×

zbMATH — the first resource for mathematics

Homotopy Batalin-Vilkovisky algebras. (English) Zbl 1258.18005
The main purpose of this article is to give an explicit description of a small quasi-free resolution, the Koszul resolution, of the operad of Batalin-Vilkovisky algebras.
Recall that a Batalin-Vilkovisky algebra is a graded commutative algebra \(A\) equipped with a unary operator \(\Delta: A\rightarrow A\), homogeneous of degree \(1\), satisfying \(\Delta^2 = 0\), and so that the operation \[ \lambda(x_1,x_2) = \Delta(x_1 x_2) - (\Delta(x_1) x_2 + x_1\Delta(x_2)){(*)} \] satisfies the relations of an odd Poisson bracket (a Gerstenhaber bracket) on \(A\). The operad of Batalin-Vilkovisky algebras \(\mathcal{BV}\) is defined by generators and relations, with a two-ary generating operation \(\mu\in\mathcal{BV}(2)\) representing the product underlying a Batalin-Vilkovisky structure, a second two-ary generating operation \(\lambda\in\mathcal{BV}(2)\) representing the Gerstenhaber bracket, and a unary generating operation \(\Delta\in\mathcal{BV}(1)\) which represents the Batalin-Vilkovisky operator.
The Koszul resolution of an operad, in the sense of V. Ginzburg and M. Kapranov [Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)], is a quasi-free operad \(\mathcal P_{\infty} = (\mathcal F(s^{-1}\mathcal P^{\text{<}}),d_2)\) whose differential \(d_2\) is quadratic on generators \(\mathcal P^{\text{<}}\). The definition of the operad \(\mathcal{BV}\) involves a relation (*) which is not homogeneous quadratic with respect to the composition of generating operations, since this equation involves the Gerstenhaber bracket as a linear term. The existence of this inhomogeneous relation gives an obstruction for the construction of a Koszul resolution in the above sense. The authors therefore consider a graded operad \(q\mathcal{BV}\), with the same generating operations as in the operad \(\mathcal{BV}\), but where this linear term in (*) is canceled. The crux of their construction relies on the observation that this operad \(q\mathcal{BV}\) is Koszul, and that a linear term can be added to the differential of the Koszul resolution of this graded operad to get a quasi-free operad \(\mathcal{BV}_{\infty} = (\mathcal F(s^{-1}q\mathcal{BV}^{\text{<}}),d_1+d_2)\) defining a resolution of the operad \(\mathcal{BV}\).
The authors considers the category of algebras associated to the operad \(\mathcal{BV}_{\infty}\) to get an explicit definition of the notion of a homotopy \(\mathcal{BV}\)-algebra. The authors notably use this effective approach to prove that any non-negatively graded topological vertex operator algebra in the sense of B. H. Lian and G. J. Zuckerman [Commun. Math. Phys 154, 613–646 (1993; Zbl 0780.17029)] inherits a homotopy \(\mathcal{BV}\)-algebra structure.

