Smeared Ishibashi States in CFT and AdS3/CFT2

• Smeared Ishibashi states are defined by applying an imaginary-time evolution operator to conventional Ishibashi states, endowing them with a finite norm and a tunable length scale τ.
• Their matrix elements, when normalized, resum to Verlinde fusion coefficients, providing a clear group-theoretic and path-integral insight into rational CFT fusion rules.
• They serve as variational trial states in deformed CFTs and underpin the construction of bulk-local operators in AdS3/CFT2, bridging algebraic boundary data with infrared physics.

Smeared Ishibashi states are Ishibashi states dressed by imaginary-time evolution, typically through an operator of the form $e^{-\tau H_{\rm CFT}}$ or $e^{-2\tau H_{\rm CFT}}$, so that the resulting states acquire a length scale $\tau$ and finite norm (Cardy, 2017, Fukusumi et al., 8 May 2026). They arise in several distinct but related settings: as variational trial states for relevant deformations of two-dimensional rational CFTs, as the CFT building blocks of local bulk fields in AdS$_3$/CFT$_2$, as free-field and affine-algebra generalizations in boundary $\widehat g_k$ WZW and parafermionic theories, and as an “unphysical” but natural basis for certain gapped phases dual to massless RG flows (Cardy, 2017, Castro et al., 2018, Liu, 2024, Fukusumi et al., 8 May 2026).

1. Definition from boundary-state gluing

In a 2d CFT on a circle of circumference $L$, a conformal boundary state $|\mathcal B\rangle$ satisfies

$$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$

Its general solution is a superposition of Ishibashi states, one in each Virasoro module. If $|i,N\rangle\otimes |\overline{i,N}\rangle$ is an orthonormal basis of level-$e^{-2\tau H_{\rm CFT}}$0 descendants above the primary $e^{-2\tau H_{\rm CFT}}$1, then the Ishibashi state in module $e^{-2\tau H_{\rm CFT}}$2 is

$e^{-2\tau H_{\rm CFT}}$3

The smeared state is defined by inserting a strip of width $e^{-2\tau H_{\rm CFT}}$4 in imaginary time,

$e^{-2\tau H_{\rm CFT}}$5

so that the smeared Ishibashi states carry a length scale $e^{-2\tau H_{\rm CFT}}$6 and have finite norm (Cardy, 2017).

A different normalization convention appears in the analysis of gapped phases, where one smears by

$e^{-2\tau H_{\rm CFT}}$7

and defines the smeared Ishibashi state by acting with this operator on an ordinary Ishibashi state (Fukusumi et al., 8 May 2026). This suggests that “smeared Ishibashi state” is best understood as a structural notion—an Ishibashi state dressed by finite imaginary-time evolution—rather than as a single universally normalized object.

The same terminology also appears in AdS$e^{-2\tau H_{\rm CFT}}$8/CFT$e^{-2\tau H_{\rm CFT}}$9, although there the emphasis is not on Virasoro boundary states on a strip but on endpoint states of open gravitational Wilson lines. In that setting the relevant objects are rotated Ishibashi states, and suitable smearing of their descendant expansion reproduces bulk-local smearing kernels (Castro et al., 2018).

2. Normalization, Cardy decomposition, and fusion-rule matrix elements

In diagonal rational CFTs, the normalization of smeared Ishibashi states is controlled by their relation to physical Cardy boundary states. In the $\tau$0 diagonal minimal series, the physical states are related to Ishibashi states by the modular $\tau$1-matrix, and inversion expresses each Ishibashi state as a linear combination of Cardy states. Using the large-$\tau$2 behavior of overlaps

$\tau$3

one obtains

$\tau$4

and hence the normalized smeared Ishibashi state

$\tau$5

These formulas make precise the statement that smearing regularizes the otherwise non-normalizable Ishibashi construction (Cardy, 2017).

A central result is that matrix elements of bulk primaries between normalized smeared Ishibashi states resum to fusion coefficients. For a bulk primary $\tau$6 of scaling dimension $\tau$7, the normalized matrix element is

$\tau$8

where $\tau$9 is the Verlinde fusion coefficient (Cardy, 2017). In the formulation summarized in that work, the full tower of descendants “has resummed itself into the integer fusion coefficient $_3$0.”

In the $_3$1 diagonal minimal model, the central charge and conformal dimensions are

$_3$2

with primaries labeled by Kac indices $_3$3 (Cardy, 2017). Explicit modular $_3$4-matrix formulae are available there, so the smeared Ishibashi matrix elements can be written completely explicitly.

