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Noninvertible T-duality defect for the compact boson

Physical scope

Conditional topological-defect construction in the idealized compact-boson two-dimensional QFT under the stated flat-gauging/SymTFT assumptions.

Statement

Under the source's compact-boson and gauging conventions, gauging $U (1)_{w}$ with flat connections and then the period- $2 π / R$ subgroup $Z \subset R$ gives a topological radius-changing interface $I_{R} : C_{R} \to C_{1 / R}$ . Composing it with the invertible radius-duality interface $U_{1 / R \to R} : C_{1 / R} \to C_{R}$ gives the endodefect $N_{R} = U_{1 / R \to R} \circ I_{R} : C_{R} \to C_{R}$ . Here $R > 0$ is the dimensionless radius in Argurio et al.'s convention. It is related to the local Ginsparg convention by $R = \sqrt{2} r$ , so $R = 1$ corresponds to $r = 1 / \sqrt{2}$ . On a genuine primary $V_{n, w}$ , $n, w \in Z$ , the source-reported charge-label action of this defect, up to a nonzero overall normalization, is $N_{R} : V_{n, w} ⟼ {\begin{cases} V_{R^{2} w, n / R^{2}}, & R^{2} w \in Z and n / R^{2} \in Z, \\ a non-genuine defect endpoint, & otherwise . \end{cases}$ For $R^{2} = p / q$ in lowest positive terms, with $p, q \in Z_{> 0}$ coprime, the finite anomaly-free gauging subgroup is $G_{R} ≅ Z_{p} \times Z_{q}$ and has order $p q$ . For the exact rational fusion claim, assume the $G_{R}$ -generated line sector is finite semisimple with the group lines pairwise non-isomorphic simples, and choose the rational normalization $N_{R} = U_{R} \circ W_{G_{R}}$ , where $W_{G_{R}} : C_{R} \to C_{R} / G_{R}$ is the standard endpoint-framed finite gauging wall and $U_{R} : C_{R} / G_{R} \to C_{R}$ is the selected invertible self-duality interface. No extra scalar is included in this definition. The source identifies its two-step and finite-rational constructions only up to normalization; this item adopts the finite-wall normalization at rational radii and does not claim that the source proves the exact scalar match. Under the self-duality identification of the group lines with $g_{a, b} V_{n, w} = e^{2 π i (a n / p + b w / q)} V_{n, w}$ , $N_{R} \otimes {\overset{ˉ}{N}}_{R} ≅ ⨁_{g \in G_{R}} g .$ For the operator interpretation let $V_{R} = ⨁_{(n, w) \in Z^{2}} C V_{n, w}$ be the algebraic primary-label space. Write $ρ (g_{a, b}) \in {End}_{C} (V_{R})$ for the displayed phase action, and interpret the direct-sum line by the sum of these actions. Its operator is $p q P_{p, q}$ , where $P_{p, q} = (p q)^{- 1} \sum_{a, b} ρ (g_{a, b})$ is the projection onto the span of labels with $p ∣ n, q ∣ w$ . Thus the defect is noninvertible for every $R \neq 1$ ; at $R = 1$ it is the invertible T-duality automorphism. No absolute fusion coefficient for irrational $R^{2}$ is asserted.

This is a conditional physics deduction. The two-step flat-gauging/SymTFT construction and its rational self-duality identification are stated inputs below, not consequences of the compact-boson operator algebra alone.

Statement references

Adopted compact-boson flat-gauging and rational self-duality model

Compact free boson as a full local CFT

Compact-boson momentum-winding lattice and cocycle

The compact-boson lattice is even, unimodular, and mutually local

Fusion and multifusion categories

Topological defects, interfaces, orientation, and fusion conventions

Finite-group gauging wall conventions

Fusion of a normalized finite-group gauging wall with its reverse

Argurio et al., §§2.2–3.1, 4.2

Ginsparg, §§8.1, 8.7

Mnev–Reshetikhin, §§6.1.5–6.1.6

Argurio–Galati–Godechal, §§2–2.1, 3.4, 4.3–4.4

Facts & Assumptions

Given: A dimensionless radius $R > 0$ , the compact-boson full-local-CFT data at $r = R / \sqrt{2}$ , its genuine primary labels $(n, w) \in Z^{2}$ , and the adopted source gauging construction.

[F1]

Compact free boson as a full local CFT: in the Ginsparg convention $p_{L} = n / (2 r) + w r$ , $p_{R} = n / (2 r) - w r$ , and $h = p_{L}^{2} / 2 + N_{o s c}$ , $\overset{ˉ}{h} = p_{R}^{2} / 2 + {\overset{ˉ}{N}}_{o s c}$ , with nonnegative integral oscillator levels.

[F2]

Compact-boson momentum-winding lattice and cocycle: the genuine charge labels are $(n, w) \in Z^{2}$ , and in this local normalization the vertex cocycle is $ε ((n, w), (n^{'}, w^{'})) = (- 1)^{n w^{'}}$ .

