Codes, vertex operators and topological modular forms

We describe a new link between the theory of topological modular forms and representations of vertex operator algebras obtained by certain lattices. The construction is motivated by the arithmetic Whitehead tower of the orthogonal groups. The tower discloses the role of codes in representation theory.

[1]  R. Borcherds.  Vertex algebras, Kac-Moody algebras, and the Monster. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Christopher L. Douglas,et al.  Topological Modular Forms , 2014 .

[3]  Christopher J. Schommer-Pries.  Central extensions of smooth 2–groups and a finite-dimensional string 2–group , 2009, 0911.2483.

[4]  M. Hopkins.  Algebraic topology and modular forms , 2002, math/0212397.

[5]  C. Rezk,et al.  Topological Modular Forms of Level 3 , 2008, 0812.2009.

[6]  J. Rosenberg,et al.  Algebraic K-Theory and Its Applications , 1995 .

[7]  J. Franke.  On the Construction of Elliptic Cohomology , 2006 .

[8]  T. Lawson,et al.  Topological modular forms with level structure , 2013, 1312.7394.

[9]  G. Laures.  THE TOPOLOGICAL q-EXPANSION PRINCIPLE☆ , 1999 .

[10]  V. Kac.  Vertex algebras for beginners , 1997 .

[11]  J. Lepowsky,et al.  Vertex Operator Algebras and the Monster , 2011 .

[12]  J. Stembridge.  The projective representations of the hyperoctahedral group , 1992 .

[13]  G. Laures.  K(1)-local topological modular forms , 2004 .

[14]  Chongying Dong,et al.  Vertex Algebras Associated with Even Lattices , 1993 .

[15]  K. Ormsby,et al.  On the homotopy of Q(3) and Q(5) at the prime 2 , 2012, 1211.0076.

[16]  P. Landweber,et al.  Periodic Cohomology Theories Defined by Elliptic Curves , 2004 .

[17]  F. Hirzebruch.  Elliptic Genera of Level N for Complex Manifolds , 1988 .

[18]  Elliptic genera of toric varieties and applications to mirror symmetry , 1999, math/9904126.

[19]  W. Stothers,et al.  On subgroups ofGL(n, A) which are generated by commutators , 1974 .

[20]  G. Segal.  The Definition of Conformal Field Theory , 1988 .

[21]  G. Laures.  ON COBORDISM OF MANIFOLDS WITH CORNERS , 2000 .

[22]  F. Williams,et al.  A Window into Zeta and Modular Physics , 2014 .

[23]  T. Johnson-Freyd,et al.  H4(Co0; Z) = Z/24 , 2018, International Mathematics Research Notices.

[24]  H. Samelson,et al.  Topology of Lie groups , 1952 .

[25]  T. Berger,et al.  Manifolds and modular forms , 1992 .

[26]  J. Brylinski.  Representations of loop groups, Dirac operators on loop space, and modular forms , 1990 .

[27]  N. J. A. Sloane,et al.  Generalizations of Gleason's theorem on weight enumerators of self-dual codes , 1972, IEEE Trans. Inf. Theory.

[28]  Andrew H. Baker.  Elliptic Genera of Level N and Elliptic Cohomology , 1994 .

[29]  Friedrich Hirzebruch,et al.  The ring of Hilbert modular forms for real quadratic fields in small discriminant , 1977 .

[30]  J. Lurie.  A Survey of Elliptic Cohomology , 2009 .

[31]  Narthana S. Epa,et al.  Platonic and alternating 2-groups , 2017, Higher Structures.