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Momentum-winding exchange at the self-dual radius

Statement

In the Ginsparg normalization, the positive self-dual radius is $r_{*} = 1 / \sqrt{2}$ . At $r_{*}$ the momentum state $q_{m} = (1, 0)$ and the winding state $q_{w} = (0, 1)$ both have weights $(h, \overset{ˉ}{h}) = (1 / 4, 1 / 4)$ and integral spin zero. Radius duality exchanges them. Their leading OPE is $V_{1, 0} (z, \overset{ˉ}{z}) V_{0, 1} (w, \overset{ˉ}{w}) = - (z - w)^{1 / 2} (\overset{ˉ}{z} - \overset{ˉ}{w})^{- 1 / 2} V_{1, 1} (w, \overset{ˉ}{w}) + descendants$ ; the cocycle sign cancels the exchange branch phase, so the full fields are mutually local.

Statement references

Compact-boson momentum-winding lattice and cocycle

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

Ginsparg, Section 8.1

Mnev–Reshetikhin, Sections 6.1.5–6.1.6

Facts & Assumptions

Given: The charge lattice, cocycle, vertex operators, and radius-duality map of the compact-boson data.

[F1]

Compact-boson momentum-winding lattice and cocycle defines $p_{L} = m / (2 r) + n r$ , $p_{R} = m / (2 r) - n r$ , $ε ((m, n), (m^{'}, n^{'})) = (- 1)^{m n^{'}}$ , and the lattice vertex fields.

[F2]

The compact-boson lattice is even, unimodular, and mutually local proves the spin formula $h - \overset{ˉ}{h} = m n$ , the OPE factor $ε (q, q^{'})$ , and cancellation of the cocycle and exchange phases.

[F3]

The compact-boson lattice is even, unimodular, and mutually local proves the duality $r \mapsto (2 r)^{- 1}$ , $(m, n) \mapsto (n, m)$ with phase $(- 1)^{m n}$ .

Proof

Given: $r > 0$ is the self-dual radius and $q_{m} = (1, 0)$ , $q_{w} = (0, 1)$ .

1.1givenF3algebra

A positive fixed point of $r \mapsto (2 r)^{- 1}$ satisfies $r = (2 r)^{- 1}$ , hence $2 r^{2} = 1$ . Positivity selects the unique solution $r_{*} = 1 / \sqrt{2}$ .

2.1F1F2step 1.1algebra

At $r_{*}$ , [F1] gives $p_{L} (q_{m}) = p_{R} (q_{m}) = 1 / \sqrt{2}$ and $p_{L} (q_{w}) = 1 / \sqrt{2}$ , $p_{R} (q_{w}) = - 1 / \sqrt{2}$ . With no oscillator excitations, each weight is half the square of the corresponding charge, so both states have $(h, \overset{ˉ}{h}) = (1 / 4, 1 / 4)$ . The spin is $h - \overset{ˉ}{h} = m n = 0$ for each label.

3.1F1F2step 2.1algebra

For $q_{m} = (1, 0)$ and $q_{w} = (0, 1)$ , [F1] gives $ε (q_{m}, q_{w}) = - 1$ and $ε (q_{w}, q_{m}) = 1$ . Their left and right charge products are $1 / 2$ and $- 1 / 2$ , so the stated leading OPE follows directly from the vertex formula. Set $λ_{*} (q) = (p_{L} (q; r_{*}), p_{R} (q; r_{*})) \in R^{2}$ . The Lorentzian pairing is $b : = B (λ_{*} (q_{m}), λ_{*} (q_{w})) = 1 / 2 - (- 1 / 2) = 1$ ; exchanging the radial order contributes $e^{π i b} = - 1$ , which is canceled by the cocycle ratio $ε (q_{m}, q_{w}) / ε (q_{w}, q_{m}) = - 1$ . Thus the continued full fields commute.

4.1F1F3step 1.1step 2.1step 3.1algebra∎

The duality map sends $(r_{*}, q_{m})$ to $(r_{*}, q_{w})$ and vice versa because $r_{*}$ is fixed and $(m, n) \mapsto (n, m)$ . Its phase $(- 1)^{m n}$ equals $+ 1$ on both labels. The duality therefore exchanges the two equal-weight sectors and, by [F3], preserves their vertex OPE data.

Remarks

The two sectors are degenerate but distinct: one carries one unit of momentum and the other one unit of winding. This example uses the dimensionless worldsheet radius convention and assigns no dimensional target-space length.

Momentum-winding exchange at the self-dual radius

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources