Extra Dimensions and Topology

1. Extra Dimensions and Topology C.T. Hill, Fermilab

2. Topology/Anomalies in Gauge Field Theories dictated by the Dirac Monopole P.A.M. Dirac, "Quantized Singularities in the Electromagnetic Field", Proceedings of the Royal Society, A133 (1931) pp 60-72.

3. Topology/Anomalies in Gauge Field Theories dictated by the Dirac Monopole P.A.M. Dirac, "Quantized Singularities in the Electromagnetic Field", Proceedings of the Royal Society, A133 (1931) pp 60-72.

4. Topology/Anomalies in Gauge Field Theories dictated by the Dirac Monopole P.A.M. Dirac, "Quantized Singularities in the Electromagnetic Field", Proceedings of the Royal Society, A133 (1931) pp 60-72. Solenoid Bohm-Aharanov phase: Stoke’s Theorem: Electric charge quantization

5. Discard the Monopole Consider solenoid as a closed string:

6. J = cF* is a “conserved topological current” in D=3 E&M in absence of “open strings” (monopoles) This “current” does NOT come from a Noether variation of the action of D=3 E&M

7. Turn on Chern-Simons term: Chern-Simons terms exist in any odd D.

8. Chern-Simons term: Generates the conserved current ala Noether:

9. Solenoid loop becomes a physical current loop when Chern-Simons term is present

10. Go from D=3 to D=1+2 photon solenoid is now a world-line of a charged particle Must have charge e! time space Chern-Simons term coefficient is determined by demanding world-line charge is e

11. Gauss linking of two solenoid strings: (1) (2) Chern-Simons term coefficient is determined by demanding action under linking shifts by multiple of

12. Exchange two solenoid particles = half of a Gauss linking time space solenoid is a fermion for odd N

13. Slice through the world-line time space instanton

14. Compactify to D=2 on orbifold Bulk photons: Orbifold: eats on branes eats R

15. Chern-Simons term induces anomalies on branes I II

16. Need Fermions (or Axions) on branes to cancel the Chern-Simons anomalies Left-handed anomaly Right-handed anomaly I II

17. Weyl fermion anomaly coefficient (D=2 loop diagram) is determined by D=3 Chern-Simons coefficient: I II Left-handed anomaly Right-handed anomaly

18. Chern-Simons term leads to new bulk interactions that violate KK-mode parity (T-parity): Topological mass mixing of levels

19. Summary: • Dirac Monopole has solenoid singularity • Closed loop of solenoid is a string in D=3 • Conserved topological current dA • Chern-Simons term, AdA, generates topological current • Chern-Simons term promotes solenoid to a current loop • Solenoid is a charged particle world-line in D=1+2 • Its charge determines the Chern-Simons coefficient • Particle is a fermion for odd N • Spacelike slice on D=2 yields instanton • Compactify: fermions/axions on branes cancel anomalies • Consistent anomalies on boundary branes determined • Weyl fermion consistent anomaly coefficient from D=2 loop is dictated by D=3 Chern-Simons term coefficient • Violates KK-mode parity (T-parity); locks KK-modes to definite space parity

20. Topology at the D=4 and D=5 Interface

21. D=5 Gauge Theories time space instanton

22. D=5 QED: Generalized Dirac Solenoid Construction: Dirac Branes Dirac-2 brane generalization of solenoid: A “stack” of current loops in xy plane yields zwt Dirac-2 brane carrying time intersection of two Dirac-2 branes is a world-line with current space

23. D=5 QED: Generalized Dirac Solenoid Construction: QED Chern-Simons term: world-line becomes charged: time space D=4 instanton is a “collision” of two solenoid world sheets

24. Compute Chern-Simons Coefficient: (D-1)/2 Intersecting (D-3)-Dirac-branes in any odd D:

25. Yang-Mills in D=5 Dirac Solenoid “instantonic soliton” Topology in the bulk: Gauge field solitons in extra dimensions.C.T. Hill , Pierre RamondNucl.Phys.B596:243-258,2001. Topological solitons from deconstructed extra dimensions.C.T. Hill Published in Phys.Rev.Lett.88:041601,2002. Singlet: Adjoint: Currents generated by D=5 Chern-Simons term

26. Yang-Mills D=5 Chern-Simons term The instanton is the most important topological object in Yang-Mills field theories In D=5, the instanton becomes a stable, static, soliton Compactify to D=4 on S1, Instanton becomes the Skyrmion (Fermi statistics, Nc=3); Chern-Simons becomes WZW term.