[F3]

The compact-boson lattice is even, unimodular, and mutually local: the supplier proves the vertex OPE/locality identities and the radius-duality OPE automorphism with charge swap and phase $(- 1)^{n w}$ .

[F4]

Topological defects, interfaces, orientation, and fusion conventions: an interface and an endodefect have the stated domain/codomain distinction; invertibility means having a two-sided fusion inverse. Dragging a local field across an invertible endodefect gives a linear bijection of the genuine local-field space, with inverse given by dragging across its inverse line.

[F5]

Fusion and multifusion categories: a fusion category is finite semisimple. For the rational subgroup only, [A2] assumes that the subcategory generated by the $G_{R}$ symmetry lines is such a category and that its $p q$ specified lines are pairwise non-isomorphic simple objects. No finite-fusion-category claim is made for the irrational-radius continuous construction.

[F6]

Finite-group gauging wall conventions: the finite gauging is untwisted and its framed flat-bundle groupoid is weighted by $1 / ∣ Aut ∣$ ; an interval sector with relative holonomy $g$ and fixed endpoint trivializations has no nonidentity endpoint-preserving automorphism and hence has weight one.

[F7]

Fusion of a normalized finite-group gauging wall with its reverse: under its finite semisimple line-sector and pairwise-distinct-simple-line hypotheses, fusing the gauging wall and its reverse gives one copy of each group line indexed by the finite group.

[A1]

Adopted compact-boson flat-gauging and rational self-duality model: adopt the source's noncompact SymTFT/two-step flat-gauging construction. First gauge $U (1)_{w}$ with flat connections, producing the noncompact intermediate scalar theory; then gauge the period- $2 π / R$ subgroup $Z \subset R$ of its momentum-shift symmetry, producing the radius-changing topological interface $I_{R} : C_{R} \to C_{1 / R}$ . Compose with the standard invertible radius-duality interface $U_{1 / R \to R} : C_{1 / R} \to C_{R}$ . The source identifies this composite as a topological line and gives its charge-label action on genuine and non-genuine vertex endpoints up to an overall prefactor. The functional-measure foundation and, for irrational $R^{2}$ , an absolute normalization of this construction are not claimed as proved here.

[A2]

Adopted compact-boson flat-gauging and rational self-duality model: For reduced rational $R^{2} = p / q$ , with $p, q \in Z_{> 0}$ coprime, adopt the source's anomaly-free finite subgroup gauging and self-duality interface $U_{R} : C_{R} / G_{R} \to C_{R}$ . Let $G_{R} = {g_{a, b} : 0 \leq a < p, 0 \leq b < q} ≅ Z_{p} \times Z_{q}$ , acting by $g_{a, b} V_{n, w} = e^{2 π i (a n / p + b w / q)} V_{n, w}$ , and assume $U_{R}$ maps each gauging symmetry line $L_{a, b}$ to this line $g_{a, b}$ . Assume the $G_{R}$ -generated topological-line subcategory is finite semisimple and the $g_{a, b}$ are pairwise non-isomorphic simple objects. Normalize the rational defect exactly as $N_{R} = U_{R} \circ W_{G_{R}}$ , with $W_{G_{R}} : C_{R} \to C_{R} / G_{R}$ the endpoint-framed wall of Finite-group gauging wall conventions; there is no additional scalar. Argurio et al. §3.1, equation (3.20), identifies the continuous two-step and rational discrete constructions only up to normalization, and §4.2, equations (4.19)–(4.25), derives the group-line sum up to proportionality and states that modular-bootstrap/locality fixes the rational normalization. The exact finite-wall normalization here is therefore an explicit adopted rational input, not a claim that those source passages prove the scalar match. No continuous-gauging normalization is inferred.

Proof

1.1givenF1F2algebra

Set $r = R / \sqrt{2}$ in [F1]. Then the primary weights are $h_{n, w} = \frac{1}{4} {(\frac{n}{R} + w R)}^{2}, {\overset{ˉ}{h}}_{n, w} = \frac{1}{4} {(\frac{n}{R} - w R)}^{2} .$ The source field $V_{n, w}$ is the local lattice field with momentum label $n$ and winding label $w$ .

1.2givenA1F4algebra

The first $U (1)_{w}$ gauging alone produces a noncompact scalar theory; after the second, period- $2 π / R$ gauging, [A1] supplies the topological radius-changing interface $I_{R} : C_{R} \to C_{1 / R}$ . Composing with the standard invertible radius-duality interface $U_{1 / R \to R}$ gives $N_{R} = U_{1 / R \to R} \circ I_{R} : C_{R} \to C_{R}$ , an endodefect. Its topologicality is adopted in [A1] under the convention [F4].

