Interfaces, Folding, and Fusion

A conformal interface joins two CFTs while preserving conformal transformations tangent to the joining surface. Folding reflects one side and converts the interface into a boundary condition of a product theory. This equivalence is exact only after orientation, stress-tensor normalization, anomalies, and operator conjugation are translated. Bringing two interfaces together is a separate limiting operation and is generally singular.

Required background. Conformal boundaries and defects provide the preserved subgroup and local data. Fusion, junctions, and endpoints provide the general composition problem.

Helpful background. Boundaries, interfaces, and domain walls distinguish interface types in QFT. Chiral blocks, sewing, and modular invariance provide the rational two-dimensional modular input used in the final section.

Interface data and stress flow

Place CFT$_1$ at $y>0$ and CFT$_2$ at $y<0$, using one coordinate $y$ increasing from side 2 to side 1. Away from $y=0$, each stress tensor is conserved. The distribution

$$T_{\mathrm{tot}}^{\mu\nu} =\Theta(y)T_1^{\mu\nu} +\Theta(-y)T_2^{\mu\nu}$$

has

$$\partial_\mu T_{\mathrm{tot}}^{\mu a} =\delta(y)\bigl(T_1^{ya}-T_2^{ya}\bigr),$$

so invariance under translations tangent to the interface requires continuity of tangential momentum flux, possibly after adding an intrinsic interface stress tensor. Normal translation is broken; the jump of normal momentum is represented by the displacement operator rather than set to zero. A transparent or topological interface satisfies stronger conditions, but those are extra dynamical statements.

Interface local data include:

• ambient operator spectra and normalizations on both sides;
• interface primaries and their parallel conformal representations;
• bulk-to-interface coefficients for operators incident from either side;
• reflection and transmission observables in a declared stress-tensor convention;
• a displacement normalization; and
• anomaly and global-symmetry matching conditions.

In two dimensions, stress-tensor one-point functions define useful reflection and transmission coefficients under unitarity and common-normalization assumptions. They do not reduce a generic higher-dimensional interface to a single probability Quella, Runkel, and Watts 2007, §§2–3.

Folding with orientation visible

Reflect side 2 by

$$(\mathbf x,y)\longmapsto(\mathbf x,-y).$$

The interface becomes a boundary condition for

$$\mathrm{CFT}_1\otimes\overline{\mathrm{CFT}}_2,$$

where the bar records orientation reversal. It affects parity-odd terms, chiral labels, anomaly signs, and the normal component of tensor operators. The interface gluing equation becomes a boundary Ward identity of the product theory. Operator insertions from side 2 are reflected and conjugated before their product-theory quantum numbers are read.

This map gives a practical dictionary:

Unfolded interface	Folded boundary
Operator incident from side 1	Bulk operator in factor 1
Operator incident from side 2	Reflected operator in the orientation-reversed factor 2
Interface primary	Boundary primary of the product theory
Equal tangential stress flux	Boundary conservation for the diagonal translations
Normal stress jump	Product-theory displacement insertion
Transparent interface	Permutation-type boundary condition when the two factors match
Factorizing interface	Product of independent boundary conditions

Folding preserves correlators only when the path-integral measure and local counterterms are translated with the fields. If the two theories have incompatible gravitational or global anomalies, a putative gluing condition may require additional interface degrees of freedom or may not exist.

The free scalar at the two extremes

Take canonically normalized copies of the same free scalar on the two sides. A transparent interface identifies the limiting fields and fluxes:

$$\phi_1|=\phi_2|, \qquad \partial_y\phi_1|=\partial_y\phi_2|,$$

where the same unfolded $y$ coordinate is used on both sides. Correlators continue through the interface with the full-space Green function and no reflected image term. After folding, the normal derivative of the reflected second field changes sign; the gluing becomes a permutation boundary condition for the two scalar factors.

A factorizing interface has zero transmission. Each side ends on its own boundary condition. The same-side propagator is then

$$G_\sigma(x,x') =\kappa_d\left[ \lvert x-x'\rvert^{2-d} +\sigma\lvert x-\bar x'\rvert^{2-d} \right], \qquad \sigma=+1\ \mathrm N,\quad -1\ \mathrm D,$$

and cross-interface connected correlators vanish. Thus the image signs represent complete reflection with different boundary operator content; they are not transmission amplitudes for a generic interface. Intermediate linear gluing can mix fields and derivatives, but reflection and transmission depend on action normalization and positivity constraints and must be derived from that gluing problem.

Fusion is a regulated limit

Represent an interface from theory 1 to 2 as an operator $\mathcal I_{12}$ between Hilbert spaces. A regulated composition through theory 2 is

$$\mathcal I_{12}\star_\epsilon\mathcal I_{23} =\mathcal I_{12}\, e^{-\epsilon H_2}\, \mathcal I_{23}, \qquad \epsilon>0.$$

The factor $e^{-\epsilon H_2}$ separates the interfaces. As $\epsilon\to0^+$, high-energy states of the middle theory can generate:

• divergent interface tension or identity terms;
• mixing among interface operators;
• logarithms and an interface RG flow;
• a sum of interfaces rather than a single simple object; or
• dependence on finite counterterms.

