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

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## Extra Dimensions and Topology

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**Extra Dimensions and Topology**C.T. Hill, Fermilab**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.**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.**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**Discard the Monopole**Consider solenoid as a closed string:**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**Turn on Chern-Simons term:**Chern-Simons terms exist in any odd D.**Chern-Simons term:**Generates the conserved current ala Noether:**Solenoid loop becomes a physical current loop when**Chern-Simons term is present**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**Gauss linking of two solenoid strings:**(1) (2) Chern-Simons term coefficient is determined by demanding action under linking shifts by multiple of**Exchange two solenoid particles**= half of a Gauss linking time space solenoid is a fermion for odd N**Slice through the world-line**time space instanton**Compactify to D=2 on orbifold**Bulk photons: Orbifold: eats on branes eats R**Need Fermions (or Axions) on branes**to cancel the Chern-Simons anomalies Left-handed anomaly Right-handed anomaly I II**Weyl fermion anomaly coefficient**(D=2 loop diagram) is determined by D=3 Chern-Simons coefficient: I II Left-handed anomaly Right-handed anomaly**Chern-Simons term leads to new bulk interactions that**violate KK-mode parity (T-parity): Topological mass mixing of levels**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**D=5 Gauge Theories**time space instanton**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**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**Compute Chern-Simons Coefficient:**(D-1)/2 Intersecting (D-3)-Dirac-branes in any odd D:**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**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.**D=5 Chern-Simons term on orbifold**CTH, Phys.Rev.D73:085001,2006. Consider U(1): R Chern-Simons anomalies on Branes:**CompactifiedLocalGauge Theories in D=5**Bulk gauge bosons: Orbifold eaten eaten b.c. on branes R**Chirally delocalized Electrons on branes:**Orbifold Wilson line mass term:**Full Gauge transformation in D=5:**Consistent Anomalies on Branes anomaly coefficients match Chern-Simons term Anomalies Cancel!**An aside: Anomalies are very tricky:**Adler 1969 (QED): Adler’s result is 1/3 Feynman diagram and 2/3 counterterm. (“covariant anomaly”)**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.**Chern-Simons term**(together with fermionic Dirac determinant) generates a new class of interactions**Separate the A5, d5 components:**Pass to A5 = 0 Gauge Interactionviolates KK-mode parity**Orbifold mode expansion:**+ Fermionic Dirac Determinant effective interaction: Full Effective Theory**Compute KK-Mode/T-parity**violating decay CTH, Phys.Rev.D73:085001,2006. = 0 gauge invariance (Landau-Yang theorem)**Holographic QCD**Yang-Mills (SU(3) quark flavor) in Bulk with Chern-Simons Term: Gauge transformation: Consistent anomaly; will cancel against fermion anomalies:**QCD “Re-Construction”**eaten eaten on branes R**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**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.**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]**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]**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]**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]

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