Boundary Entropy and Defect Monotonicity

Boundary and defect contributions to partition functions and entanglement can contain universal information, but the raw quantities are ultraviolet divergent and their monotonicity depends strongly on defect dimension and on whether the ambient bulk remains critical. This page separates definitions, proven inequalities, conjectural extensions, and counterexamples before applying any endpoint comparison.

Required background. Conformal boundaries and defects provide the defect geometry and local data. Ultraviolet and infrared fixed points provide the meaning of an endpoint RG comparison.

Helpful background. Interfaces, folding, and fusion provide the two-dimensional boundary-state normalization. The displacement operator supplies shape-response data. Monotonicity theorems and flow constraints compare the wider family of bulk and local-RG theorems.

Evidence cutoff: 2026-08-09. The theorem-status statements below were checked against the cited primary literature through this date. They are deliberately conditional: a change in hypotheses, a correction, or a new counterexample can change which row applies without changing the definitions.

Boundary entropy in two dimensions

For a unitary two-dimensional BCFT on a long cylinder with conformal boundary conditions $a$ and $b$, the closed-channel vacuum contribution has the asymptotic form

$$\log Z_{ab} =\frac{\pi c L}{6\beta} +\log g_a+\log g_b +o(1) \qquad(L/\beta\to\infty),$$

in the standard cylinder convention. The boundary entropy is

$$s_a=\log g_a, \qquad g_a=\langle0\lvert a\rangle,$$

after fixing the bulk vacuum and boundary-state normalization. Local boundary counterterms can shift extensive terms, but not this fixed-point constant in the stated BCFT setup.

The $g$-theorem says

$$g_{\mathrm{UV}}\geq g_{\mathrm{IR}}$$

for a local unitary boundary RG flow with the bulk held at a two-dimensional CFT and conformal UV and IR boundary endpoints. The thermal argument identifies the fixed-point quantity Affleck and Ludwig 1991; the gradient-formula proof requires boundary locality and suitable infrared behavior Friedan and Konechny 2004; and the zero-temperature entropic proof uses relative entropy and its monotonicity Casini, Salazar Landea, and Torroba 2016, §§2–4. None of these results licenses a comparison between arbitrary boundaries in different bulk theories.

Ising boundary magnetic-field flow

For the diagonal Ising CFT, a Cardy boundary state has Cardy 1989, §4

$$g_a=\frac{S_{a0}}{\sqrt{S_{00}}}.$$

With the usual Ising modular matrix,

$$g_{\mathrm{free}}=1, \qquad g_{\mathrm{fixed}\,+} =g_{\mathrm{fixed}\,-} =\frac1{\sqrt2}.$$

A boundary magnetic field drives the free boundary to one of the fixed boundaries, so

$$\Delta\log g =\log g_{\mathrm{IR}}-\log g_{\mathrm{UV}} =-\frac12\log2<0.$$

This is an endpoint check inside a unitary rational BCFT. It does not prove a monotone formula for higher-codimension defects, nor does it use the image-sign scalar of $d>2$.

Renormalized defect free energy

For a defect $\mathcal D$ on a background $M$, define the relative generating functional

$$F_{\mathcal D} =-\log\frac{Z[M;\mathcal D]}{Z[M]}.$$

It contains local divergences supported on $\mathcal D$:

$$F_{\mathcal D}^{\mathrm{div}} =\int_{\mathcal D}d^p x\,\sqrt\gamma \left( c_0\Lambda^p+c_1\Lambda^{p-2}\mathcal R+\cdots \right).$$

The ellipsis includes extrinsic-curvature and background-field invariants allowed by the symmetries. A universal quantity must survive the allowed local counterterms. For an even-dimensional defect, a logarithmic coefficient is tied to a defect Weyl anomaly. For an odd-dimensional spherical defect, a suitably defined finite part can be universal. Entanglement and sphere-free-energy definitions can differ by a universal constant at codimension greater than one, so they must not be interchanged silently Kobayashi et al. 2019, §§2–3.

The subtraction is part of the observable. Writing only $-\log Z_{\mathcal D}$ does not identify a universal number, because bulk normalization, defect tension, curvature counterterms, and zero modes remain mixed.