A common misconception is that smearing merely regularizes overlaps. The fusion-rule formula shows a stronger statement: the smeared matrix elements encode rational-CFT fusion data directly, and in the presentation summarized in (Cardy, 2017) this provides a direct path-integral proof of the Verlinde formula.

3. Variational role in relevant deformations and free-energy bounds

Smeared boundary states and smeared Ishibashi states were proposed as variational approximations to the ground state of a CFT deformed by relevant bulk operators,

$_3$5

In the physical Cardy basis the trial energy is diagonal in the boundary label $_3$6 and is minimized over both $_3$7 and $_3$8: $_3$9 This reproduces the correct scaling $_2$0 and yields a simple phase diagram near the critical point, although within this ansatz all transitions appear as first-order (Cardy, 2017).

The same formalism gives rigorous upper bounds on the universal amplitude of the ground-state free-energy density $_2$1. In $_2$2 spatial dimensions one obtains

$_2$3

and in 2d minimal models this becomes an explicit bound written in terms of the modular $_2$4-matrix entries and the couplings $_2$5 (Cardy, 2017). The significance of smeared Ishibashi states in this context is that the variational problem can be formulated either in the physical Cardy basis or, equivalently, in the smeared Ishibashi basis.

The variational interpretation also clarifies what smearing accomplishes physically. The factor $_2$6 suppresses high-energy descendants by a Boltzmann weight and converts a conformal boundary state into a scale-dependent trial state for a massive theory. A plausible implication is that smeared Ishibashi states interpolate between purely algebraic boundary-state data and the infrared energetics of relevant deformations.

4. Rotated and smeared Ishibashi states in AdS$_2$7/CFT$_2$8

In AdS$_2$9 gravity written in Chern–Simons form, an open gravitational Wilson line from $\widehat g_k$0 to $\widehat g_k$1 is evaluated between endpoint states $\widehat g_k$2 in a representation $\widehat g_k$3,

$\widehat g_k$4

The endpoint states are identified with rotated Ishibashi states $\widehat g_k$5, defined by

$\widehat g_k$6

and these states furnish an over-complete coherent-state basis in $\widehat g_k$7 (Castro et al., 2018). Choosing $\widehat g_k$8 and $\widehat g_k$9 selects which rotated Ishibashi state sits at each endpoint of the Wilson line.

When the endpoints are sent to the boundary, the Wilson-line operator produces a non-local boundary insertion. In Poincaré AdS$L$0, the descendant expansion of the Ishibashi sum can be rewritten as a smearing integral,

$L$1

with kernel, up to an overall normalization,

$L$2

Equivalently, expanding in derivatives reproduces the series of $L$3 with coefficients $L$4 and alternating sign $L$5 (Castro et al., 2018).

The corresponding two-point function of smeared boundary insertions reproduces the bulk-to-bulk propagator, and the same answer emerges directly from the Wilson line: $L$6 where $L$7 is the geodesic distance in the effective metric

$L$8

in the notation of the summary (Castro et al., 2018). In this formulation, suitably smeared Ishibashi states in the CFT provide exactly the building blocks for local bulk fields in AdS$L$9.

The comparison with HKLL is explicit. Earlier constructions of local bulk operators proceed by choosing a particular smearing of $|\mathcal B\rangle$0 over a spacelike or complexified region and rely heavily on the bulk wave equation in each background. By contrast, the Wilson-line plus rotated-Ishibashi construction is described as purely group-theoretic and gauge-invariant: one uses flat $|\mathcal B\rangle$1 connections and algebraic overlaps in the infinite-dimensional representation, without solving bulk differential equations (Castro et al., 2018).

5. Free-field, affine, and parafermionic generalizations

A free-field extension of Ishibashi-state technology has been developed for boundary $|\mathcal B\rangle$2 WZW theories and $|\mathcal B\rangle$3 parafermionic theories. The starting point is the free-field resolution conjecture for dominant irreducible $|\mathcal B\rangle$4 highest-weight modules $|\mathcal B\rangle$5, realized as the zeroth cohomology of a complex of Fock modules $|\mathcal B\rangle$6 built from affine screening currents (Liu, 2024). The same free-field complex, projected to $|\mathcal B\rangle$7-singlets, yields the parafermion modules $|\mathcal B\rangle$8.

