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Rational noninvertible T-duality at R squared equals two

Physical scope

A conditional finite-gauging example in the idealized compact-boson two-dimensional QFT at dimensionless radius $R = \sqrt{2}$ .

Statement

At $R = \sqrt{2}$ , so $R^{2} = 2 / 1$ , the rational finite gauging subgroup has order two. The source's self-duality identification labels its nontrivial symmetry line by $g$ , which shifts the compact scalar by $π$ and acts as $g V_{n, w} = (- 1)^{n} V_{n, w}$ . Assume the line subcategory generated by $1$ and $g$ is finite semisimple and that these are non-isomorphic simple objects. Use the rationally normalized defect $N_{R} = U_{R} \circ W_{Z_{2}}$ , where $W_{Z_{2}} : C_{R} \to C_{R} / Z_{2}$ is the endpoint-framed finite gauging wall and $U_{R} : C_{R} / Z_{2} \to C_{R}$ is the selected invertible self-duality interface. With this normalization, $N_{R} \otimes {\overset{ˉ}{N}}_{R} ≅ 1 \oplus g .$ On the algebraic primary-label space $V_{R} = ⨁_{(n, w) \in Z^{2}} C V_{n, w}$ , let $ρ (g) V_{n, w} = (- 1)^{n} V_{n, w}$ and $P = (I_{V_{R}} + ρ (g)) / 2$ . The sum of the two line actions is $I_{V_{R}} + ρ (g) = 2 P$ . The projection $P$ annihilates the genuine field $V_{1, 0}$ , and the charge action sends that field to a non-genuine endpoint. An invertible endodefect acts bijectively on genuine local fields, so this rational-radius duality line is noninvertible. In general its charge action is $V_{n, w} \mapsto V_{2 w, n / 2}$ when $n$ is even and to a non-genuine endpoint when $n$ is odd.

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

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., §3.1, §4.2

Ginsparg, §§8.1, 8.7

Facts & Assumptions

Given: The compact-boson CFT at dimensionless radius $R = \sqrt{2}$ , its genuine primary fields $V_{n, w}$ , $n, w \in Z$ , and the adopted two-step flat-gauging construction.

[F1]

Compact free boson as a full local CFT: with $R = \sqrt{2} r$ , the primary weights are $h = \frac{1}{4} (n / R + w R)^{2}$ and $\overset{ˉ}{h} = \frac{1}{4} (n / R - w R)^{2}$ .

[F2]

Compact-boson momentum-winding lattice and cocycle: genuine primary labels are integral pairs $(n, w) \in Z^{2}$ .

[F3]

Topological defects, interfaces, orientation, and fusion conventions: an invertible endodefect has a two-sided fusion inverse. Dragging a genuine local field across an invertible endodefect and then across its inverse returns the original field; consequently, the endodefect acts bijectively on the genuine local-field space. A non-genuine endpoint is not an element of that space.

[F4]

Finite-group gauging wall conventions: the finite gauging is untwisted and the framed flat-bundle groupoid uses weight $1 / ∣ Aut ∣$ ; each relative-holonomy sector on the fused interval has trivial endpoint-preserving automorphism group and weight one.

[F5]

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

[A1]

Adopted compact-boson flat-gauging and rational self-duality model: adopt the source's two-step flat gauging followed by T-duality and its field-label action $(n, w) \mapsto (R^{2} w, n / R^{2})$ when the output is integral; otherwise the line ends on a non-genuine sector. This is a conditional model construction, not a derivation of a continuum functional measure.

[A2]

Adopted compact-boson flat-gauging and rational self-duality model: At $R^{2} = 2$ , the source's finite subgroup gauging and selected self-duality interface $U_{R}$ identify its nontrivial gauging line with $g V_{n, w} = (- 1)^{n} V_{n, w}$ . Normalize the rational defect exactly as $N_{R} = U_{R} \circ W_{Z_{2}}$ , with no extra scalar, using Finite-group gauging wall conventions. Assume the line category generated by $1, g$ is finite semisimple and these two lines are non-isomorphic simple objects. Argurio et al. §3.1 identifies the continuous and finite constructions only up to normalization; §4.2 discusses fixing the rational normalization by modular-bootstrap/locality. The exact finite-wall normalization here is adopted as the rational model input and is not claimed to be proved by those source passages.

Proof

1.1givenF1F2algebra

At $R = \sqrt{2}$ , [F1] gives $h_{n, w} = \frac{(n + 2 w)^{2}}{8}, {\overset{ˉ}{h}}_{n, w} = \frac{(n - 2 w)^{2}}{8} .$ For the genuine input $V_{1, 0}$ , these weights are both $1 / 8$ .

1.2givenA2F2algebra

The induced generator satisfies $ρ (g)^{2} V_{n, w} = (- 1)^{2 n} V_{n, w} = V_{n, w}$ . Thus $P = (I + ρ (g)) / 2$ satisfies $P^{2} = (I + 2 ρ (g) + ρ (g)^{2}) / 4 = (I + ρ (g)) / 2 = P$ . Its action is $P V_{n, w} = V_{n, w}$ for even $n$ and $P V_{n, w} = 0$ for odd $n$ , so in particular $P V_{1, 0} = 0$ .

2.1givenA1F1F2step 1.1algebra

Applying [A1] at $R^{2} = 2$ sends $V_{n, w}$ to $V_{2 w, n / 2}$ exactly when $n$ is even: $2 w$ is integral for every $w \in Z$ , and $n / 2$ is integral precisely for even $n$ . When $n$ is odd the winding endpoint is not genuine. For integral outputs, substitution into the weights of step 1.1 gives unchanged $h$ and $\overset{ˉ}{h}$ , because $n^{'} / R + w^{'} R = n / R + w R$ and $n^{'} / R - w^{'} R = - (n / R - w R)$ .

3.1givenA1A2F2F3F4F5algebra∎

By [A2], fusing $N_{R}$ with its orientation reverse cancels the adjacent inverse T-duality interfaces and leaves $W_{Z_{2}}$ fused with its orientation reverse. Applying [F4] and [F5] gives $N_{R} \otimes {\overset{ˉ}{N}}_{R} ≅ 1 \oplus g$ ; the sum of these line actions on $V_{R}$ is $2 P$ . More directly, [A1] at $(n, w) = (1, 0)$ sends the nonzero genuine local field $V_{1, 0}$ to the non-genuine endpoint with charge label $(0, 1 / 2)$ . Thus the defect action does not preserve the genuine local-field space. If $N_{R}$ were invertible, [F3] says its action on that space would be bijective, which is impossible. Therefore $N_{R}$ is noninvertible.

Remarks

The normalization is the finite-group gauging convention, not the undetermined absolute proportionality coefficient of the irrational continuous-gauging construction. The diagram below records the stated rational fusion channel; the two incoming labels specify orientation and the direct-sum output has two simple channels. It is explanatory only; the proof above uses the explicit group action and projector.

Rational noninvertible T-duality at R squared equals two

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

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