MSC:
18D50 Operads (MSC2010)
18G55 Nonabelian homotopical algebra (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B69 Vertex operators; vertex operator algebras and related structures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] F. Akman, On some generalizations of Batalin-Vilkovisky algebras. J. Pure Appl. Algebra 120 (1997), 105-141. · Zbl 0885.17020 · doi:10.1016/S0022-4049(96)00036-9
[2] F. Akman and L. M. Ionescu, Higher derived brackets and deformation theory. I. J. Homotopy Relat. Struct. 3 (2008), 385-403. · Zbl 1287.17040 · arxiv:math/0504541v2 · emis:journals/JHRS/volumes/2008/n1a14/abstract.htm
[3] G. Barnich, R. Fulp, T. Lada, and J. Stasheff, The sh Lie structure of Poisson brackets in field theory. Comm. Math. Phys. 191 (1998), 585-601. · Zbl 0951.37035 · doi:10.1007/s002200050278 · arxiv:hep-th/9702176
[4] A. Beilinson and V. Drinfeld, Chiral algebras . Amer. Math. Soc. Colloq. Publ. 51, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1138.17300
[5] C. Berger and I. Moerdijk, Axiomatic homotopy theory for operads. Comment. Math. Helv. 78 (2003), 805-831. · Zbl 1041.18011 · doi:10.1007/s00014-003-0772-y · arxiv:math/0206094
[6] K. Bering, P. H. Damgaard, and J. Alfaro, Algebra of higher antibrackets. Nuclear Phys. B 478 (1996), 459-503. · Zbl 0925.81398 · doi:10.1016/0550-3213(96)00401-4
[7] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topologi- cal spaces . Lecture Notes in Math. 347, Springer-Verlag, Berlin 1973. · Zbl 0285.55012
[8] A. Braverman and D. Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. Algebra 181 (1996), 315-328. · Zbl 0860.17002 · doi:10.1006/jabr.1996.0122 · arxiv:hep-th/9411113
[9] E. H. Brown, Jr., Twisted tensor products. I. Ann. of Math. (2) 69 (1959), 223-246. · Zbl 0199.58201 · doi:10.2307/1970101
[10] A. S. Cattaneo and F. Schätz, Equivalences of higher derived brackets. J. Pure Appl. Algebra 212 (2008), 2450-2460. · Zbl 1177.53073 · doi:10.1016/j.jpaa.2008.03.013
[11] M. Chas and D. Sullivan, String topology. Preprint 1999. · arxiv.org
[12] K. Costello, Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210 (2007), 165-214. · Zbl 1171.14038 · doi:10.1016/j.aim.2006.06.004 · arxiv:math/0412149
[13] V. Dolgushev, D. Tamarkin, and B. Tsygan, The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal. J. Noncommut. Geom. 1 (2007), 1-25. · Zbl 1144.18007 · doi:10.4171/JNCG/1 · www.ems-ph.org · arxiv:math/0605141
[14] W. G. Dwyer and J. Spaliński, Homotopy theories and model categories. In Hand- book of algebraic topology , North-Holland, Amsterdam 1995, 73-126. · Zbl 0869.55018
[15] D. Eisenbud, The geometry of syzygies . Graduate Texts in Math. 229, Springer-Verlag, New York 2005. · Zbl 1066.14001 · doi:10.1007/b137572
[16] E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves . Math. Surveys Monogr. 88, 2nd ed., Amer. Math. Soc., Providence, RI 2004. · Zbl 1106.17035
[17] B. Fresse, Koszul duality of operads and homology of partition posets. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K -theory, Contemp. Math. 346, Amer. Math. Soc., Providence, RI, 2004, 115-215. · Zbl 1077.18007 · arxiv:math/0301365
[18] B. Fresse, Théorie des opérades de Koszul et homologie des algèbres de Poisson. Ann. Math. Blaise Pascal 13 (2006), 237-312. · Zbl 1141.55006 · doi:10.5802/ambp.219 · numdam:AMBP_2006__13_2_237_0 · eudml:10532
[19] I. Gálvez, V. Gorbounov, and A. Tonks, Homotopy Gerstenhaber structures and vertex algebras. Appl. Categ. Structures 18 (2010), 1-15. · Zbl 1268.17030 · doi:10.1007/s10485-008-9170-3 · arxiv:math/0611231
[20] M. Gerstenhaber, The cohomology structure of an associative ring. Ann. of Math. (2) 78 (1963), 267-288. · Zbl 0131.27302 · doi:10.2307/1970343
[21] M. Gerstenhaber, On the deformation of rings and algebras. Ann. of Math. (2) 79 (1964), 59-103. · Zbl 0123.03101 · doi:10.2307/1970484 · www.jstor.org
[22] M. Gerstenhaber and A. A. Voronov, Homotopy G-algebras and moduli space operad. Internat. Math. Res. Notices (1995), 141-153. · Zbl 0827.18004 · doi:10.1155/S1073792895000110 · arxiv:hep-th/9409063
[23] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159 (1994), 265-285. · Zbl 0807.17026 · doi:10.1007/BF02102639
[24] E. Getzler, Two-dimensional topological gravity and equivariant cohomology. Comm. Math. Phys. 163 (1994), 473-489. · Zbl 0806.53073 · doi:10.1007/BF02101459 · arxiv:hep-th/9305013
[25] Ezra Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint 1994. · arxiv.org
[26] J. Giansiracusa and P. Salvatore, Formality of the framed little 2-discs operad and semidi- rect products. In Homotopy theory of function spaces and related topics , Contemp. Math. 519, Amer. Math. Soc., Providence, RI, 2010, 115-121 · Zbl 1209.18008 · arxiv:0911.4428
[27] G. Ginot, Homologie et modèle minimal des algèbres de Gerstenhaber. Ann. Math. Blaise Pascal 11 (2004), 95-127. · Zbl 1139.16301 · doi:10.5802/ambp.187 · numdam:AMBP_2004__11_1_95_0 · eudml:10501
[28] V. Ginzburg, Calabi-Yau algebras. Preprint 2006. · arxiv.org
[29] V. Ginzburg and M. Kapranov, Koszul duality for operads. Duke Math. J. 76 (1994), 203-272. · Zbl 0855.18006 · doi:10.1215/S0012-7094-94-07608-4
[30] P. G. Goerss and M. J. Hopkins, André-Quillen (co)-homology for simplicial algebras over simplicial operads. In Une dégustation topologique: homotopy theory in the Swiss Alps (Arolla, 1999), Contemp. Math. 265, Amer. Math. Soc., Providence, RI, 2000, 41-85. · Zbl 0999.18009
[31] V. Gorbounov, F. Malikov, and V. Schechtman, Gerbes of chiral differential operators. III. In The orbit method in geometry and physics (Marseille, 2000), Progr. Math. 213, Birkhäuser, Boston 2003, 73-100. · Zbl 1106.17036 · arxiv:math/0005201
[32] V. Gorbounov, F. Malikov, and V. Schechtman, Gerbes of chiral differential operators. II. Vertex algebroids. Invent. Math. 155 (2004), 605-680. · Zbl 1056.17022 · doi:10.1007/s00222-003-0333-4
[33] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5-361. · Zbl 0153.22301 · numdam:PMIHES_1967__32__5_0 · eudml:103873
[34] V. Hinich, Homological algebra of homotopy algebras. Comm. Algebra 25 (1997), 3291-3323. · Zbl 0894.18008 · doi:10.1080/00927879708826055 · arxiv:q-alg/9702015
[35] J. Hirsh and J. Millès, Curved Koszul duality theory. Math. Ann. , to appear, · Zbl 1276.18009 · dx.doi.org · arxiv.org
[36] Y.-Z. Huang, Two-dimensional conformal geometry and vertex operator algebras . Progr. Math. 148, Birkhäuser, Boston 1997. · Zbl 0884.17021
[37] Y.-Z. Huang, A functional-analytic theory of vertex (operator) algebras. II. Comm. Math. Phys. 242 (2003), 425-444. · Zbl 1082.17013 · doi:10.1007/s00220-003-0949-7
[38] V. Kac, Vertex algebras for beginners . Univ. Lecture Ser. 10, 2nd ed., Amer. Math. Soc., Providence, RI, 1998. · Zbl 0924.17023
[39] M. Kapranov and Y. Manin, Modules and Morita theorem for operads. Amer. J. Math. 123 (2001), 811-838. · Zbl 1001.18004 · doi:10.1353/ajm.2001.0033 · muse.jhu.edu · arxiv:math/9906063
[40] R. M. Kaufmann, A proof of a cyclic version of Deligne’s conjecture via cacti. Math. Res. Lett. 15 (2008), 901-921. · Zbl 1161.55001 · doi:10.4310/MRL.2008.v15.n5.a7 · arxiv:math/0403340
[41] T. Kimura, A. A. Voronov, and G. J. Zuckerman, Homotopy Gerstenhaber algebras and topological field theory. In Operads: Proceedings of Renaissance Conferences (Hart- ford, CT/Luminy, 1995), Contemp. Math. 202, Amer. Math. Soc., Providence, RI, 1997, 305-333. · Zbl 0881.55016
[42] M. Kontsevich, Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66 (2003), 157-216. · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf · arxiv:q-alg/9709040
[43] M. Kontsevich and Y. Soibelman, Notes on A1-algebras, A1-categories and non- commutative geometry. In Homological mirror symmetry , Lecture Notes in Phys. 757, Springer, Berlin 2009, 153-219. · Zbl 1202.81120 · doi:10.1007/978-3-540-68030-7_6
[44] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie. Astérisque , Hors Sér. 1985, 257-271. · Zbl 0615.58029
[45] O. Kravchenko, Deformations of Batalin-Vilkovisky algebras. In Poisson geometry (Warsaw, 1998), Banach Center Publ. 51, Polish Acad. Sci., Warsaw 2000, 131-139. · Zbl 1015.17029 · journals.impan.gov.pl · eudml:209024
[46] B. H. Lian and G. J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory. Comm. Math. Phys. 154 (1993), 613-646. · Zbl 0780.17029 · doi:10.1007/BF02102111
[47] J.-L. Loday and B. Vallette, Algebraic operads . Grundlehren Math. Wiss. 346, Springer- Verlag, Berlin 2012. · Zbl 1260.18001 · doi:10.1007/978-3-642-30362-3
[48] S. MacLane, Categorical algebra. Bull. Amer. Math. Soc. 71 (1965), 40-106. · Zbl 0161.01601 · doi:10.1090/S0002-9904-1965-11234-4
[49] F. Malikov, V. Schechtman, and A. Vaintrob, Chiral de Rham complex. Comm. Math. Phys. 204 (1999), 439-473. · Zbl 0952.14013 · doi:10.1007/s002200050653 · arxiv:math/9803041
[50] Yu. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces . Amer. Math. Soc. Colloq. Publ. 47, Amer. Math. Soc., Providence, RI, 1999. · Zbl 0952.14032
[51] M. Markl, Distributive laws and Koszulness. Ann. Inst. Fourier ( Grenoble) 46 (1996), 307-323. · Zbl 0853.18005 · doi:10.5802/aif.1516 · numdam:AIF_1996__46_2_307_0 · eudml:75180
[52] M. Markl, S. Shnider, and J. Stasheff, Operads in algebra, topology and physics . Math. Surveys Monogr. 96, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1017.18001
[53] W. S. Massey, Some higher order cohomology operations. In Symposium internacional de topología algebraica International symposium on algebraic topology , Universidad Na- cional Autónoma de México and UNESCO, Mexico City 1958, 145-154. · Zbl 0123.16103
[54] W. S. Massey, Higher order linking numbers. In Conf. on algebraic topology (Univ. of Illinois at Chicago Circle, Chicago, Ill., 1968), Univ. of Illinois at Chicago Circle, Chicago 1969, 174-205. · Zbl 0212.55904
[55] J. P. May, The geometry of iterated loop spaces . Springer-Verlag, Berlin 1972. Lectures Notes in Math. 271. · Zbl 0244.55009
[56] L. Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology. Bull. Soc. Math. France 137 (2009), 277-295. · Zbl 1180.16007 · smf.emath.fr · arxiv:0711.1946
[57] S. A. Merkulov, Operads, deformation theory and F -manifolds. In Frobenius manifolds , Aspects Math. E36, Vieweg, Wiesbaden 2004, 213-251. · Zbl 1075.53094
[58] S. Merkulov and B. Vallette, Deformation theory of representations of prop(erad)s. I. J. Reine Angew. Math. 634 (2009), 51-106. · Zbl 1187.18006 · doi:10.1515/CRELLE.2009.069
[59] S. Merkulov and B. Vallette, Deformation theory of representations of prop(erad)s. II. J. Reine Angew. Math. 636 (2009), 123-174. · Zbl 1191.18003 · doi:10.1515/CRELLE.2009.084 · arxiv:0707.0889
[60] J. Milles, André-Quillen cohomology of algebras over an operad. Adv. Math. 226 (2011), 5120-5164. · Zbl 1218.18007 · doi:10.1016/j.aim.2011.01.002
[61] L. E. Positsel0skii, Nonhomogeneous quadratic duality and curvature. Funktsional. Anal. i Prilozhen. 27 (1993), no. 3, 57-66; English transl. Funct. Anal. Appl. 27 (1993), 197-204. · Zbl 0826.16041 · doi:10.1007/BF01768662
[62] A. Polishchuk and L. Positselski, Quadratic algebras . Univ. Lecture Ser. 37, Amer. Math. Soc., Providence, RI, 2005. · Zbl 1145.16009
[63] S. B. Priddy, Koszul resolutions. Trans. Amer. Math. Soc. 152 (1970), 39-60. · Zbl 0261.18016 · doi:10.2307/1995637
[64] R. Ree, Lie elements and an algebra associated with shuffles. Ann. of Math. (2) 68 (1958), 210-220. · Zbl 0083.25401 · doi:10.2307/1970243
[65] P. Salvatore and N. Wahl, Framed discs operads and Batalin-Vilkovisky algebras. Quart. J. Math. Oxford Ser. (2) 54 (2003), 213-231. · Zbl 1072.55006 · doi:10.1093/qmath/hag012 · arxiv:math/0106242
[66] G. Segal, The definition of conformal field theory. In Topology, geometry and quantum field theory , London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cam- bridge 2004, 421-577. · Zbl 1372.81138
[67] J.-P. Serre, Gèbres. Enseign. Math. (2) 39 (1993), 33-85. · Zbl 0810.16039
[68] P. Ševera, Formality of the chain operad of framed little disks. Lett. Math. Phys. 93 (2010), 29-35. · Zbl 1207.55008 · doi:10.1007/s11005-010-0399-z · arxiv:0902.3576
[69] J. D. Stasheff, Homotopy associativity of H -spaces. I, II. Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293-312. · Zbl 0114.39402 · doi:10.2307/1993608
[70] D. Tamarkin, Another proof of M. Kontsevich formality theorem for Rn. Doctoral thesis, Penn. State University, 1999. · arxiv.org
[71] D. Tamarkin, Action of the Grothendieck-Teichmüller group on the operad of Gersten- haber algebras. Preprint 2002. · arxiv.org
[72] D. Tamarkin, Quantization of Lie bialgebras via the formality of the operad of little disks. Geom. Funct. Anal. 17 (2007), 537-604. · Zbl 1163.17024 · doi:10.1007/s00039-007-0591-1
[73] D. Tamarkin and B. Tsygan, Noncommutative differential calculus, homotopy BV al- gebras and formality conjectures. Methods Funct. Anal. Topology 6 (2000), 85-100. · Zbl 0965.58010
[74] T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products. Ann. Inst. Fourier ( Grenoble) 58 (2008), 2351-2379. · Zbl 1218.16004 · doi:10.5802/aif.2417 · numdam:AIF_2008__58_7_2351_0 · eudml:10381 · arxiv:math/0210150
[75] T. Tradler and M. Zeinalian, On the cyclic Deligne conjecture. J. Pure Appl. Algebra 204 (2006), 280-299. · Zbl 1147.16012 · doi:10.1016/j.jpaa.2005.04.009 · arxiv:math/0404218
[76] B. Vallette, A Koszul duality for PROPs. Trans. Amer. Math. Soc. 359 (2007), 4865-4943. · Zbl 1140.18006 · doi:10.1090/S0002-9947-07-04182-7 · arxiv:math/0411542
[77] B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture. J. Reine Angew. Math. 620 (2008), 105-164. · Zbl 1159.18001 · doi:10.1515/CRELLE.2008.051 · arxiv:math/0609002
[78] P. Van der Laan, Operads up to homotopy and deformations of operad maps. Preprint 2002. · arxiv.org
[79] P. Van der Laan, Coloured Koszul duality and strongly homotopy operads. Preprint 2003. · arxiv.org
[80] T. Voronov, Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202 (2005), 133-153. · Zbl 1086.17012 · doi:10.1016/j.jpaa.2005.01.010 · arxiv:math/0304038
[81] C. A. Weibel, An introduction to homological algebra . Cambridge Stud. Adv. Math. 38, Cambridge University Press, Cambridge 1994. · Zbl 0797.18001
[82] S. O. Wilson, Free Frobenius algebra on the differential forms of a manifold. Preprint 2007. · arxiv.org
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.