Theorem status by defect dimension

The table below is the semantic comparison required before an endpoint claim. “Channel” names the representation or information-theoretic construction that defines the quantity, not an OPE channel.

Codimension	Boundary or defect	Preserved symmetry	Normalization	Channel	Theorem status at the evidence cutoff	Evidence
$q=1$ in bulk $d=2$	Conformal boundary, $p=1$	$SO(2,1)$	Cylinder vacuum overlap $g_a=\langle0\lvert a\rangle$	Annulus closed-channel vacuum or relative entropy	Proven $g_{\mathrm{UV}}\geq g_{\mathrm{IR}}$ for local unitary boundary flow with fixed critical bulk and conformal endpoints	Affleck–Ludwig 1991; Casini et al. 2016
$q=d-1$	Line defect, $p=1$, in a $d$-dimensional CFT	$SO(2,1)\times SO(d-1)$	Canonical defect entropy defined from the circular defect partition function	Defect entropy/gradient relation	Proven irreversibility for unitary line-defect flows with the ambient CFT fixed and the stated locality assumptions	Cuomo, Komargodski, and Raviv-Moshe 2022
$q=d-2$	Surface defect, $p=2$; includes a boundary in bulk $d=3$	$SO(3,1)\times SO(q)$	Euler-density anomaly coefficient $b$ in a fixed trace-anomaly convention	Defect dilaton or relative entropy	Proven $b_{\mathrm{UV}}\geq b_{\mathrm{IR}}$ for unitary defect-localized flow with conformal ambient bulk	Jensen and O’Bannon 2016; Casini et al. 2019
$q=d-3$	Three-dimensional planar defect, $p=3$	$SO(4,1)\times SO(q)$	Universal defect contribution isolated by relative entropy	Null-cone relative-entropy construction using QNEC	A defect $F$-type irreversibility theorem is established under unitarity, Lorentz invariance, fixed ambient CFT, and the construction’s endpoint assumptions	Casini, Salazar Landea, and Torroba 2023, §§4–5
General $q=d-p$	Spherical conformal defect	$SO(p+1,1)\times SO(q)$	Regularized sphere free-energy increment	Sphere partition function	A general monotonicity proposal has supporting field-theory and holographic examples; those examples are not a proof for every $p,q$ and flow	Kobayashi et al. 2019; Nishioka and Sato 2021
General	Defect while the ambient bulk also flows	Endpoint symmetry can change	Must include bulk and defect counterterms in one convention	Coupled bulk–defect RG	No unrestricted extension of the fixed-bulk $b$-theorem: perturbative examples violate the naive inequality	Shachar, Sinha, and Smolkin 2024, §§3–5

The QNEC construction also gives qualified relations for defect dimensions up to four and a relative-entropy coefficient for higher dimensions. Those statements use a particular planar geometry, Lorentzian continuation, and subtraction. They should be applied from the precise theorem rather than summarized as “every defect free energy decreases” Casini, Salazar Landea, and Torroba 2023, §§1 and 5.

Free scalar boundary flow: what the images establish

For the canonical massless scalar on $y\geq0$, add a boundary quadratic coupling

$$S_{\partial} =\frac{c}{2}\int_{y=0}d^{d-1}x\,\phi^2.$$

With a fixed normal convention, variation gives a Robin condition

$$\partial_y\phi=c\,\phi$$

up to the corresponding outward-normal sign. The coupling has mass dimension one. The endpoint $c=0$ is Neumann, while the large-$c$ infrared boundary limit suppresses $\phi|$ and approaches Dirichlet. Their exact planar correlators have image signs $+1$ and $-1$.

This establishes the endpoint boundary conditions and their local CFT data. It does not by itself compute a universal monotone:

• the planar determinant contains boundary-volume divergences;
• curvature counterterms are needed on a sphere;
• a compact Neumann scalar has a constant zero mode that requires an explicit prescription;
• the universal part depends on $d$ and on the chosen theorem; and
• a simultaneous bulk mass flow would leave the fixed-ambient-CFT hypotheses.

