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Commutative Algebra pp 25–56Cite as

Some Applications of Commutative Algebra to String Theory

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Abstract

String theory was first introduced as a model for strong nuclear interactions, then reinterpreted as a model for quantum gravity, and then all fundamental physics.

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Notes

  1. 1.

    The basic idea of a proof was proposed in [6] and further studied in [4, 7, 8]. The current physics proofs only say that D-brane category has the derived category as a full subcategory.

  2. 2.

    In physics language this is phrased as coupling a topological field theory to topological gravity. The term TCFT actually means something quite different in the physics literature [10].

  3. 3.

    In the context of this chapter, this condition is automatically satisfied.

  4. 4.

    The Macaulay 2 variable X 1 is our − p. Note that Macaulay 2 views complexes in terms of homology rather than cohomology and so X 1 has R charge − 2.

  5. 5.

    Let us assume the Calabi–Yau threefold is simply connected and the cohomology is torsion-free.

  6. 6.

    This is proven by showing that it is incompatible with any weight order.

References

  1. Grayson, D.R., Stillman, M.E.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

  2. Atiyah, M.: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1989)

    Google Scholar 

  3. Lurie, J.: On the classification of topological field theories. In: Current Developments in Mathematics, 2008, pp. 129–280. International Press, Somerville (2009). arXiv:0905.0465.

    Google Scholar 

  4. Douglas, M.R., Gross, M. (eds.): Dirichlet branes and mirror symmetry. Clay Mathematical Monographs, vol. 4. American Mathematical Society, New York, Clay Mathematics Institute, Cambridge (2009)

    Google Scholar 

  5. Witten, E.: Mirror manifolds and topological field theory. In: Yau, S.-T. (ed.) Essays on Mirror Manifolds. International Press, Somerville (1992). hep-th/9112056

    Google Scholar 

  6. Douglas, M.R.: D-Branes, categories and N=1 supersymmetry. J. Math. Phys. 42, 2818–2843 (2001). hep-th/0011017

    Google Scholar 

  7. Aspinwall, P.S., Lawrence, A.E.: Derived categories and zero-brane stability. J. High Energy Phys. 08, 004 (2001). hep-th/0104147

    Google Scholar 

  8. Aspinwall, P.S.: D-Branes on Calabi–Yau manifolds. In: Maldacena, J.M. (ed.) Progress in String Theory. TASI 2003 Lecture Notes, pp. 1–152. World Scientific, Singapore (2005). hep-th/0403166

    Google Scholar 

  9. Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Intersection Floer Theory—Anomaly and Obstruction. (2000). http://www.math.kyoto-u.ac.jp/~fukaya/fukaya.html

  10. Dijkgraaf, R., Verlinde, H.L., Verlinde, E.P.: Notes on topological string theory and 2-D quantum gravity. In Green, M., et al. (eds.) String Theory and Quantum Gravity: Proceedings. World Scientific, Singapore (1991). Proceedings of Trieste Spring School: String Theory and Quantum Gravity, Trieste, Italy, Apr 23–1 May (1990)

    Google Scholar 

  11. Costello, K.: Topological conformal field theories and Calabi–Yau categories. Adv. Math. 210, 165–214 (2007). arXiv:math/0412149

    Google Scholar 

  12. Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167, 615–667 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Oda, T., Park, H.S.: Linear Gale transforms and Gelfand–Kapranov–Zelevinskij decompositions. Tôhoku Math. J. 43, 375–399 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cox, D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4, 17–50 (1995). alg-geom/9210008

    Google Scholar 

  15. Cox, D.A., Little, J.B., Schenck, H.K.: Toric varities. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2010)

    Google Scholar 

  16. Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  17. Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. alg-geom/9506012

    Google Scholar 

  18. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B403, 159–222 (1993). hep-th/9301042

    Google Scholar 

  19. Herbst, M., Hori, K., Page, D.: Phases of N = 2 theories in 1 + 1 dimensions with boundary. arXiv:0803.2045

    Google Scholar 

  20. Pantev, T., Sharpe, E.: String compactifications on Calabi–Yau stacks. Nucl. Phys. B733, 233–296 (2006). hep-th/0502044

    Google Scholar 

  21. Segal, E.: Equivalence between GIT quotients of Landau–Ginzburg B-models. Comm. Math. Phys. 304, 411–432 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer, New York (2005)

    Google Scholar 

  23. Aspinwall, P.S.: Topological D-branes and commutative algebra. Commun. Num. Theory. Phys. 3, 445–474 (2009). hep-th/0703279

    Google Scholar 

  24. Eisenbud, D., Mustaţǎ, M., Stillman, M.: Cohomology on toric varieties and local cohomology with monomial supports. J. Symbolic Comput. 29, 583–600 (2000). arXiv:math/0001159

    Google Scholar 

  25. Van den Bergh, M.: Non-commutative crepant resolutions. In: The Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo 2002, pp. 749–770. Springer, Berlin (2004). arXiv:math/0211064

    Google Scholar 

  26. Aspinwall, P.S.: D-Branes on toric Calabi–Yau varieties. arXiv:0806.2612

    Google Scholar 

  27. Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. In: Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin. Vol. II. Progress in Mathematics, vol. 270. pp. 503–531. Birkhäuser, Boston (2009). math.AG/ 0506347

    Google Scholar 

  28. Avramov, L.L., Grayson, D.R.: Resolutions and cohomology over complete intersections. In: Eisenbud, D., et al. (eds.) Computations in Algebraic Geometry with Macaulay 2, Algorithms and Computations in Mathematics, vol. 8, pp. 131–178. Springer, New York (2001)

    Chapter  Google Scholar 

  29. Gulliksen, T.H.: A change of ring theorem with applications to Poincare series and intersection multiplicity. Math. Scand. 34, 167–183 (1974)

    MathSciNet  MATH  Google Scholar 

  30. Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260, 35–64 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. arXiv:1203.6643

    Google Scholar 

  32. Kapustin, A., Li, Y.: D-branes in Landau–Ginzburg models and algebraic geometry. J. High Energy Phys. 12 005 (2003). hep-th/0210296

    Google Scholar 

  33. Hori, K., Walcher, J.: D-branes from matrix factorizations. Compt. Rendus Phys. 5, 1061–1070 (2004). hep-th/0409204

    Google Scholar 

  34. Ashok, S.K., Dell’Aquila, E., Diaconescu, D.-E., Florea, B.: Obstructed D-branes in Landau–Ginzburg orbifolds. Adv. Theor. Math. Phys. 8, 427–472 (2004). hep-th/0404167

    Google Scholar 

  35. Aspinwall, P.S.: The Landau–Ginzburg to Calabi–Yau dictionary for D-branes, J. Math. Phys. 48, 082304 (2007). arXiv:hep-th/0610209

    Google Scholar 

  36. Vafa, C., Warner, N.: Catastrophes and the classification of conformal theories. Phys. Lett. 218B, 51–58 (1989)

    MathSciNet  Google Scholar 

  37. Kapustin, A., Rozansky, L.: On the relation between open and closed topological strings. Commun. Math. Phys. 252, 393–414 (2004). arXiv:hep-th/0405232

    Google Scholar 

  38. Dyckerhoff, T.: Compact generators in categories of matrix factorizations. Duke Math. J. 159, 223–274 (2011). arXiv:0904.4713

    Google Scholar 

  39. Joyce, D.: Lectures on Calabi–Yau and special Lagrangian geometry. In: Global Theory of Minimal Surfaces. Clay Mathematical Proceedings, vol. 2. American Mathematical Society, Providence (2005). math.DG/0108088

    Google Scholar 

  40. Douglas, M.R., Fiol, B., Römelsberger, C.: Stability and BPS branes. J. High Energy Phys. 0509, 006 (2005). hep-th/0002037

    Google Scholar 

  41. Aspinwall, P.S., Douglas, M.R.: D-brane stability and monodromy. J. High Energy Phys. 05, 031 (2002), hep-th/0110071

    Google Scholar 

  42. Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2) 166, 317–345 (2007). math.AG/0212237

    Google Scholar 

  43. Gel’fand, I.M., Zelevinskiĭ A.V., Kapranov, M.M.: Hypergeometric functions and toral (toric) manifolds. Func. Anal. Appl. 23, 94–106 (1989). (from Russian in Funk. Anal. Pril. 23)

    Google Scholar 

  44. Gelfand, I.M., Kapranov, M.M., Zelevinski, A.V.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Basel (1994)

    MATH  Google Scholar 

  45. Batyrev, V.V.: Variations of the mixed hodge structure of affine hypersurfaces in algebraic tori. Duke Math. J. 69, 349–409 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. Aspinwall, P.S., Greene, B.R., Morrison, D.R.: The monomial-divisor mirror map. Internat. Math. Res. Notices 319–33 (1993). alg-geom/9309007

    Google Scholar 

  47. Hosono, S., Lian, B.H., Yau, S.-T.: GKZ-generalized hypergeometric systems in mirror symmetry of Calabi-Yau hypersurfaces. Comm. Math. Phys. 182, 535–577 (1996). alg-geom/ 9511001

    Google Scholar 

  48. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  49. Eisenbud, D.: Commutative algebra with a view towards algebraic geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (2004)

    Google Scholar 

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Acknowledgements

I thank R. Plesser for many useful discussions. This work was partially supported by NSF grant DMS–0905923. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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  1. Mathematics Department, Duke University, Durham, USA

    Paul S. Aspinwall

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Correspondence to Paul S. Aspinwall .

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  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Aspinwall, P.S. (2013). Some Applications of Commutative Algebra to String Theory. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_2

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