# Workshop on Matrix Factorizations – References

### Background and foundations

A matrix factorization of an element w in a commutative ring S is a pair of free S-modules F,G and maps A: F → G, B: G → F such that AB = w⋅1_{G}and BA = w⋅1

_{F}. In what follows we will always assume that S is a regular, e.g. a polynomial or power series ring over a field.

Matrix factorizations were introduced by Eisenbud in his 1980 paper "Homological algebra on a complete intersection..." He showed that the free resolution of every finitely generated module over R = S/(w) is eventually given by a matrix factorization. In particular every such resolution is eventually 2-periodic. This in turn allowed him to show that matrix factorizations describe all maximal Cohen-Macaulay modules without free summands. Since this groundbreaking work matrix factorizations have been a common tool in commutative algebra.

Buchweitz introduced the notion of the stable derived category of a ring in 1986. In an unpublished, but now-famous, manuscript he showed that the homotopy category of matrix factorizations gives one of four equivalent descriptions of the stable derived category of a hypersurface ring. This work was generalized by Orlov to a large class of schemes.

1. Ragnar-Olaf Buchweitz. Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. Unpublished manuscript, available at https://tspace.library.utoronto.ca/handle/1807/16682, 1987.

2. Ragnar Olaf Buchweitz, Gert Martin Greuel, and Frank Olaf Schreyer. Cohen-Macaulay modules on hypersurface singularities. II. Invent. Math., 88(1):165–182, 1987.

3. Ragnar-Olaf Buchweitz, David Eisenbud, and Jürgen Herzog. Cohen-Macaulay modules on quadrics. In Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), volume 1273 of Lecture Notes in Math., pages 58–116. Springer, Berlin, 1987.

4. Yu. A. Drozd and G.-M. Greuel. Cohen-Macaulay module type. Compositio Math., 89(3):315–338, 1993.

5. David Eisenbud. Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc., 260(1):35–64, 1980.

6. Werner Geigle and Helmut Lenzing. A class of weighted projective curves arising in representation theory of finite-dimensional algebras. In Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), volume 1273 of Lecture Notes in Math., pages 265–297. Springer, Berlin, 1987.

7. Melvin Hochster. The dimension of an intersection in an ambient hypersurface. In Algebraic geometry (Chicago, Ill., 1980), volume 862 of Lecture Notes in Math., pages 93–106. Springer, Berlin, 1981.

8. Craig Huneke and Roger Wiegand. Tensor products of modules and the rigidity of Tor. Math. Ann., 299(3):449–476, 1994.

9. Horst Knörrer. Cohen-Macaulay modules on hypersurface singularities. I. Invent. Math., 88(1):153–164, 1987.

10. Dmitry Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262, 2004.

11. Dmitry Orlov. Triangulated categories of singularities, and equivalences between Landau-Ginzburg models.Mat. Sb., 197(12):117–132, 2006.

12. Dmitri Orlov. Derived categories of coherent sheaves and triangulated categories of singularities. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, volume 270 of Progr. Math., pages 503–531. Birkhäuser Boston Inc., Boston, MA, 2009.

13. Yuji Yoshino. Cohen-Macaulay modules over Cohen-Macaulay rings, volume 146 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1990.

### New developments in commutative algebra and algebraic geometry

1. Matthew Ballard, David Favero, and Ludmil Katzarkov. Orlov spectra: bounds and gaps, 2010. arXiv:1012.0864.

2. Ragnar-Olaf Buchweitz and Graham J. Leuschke. Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant. Amer. J. Math., 129(4):943–981, 2007.

3. Ragnar-Olaf Buchweitz, Graham J.Leuschke, and Michel Van den Bergh. Non-commutative desingularization of determinantal varieties I. Invent. Math., 182(1):47–115, 2010.

