The beginning of the endpoint bootstrap for conformal line defects

TL;DR

The paper develops a numerical conformal bootstrap framework for endable conformal line defects by incorporating defect endpoints into four-point functions and employing crossing relations that mix bulk and defect data with positivity. Applied to the 3d Ising CFT pinning-field defect, it yields tight, cross-validated bounds on defect data, including a well-defined island for the leading endpoint dimension Δ_0^{0+} and robust bounds on the defect g-function and defect-changing operator dimensions. A key conclusion is that the Z2-symmetric long-range-ordered defect D^+ ⊕ D^- cannot be stable, as the leading domain-wall operator Δ_0^{+-} is constrained to be less than 1 under reasonable assumptions, implying domain-wall proliferation without fine-tuning. The work also provides perturbative checks in D = 4 − ε and cross-validates with fuzzy-sphere results, outlining a path to systematic improvements and applications to other defect theories. Overall, the approach establishes a rigorous, data-informed framework to bound and infer defect data in higher-dimensional CFTs, with implications for understanding ordering on defects and for extending bootstrap to defect sectors.

Abstract

A challenge in the study of conformal field theory (CFT) is to characterize the possible defects in specific bulk CFTs. Given the success of numerical bootstrap techniques applied to the characterization of bulk CFTs, it is desirable to develop similar tools to study conformal defects. In this work, we successfully demonstrate this possibility for endable conformal line defects. We achieve this by incorporating the endpoints of a conformal line defect into the numerical four-point bootstrap and exploit novel crossing symmetry relations that mix bulk and defect CFT data in a way that further possesses positivity, so that rigorous numerical bootstrap techniques are applicable. We implement this approach for the pinning field line defect of the $3d$ Ising CFT, obtaining estimates of its defect CFT data that agree well with other recent estimates, particularly those obtained via the fuzzy sphere regularization. An interesting consequence of our bounds is nearly rigorous evidence that the $\mathbb Z_2$-symmetric defect exhibiting long range order obtained as a direct sum of two conjugate pinning field defects is unstable to domain wall proliferation.

Figures (17)

• Figure 1: An illustration of the four types of defect-changing operators considered in this work.
• Figure 2: A graphical representation of the set of correlation functions involved in our numerical bootstrap calculations. We study correlation functions that involve four pinning field endpoint operators, two endpoint operators with two bulk operators, and four bulk operators. In all cases, the operators are inserted collinearly and all non-trivial line defects are straight. We suppress explicitly illustrating the trivial defect for clarity.
• Figure 3: A graphical depiction of the central crossing symmetry relation exploited in this work for four-point functions of endpoints of non-trivial line defects of (possibly) distinct types (indicated by the red and blue lines). Schematically, the crossing transformation acts as a cyclic permutation of the endpoints. This leads to a direct relationship between the expansion of the four-point function in conformal blocks corresponding to the endpoints fusing either to bulk operators (left) or defect-changing operators (right).
• Figure 4: Universal bounds on stable $(\Delta_1^{++} \ge 1)$ conformal line defects that explicitly break a $\mathbb Z_2$ symmetry in $D=2,3,4$ compared with our most general bootstrap bounds for the $3d$ Ising pinning field defect. The shaded regions are allowed by bootstrap at derivative order $\Lambda = 45$. For the $D=3$ Ising bootstrap bounds, we only plot the island expected to contain the pinning field defect created by a bulk $\sigma$ perturbation. There is a larger allowed region beyond the island that we do not show for clarity. Left: Universal lower and upper bounds on the defect $g$-function. For $D=2$ we do not find a non-trivial lower bound. The $D=2$ Ising point lies at $(\Delta_0^{0+},g)_{2d} = (1/32,1/2)$. In the $D=4$ Gaussian theory $\Delta_0^{0+}$ may take any positive real value with $g_{4d} = 1$. For the $3d$ Ising bootstrap result we plot the allowed region where only $\Delta_1^{++} \ge 1$ is assumed. Right: Universal bounds on the leading domain wall primary operator $\Delta_0^{+-}$. In $D=2$ Ising we have $\Delta_0^{+-} = 1$ and in the $D=4$ Gaussian fixed point we have $\Delta_0^{+-} = 4\Delta_0^{0+}$. The $D=3$ Ising bootstrap island is computed assuming $\Delta_1^{++} \ge 1$ and $\Delta_1^{+-} \ge 1$.
• Figure 5: Bootstrap allowed regions for the scaling dimension of the leading domain wall primary $\Delta_0^{+-}$ and the defect $g$ function at $\Lambda = 45$, each computed as a function of the scaling dimension of the leading endpoint operator $\Delta_0^{0+}$. For comparison, we show in the light shaded regions estimates of the same quantities computed via the fuzzy sphere regularization technique Zhou:2023fqu. The bounds are obtained using non-generic gap assumptions in the defect-changing sectors. In each bootstrap bound it is assumed that $\Delta_1^{++} \ge 1.4$, $\Delta_0^{+-} \ge 0.6$, and $\Delta_1^{0+} \ge \Delta_0^{0+} + 1.5$. The calculations use input from the OPE ratio bounds of Fig. \ref{['fig:end_end_sig_ep']}, discussed in Sec. \ref{['sec:OPE_ratio_bounds']}. To obtain the bound on $\Delta_0^{+-}$, we further assume that there is at most one relevant domain wall operator, i.e. $\Delta_1^{+-} \ge 1$.