, the block is a linear combination of $\langle TTTT\rangle$13 and $\langle TTTT\rangle$14; for even $\langle TTTT\rangle$15, it takes the form

$\langle TTTT\rangle$16

The paper explicitly states that this is only part of a full mixed-correlator bootstrap, because the crossed-channel decomposition is still needed (Rakshit et al., 2023).

Together, these supersymmetric results show that stress-tensor bootstrap is not tied to one kinematic or symmetry class. In maximally constrained settings, protected subsectors can reduce the unknown data dramatically and convert the universal stress-tensor multiplet into an exceptionally rigid bootstrap object.

6. Analytic heavy-light and large-$\langle TTTT\rangle$17 stress-tensor sectors

A different branch of stress-tensor bootstrap studies heavy-heavy-light-light correlators in large-$\langle TTTT\rangle$18 CFTs. In even spacetime dimension, the leading minimal-twist $\langle TTTT\rangle$19-stress-tensor contribution to the near-lightcone correlator is captured by the ansatz

$\langle TTTT\rangle$20

with

$\langle TTTT\rangle$21

Crossing determines the coefficients recursively: stress-tensor exchange fixes heavy-light double-twist anomalous dimensions and OPE coefficients at order $\langle TTTT\rangle$22, those determine the double-stress ansatz at order $\langle TTTT\rangle$23, and so on. This program yields explicit double-stress data in $\langle TTTT\rangle$24 and $\langle TTTT\rangle$25, triple-stress data in $\langle TTTT\rangle$26, and evidence for exponentiation of the near-lightcone stress-tensor sector, analogous in spirit to the 2d Virasoro vacuum block (Karlsson et al., 2019).

A complementary “back-and-forth” Lorentzian inversion program turns this into a recursive algorithm. Starting from the universal stress-tensor OPE coefficient

$\langle TTTT\rangle$27

one inverts to obtain large-spin heavy-light double-twist data; reconstructs the crossed correlator; inverts back to isolate the lowest-twist $\langle TTTT\rangle$28 family; and then iterates to $\langle TTTT\rangle$29 and higher. In general dimension, this gives a closed formula for the order-$\langle TTTT\rangle$30 anomalous dimensions of $\langle TTTT\rangle$31; in $\langle TTTT\rangle$32, it yields exact lowest-twist double-stress-tensor OPE coefficients and low-spin triple-stress-tensor data (Li, 2019).

The finite-spin refinement of this program shows that order-$\langle TTTT\rangle$33 stress-tensor exchange fixes finite-spin heavy-light anomalous dimensions in $\langle TTTT\rangle$34, and in general dimension gives a universal formula for $\langle TTTT\rangle$35. This in turn resolves the interpretation of poles in $\langle TTTT\rangle$36 appearing in lowest-twist double-stress-tensor OPE coefficients: they signal mixing with double-trace operators $\langle TTTT\rangle$37. The paper verifies the residue relation

$\langle TTTT\rangle$38

and analyzes the $\langle TTTT\rangle$39 case separately, where Virasoro symmetry implies the uniqueness of the double-stress-tensor contribution (Li et al., 2020).

A related large-$\langle TTTT\rangle$40 study of heavy-heavy-light-light correlators isolates the stress-tensor sector as the contribution of the identity, the stress tensor, and all multi-stress-tensor primaries. It bootstraps an ansatz for the lightcone functions $\langle TTTT\rangle$41, determines the higher-spin double-stress OPE coefficients through twist $\langle TTTT\rangle$42, and shows that crossing leaves unfixed only the OPE coefficients of multi-stress tensors with spin $\langle TTTT\rangle$43 and $\langle TTTT\rangle$44. In holographic CFTs, a bulk phase shift then fixes the spin-2 ambiguities, leaving only spin-0 data undetermined (Karlsson et al., 2020).

The Lorentzian inversion formula also enters a more conventional analytic-bootstrap setting. Nonperturbative-in-spin terms, exponentially suppressed at large spin but numerically important at low spin, are essential if one wants the analytically continued leading double-twist trajectory to reproduce the spin-2 stress tensor accurately. In the 3d Ising model, including these terms yields

$\langle TTTT\rangle$45

reproducing the stress-tensor twist at the $\langle TTTT\rangle$46 level. In the 3d $\langle TTTT\rangle$47 model, imposing that the singlet leading trajectory contains the exact stress tensor predicts

$\langle TTTT\rangle$48

improved to

$\langle TTTT\rangle$49

with Monte Carlo inputs (Albayrak et al., 2019).

This analytic line of work shows that “stress tensor bootstrap” is not limited to direct semidefinite studies of $\langle TTTT\rangle$50. It also includes recursive reconstruction of universal multi-stress sectors and precision low-spin constraints extracted from large-spin analyticity.

7. Form-factor and non-CFT extensions

In planar $\langle TTTT\rangle$51 supersymmetric Yang–Mills theory, “stress-tensor bootstrap” often refers to a different class of problems: the bootstrap of form factors of the chiral stress-tensor multiplet. One major result bootstraps the three-point MHV form factor through six, seven, and eight loops. The object actually bootstrapped is the BDS-like normalized quantity $\langle TTTT\rangle$52, using a function space $\langle TTTT\rangle$53 characterized by symbol-letter constraints, integrability, extended-Steinmann-like pair restrictions, a genuinely new triple-adjacency rule, a coaction principle on first coproduct entries, multiple-final-entry conditions, and near-collinear FFOPE data. Through eight loops, the answer is fixed by the $\langle TTTT\rangle$54 data with $\langle TTTT\rangle$55, and the paper interprets the resulting structure through an antipodal duality with the six-point amplitude (Dixon et al., 2022).

The program extends beyond MHV. A later work bootstraps the two-loop four-point NMHV ratio function for the chiral stress-tensor form factor at symbol level. Starting from a finite one-mass two-loop integral space, it imposes finiteness, Galois symmetry, parity, dihedral symmetry, spurious-pole cancellation, ordinary collinear behavior, and finally triple-collinear consistency, which is the decisive condition fixing the remaining parameters. The final symbol contains $\langle TTTT\rangle$56 letters, all drawn from the previously identified $\langle TTTT\rangle$57-letter alphabet for the four-point MHV stress-tensor form factor. This is the first multi-loop non-MHV stress-tensor form factor obtained in this way (He et al., 27 May 2026).

There is also a non-conformal, spectral version of stress-tensor bootstrap for gapped QFTs. In that framework, one studies the Wightman two-point function of the stress tensor, decomposes the spectral density into the trace part $\langle TTTT\rangle$58 and the spin-2 part $\langle TTTT\rangle$59, and uses positivity,

$\langle TTTT\rangle$60

together with semidefinite positivity matrices mixing stress-tensor-created states, two-particle form factors, and partial-wave S-matrix elements. In $\langle TTTT\rangle$61, the UV and IR stress-tensor central charges are encoded in the asymptotics of $\langle TTTT\rangle$62; in $\langle TTTT\rangle$63, one instead obtains the exact sum rule

$\langle TTTT\rangle$64

re-deriving the $\langle TTTT\rangle$65 $\langle TTTT\rangle$66-theorem (Karateev, 2020).

These broader usages are not standard spinning-correlator conformal bootstrap. They nevertheless belong to the history of the subject because they retain the same basic idea: the stress tensor or stress-tensor multiplet is treated as a universal observable whose analyticity, positivity, and consistency conditions constrain the admissible theory space.