Free-scalar calculations on conformally related hyperbolic and spherical backgrounds support the regularized defect-free-energy proposal for specified Neumann-to-Dirichlet-type flows, with the boundary conditions and zero-mode treatment stated explicitly Nishioka and Sato 2021, §§2–5. Such model checks remain distinct from a dimension-independent theorem.

Stationarity and endpoint checks

A useful endpoint quantity need not be stationary to first order under every defect perturbation. Conversely, a gradient formula can imply stationarity only after operator normalization, contact terms, and beta-function coordinates are fixed. When applying a theorem:

1. identify the ambient theory and confirm it remains at the same CFT;
2. specify defect dimension $p$, codimension $q$, and the preserved symmetries;
3. define the renormalized quantity, including all subtractions and zero modes;
4. state unitarity, reflection-positivity, locality, Lorentz-invariance, and endpoint assumptions;
5. check whether the result is a theorem, a perturbative check, a holographic proof in a model class, or a conjecture; and
6. compare only the UV and IR quantities defined in the same convention.

Failure tests

Raw-partition-function test. Add an allowed local defect counterterm. If the proposed “entropy” changes, it was not yet universal.

Bulk-flow test. Let a bulk coupling run. The fixed-bulk $g$- or $b$-theorem cannot be applied; the cited perturbative counterexamples show that the naive extension can fail.

Dimension test. Replace a line defect by a surface defect while keeping the same symbol for the monotone. The universal term and theorem change.

Zero-mode test. Put the Neumann scalar on a compact background. If the determinant includes the constant mode without a prescription, the endpoint free energy is undefined.

Status test. A sphere-free-energy decrease in several examples is evidence for a proposal, not a proof for all defects. Label the conclusion accordingly.

Defect anomaly coefficients and their scheme-independent combinations are developed on Boundary and defect Weyl anomalies. General flow comparisons belong to Monotonicity theorems and flow constraints.

References

• Affleck, Ian, and Andreas W. W. Ludwig. “Universal Noninteger ‘Ground-State Degeneracy’ in Critical Quantum Systems.” Physical Review Letters 67 (1991): 161–164. DOI.
• Cardy, John L. “Boundary Conditions, Fusion Rules and the Verlinde Formula.” Nuclear Physics B 324 (1989): 581–596. DOI.
• Casini, Horacio, Ignacio Salazar Landea, and Gonzalo Torroba. “The $g$-Theorem and Quantum Information Theory.” Journal of High Energy Physics 10 (2016): 140. DOI. Open PDF
• Casini, Horacio, Ignacio Salazar Landea, and Gonzalo Torroba. “Irreversibility in Quantum Field Theories with Boundaries.” Journal of High Energy Physics 04 (2019): 166. DOI. Open PDF
• Casini, Horacio, Ignacio Salazar Landea, and Gonzalo Torroba. “Irreversibility, QNEC, and Defects.” Journal of High Energy Physics 07 (2023): 004. DOI. Open PDF
• Cuomo, Gabriel, Zohar Komargodski, and Avia Raviv-Moshe. “Renormalization Group Flows on Line Defects.” Physical Review Letters 128 (2022): 021603. DOI. Open PDF
• Friedan, Daniel, and Anatoly Konechny. “On the Boundary Entropy of One-Dimensional Quantum Systems at Low Temperature.” Physical Review Letters 93 (2004): 030402. DOI. Open PDF
• Jensen, Kristan, and Andy O’Bannon. “A Constraint on Defect and Boundary Renormalization Group Flows.” Physical Review Letters 116 (2016): 091601. DOI. Open PDF
• Kobayashi, Nozomu, Tatsuma Nishioka, Yoshiki Sato, and Kento Watanabe. “Towards a $C$-Theorem in Defect CFT.” Journal of High Energy Physics 01 (2019): 039. DOI. Open PDF
• Nishioka, Tatsuma, and Yoshiki Sato. “Free Energy and Defect $C$-Theorem in Free Scalar Theory.” Journal of High Energy Physics 05 (2021): 074. DOI. Open PDF
• Shachar, Tom, Ritam Sinha, and Michael Smolkin. “The Defect $b$-Theorem under Bulk RG Flows.” Journal of High Energy Physics 09 (2024): 057. DOI. Open PDF