4. Igor Burban and Yuriy Drozd. Maximal Cohen-Macaulay modules over non-isolated surface singularities. arXiv:1002.3042.

5. Olgur Celikbas and Mark E. Walker. Hochster’s theta pairing and algebraic equivalence. arXiv:1011.4886.

6. Andrew Crabbe and Graham J. Leuschke. Wild hypersurfaces. arXiv:1008.2465.

7. Hailong Dao. Some observations on local and projective hypersurfaces. Math. Res. Lett., 15(2):207–219, 2008.

8. Hailong Dao. Remarks on non-commutative crepant resolutions of complete intersections. Adv. Math., 224(3):1021–1030, 2010.

9. Hailong Dao and Craig Huneke. Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings. arXiv:1005.5359.

10. W. Frank Moore, Greg Piepmeyer, Sandra Spiroff, and Mark E. Walker. Hochster’s theta invariant and the Hodge-Riemann bilinear relations, to appear in Adv. Math. arxiv:0910.1289v2.

### New developments in representation theory

1. Igor Burban, Osamu Iyama, Bernhard Keller, and Idun Reiten. Cluster tilting for one-dimensional hypersurface singularities. Adv. Math., 217(6):2443–2484, 2008.

2. Osamu Iyama and Michael Wemyss. Auslander-reiten duality and maximal modifications for non-isolated singularities. arXiv:1007.1296.

3. Hiroshige Kajiura, Kyoji Saito, and Atsushi Takahashi. Matrix factorization and representations of quivers. II. Type ADE case. Adv. Math., 211(1):327–362, 2007.

4. Hiroshige Kajiura, Kyoji Saito, and Atsushi Takahashi. Triangulated categories of matrix factorizations for regular systems of weights with ϵ = -1. Adv. Math., 220(5):1602–1654, 2009.

5. Dirk Kussin, Helmut Lenzing, and Hagen Meltzer. Nilpotent operators and weighted projective lines. arXiv:1002.3797.

6. Helmut Lenzing and Jos Antonio de la Pea. Extended canonical algebras and fuchsian singularities. arXiv:math/0611532.

7. Helmut Lenzing and José Antonio de la Peña. Spectral analysis of finite dimensional algebras and singularities. In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pages 541–588. Eur. Math. Soc., Zürich, 2008.

### Hochschild homology of the homotopy category of matrix factorizations and other ideas inspired by physics.

It was conjectured that the Hochschild (co)homology of the dg-category of matrix factorizations of an isolated singularity is the Jacobi ring of the singularity. This was proven by Dyckerhoff; see also Segal. Using this calculation Polishchuk and Vaintrob defined Chern characters of matrix factorizations which take values in the Jacobi ring. They then showed that the pairing on Chern characters given by applying the residue map after multiplication in the Jacobi ring, can be calculated in terms of the entries of the corresponding matrix factorizations. They referred to this as a "Hirzebruch-Riemann-Roch" formula. Dyckerhoff and Murfet have given two additional approaches and proofs to the HRR theorem: one using topological quantum field theory in (3) and the other an elementary approach in (4) using computations inpspired by knot theory.

1. Andrei Caldararu and Junwu Tu. Curved A-infinity algebras and Landau-Ginzburg models. arxiv:1007.2679.

2. Tobias Dyckerhoff. Compact generators in categories of matrix factorizations. arxiv:0904.4713v4.

3. Tobias Dyckerhoff and Daniel Murfet. The Kapustin-Li formula revisited. arxiv:1004.0687v1.

4. Tobias Dyckerhoff and Daniel Murfet. Pushing forward matrix factorisations. arxiv:1102.2957v1.

5. Alexander I. Efimov. Homological mirror symmetry for curves of higher genus. arXiv:0907.3903.

6. Daniel Murfet. Residues and duality for singularity categories of isolated Gorenstein singularities. arxiv:0912.1629v2.

7. Alexander Polishchuk and Arkady Vaintrob. Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations. arxiv:1002.2116.

8. Ed Segal. The closed state space of affine Landau-Ginzburg B-models. arxiv:0904.1339v1.

9. Junwu Tu. Matrix factorizations via Koszul duality. arxiv:1009.4151.

### Non-affine matrix factorizations

1. Kevin H. Lin and Daniel Pomerleano. Global matrix factorizations. arXiv:1101.5847.

2. Dmitri Orlov. Matrix factorizations for nonaffine LG-models. arXiv:1101.4051.

3. Alexander Polishchuk and Arkady Vaintrob. Matrix factorizations and singularity categories for stacks. arxiv:1011.4544.

4. Leonid Positselski. Coherent analogues of matrix factorizations and relative singularity categories. arXiv:1102.0261.

5. Anatoly Preygel. Thom-Sebastiani duality for matrix factorizations. arXiv:1101.5834.

6. Ed Segal. Equivalences between GIT quotients of Landau-Ginzburg B-models. arxiv:0910.5534v3.

7. Ian Shipman. A geometric approach to Orlov’s theorem. arXiv:1012.5282.

### Physics and matrix factorizations (Contributed by Nils Carqueville)

In mathematical physics, matrix factorisations most prominently appear as supersymmetric boundary conditions (D-branes) and defect lines in topological Landau-Ginzburg models [10,23,30]. These two-dimensional quantum field theories are intimately connected to conformal field theories and play an important role in the worldsheet approach to string theory. Recently matrix factorisations also arose in the study of Rozansky-Witten models [26] and quiver gauge theories [3].More precisely, matrix factorisations of a polynomial W describe D-branes in the LG model with potential W, and the morphisms in the triangulated category MF(W) describe open strings between branes. This boundary sector is complemented by the bulk sector which is the Jacobian of W, and if W is an isolated singularity then Jac(W) together with MF(W) satisfy all the axioms of an open/closed 2dTFT, see [19,24,25] and work by Murfet and Polishchuk/Vaintrob. Similarly, matrix factorisations of W-W' describe topological defect lines between LG models with potentials W and W' [11,12,25]. Thus it is natural to study and discover the rich structure of the bicategory of all LG models whose objects are potentials W and whose 1-morphisms are defect categories MF(W-W') [15,16].

Landau-Ginzburg models are closely related to N=2 superconformal field theories which are viewed as the endpoints of the so-called renormalisation group flow. Understanding this CFT/LG correspondence in full is very challenging, but detailed comparisons of the topological subsectors (using vertex operator algebraic and categorial descriptions of the CFT side and matrix factorisations on the LG side) have successfully tested the correspondence on many levels, see e.g. [1,6,7,8,17,27]. This duality has been made rigorous for certain models by the work of Orlov and [18], and taking it as a given affords an efficient setting to tackle problems such as deformation theory or CFT moduli spaces, see e.g. [9,13,21,22,28]. The correspondence also leads to interesting conjectures (and sometimes even their proofs) on the structure of matrix factorisations [9,15,16,18].

Perturbative string theory can very roughly be characterised as the study of not one single conformal field theory, but of the whole moduli space of CFTs. It is a general result that in the open topological subsector, string theoretic amplitudes are equivalently encoded in a Calabi-Yau A-infinity-structure [20]. Hence in this sense passing from TFT to string theory means to endow MF(W) with the correct A-infinity-structure. Once this is achieved one can claim to have solved the theory completely, and as a corollary one immediately obtains information about interactions in the associated low-energy effective field theory in four spacetime dimensions, see [2,14] and also [4,5,29] for related ideas.

1. S. K. Ashok, E. Dell’Aquila, and D.-E. Diaconescu, Fractional Branes in Landau-Ginzburg Orbifolds, Adv. Theor. Math. Phys. 8 (2004), 461-513, arXiv:hep-th/0401135.

2. P. S. Aspinwall, Topological D-Branes and Commutative Algebra, arXiv:hep-th/0703279.

3. P. S. Aspinwall and D. R. Morrison, Quivers from Matrix Factorizations, arXiv:1005.1042.

4. M. Baumgartl, I. Brunner, and M. R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP 0707 (2007), 061, arXiv:0704.2666.

5. M. Baumgartl, I. Brunner, and M. Soroush, D-brane Superpotentials: Geometric and Worldsheet Approaches, Nucl. Phys. B 843 (2011), 602-637, arXiv:1007.2447.

6. N. Behr and S. Fredenhagen, D-branes and matrix factorisations in supersymmetric coset models, JHEP 1011 (2010), 136, arXiv:1005.2117.

7. I. Brunner and M. R. Gaberdiel, Matrix factorisations and permutation branes, JHEP 0507 (2005), 012, arXiv:hep-th/0503207.

8. I. Brunner and M. R. Gaberdiel, The matrix factorisations of the D-model, J. Phys. A 38 (2005), 7901‚ arXiv:hep-th/0506208.

9. I. Brunner and M. Herbst, Orientifolds and D-branes in N=2 gauged linear sigma models, arXiv:0812.2880.

10. I. Brunner, M. Herbst, W. Lerche, and B. Scheuner, Landau-Ginzburg realization of open string TFT, JHEP 0611 (2003), 043, arXiv:hep-th/0305133.

11. I. Brunner and D. Roggenkamp, B-type defects in Landau-Ginzburg models, JHEP 0708 (2007), 093, arXiv:0707.0922.

12. I. Brunner and D. Roggenkamp, Defects and bulk perturbations of boundary Landau-Ginzburg orbifolds, JHEP 0804 (2008), 001, arXiv:0712.0188.

13. I. Brunner, H. Jockers, and D. Roggenkamp, Defects and D-Brane monodromies, JHEP 0804 (2008), 001, arXiv:0806.4734.

14. N. Carqueville, Matrix factorisations and open topological string theory, JHEP 07 (2009), 005, arXiv:0904.0862.

15. N. Carqueville and I. Runkel, On the monoidal structure of matrix bifactorisations, J. Phys. A: Math. Theor. 43 (2010) 275401, arXiv:0909.4381.

16. N. Carqueville and I. Runkel, Rigidity and defect actions in Landau-Ginzburg models, arXiv:1006.5609.

17. H. Enger, A. Recknagel, and D. Roggenkamp, Permutation branes and linear matrix factorisations, JHEP 0601 (2006), 087, arXiv:hep-th/0508053.

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

19. M. Herbst and C. I. Lazaroiu, Localization and traces in open-closed topological Landau-Ginzburg models, JHEP 0505 (2005), 044, arXiv:hep-th/0404184.

20. M. Herbst, C. I. Lazaroiu, and W. Lerche, Superpotentials, A-infinity Relations and WDVV Equations for Open Topological Strings, JHEP 0502 (2005), 071, arXiv:hep-th/0402110.

21. K. Hori and J. Walcher, F-term equations near Gepner points, JHEP 0501 (2005), 008, arXiv:hep-th/0404196.

22. H. Jockers, D-brane monodromies from a matrix-factorization perspective, JHEP 0702 (2007), 006, arXiv:hep-th/0612095.

23. A. Kapustin and Y. Li, D-branes in Landau-Ginzburg Models and Algebraic Geometry, JHEP 0312 (2003), 005, arXiv:hep-th/0210296.

24. A. Kapustin and Y. Li, Topological Correlators in Landau-Ginzburg Models with Boundaries, Adv. Theor. Math. Phys. 7 (2004), 727‚Äì749, arXiv:hep-th/0305136.

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

26. A. Kapustin, L. Rozansky, and N. Saulina, Three-dimensional topological field theory and symplectic algebraic geometry I, arXiv:0810.5415.

27. C. A. Keller and S. Rossi, Boundary states, matrix factorisations and correlation functions for the E-models, JHEP 0703 (2007), 038, arXiv:hep-th/0610175.

28. J. Knapp and H. Omer, Matrix Factorizations, Minimal Models and Massey Products, JHEP 0605 (2006), 064, arXiv:hep-th/0604189.

29. J. Knapp and E. Scheidegger, Towards Open String Mirror Symmetry for One-Parameter Calabi-Yau Hypersurfaces, Adv. Theor. Math. Phys. Volume 13, Number 4 (2009), 991-1075, arXiv:0805.1013.

30. C. I. Lazaroiu, On the boundary coupling of topological Landau-Ginzburg models, JHEP 0505 (2005), 037, arXiv:hep-th/0312286.