Batalin-Vilkovisky Algebraと関連した概念

Batalin-Vilkovisky algebra は, 元々 Batalin と Vilkovisky の [BV77] で考えられた構造が元になっているようである。

代数的トポロジーとの関係では, まずは2重ループ空間のホモロジーがある。 Gerstenhaber は, [Ger63] で Hochschild cohomology が, ある代数的構造 (Gerstenhaber algebra) を持つことを示したが, 後にその代数的構造は, \(S^1\) の作用を持つ \(2\)重ループ空間のホモロジーが持つ代数的構造, つまり Batalin-Vikovisky algebra から来ていることがわかり, また Batalin-Vilkovisky algebra は framed disk operad のホモロジー上の algebra と同等であることが分った。

他にも様々なところに現われる。Drummond-Cole と Vallette の [DV13] や Ward の [War] の Introduction にある list を見るとよい。

• vertex algebra と vertex operator algebra [Bor86; LZ93]
• Lie algebra の Chevalley-Eilenberg cohomology [Kos85]
• \(A_{\infty }\)-algebra の bar construction [TTW]
• Gromov-Witten invariant [BK98; Man99; LS]
• chiral algebra [BD04; FB04]
• symmetric algebra の Hochschild cohomology [TT00; Tra08; Gin; Men09]
• cyclic Deligne conjecture [Kau08; TZ06; Cos07; BB09]
• noncommutative differential operator [GS10]
• \(S^1\) の作用を持つ空間の2重ループ空間のホモロジーや topological conformal field theory [Get94]
• string topology [CS]
• Batalin と Vilkovisky による quantization [BV77; BV81; Wit90; Sch93; Rog09]
• BRST cohomology [LZ93; Sta98]
• string theory [Wit92; WZ92; Zwi93; PS94]

Drummond-Cole と Vallette の [DV13] にも書かれているように, これらの Batalin-Vilkovisky algebra は, 何かの homology になっていることが多い。 そこで, homology を取る前の構造として, homotopy Batalin-Vilkovisky algebra という構造が考えられている。

• homotopy Batalin-Vilkovisky algebra

Ward は [War] で (homotopy) Batalin-Vilkovisky algebra の構造の元になっている構造が, 多くの場合 cyclic operad を用いて表せることを指摘している。

References

[BB09]

M. A. Batanin and C. Berger. “The lattice path operad and Hochschild cochains”. In: Alpine perspectives on algebraic topology. Vol. 504. Contemp. Math. Providence, RI: Amer. Math. Soc., 2009, pp. 23–52. arXiv: 0902. 0556. url: http://dx.doi.org/10.1090/conm/504/09874.

[BD04]

Alexander Beilinson and Vladimir Drinfeld. Chiral algebras. Vol. 51. American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society, 2004, p. vi 375. isbn: 0-8218-3528-9.

[BK98]

Sergey Barannikov and Maxim Kontsevich. “Frobenius manifolds and formality of Lie algebras of polyvector fields”. In: Internat. Math. Res. Notices 4 (1998), pp. 201–215. arXiv: alg-geom/9710032. url: http://dx.doi.org/10.1155/S1073792898000166.

[Bor86]

Richard E. Borcherds. “Vertex algebras, Kac-Moody algebras, and the Monster”. In: Proc. Nat. Acad. Sci. U.S.A. 83.10 (1986), pp. 3068–3071. url: http://dx.doi.org/10.1073/pnas.83.10.3068.

[BV77]

I.A. Batalin and G.S. Vilkovisky. “Relativistic \(S\)-matrix of dynamical systems with bosons and fermion constraints”. In: Phys. Lett. 69B (1977), pp. 309–312.

[BV81]

I. A. Batalin and G. A. Vilkovisky. “Gauge algebra and quantization”. In: Phys. Lett. B 102.1 (1981), pp. 27–31. url: https://doi.org/10.1016/0370-2693(81)90205-7.

[Cos07]

Kevin Costello. “Topological conformal field theories and Calabi-Yau categories”. In: Adv. Math. 210.1 (2007), pp. 165–214. arXiv: math/ 0412149. url: http://dx.doi.org/10.1016/j.aim.2006.06.004.

[CS]

Moira Chas and Dennis Sullivan. String Topology. arXiv: math/ 9911159.

[DV13]

Gabriel C. Drummond-Cole and Bruno Vallette. “The minimal model for the Batalin-Vilkovisky operad”. In: Selecta Math. (N.S.) 19.1 (2013), pp. 1–47. arXiv: 1105.2008. url: http://dx.doi.org/10.1007/s00029-012-0098-y.

[FB04]

Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves. Second. Vol. 88. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 2004, pp. xiv+400. isbn: 0-8218-3674-9.

[Ger63]

Murray Gerstenhaber. “The cohomology structure of an associative ring”. In: Ann. of Math. (2) 78 (1963), pp. 267–288.

[Get94]

E. Getzler. “Batalin-Vilkovisky algebras and two-dimensional topological field theories”. In: Comm. Math. Phys. 159.2 (1994), pp. 265–285. arXiv: hep-th/9212043. url: http://projecteuclid.org/euclid.cmp/1104254599.

[Gin]

Victor Ginzburg. Calabi-Yau algebras. arXiv: math/0612139.

[GS10]

Victor Ginzburg and Travis Schedler. “Differential operators and BV structures in noncommutative geometry”. In: Selecta Math. (N.S.) 16.4 (2010), pp. 673–730. arXiv: 0710.3392. url: http://dx.doi.org/10.1007/s00029-010-0029-8.

[Kau08]

Ralph M. Kaufmann. “A proof of a cyclic version of Deligne’s conjecture via cacti”. In: Math. Res. Lett. 15.5 (2008), pp. 901–921. arXiv: math/0403340.

[Kos85]

Jean-Louis Koszul. “Crochet de Schouten-Nijenhuis et cohomologie”. In: Astérisque Numero Hors Serie (1985). The mathematical heritage of Élie Cartan (Lyon, 1984), pp. 257–271.

[LS]

A. Losev and S. Shadrin. From Zwiebach invariants to Getzler relation. arXiv: math/0506039.

[LZ93]

Bong H. Lian and Gregg J. Zuckerman. “New perspectives on the BRST-algebraic structure of string theory”. In: Comm. Math. Phys. 154.3 (1993), pp. 613–646. arXiv: hep-th/9211072. url: http://projecteuclid.org/euclid.cmp/1104253081.

[Man99]

Yuri I. Manin. Frobenius manifolds, quantum cohomology, and moduli spaces. Vol. 47. American Mathematical Society Colloquium Publications. Providence, RI: American Mathematical Society, 1999, pp. xiv+303. isbn: 0-8218-1917-8.

[Men09]

Luc Menichi. “Batalin-Vilkovisky algebra structures on Hochschild cohomology”. In: Bull. Soc. Math. France 137.2 (2009), pp. 277–295. arXiv: 0711.1946.

[PS94]

Michael Penkava and Albert Schwarz. “On some algebraic structures arising in string theory”. In: Perspectives in mathematical physics. Conf. Proc. Lecture Notes Math. Phys., III. Int. Press, Cambridge, MA, 1994, pp. 219–227.

[Rog09]

Claude Roger. “Gerstenhaber and Batalin-Vilkovisky algebras; algebraic, geometric, and physical aspects”. In: Arch. Math. (Brno) 45.4 (2009), pp. 301–324.

[Sch93]

Albert Schwarz. “Geometry of Batalin-Vilkovisky quantization”. In: Comm. Math. Phys. 155.2 (1993), pp. 249–260. url: http://projecteuclid.org/euclid.cmp/1104253279.

[Sta98]

Jim Stasheff. “The (secret?) homological algebra of the Batalin-Vilkovisky approach”. In: Secondary calculus and cohomological physics (Moscow, 1997). Vol. 219. Contemp. Math. Amer. Math. Soc., Providence, RI, 1998, pp. 195–210. url: https://doi.org/10.1090/conm/219/03076.

[Tra08]

Thomas Tradler. “The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products”. In: Ann. Inst. Fourier (Grenoble) 58.7 (2008), pp. 2351–2379. arXiv: math/0210150. url: http://aif.cedram.org/item?id=AIF_2008__58_7_2351_0.

[TT00]

D. Tamarkin and B. Tsygan. “Noncommutative differential calculus, homotopy BV algebras and formality conjectures”. In: Methods Funct. Anal. Topology 6.2 (2000), pp. 85–100. arXiv: math/0002116.

[TTW]

John Terilla, Thomas Tradler, and Scott O. Wilson. Homotopy DG algebras induce homotopy BV algebras. arXiv: 1106.1856.

[TZ06]

Thomas Tradler and Mahmoud Zeinalian. “On the cyclic Deligne conjecture”. In: J. Pure Appl. Algebra 204.2 (2006), pp. 280–299. arXiv: math/0404218. url: http://dx.doi.org/10.1016/j.jpaa.2005.04.009.

[War]

Benjamin C. Ward. Maurer-Cartan Elements and Cyclic Operads. arXiv: 1409.5709.

[Wit90]

Edward Witten. “A note on the antibracket formalism”. In: Modern Phys. Lett. A 5.7 (1990), pp. 487–494. url: https://doi.org/10.1142/S0217732390000561.

[Wit92]

Edward Witten. “The \(N\) matrix model and gauged WZW models”. In: Nuclear Phys. B 371.1-2 (1992), pp. 191–245. url: https://doi.org/10.1016/0550-3213(92)90235-4.

[WZ92]

Edward Witten and Barton Zwiebach. “Algebraic structures and differential geometry in two-dimensional string theory”. In: Nuclear Phys. B 377.1-2 (1992), pp. 55–112. url: http://dx.doi.org/10.1016/0550-3213(92)90018-7.

[Zwi93]

Barton Zwiebach. “Closed string field theory: quantum action and the Batalin-Vilkovisky master equation”. In: Nuclear Phys. B 390.1 (1993), pp. 33–152. url: http://dx.doi.org/10.1016/0550-3213(93)90388-6.