# 2-Divisible groups over Z by Abrashkin V.A. PDF

By Abrashkin V.A.

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In any case, deformation quantization gets around all of these questions in a rather peculiar way: the series expansions in h one obtains are usually formal in the sense that they do not converge, not just when applied to badly behaved functions but for any sufficiently large class of functions. , for generic values of h) do not define algebras of bounded operators, and do not even admit representations on Hilbert space of the sort which explicitly or implicitly lay behind many of our considerations (Fedosov, 1996; Rieffel, 1998).

This appears to be connected to the fact that 1 (T 2 ) ϭZ2 0, and a nice string theory explanation of how this changes the problem can be found in Seiberg and Witten (1999); because a string can wind about 1 , the phase space of an open string on T 2 is in fact noncompact. In any case, deformation quantization gets around all of these questions in a rather peculiar way: the series expansions in h one obtains are usually formal in the sense that they do not converge, not just when applied to badly behaved functions but for any sufficiently large class of functions.

It will also lead to new theories: noncommutative string theories, and even more exotic theories such as open membrane (OM) theory. After reviewing a range of arguments which lead to noncommutative gauge theory, we focus on its origins from the string world-sheet, following Seiberg and Witten (1999), who were the first to precisely state the limits involved. We also describe related arguments in topological string theory, originating in work of Kontsevich (1997). We then give the string theory pictures for the solitonic physics of Sec.

### 2-Divisible groups over Z by Abrashkin V.A.

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