In Fock space, an Ishibashi state glued by an automorphism $|\mathcal B\rangle$9 is written as

$$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$0

and by summing with alternating Weyl signs one obtains the $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$1-Ishibashi state in the irreducible module $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$2 (Liu, 2024). The automorphism $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$3 is constrained by Virasoro-gluing, Cartan-gluing, and ladder-gluing conditions; the solutions compatible with all three are, up to Weyl conjugation, the trivial automorphism, the charge-conjugation automorphism, and in some algebras a small discrete family of outer automorphisms.

The free-field formalism makes it natural to define an extended smeared Ishibashi state by integrating over continuous bosonic momenta,

$$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$4

and then imposing BRST or resolution constraints that localize $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$5 onto the affine-Weyl orbit $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$6 (Liu, 2024). Equivalently, with a smearing function $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$7 on $$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$8, one sets

$$(L_n-\bar L_{-n})\,|\mathcal B\rangle=0,\qquad \forall\, n\in\mathbb Z.$$9

In practice one picks $|i,N\rangle\otimes |\overline{i,N}\rangle$0 to be a Gaussian or other regulator so that all sums and integrals converge rapidly.

This construction is significant because it extends the notion of smearing beyond Virasoro Ishibashi states in rational minimal models. It incorporates mode expansions, screening-insertion rules, and disk one- and two-point functions in a setting where affine and coset symmetries are explicit. A plausible implication is that smearing can be organized at the level of free-field resolution data, rather than only at the level of already-projected irreducible modules.

6. “Unphysical” boundary states, gapped phases, and noninvertible symmetry breaking

In ordinary boundary CFT, Ishibashi states are not themselves physical boundary conditions because their overlap decompositions involve modular $|i,N\rangle\otimes |\overline{i,N}\rangle$1-matrix entries $|i,N\rangle\otimes |\overline{i,N}\rangle$2, which are not generally non-negative integers. Cardy states repair this by forming non-negative integer linear combinations of Ishibashi states. In the language of the 2026 analysis, smeared Ishibashi states are therefore “unphysical” in boundary critical phenomena (Fukusumi et al., 8 May 2026).

That same work argues that this conclusion changes for gapped bulk phases obtained from strongly deformed boundary strips. There one trades conformal invariance for a physical mass gap, and the requirement of integer open-channel multiplicities is relaxed. In that setting, the smeared Ishibashi states provide a basis for the low-energy Hilbert space even though they are not admissible as exact BCFT boundary conditions (Fukusumi et al., 8 May 2026). This addresses a frequent source of confusion: “unphysical” here means “not a conformal boundary condition of critical BCFT,” not “mathematically ill-defined” or “irrelevant to bulk physics.”

The central example is the dual massive flow associated with the tricritical Ising $|i,N\rangle\otimes |\overline{i,N}\rangle$3 Ising $|i,N\rangle\otimes |\overline{i,N}\rangle$4 massless flow. The flow preserves an Ising-fusion-ring symmetry $|i,N\rangle\otimes |\overline{i,N}\rangle$5. Among the six tricritical idempotents $|i,N\rangle\otimes |\overline{i,N}\rangle$6, three are preserved under the massless embedding and appear in the Cardy basis of the tricritical boundary CFT dual to the massless flow; the remaining three form the set $|i,N\rangle\otimes |\overline{i,N}\rangle$7 and are “Higgsed” in the massless flow. In the dual massive flow, the gapped-phase Hilbert space is spanned precisely by the smeared Ishibashi states associated with these unpreserved idempotents (Fukusumi et al., 8 May 2026).

Within this basis, bulk one-point functions of CFT primaries are computed by fusion and quantum-dimension data: $|i,N\rangle\otimes |\overline{i,N}\rangle$8 The formula is presented as the exact analog of Cardy’s formula for boundary one-point functions, but now written in the Ishibashi basis and applied to a gapped bulk rather than a critical boundary (Fukusumi et al., 8 May 2026).

The broader interpretation is in terms of spontaneous breaking of non-group-like, or noninvertible, symmetries. When the symmetry algebra is a non-group fusion ring, the module of ground states can lie outside the span of Cardy states for any boundary CFT with that symmetry, yet the smeared Ishibashi basis still transforms correctly under the preserved subring and produces a multiplet of degenerate vacua linked by fusion (Fukusumi et al., 8 May 2026). This suggests that smeared Ishibashi states are not merely a regularization device; in some phases they are the natural state basis singled out by duality and symmetry breaking.