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One-instanton SU(2) sign cancelled by a mod-two-index bulk

Physical scope

This worked example uses one specified Euclidean SU(2) Weyl-doublet inverse- determinant-line model, the explicitly proved round index-one geometry, the central order-two test, and its mod-two-index bulk response. It does not identify this line with a fermionic Pfaffian line. The finite BC2 calculation is a normalization check, not a continuum discretization or a general anomaly classification.

Example

Assume full AC and take the explicit unit round spin $S^{4}$ background with $E = S^{+}$ and its positive-spin Levi-Civita connection, one massless left-handed Weyl doublet of a nondynamical $S U (2)$ global symmetry, and the order-two central transformation $z = - I$ from the global-anomaly test. Use the inverse determinant line $L_{A} = (\det_{C} \ker D_{A}^{+})^{*} \otimes \det_{C} coker D_{A}^{+}$ ; this example makes no identification with the Pfaffian line used in a pseudoreal fermion path-integral presentation. The local Hodge calculation gives complex chiral index $1$ and Chern number $- 1$ . Pair this boundary line with the assigned bulk line, which is its dual, adopting $(- 1)^{{ind}_{2} D_{5}}$ as the closed phase and as the bulk-line holonomy on the mapping torus with product orientation and periodic total spinors. The boundary line has character $χ (1) = 1$ , $χ (z) = - 1$ ; the mapping-torus mod-two index is $1$ , so the bulk character is $χ^{- 1} = χ$ and the combined character is trivial. With $F_{A}$ anti-Hermitian, use $ch (E) = {tr}_{2} \exp (- F_{A} / (2 π i))$ , $p_{1} (T M) = - (8 π^{2})^{- 1} {tr}_{R} (R^{T M} \land R^{T M})$ , and ${\hat{A}}_{4} = - p_{1} (T M) / 24$ ; the adopted index-density polynomial $I_{6} = [\hat{A} (T M) ch (E)]_{6}$ is zero. This algebraic characteristic-polynomial calculation alone does not assert vanishing of an infinitesimal BRST variation. For the finite normalization check take a single-vertex complex $K$ and the ordinary field groupoid $F (K) = B C_{2}$ , with one object and automorphism group $C_{2}$ . The combined unit weight is arrow-invariant and its normalized groupoid sum is $1 / 2$ . Without the bulk, the odd boundary character is not a scalar invariant weight and no such finite gauging sum is defined. For two identical copies the boundary character itself is trivial and the same normalized groupoid sum applies. This example tests only the stated loop and does not establish that even copies have no other anomalies.

Example references

An explicit round-S4 spin twist has charge one and chiral index one

Periodic real product kernel and central action on the inverse determinant line

The Axiom of Choice

Wang, Wen, and Witten, §2.2

Freed, Lecture 11, §§11.1–11.2

Facts & Assumptions

Given: The explicit round index-one SU(2) inverse-determinant-line model with full AC, the order-two line character, the adopted bulk-line holonomy, and the finite single-vertex groupoid specified in the Example.

[F1]

A global SU(2) determinant-line phase and its mod-two inflow cancellation computes boundary and adopted bulk phases for the same model and loop, including the actually proved round-sphere index-one instance.

[F2]

Determinant-line holonomy for a global anomaly test defines the line holonomy as a unit-modulus scalar for the chosen regularization.

[F3]

Relative boundary theory and invertible anomaly inflow requires inverse boundary/bulk holonomies for cancellation.

[F4]

Finite groupoid cardinality and homotopy-fiber integration defines the normalized sum over isomorphism classes with reciprocal automorphism weights for an invariant function on a finite groupoid.

[F5]

An explicit round-S4 spin twist has charge one and chiral index one is a draft local mathematical lemma with a complete authored proof under full AC; it proves the actual round positive-spin twist has kernel dimension one, cokernel zero and Chern number $- 1$ .

[F6]

Periodic real product kernel and central action on the inverse determinant line is a draft local mathematical lemma with a complete authored proof under full AC; it proves the periodic-product $J$ -fixed real kernel has dimension one and the central inverse-determinant action is $- 1$ .

[A2]

The Axiom of Choice is retained through these mathematical suppliers.

[A1]

$C_{2} = {1, z}$ with $z^{2} = 1$ ; the boundary line is the one-dimensional character $χ (z) = - 1$ , while the inverse bulk character is $χ^{- 1}$ .

Verification

1.1givenA1A2F1F2F5F6algebra

By [A2], [F5] and [F6], the concrete background has index one, real product-kernel parity one and central line action $- 1$ , so the specified physical test [F1] applies. Since $z^{2} = 1$ and $χ (z)^{2} = (- 1)^{2} = 1$ , the assignment $χ (1) = 1$ , $χ (z) = - 1$ is a character of $C_{2}$ . Its inverse equals itself, so the tensor product of the boundary and bulk lines has the trivial character.

2.1givenF4step 1.1algebra

The single-vertex groupoid $B C_{2}$ has exactly one isomorphism class and its automorphism group has order $2$ . The combined unit weight is constant on its arrows by step 1.1, so [F4] gives $Z = 1 / 2$ . The boundary-only character is not arrow-invariant, so it is not an admissible scalar weight for that sum.

3.1givenF1F3step 1.1step 2.1algebra∎

For two identical boundary copies the character is $χ^{2} = 1$ , and the mod-two bulk phase is also $(- 1)^{2 \cdot 1} = 1$ . Thus the same finite groupoid sum has unit invariant weight and value $1 / 2$ ; this verifies cancellation for the stated test without asserting absence of any other anomaly.

One-instanton SU(2) sign cancelled by a mod-two-index bulk

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources