×

Non-birational twisted derived equivalences in abelian GLSMs. (English) Zbl 1231.14035

The authors argue that A. Kuznetsov’s homological projective duality [Publ.Math., Inst.Hautes Étud.Sci.105, 157–220 (2007; Zbl 1131.14017)] can be used to explain dualities between non-birational Kähler phases of gauged linear sigma models (GLSMs).
Physicists expected in the past that different geometric Kähler phases of the same GLSM are birational one to another. However, the geometry at the Landau-Ginzburg point of the GLSM for the complete intersection of four quadrics in \({\mathbb P}^7\) is a branched double cover of \({\mathbb P}^3\) with a singular octic branch locus. Such a double cover is not birational to the complete intersection of quadrics in \({\mathbb P}^7\).
In the first half of this paper, this example is studied in detail. Physical arguments are provided which support the view of the authors that the double octic should be replaced by a non-commutative space (in the sense of Kontsevich), represented by the derived category of coherent sheaves which are modules over the sheaf of even parts of Clifford algebras over \({\mathbb P}^3\).
The authors also analyse GLSMs for other complete intersections of quadrics and of higher degree hypersurfaces in projective spaces; some of these examples involve non-Calabi-Yau 3-folds. Based on these examples it is conjectured that Kuznetsov’s homological projective duality can be applied to GLSMs in general.

MSC:

14J81 Relationships between surfaces, higher-dimensional varieties, and physics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
32L81 Applications of holomorphic fiber spaces to the sciences
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81R60 Noncommutative geometry in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Witten E.: Phases of N = 2 theories in two dimensions. Nucl Phys. B 403, 159–222 (1993) · Zbl 0910.14020 · doi:10.1016/0550-3213(93)90033-L
[2] Hellerman S., Henriques A., Pantev T., Sharpe E., Ando M.: Cluster decomposition, T-duality, and gerby CFT’s. Adv. Theo. Math. Phys. 11, 751–818 (2007) · Zbl 1156.81039
[3] Hori K., Tong D.: Aspects of non-Abelian gauge dynamics in two-dimensional N = (2, 2) theories. JHEP 0705, 079 (2007) · doi:10.1088/1126-6708/2007/05/079
[4] Sharpe E.: String orbifolds and quotient stacks. Nucl. Phys. B627, 445–505 (2002) · Zbl 0991.83045 · doi:10.1016/S0550-3213(02)00039-1
[5] Vafa C., Witten E.: On orbifolds with discrete torsion. J. Geom. Phys. 15, 189–214 (1995) · Zbl 0816.53053 · doi:10.1016/0393-0440(94)00048-9
[6] Donagi R., Sharpe E.: GLSMs for partial flag manifolds. J. Geom. Phys. 58, 1662–1692 (2008) · Zbl 1218.81091 · doi:10.1016/j.geomphys.2008.07.010
[7] Reid, M.: The complete intersection of two or more quadrics. Ph.D. thesis, Trinity College, Cambridge, 1972, available at http://www.warwick.ac.uk/\(\sim\)masda/3folds/qu.pdf
[8] Griffiths P., Harris J.: Principles of Algebraic Geometry. John Wiley & Sons, New York (1978) · Zbl 0408.14001
[9] Clemens H.: Double solids. Adv. in Math. 47, 107–230 (1983) · Zbl 0509.14045 · doi:10.1016/0001-8708(83)90025-7
[10] Cynk, S., Meyer, C.: Geometry and arithmetic of certain double octic Calabi-Yau manifolds. http://arxiv.org/abs/math/0304121v1[math.AG] , 2003
[11] Căldăraru A., Katz S., Sharpe E.: D-branes, B fields, and Ext groups. Adv. Theor. Math. Phys. 7, 381–404 (2004)
[12] Gross, M., Pavanelli, S.: A Calabi-Yau threefold with Brauer group (Z 8)2. http://arxiv.org/abs/math/0512182v1[math.AG] , 2005 · Zbl 1127.14036
[13] Gross, M.: Private communication, September 27, 2006
[14] Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. http://arxiv.org/abs/math/0510670v1[math.AG] , 2005 · Zbl 1168.14012
[15] Sharpe E.: D-branes, derived categories, and Grothendieck groups. Nucl. Phys. B561, 433–450 (1999) · Zbl 1028.81519 · doi:10.1016/S0550-3213(99)00535-0
[16] Douglas M.: D-branes, categories, and N = 1 supersymmetry. J. Math. Phys. 42, 2818–2843 (2001) · Zbl 1036.81027 · doi:10.1063/1.1374448
[17] Sharpe, E.: Lectures on D-branes and sheaves. Writeup of lectures given at the Twelfth Oporto meeting on ”Geometry, Topology, and Physics,” and at the Adelaide Workshop ”Strings and Mathematics 2003,” http://arxiv.org/abs/hep-th/0307245v2 , 2003
[18] Pantev, T., Sharpe, E.: Notes on gauging noneffective group actions. http://arxiv.org/abs/hep-th/0502027v2 , 2005
[19] Pantev T., Sharpe E.: String compactifications on Calabi-Yau stacks. Nucl. Phys. B733, 233–296 (2006) · Zbl 1119.81091 · doi:10.1016/j.nuclphysb.2005.10.035
[20] Pantev T., Sharpe E.: GLSM’s for gerbes (and other toric stacks). Adv. Theor. Math. Phys. 10, 77–121 (2006) · Zbl 1119.14038
[21] Sharpe, E.: Derived categories and stacks in physics. http://arxiv.org/abs/hep-th/0608056v2 , 2006 · Zbl 1166.81360
[22] Kuznetsov, A.: Homological projective duality. http://arxiv.org/abs/math/0507292v1[math.AG] , 2005
[23] Kuznetsov, A.: Homological projective duality for Grassmannians of lines. http://arxiv.org/abs/math/0610957v1[math.AG] , 2006
[24] Harris, J.: Algebraic geometry: a first course. Grad. Texts in Math. 133, New York: Springer-Verlag, 1992 · Zbl 0779.14001
[25] Căldăraru, A.: Derived categories of twisted sheaves on elliptic threefolds. http://arxiv.org/abs/math/0012083v3[math.AG] , 2001
[26] Căldăraru, A.: N. Addington: Work in progress
[27] Kapustin A., Li Y.: D-branes in Landau-Ginzburg models and algebraic geometry. JHEP 0312, 005 (2003) · Zbl 1058.81061 · doi:10.1088/1126-6708/2003/12/005
[28] Kuznetsov, A.: Private communication
[29] Mukai S.: Moduli of vector bundles on K3 surfaces, and symplectic manifolds. Sugaku Expositions 1, 139–174 (1988) · Zbl 0685.14021
[30] Kuznetsov, A.: Private communication, January 29, 2007
[31] Căldăraru, A.: To appear
[32] Sharpe E.: Discrete torsion. Phys. Rev. D68, 126003 (2003) · Zbl 1028.83521
[33] Sharpe E.: Recent developments in discrete torsion. Phys. Lett. B498, 104–110 (2001) · Zbl 0972.81157
[34] Căldăraru, A., Giaquinto, A., Witherspoon, S.: Algebraic deformations arising from orbifolds with discrete torsion. http://arxiv.org/abs/math/0210027v2[math.KT] , 2003
[35] Melnikov I., Plesser R.: A-model correlators from the Coulomb branch. JHEP 0602, 044 (2006) · doi:10.1088/1126-6708/2006/02/044
[36] Bertram, A.: Private communication, January 5, 2007
[37] Beauville, A.: Complex Algebraic Surfaces. Second edition, Cambridge: Cambridge University Press, 1996 · Zbl 0849.14014
[38] Iyer, J., Simpson, C.: A relation between the parabolic Chern characters of the de Rham bundles. http://arxiv.org/abs/math/0603677v2[math.AG] , 2006
[39] Adams A., Polchinski J., Silverstein E.: Don’t panic! Closed string tachyons in ALE space-times. JHEP 0110, 029 (2001) · doi:10.1088/1126-6708/2001/10/029
[40] Harvey, J., Kutasov, D., Martinec, E., Moore, G.: Localized tachyons and RG flows. http://arxiv.org/abs/hep-th/0111154v2 , 2001
[41] Martinec, E., Moore, G.: On decay of K theory. http://arxiv.org/abs/hep-th/0212059v1 , 2002
[42] Kuznetsov, A.: Private communication, March 10, 2007
[43] Orlov, D.: Triangulated categories of singularities and D-branes in Landau-Ginzburg models. http://arxiv.org/abs/math/0302304v2[math.AG] , 2004 · Zbl 1101.81093
[44] Orlov, D.: Triangulated categories of singularities and equivalences between Landau-Ginzburg models. http://arxiv.org/abs/math/0503630v1[math.AG] , 2005
[45] Orlov, D.: Derived categories of coherent sheaves and triangulated categories of singularities. math.AG/0503632 · Zbl 1200.18007
[46] Pantev, T., Sharpe, E.: Work in progress
[47] Guffin, J., Sharpe, E.: To appear
[48] Kontsevich, M.: Course on non-commutative geometry. ENS, 1998 Lecture notes at http://www.math.uchicago.edu/\(\sim\)mitya/langlands/html
[49] Kontsevich, M.: Talk at the ”Hodge centennial conference,” Edinburgh, 2003
[50] Soibelman, Y.: Lectures on deformation theory and mirror symmetry. IPAM, 2003, http://www.math.ksu.edu/\(\sim\)soibel/ipam-final.ps , 2003
[51] Costello, K.: Topological conformal field theories and Calabi-Yau categories. http://arxiv.org/abs/math/0412149v7[math.QA] , 2006 · Zbl 1171.14038
[52] Toën, B., Vaquie, M.: Moduli of objects in dg-categories. http://arxiv.org/abs/math/0503269v5[math.AG] , 2007
[53] Bondal A., van den Bergh M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3, 1–36 (2003) · Zbl 1135.18302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.