27. D=5 Chern-Simons term on orbifold CTH, Phys.Rev.D73:085001,2006. Consider U(1): R Chern-Simons anomalies on Branes:

28. CompactifiedLocalGauge Theories in D=5 Bulk gauge bosons: Orbifold eaten eaten b.c. on branes R

29. Chirally delocalized Electrons on branes: Orbifold Wilson line mass term:

30. Full Gauge transformation in D=5: Consistent Anomalies on Branes anomaly coefficients match Chern-Simons term Anomalies Cancel!

31. An aside: Anomalies are very tricky: Adler 1969 (QED): Adler’s result is 1/3 Feynman diagram and 2/3 counterterm. (“covariant anomaly”)

32. An aside: Anomalies are very tricky: Adler 1969 (QED): Adler’s result is 1/3 Feynman diagram and 2/3 counterterm. (“covariant anomaly”) Pure LH spinor result is pure Feynman diagram, unambiguous no counterterms “consistent anomaly” Our physical set-up is a UV completion that explains the origin of the counterterm coming from Chern-Simons term CTH, Phys.Rev.D73:085001,2006.

33. Chern-Simons term (together with fermionic Dirac determinant) generates a new class of interactions

34. Separate the A5, d5 components: Pass to A5 = 0 Gauge Interactionviolates KK-mode parity

35. Orbifold mode expansion: + Fermionic Dirac Determinant effective interaction: Full Effective Theory

36. Compute KK-Mode/T-parity violating decay CTH, Phys.Rev.D73:085001,2006. = 0 gauge invariance (Landau-Yang theorem)

37. Holographic QCD Yang-Mills (SU(3) quark flavor) in Bulk with Chern-Simons Term: Gauge transformation: Consistent anomaly; will cancel against fermion anomalies:

38. QCD “Re-Construction” eaten eaten on branes R

39. Yang-Mills flavor in D=5 A5 zero-mode -> mesons fpi = 1/R; Wilson line mass term <-> chiral condensate Quark Chiral delocalization requires a Chern-Simons term; anomaly matching, quantization Chern-Simons term implies bulk plus holographic interactions amongst KK-modes effective D=4 interaction, Large mq limit -> Fermionic Dirac determinant modifies effective interaction; maintains gauge invariance Derivation of the full Wess-Zumino-Witten term directly from the compactified D=5 Yang-Mills flavor theory

40. Obtain the full Wess-Zumino-Witten Term C. T. H., C. K. ZachosPhys.Rev.D71:046002,2005 C. T. H. Phys.Rev.D73:126009,2006.

41. Wess-Zumino-Witten Term Interactions of ageneric Little Higgs with Axial vector B T-Parity Violation by Anomalies C. T. Hill, Richard J. Hill . arXiv:0705.0697 [hep-ph]

42. Wess-Zumino-Witten Term Interactions of ageneric Little Higgs with Axial vector B T-Parity Violation by Anomalies C. T. Hill, Richard J. Hill . arXiv:0705.0697 [hep-ph]

43. Applications of Topology to Phenomenology: (1) Little Higgs Theories. (Richard Hill’s talk today) WZW term contains new physics, e.g. B -> ZZ, WW, etc. Violates T-parity; determines number of pre-quark colors, N (2) R-S Models!!! (3) A WZW Term for the Goldstone-Wilczek Current (4) AdS-CFT, gravity, Taub-NUT instantonic soliton, etc. .... Topological Physics of Little Higgs Bosons.C.T. Hill, Richard J. Hill (Fermilab) . e-Print: hep-ph/0701044 T-Parity Violation by Anomalies C. T. Hill, Richard J. Hill . arXiv:0705.0697 [hep-ph]

44. D=4 Anomalies are best viewed from perspective of D=5 Anomaly Consistency, Holography ANOMALIES, CHERN-SIMONS TERMS AND CHIRAL DELOCALIZATION IN EXTRA DIMENSIONS. FERMILAB-PUB-06-010-T (Jan 2006) 46p. [HEP-TH 0601154] CTH, Phys.Rev.D73:085001,2006. LECTURE NOTES FOR MASSLESS SPINOR AND MASSIVE SPINOR TRIANGLE DIAGRAMS. CTH, FERMILAB-TM-2341-T (Jan 2006) 17p. [HEP-TH 0601155]