# zbMATH — the first resource for mathematics

Hori-mological projective duality. (English) Zbl 1427.14042
In the article under review, the authors establish two goals: 1) prove a version of homological projective duality (HPD) for Pfaffian varieties as proposed by A. Kuznetsov [Publ. Math., Inst. Hautes Étud. Sci. 105, 157–220 (2007; Zbl 1131.14017)] and 2) prove a duality between two quantum field theories called gauged linear sigma models (GLSM) as proposed by K. Hori [J. High Energy Phys. 2013, No. 10, Paper No. 121, 74 p. (2013; Zbl 1342.81635)].
For the first result the authors build on their previous work [N. Addington et al., Algebr. Geom. 2, No. 3, 332–364 (2015; Zbl 1322.14037)], [E. Segal and R. Thomas, J. Reine Angew. Math. 743, 245–259 (2018; Zbl 1454.14051)] and Kuznetzov’s own work on the homological projective duality of line Grassmannians (a special case of Pfaffian varieties). The version of HPD they prove is for a non-commutative crepant resolutions (as constructed by Š. Špenko and M. Van den Bergh [Invent. Math. 210, No. 1, 3–67 (2017; Zbl 1375.13007)]), the reason being that in general Pfaffian varieties are very singular, as opposed to the smooth case of Grassmannians treated by Kuznetzov.
The second result relies firstly on providing a mathematical, and algebro-geometric in nature, definition of B-branes for the gauged linear sigma model, which seems to be novel in the physics literature. Their definition of B-brane recovers the non-commutative crepant resolution of Pfaffian as an example of the general theory of Spenko and Van den Bergh. Finally, the duality of quantum field theories seems to be a consequence of HPD as proved in the first part.
The authors concentrated in the case of an odd-dimensional vector space and the stronger results are proved in this case, as the even-dimensional case doesn’t lend itself to a physical interpretation.
The article has an extensive introduction that explain the main construction used in the article and also include a sketch of the proof of the main duality result.

##### MSC:
 14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 16E35 Derived categories and associative algebras 16S38 Rings arising from noncommutative algebraic geometry
Full Text:
##### References:
 [1] N. Addington, W. Donovan, and E. Segal, The Pfaffian-Grassmannian equivalence revisited, Algebr. Geom. 2 (2015), no. 3, 332-364. · Zbl 1322.14037 [2] M. Ballard, D. Deliu, D. Favero, M. U. Isik, and L. Katzarkov, Homological projective duality via variation of geometric invariant theory quotients, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 1127-1158. · Zbl 1400.14048 [3] M. Ballard, D. Favero, and L. Katzarkov, Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746 (2019), 235-303. · Zbl 1432.14015 [4] A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1-36. · Zbl 1135.18302 [5] M. Brion, Sur les modules de covariants, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), no. 1, 1-21. · Zbl 0781.13002 [6] A. I. Efimov and L. Positselski, Coherent analogues of matrix factorizations and relative singularity categories, Algebra Number Theory 9 (2015), no. 5, 1159-1292. · Zbl 1333.14018 [7] D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Grad. Texts in Math. 150, Springer, New York, 1995. · Zbl 0819.13001 [8] W. Fulton and J. Harris, Representation Theory: A First Course, Grad. Texts in Math. 129, Springer, New York, 1991. · Zbl 0744.22001 [9] D. Halpern-Leistner, The derived category of a GIT quotient, J. Amer. Math. Soc. 28 (2015), no. 3, 871-912. · Zbl 1354.14029 [10] R. Hartshorne, Residues and Duality, with appendix “Cohomologie à support propre et construction du foncteur $$f^!