Consequently, “bring the interfaces together” is not a definition of fusion without a regulator and subtraction prescription. In the compact free boson, explicit conformal interfaces exhibit such singular terms; topological interfaces have a particularly controlled composition, while generic conformal ones require renormalization Bachas and Brunner 2008, §§3–5.

An interface is topological when its position can be deformed without changing separated correlators, equivalently when the appropriate displacement vanishes after null states are removed. It is invertible only if its fusion with an oppositely oriented interface gives the identity interface with no additional summands. Topological does not imply invertible: noninvertible topological defects can have nontrivial fusion multiplicities.

Bounded two-dimensional boundary-state specialization

The rest of this page applies only to a two-dimensional CFT with a specified chiral algebra and a discrete rational set of modules. It is not a construction for a generic codimension-$q$ defect.

A conformal boundary state obeys the Virasoro gluing condition

$$(L_n-\bar L_{-n})|a\rangle=0.$$

For every compatible left–right module, an Ishibashi state $|j\rangle\!\rangle$ solves this equation inside that module. It is a formal closed-channel state, not by itself a physical boundary condition Ishibashi 1989. Expand

$$|a\rangle=\sum_j B_a^{\,j}|j\rangle\!\rangle.$$

The annulus with boundary conditions $a,b$ has an open-channel description

$$Z_{ab}(q) =\operatorname{Tr}_{\mathcal H_{ab}} q^{L_0-c/24} =\sum_i n_{ab}^{\,i}\chi_i(q), \qquad n_{ab}^{\,i}\in\mathbb Z_{\geq0},$$

and a closed-channel description

$$Z_{ab}(q) =\langle a| \widetilde q^{\frac12(L_0+\bar L_0-c/12)} |b\rangle =\sum_j(B_a^{\,j})^*B_b^{\,j}\chi_j(\widetilde q).$$

$q$ and $\widetilde q$ are related by the modular $S$ transformation. If

$$\chi_j(\widetilde q)=\sum_i S_{ji}\chi_i(q),$$

open–closed consistency requires

$$n_{ab}^{\,i} =\sum_j(B_a^{\,j})^*B_b^{\,j}S_{ji} \in\mathbb Z_{\geq0}.$$

For a diagonal rational theory, the Cardy solution

$$B_a^{\,j}=\frac{S_{aj}}{\sqrt{S_{0j}}}$$

turns these multiplicities into fusion coefficients, with conjugations determined by the labeling convention Cardy 1989, pp. 581–596. This conclusion assumes a nondegenerate modular matrix, compatible left–right pairing, complete boundary spectrum, and the selected chiral-algebra gluing. It does not follow from Virasoro symmetry alone in a nonrational or continuous-spectrum theory.

Di Francesco, Mathieu, and Sénéchal work through the cylinder channel exchange and its relation to the Verlinde formula in Di Francesco, Mathieu, and Sénéchal 1997, §11.3.2, pp. 422–426.

Failure tests

Orientation test. Fold and unfold one parity-odd operator. If its sign or chiral representation does not return, the product-theory dictionary is incomplete.

Stress-flux test. Integrate tangential momentum conservation across a narrow pillbox. A discontinuity without an intrinsic contribution violates the interface Ward identity.

Fusion test. Keep $\epsilon>0$ until all divergent identity and relevant interface terms are classified. A finite answer obtained by simply setting $\epsilon=0$ is not regulator independent.

Topologicality test. Move the interface across separated insertions. Vanishing reflection alone does not prove invariance under arbitrary deformation or invertibility under fusion.

Cardy test. Modular-transform the closed-channel amplitude and inspect every $n_{ab}^{\,i}$. Nonintegral or negative multiplicities invalidate the proposed rational boundary state.

The universal endpoint quantities associated with interface or boundary flows are treated on Boundary entropy and defect monotonicity. General anomaly matching is treated on Boundary and defect Weyl anomalies.

References

• Bachas, Constantin, and Ilka Brunner. “Fusion of Conformal Interfaces.” Journal of High Energy Physics 02 (2008): 085. DOI. Open PDF
• Cardy, John L. “Boundary Conditions, Fusion Rules and the Verlinde Formula.” Nuclear Physics B 324 (1989): 581–596. DOI.
• Di Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. Graduate Texts in Contemporary Physics. New York: Springer, 1997. DOI.
• Ishibashi, Nobuyuki. “The Boundary and Crosscap States in Conformal Field Theories.” Modern Physics Letters A 4 (1989): 251–264. DOI.
• Quella, Thomas, Ingo Runkel, and Gérard M. T. Watts. “Reflection and Transmission for Conformal Defects.” Journal of High Energy Physics 04 (2007): 095. DOI. Open PDF