1.3givenA1F2algebra

Let $R^{2} = p / q$ with $\gcd (p, q) = 1$ . Both output labels $n^{'} = p w / q$ and $w^{'} = q n / p$ are integers exactly when $q ∣ w$ and $p ∣ n$ . For example, $q ∣ p w$ implies $q ∣ w$ : choose integers $u, v$ with $u p + v q = 1$ and multiply by $w$ ; the converse divisibility and the corresponding $p ∣ n$ statement follow by the same argument. Therefore the genuine-output sublattice is precisely the image selected by $P_{p, q}$ .

2.1givenA1step 1.1algebra

If the output labels in [A1] are integral, put $n^{'} = R^{2} w$ , $w^{'} = n / R^{2}$ . Then $n^{'} / R + w^{'} R = R w + n / R$ and $n^{'} / R - w^{'} R = R w - n / R = - (n / R - w R)$ . Substitution in step 1.1 proves that $h$ and $\overset{ˉ}{h}$ are separately preserved; the oscillator levels are unchanged. Consequently the conformal spin $h - \overset{ˉ}{h}$ is preserved.

2.2givenA2F2F3F6step 1.1algebra

At $R = 1$ , the charge map is $A (n, w) = (w, n)$ . The phase $χ (n, w) = (- 1)^{n w}$ makes it an OPE automorphism: using [F2], $χ (n, w) χ (n^{'}, w^{'}) ε (A (n, w), A (n^{'}, w^{'})) = ε ((n, w), (n^{'}, w^{'})) χ (n + n^{'}, w + w^{'}),$ since both exponents agree modulo $2$ . Its square is the identity because $A^{2} = 1$ and $χ (n, w) χ (w, n) = 1$ . To verify line invertibility, take $p = q = 1$ in [A2]: $G_{R}$ is trivial, its standard wall is the identity by [F6], and $U_{R}$ is an invertible self-duality interface. Hence $N_{R} = U_{R}$ is invertible, with the charge-swap action above.

2.3givenA2F2F5F6F7step 1.3algebra

Define $P_{p, q} = (p q)^{- 1} \sum_{a = 0}^{p - 1} \sum_{b = 0}^{q - 1} ρ (g_{a, b})$ on $V_{R}$ . On $V_{n, w}$ , the two finite geometric sums are $p$ when $p ∣ n$ and $0$ otherwise, and $q$ when $q ∣ w$ and $0$ otherwise. Hence $P_{p, q}$ is the projection onto the genuine-output sublattice and $P_{p, q}^{2} = P_{p, q}$ . By [A2], $N_{R}$ is the normalized composite of the gauging wall and the invertible self-duality interface. In the fusion $N_{R} \otimes {\overset{ˉ}{N}}_{R}$ , the adjacent duality interfaces cancel, leaving the gauging wall fused with its orientation reverse; [F6] and [F7], using the finite semisimple line sector in [F5], give $N_{R} \otimes {\overset{ˉ}{N}}_{R} ≅ ⨁_{a, b} g_{a, b}$ . The sum of the induced operators on $V_{R}$ is $p q P_{p, q}$ . If $p q > 1$ , either $p > 1$ and $P_{p, q} V_{1, 0} = 0$ , or $p = 1 < q$ and $P_{p, q} V_{0, 1} = 0$ . These are nonzero genuine local fields by [F2], so the fusion is not the identity.

3.1givenA1F2F4step 1.3step 2.2algebra∎

If $R \neq 1$ and $R^{2} = p / q$ is reduced, then $(p, q) \neq (1, 1)$ . If $p > 1$ , the genuine field $V_{1, 0}$ maps by [A1] to label $(0, q / p)$ , which is nonintegral; if $p = 1$ , then $q > 1$ and $V_{0, 1}$ maps to $(1 / q, 0)$ , which is nonintegral. If $R^{2}$ is irrational, $V_{1, 0}$ maps to $(0, 1 / R^{2})$ , also nonintegral. In each case a nonzero genuine local field is sent to a non-genuine endpoint, so the defect action does not map the local-field space to itself. If the endodefect were invertible, [F4] would give a bijection of that space, a contradiction. Thus the defect is noninvertible for every $R \neq 1$ . At $R = 1$ , step 2.2 proves invertibility.

Remarks

Argurio et al. §§3.1 and 4.2 give the two-step construction and charge action, but leave the absolute normalization of the irrational-radius fusion unresolved; they state that the continuous case requires a twisted-Hilbert-space analysis not supplied there. The rational coefficient is computed conditionally from the finite-gauging wall normalization [F6] and the finite-holonomy count [F7], with the source's rational self-duality identification [A2]. This item does not derive the continuous functional gauge-field integrals from a rigorously constructed continuum measure. Those are the exact unresolved proof obligations for Step 1 readiness.

The following string diagram is shorthand for the rational fusion rule: two oppositely oriented gauging lines fuse to the direct sum of the $p q$ group lines. The upward output denotes the direct-sum object; the vertex denotes fusion. It records the stated orientation and direct-sum channel convention only, and no proof step depends on it.

No claim identifies all irrational-radius defect families with a finite fusion category or with a Tambara–Yamagami category.

Noninvertible T-duality defect for the compact boson

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources