Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 034, 31 pages      arXiv:0712.3670      http://dx.doi.org/10.3842/SIGMA.2009.034
Contribution to the Special Issue on Kac-Moody Algebras and Applications

Homological Algebra and Divergent Series

Vassily Gorbounov a and Vadim Schechtman b
a) Department of Mathematical Sciences, King's College, University of Aberdeen, Aberdeen, AB24 3UE, UK
b) Laboratoire de Mathématiques Emile Picard, Université Paul Sabatier, Toulouse, France

Received October 01, 2008, in final form March 04, 2009; Published online March 24, 2009

Abstract
We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits.

Key words: Koszul resolution; Koszul duality, divergent series.

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References

  1. Aisaka Yu., Arroyo E.A., Berkovits N., Nekrasov N., Pure spinor partition function and the massive superstring spectrum, J. High Energy Phys. 2008 (2008), no. 8, 050, 72 pages, arXiv:0806.0584.
  2. Beauville A., Laszlo Y., Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), 385-419, alg-geom/9309003.
  3. Beilinson A., Drinfeld V., Chiral algebras, American Mathematical Society Colloquium Publications, Vol. 51, American Mathematical Society, Providence, RI, 2004.
  4. Beilinson A.A., Goncharov A.B., Schechtman V.V., Varchenko A.N., Aomoto dilogarithmes, mixed Hodge structures, and motivic cohomology of pairs of triangles on the plane, in The Grothendieck Festschrift, Vol. I, Progr. Math., Vol. 86, Birkhäuser Boston, Boston, MA, 1990, 135-172.
  5. Berkovits N., Nekrasov N.A., The character of pure spinors, Lett. Math. Phys. 74 (2005), 75-109, hep-th/0503075.
  6. Bezrukavnikov R., Koszul property and Frobenius splitting of Schubert varieties, alg-geom/9502021.
  7. Cohen R.L., Jones J.D.S., A homotopy theoretic realisation of string topology, Math. Ann. 324 (2002), 773-798, math.GT/0107187.
  8. Chas M., Sullivan D., String topology, math.GT/9911159.
  9. Brandt A.J., The free Lie ring and Lie representations of the full linear group, Trans. Amer. Math. Soc. 56 (1944), 528-536.
  10. Drinfeld V.G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(`Q/Q), Algebra i Analiz 2 (1990), no. 4, 149-181 (English transl.: Leningrad Math. J. 2 (1991), no. 4, 829-860).
  11. Gauss C.F., Disquisitiones generales de congruentis, Analysis residuorum. Caput octavum, Collected Works, Vol. 2, Georg Olms Verlag, Hildersheim - New York, 1973, 212-242.
  12. Golyshev V., The canonical strip. I, arXiv:0903.2076.
  13. Gorodentsev A.L., Khoroshkin A.S., Rudakov A.N., On syzygies of highest weight orbits, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 2007, 79-120, math.AG/0602316.
  14. Hardy G.H., Divergent series, The Clarendon Press, Oxford, 1949.
  15. Hirzebruch F., Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9, Springer-Verlag, Berlin - Göttingen - Heidelberg, 1956.
  16. Hirzebruch F., Private communication, 2008.
  17. Loday J.-L., Cyclic homology, Springer-Verlag, Berlin, 1992.
  18. Kaufmann R.M., A proof of a cyclic version of Deligne's conjecture via cacti, math.QA/0403340.
  19. Kontsevich M., Formal (non)commutative symplectic geometry, The Gelfand Mathematical Seminars, 1990-1992, Editors L. Corwin et al., Birkhäuser Boston, Boston, MA, 1993, 173-188.
  20. Lian B.H., Zuckerman G.J., New perspectives on the BRST-algebraic structure of string theory, Comm. Math. Phys. 154 (1993), 613-646, hep-th/9211072.
  21. Moreau C., Sur les permutations circulaires distinctes, Nouv. Ann. Math. 11 (1872), 309-314.
  22. Moree P., On the average number of elements in a finite field with order or index in a prescribed residue class, Finite Fields Appl. 10 (2004), 438-463, math.NT/0212220.
  23. Movshev M., On deformations of Yang-Mills algebras, hep-th/0509119.
  24. Movshev M., Schwarz A., Algebraic structure of Yang-Mills theory, hep-th/0404183.
  25. Petrogradsky V.M., On Witt's formula and invariants for free Lie superalgebras, in Formal Power Series and Algebraic Combinatorics (Moscow, 2000), Springer, Berlin, 2000, 543-551.
  26. Polishchuk A., Positselski L., Quadratic algebras, University Lecture Series, Vol. 37, American Mathematical Society, Providence, RI, 2005.
  27. Pólya G., Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937), 145-254.
  28. Polyakov A., Gauge fields and space-time, Internat. J. Modern Phys. 17 (2002), suppl., 119-136, hep-th/0110196.
  29. Quillen D., Rational homotopy theory, Ann. Math. (2) 90 (1969), 205-295.
  30. Quillen D., Algebra cochains and cyclic cohomology, Inst. Hautes Études Sci. Publ. Math. (1988), no. 68, 139-174.
  31. Ramis J.-P., Séries divergentes et théories asymptotiques, Bull. Soc. Math. France 121 (1993), suppl., 74 pages.
  32. Tradler T., Zeinallian M., On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006), 280-299, math.QA/0404218.
  33. Weil A., Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 88, Springer-Verlag, Berlin - New York, 1976.
  34. Whittaker E.T., Watson G.N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, reprint of 4th ed. (1927), Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  35. Witt E., Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152-160.
  36. Zagier D., Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, 445-462.

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