$$” by P. Deligne, Lecture Notes in Math. 20, Springer, Berlin, 1966. · Zbl 0212.26101 [11] R. Hartshorne, Local Cohomology, Lecture Notes in Math. 41, Springer, Berlin, 1967. [12] R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121-176. · Zbl 0431.14004 [13] Y. Hirano, Derived Knörrer periodicity and Orlov’s theorem for gauged Landau-Ginzburg models, Compos. Math. 153 (2017), no. 5, 973-1007. · Zbl 1370.14019 [14] K. Hori, Duality in two-dimensional $$(2,2)$$ supersymmetric non-abelian gauge theories, J. High Energy Phys. 2013, no. 10, art. ID 121. · Zbl 1342.81635 [15] K. Hori and J. Knapp, Linear sigma models with strongly coupled phases—one parameter models, J. High Energy Phys., published online 8 November 2013. [16] K. Hori and D. Tong, Aspects of non-abelian gauge dynamics in two-dimensional $$N=(2,2)$$ theories, J. High Energy Phys. 2007, no. 5, art. ID 079. [17] S. Hosono and H. Takagi, Duality between $${S}^2\mathbb{P}^4$$ and the double quintic symmetroid, preprint, arXiv:1302.5881v2 [math.AG]. [18] M. Kapranov, On the derived category of coherent sheaves on Grassmann manifolds (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 48 (1984), no. 1, 192-202; English translation in Izv. Math. 24 (1985), no. 1, 183-192. · Zbl 0564.14023 [19] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008), no. 1, 1-91. · Zbl 1145.57009 [20] A. Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157-220. · Zbl 1131.14017 [21] A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008), no. 5, 1340-1369. · Zbl 1168.14012 [22] A. Kuznetsov, Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc. (3) 97 (2008), no. 1, 155-182. · Zbl 1168.14032 [23] A. Kuznetsov, Homological projective duality for Grassmannians of lines, preprint, arXiv:math/0610957v1 [math.AG]. [24] J. Lipman, “Lectures on local cohomology and duality” in Local Cohomology and Its Applications (Guanajuato, 1999), Lect. Notes Pure Appl. Math. 226, Dekker, New York, 2002, 39-89. · Zbl 1011.13010 [25] C. Năstăsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Math. Lib. 28, North-Holland, Amsterdam, 1982. [26] D. Orlov, Matrix factorizations for nonaffine LG-models, Math. Ann. 353 (2012), no. 1, 95-108. · Zbl 1243.81178 [27] L. Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212 (2011), no. 996. · Zbl 1275.18002 [28] J. V. Rennemo, The homological projective dual of $$\operatorname{Sym}^2\mathbb{P}({V})$$, preprint, arXiv:1509.04107v1 [math.AG]. [29] J. V. Rennemo, The fundamental theorem of homological projective duality via variation of GIT stability, preprint, arxiv:1705.01437v1 [math.AG]. [30] J. V. Rennemo and E. Segal, Addendum to Hori-mological projective duality, preprint, http://www.homepages.ucl.ac.uk/ ucaheps/ (accessed 8 July 2019). [31] J. V. Rennemo, E. Segal, and M. Van den Bergh, A non-commutative Bertini theorem, preprint, arXiv:1705.01366v1 [math.AG]. [32] E. Segal, Equivalence between GIT quotients of Landau-Ginzburg B-models, Comm. Math. Phys. 304 (2011), no. 2, 411-432. · Zbl 1216.81122 [33] E. Segal and R. Thomas, Quintic threefolds and Fano elevenfolds, J. Reine Angew. Math. 743 (2018), 245-259. [34] I. Shipman, A geometric approach to Orlov’s theorem, Compos. Math. 148 (2012), no. 5, 1365-1389. · Zbl 1253.14019 [35] Š. Špenko and M. Van den Bergh, Non-commutative resolutions of quotient singularities, to appear in Invent. Math., preprint, arXiv:1502.05240v3 [math.AG]. [36] Stacks Project Authors, The Stacks Project, http://stacks.math.columbia.edu (accessed 23 May 2019). [37] S. Sundarem, On the combinatorics of representations of $$\operatorname{Sp}(2n,\mathbb{C})$$, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1986. [38] R. P. Thomas, Notes on HPD, preprint, arXiv:1512.08985v4 [math.AG]. [39] M. Van den Bergh, “Non-commutative crepant resolutions” in The Legacy of Niels Henrik Abel (Oslo, 2002), Springer, Berlin, 2004, 749-770. · Zbl 1082.14005 [40] J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Math. 149, Cambridge Univ. Press, Cambridge, 2003. [41] A. Yekutieli and J. J. Zhang, Dualizing complexes and perverse sheaves on noncommutative ringed schemes, Selecta Math. (N.S.) 12 (2006), no. 1, 137-177. · Zbl 1137.14300 [42] A. Yekutieli and J. J. Zhang, Rigid dualizing complexes on schemes, preprint, arXiv:math/0405570v3 [math.AG].
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.