[Paper Review] Dirac Branes and Anomalies/Chern-Simons terms in any D

Christopher T. Hill|ArXiv.org|Jul 7, 2009

Black Holes and Theoretical Physics14 references3 citations

TL;DR

This paper presents a purely bosonic derivation of axial anomalies and Chern-Simons (CS) term coefficients in any odd spacetime dimension D, using Dirac branes—generalized magnetic solenoids—whose intersections form charged worldlines. By enforcing Dirac flux quantization on these intersections, the CS term coefficient is fixed unambiguously, reproducing consistent and covariant anomaly coefficients without fermion loops, revealing anomalies as topological effects from CS terms in odd D.

ABSTRACT

The Dirac quantization procedure of a magnetic monopole can be used to derive the coefficient of the D=3 Chern-Simons term through a self-consistency argument, which can be readily generalized to any odd D. This yields consistent and covariant axial anomaly coefficients on a D-1 boundary, and Chern-Simons term coefficients in D. In D=3 magnetic monopoles cannot exist if the Chern-Simons AdA term is present. The Dirac solenoid then becomes a physical closed string carrying electric current. The charge carriers on the string must be consistent with the charge used to quantize the Dirac solenoidal flux. This yields the Chern-Simons term coefficient. In higher odd D the intersection of (D-1)/2 Dirac branes yields a charged world-line permitting the consistency argument. The covariant anomaly coefficients follow readily from generalizing the counterterm. This purely bosonic derivation of anomalies is quite simple, involving semiclassical evaluation of exact integrals, like \int dAdA...dA, in the brane intersections.

Motivation & Objective

• To provide a simple, purely bosonic derivation of axial anomalies in any odd D without fermion loop calculations.
• To clarify the physical origin of Chern-Simons term coefficients in D-dimensional gauge theories.
• To show that anomalies arise holographically from the intersection of (D−1)/2 Dirac branes in odd D.
• To establish a direct link between Dirac monopole quantization and the coefficient of the CS term in D=3 and higher odd D.
• To derive both consistent and covariant anomaly coefficients using a self-consistency condition based on flux quantization and gauge invariance.

Proposed method

• Generalizes the Dirac monopole construction to D-dimensional spacetime, introducing Dirac branes as (D−2)-dimensional hypersurfaces with quantized electromagnetic flux.
• Considers the intersection of (D−1)/2 such Dirac branes, which forms a charged worldline in odd D, acting as a source for electric charge.
• Imposes the Dirac quantization condition on the flux carried by the brane intersection, requiring the induced electric charge to match the original flux-quantization condition.
• Uses the boundary variation of the Chern-Simons action under gauge transformations to derive the consistent anomaly, which matches the standard fermionic result.
• Applies the Adler-Bardeen counterterm to the Chern-Simons term to obtain the covariant anomaly, ensuring vector current conservation.
• Demonstrates that the CS term coefficient is fixed uniquely by the self-consistency of flux quantization on the brane intersection, independent of the brane core structure.

Experimental results

Research questions

• RQ1 What is the physical origin of the Chern-Simons term coefficient in D-dimensional U(1) gauge theories for odd D?
• RQ2 How can axial anomalies be derived without computing fermion loops, using only classical and topological arguments?
• RQ3 Why do magnetic monopoles not exist when a Chern-Simons term is present in D=3, and what replaces them?
• RQ4 How does the intersection of multiple Dirac branes in odd D generate a charged worldline that enforces the CS term coefficient?
• RQ5 What is the relationship between the consistent anomaly and the boundary variation of the Chern-Simons action in odd D?

Key findings

• The coefficient of the Chern-Simons term in any odd D is uniquely fixed by requiring consistency between the Dirac flux quantization condition and the electric charge induced on the intersection of (D−1)/2 Dirac branes.
• In D=3, the presence of a Chern-Simons term forbids magnetic monopoles, forcing Dirac solenoids to close into loops that carry electric current.
• The resulting electric charge on the closed Dirac brane loop must match the Dirac quantization condition, which fixes the CS term coefficient unambiguously.
• The consistent anomaly coefficient in even D−1 dimensions is derived directly from the boundary variation of the Chern-Simons term in D dimensions, matching known fermionic results.
• The covariant anomaly coefficient is obtained by adding the Adler-Bardeen counterterm to the Chern-Simons action, ensuring vector current conservation.
• The entire anomaly structure is shown to be purely bosonic in origin, with no need for fermion loop calculations, and arises from exact topological integrals like ∫dA dA … dA over brane intersections.