Advertisement

From Grothendieck’s Schemes to QCD

Chapter
  • 637 Downloads
Part of the Springer Theses book series (Springer Theses)

Abstract

In order to have a self-contained discussion about universal coefficient theorems, coefficient groups and their effects on quantum field theories some supplemental concepts must be introduced. I suppose that the concept of ring is well understood. Basically it represents a set of elements for which we can define two operations: multiplication and addition. The set is then a group for addition and a monoid for multiplication while the multiplication is distributive with respect to addition. The set can contain not only numbers but various other objects. In the theory of rings we can define the so called ideal of a ring. For a ring \((R,+,\cdot )\) we consider \((R,+)\) to be its additive group. We call a subset I its ideal if it is an additive subgroup of R that absorbs through multiplication by elements of R all the other elements.

Keywords

Gauge Group Riemann Surface Topological Space Renormalization Group Homology Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    A. Grothendieck, J. Dieudonne, Elements de geometrie algebrique. Publ. Inst. des Hautes Etudes Scientifiques, 4 (1960)Google Scholar
  2. 2.
    D. Eisenbud, J. Harris, The Geometry of Schemes (Springer, Heidelberg, 2000). ISBN 978-0-387-22639-2Google Scholar
  3. 3.
    P. Nelson, An Introduction to Schemes, Lecture Notes (University of Chicago, Chicago, 2009)Google Scholar
  4. 4.
    A. Grothendieck, Inst. des Hautes Etudes Scientiques. Pub. Math. 29(29), 95 (1966)Google Scholar
  5. 5.
    J. Walcher, Extended holomorphic anomaly and loop amplitudes in open topological strings. Nucl. Phys. B 817(3), 167 (2009)Google Scholar
  6. 6.
    A. Kanazawa, J. Zhou, Lectures on BCOV holomorphic anomaly equations, Fields Institute Monograph (2014). arXiv:1409.4105 [math.AG]
  7. 7.
    M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Comm. Math. Phys. 165(2), 311 (1994)Google Scholar
  8. 8.
    A. Hamilton, A. Lazarev, Graph cohomology classes in the Batalinilkovisky formalism. J. Geom. Phys. 59(5), 555 (2009)Google Scholar
  9. 9.
    E.F. Kurusch, D. Kreimer, Hopf algebra approach to Feynman diagram calculations. J. Phys. A: Math. Gen. 38, 50 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    R. Gilman, R.J. Holt, P. Stoler, Transition to perturbative QCD. J. Phys.: Conf. Ser. 299, 012009 (2011)ADSGoogle Scholar
  11. 11.
    D.J. Gross, F. Wilczek, Ultraviolet behavior of non-abelian gauge theories. Phys. Rev. Lett. 30(26), 1343 (1973)Google Scholar
  12. 12.
    S. Pokorski, Gauge Field Theories (Cambridge University Press, Cambridge, 1987). ISBN 0-521-36846-4Google Scholar
  13. 13.
    H.D. Politzer, Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30(26), 1346 (1973)Google Scholar
  14. 14.
    L. Brink, H.B. Nielsen, Two mass relations for mesons from string-quark duality. Nucl.Phys. B 89(1), 118 (1975)Google Scholar
  15. 15.
    C.G. Callan, Broken scale invariance in scalar field theory. Phys. Rev. D 2, 1541 (1970)Google Scholar
  16. 16.
    J.C. Collins, Renormalization (Cambridge University Press, Cambridge, 1984). ISBN 0-521-24261-4Google Scholar
  17. 17.
    M. Gell-Mann, Symmetries of Baryons and Mesons. Phys. Rev. 125(3), 1067 (1962)Google Scholar
  18. 18.
    D.J.E. Callaway, A. Rahman, Lattice gauge theory in the microcanonical ensemble. Phys. Rev. D 28(6), 1506 (1983)Google Scholar
  19. 19.
    J. Alitti, An improved determination of the ratio of W and Z masses at the CERN pp collider. Phys. Lett. B276, 354 (1992)Google Scholar
  20. 20.
    J. Polchinski, Introduction to gauge gravity duality. TASI Lectures (2010). arXiv:hep-th/1010.6134
  21. 21.
    G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories. J. High Energy Phys. (JHEP) 12, 116 (2014)Google Scholar
  22. 22.
    G. ’t Hooft, A planar diagram theory for strong interactions. Nucl. Phys. B 72(3), 461 (1974)Google Scholar
  23. 23.
    A.V. Manohar, Large N QCD, Les Houches Lectures (1998) p. 22. arXiv:hep-ph/9802419
  24. 24.
    G.W. Moore, PiTP Lectures on BPS states and Wall-Crossing in d=4, N=2 theories, PiTP Prospects in theoretical physics (2010)Google Scholar
  25. 25.
    J.F. Davis, P. Kirk, Lecture Notes in Algebraic Topology. Dept. of Math. Indiana University, Bloomington, IN 47405 (1991)Google Scholar
  26. 26.
    E. Dror, Homology spheres. Israel J. of Math. 15, 115 (1973)Google Scholar
  27. 27.
    A.T. Patrascu, Quantization, holography and the universal coefficient theorem. Phys. Rev. D 90, 045018 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    J. Harvey, G. Moore, On the algebras of BPS states. Comm. Math. Phys. 197, 489 (1998)Google Scholar
  29. 29.
    A. Grothendieck, Sur quelques points d’algebre homologique. Tohoku Math. J. 2(9), 119 (1957)Google Scholar
  30. 30.
    E. Felix, W. Heffern, Lectures on Homology and Cohomology, Florida International University Lectures, (2011-2012), p. 11Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity College LondonLondonUK

Personalised recommendations