# Presentation: 905256

## Physics of the D=5 Chern-Simons Term

-           C.T. Hill,  Fermilab

## I. QED in D=5

-  Photons propagate in bulk; Naturalness, me << 1/R, requires
-      chiral delocalization; Wilson-line mass term

- (b) Chiral delocalization requires a Chern-Simons term

- (d) Chern-Simons term implies new interactions amongst
- bulk KK-mode photons, effective D=4 interaction,

- (iii) Compute K’ -> K + gamma via effective interaction

- (ii) Large me limit -> Fermionic Dirac determinant modifies
- effective interaction; maintains gauge invariance

- (c) Anomaly-freedom implies quantized coefficient of CS term;
- “consistent anomalies” vs. “covariant anomalies”

- (i) Use “Wilson-Line Gauge Transformation” to A5 = 0

## II. Yang-Mills in D=5

-   compactify with A5 zero-mode  -> mesons   fpi  = 1/R;
-   Wilson line mass term <-> chiral condensate

- (b) Chiral delocalization requires a Chern-Simons term;
-      anomaly matching, quantization

- (d) Large mq limit -> Fermionic Dirac determinant modifies
- effective interaction; maintains gauge invariance

- (c) Chern-Simons term implies bulk and
-       holographic interactions amongst KK-modes
-       effective D=4 interaction,

- (e) Obtain effective interaction: holographic part is
-       the full Wess-Zumino-Witten term.

## (to appear in Phys. Rev. D) 

- CHIRAL DELOCALIZATION IN EXTRA DIMENSIONS.
- FERMILAB-PUB-06-010-T (Jan 2006) 46p.         [HEP-TH 0601154]
- (to appear in Phys. Rev. D)

- LECTURE NOTES FOR
- MASSLESS SPINOR AND MASSIVE SPINOR
- TRIANGLE DIAGRAMS.
- FERMILAB-TM-2341-T (Jan 2006) 17p.              [HEP-TH 0601155]

- EXACT EQUIVALENCE OF THE D=4 GAUGED
- WESS-ZUMINO-WITTEN TERM AND THE D=5
- YANG-MILLS CHERN-SIMONS TERM.
- FERMILAB-PUB-06-046-T (Mar 2006) 25p.        [HEP-TH 0603060]

## QED Chern-Simons term

- Deser, Jackiw, Templeton, Schonfeld, Siegel; Niemi, Semenoff, Y.S. Wu

- Bulk Physics:  Photon Mass Term

-  D=3:  Knot Theory               “Gauss’ Linking Theorem”

- D=5:

- Bulk Physics:  New interactions amongst  KK-modes

## Topological object: “instantonic soliton”

- Associated Conserved Topological Currents:

- Singlet:

- Adjoint:

- Deser’s Theorem
- Ramond and CTH

- These currents come from a “completion” of the Lagrangian

- Adjoint current  -  2nd Chern character:

- Singlet currents  - auxiliary characters:

## Instantonic Soliton              Skyrmion

- Chiral Lagrangian theory obtainable via deconstruction
- Bianchi ID’s, etc.:

-  Gauge currents             Chiral currents
- Chern-Simons term
-  + boundary term

- CTH, CTH & Zachos

- Mathematically exact matchings:

- WZW term

##  I.  Technically natural QED in D=5:

- Bulk:

- R

- Orbifold

## Orbifold Boundary Conditions: 

-                 (a) Horava-Witten
-                    (b) Magnetic Josephson Junction

- Spectrum:   (a)   A zero mode and KK tower
-                     (b)  No A5 zero mode
-                     (c)  All A5 modes eaten -> longitudinal dof’s

- Flipped Orbifold Boundary Conditions:
-                   (a) parity reversed Horava-Witten
-                       (b) Josephson Junction

- Spectrum:  (a)  A zero mode
-                    (b) No Amu zero mode
-                    (c) All other A5 modes eaten -> longitudinal dof’s

## Gauge transformation in D=5:

- Consistent Anomalies

## QED in D=5 requires Chern-Simons term:

## Anomaly Cancellation Condition:

- Consistent Anomalies:

## Summary of Anomalies:

- W. A. Bardeen, PR 184, 1848 (199)

## Anomalies (cont’d):

- [HEP-TH 0601155]

## Summary: Technically natural QED in D=5

- Bulk:

- Orbifold

## Pass to A5 = 0  Gauge 

- Pass to A5 = 0  Gauge

## The orbifold mode expansion

## Slide 18

## D=4  Effective Theory

## D=4  Effective Theory Current Algebra

- KK-mode anomalies:

## D=4 Effective Theory in large  m  limit

- Integrate out the Fermions:
-  Dirac Determinant effective interactions

##  Dirac Determinant effective interaction:

-  Dirac Determinant effective interaction:

## Full Effective Theory

## Compute KK-Mode decay

- c = photon

- Gauge invariant in photon

##  = 0    gauge invariance  (Landau-Yang theorem)

-  = 0    gauge invariance  (Landau-Yang theorem)

## R

- Bulk:

- Generic compactification; include B5 zero-mode

## Derivation of the full Wess-Zumino-Witten term 

- Derivation of the full Wess-Zumino-Witten term
- directly from the Yang-Mills theory

## Theory Requires Chern-Simons Term:

- Gauge transformation:

- Consistent Anomaly;  To cancel against fermion anomalies:

## B is now a tower of vector mesons comingled 

- B is now a tower of vector mesons comingled
- with the spin-0 mesons; must extract the physical mesons:

- Redefinition:

## Compactification Decomposition of Chern-Simons Term:

- Axial Gauge:

- Form notation:

- CTH and Zachos

## Integrate out the Fermions:

-  Dirac Determinant effective interactions

- + ...

- Dirac Determinant effective interaction
- Is the Bardeen Counterterm as in QED:

- “Boundary term”

## Notation:

## Slide 33

## Slide 34

##  Effective brane (holographic) interaction

-  Effective brane (holographic) interaction

- Obtain the full Wess-Zumino-Witten Term

## Normal Derivation of WZW term: 

- promote full theory of mesons to D=5.
- In D=5, a certain manifestly chirally invariant and topologically interesting Chern-Simons term occurs, which is included into the theory.
- Compactify the fifth dimension with the Chern-Simons term, back into to D=4, resulting in the Wess-Zumino term.
- Perform gauge transformations upon the resulting object, and infer how to ``integrate in'' the gauge fields by brute force and some guess work.

##  Effective bulk interaction

-  Effective bulk interaction

##  Suppose we don’t integrate out the quarks?

-  Parity symmetric redefinition field:

-  chiral currents

-  Effective theory with unintegrated massless fermions

## Envisioned applications:

- (1)  Little Higgs Theories.
- (2)  RS Models
- (3)  A WZW Term for the Goldstone-Wilczek Current
- (4) Skyrme/instanton baryogenesis/b+L violation
-  	in extra dimensional theories
- 	(which is what started this project).
- (5) Technical tool to analyze/generalize WZW term structure
- (6) Can apply to D=3 YM matching to D=2;
- (7) Any D+1 to D.
- (8) ...

## Slide 40

## Slide 41

## Slide 42

## Yang-Mills D=5 Chern-Simons term

- 3(SU(2)): instanton naturally lives on a
- D=4 Euclidean space, mapping SU(2) onto S3

- In D=5, the instanton becomes a stable, static, soliton

- Compactify to D=4,  with an A5  zero-mode,
- the instanton becomes the Skyrmion; exact correspondence with chiral lagrangians.