20,639 research outputs found
Homotopy theory of diagrams
In this paper we develop homotopy theoretical methods for studying diagrams.
In particular we explain how to construct homotopy colimits and limits in an
arbitrary model category. The key concept we introduce is that of a model
approximation. Our key result says that if a category admits a model
approximation then so does any diagram category with values in this category.
From the homotopy theoretical point of view categories with model
approximations have similar properties to those of model categories. They admit
homotopy categories (localizations with respect to weak equivalences). They
also can be used to construct derived functors by taking the analogs of fibrant
and cofibrant replacements.
A category with weak equivalences can have several useful model
approximations. We take advantage of this possibility and in each situation
choose one that suits our needs. In this way we prove all the fundamental
properties of the homotopy colimit and limit: Fubini Theorem (the homotopy
colimit -respectively limit- commutes with itself), Thomason's theorem about
diagrams indexed by Grothendieck constructions, and cofinality statements.
Since the model approximations we present here consist of certain functors
"indexed by spaces", the key role in all our arguments is played by the
geometric nature of the indexing categories.Comment: 95 pages with inde
Wrapping interactions and the genus expansion of the 2-point function of composite operators
We perform a systematic analysis of wrapping interactions for a general class
of theories with color degrees of freedom, including N=4 SYM. Wrapping
interactions arise in the genus expansion of the 2-point function of composite
operators as finite size effects that start to appear at a certain order in the
coupling constant at which the range of the interaction is equal to the length
of the operators. We analyze in detail the relevant genus expansions, and
introduce a strategy to single out the wrapping contributions, based on adding
spectator fields. We use a toy model to demonstrate our procedure, performing
all computations explicitly. Although completely general, our treatment should
be particularly useful for applications to the recent problem of wrapping
contributions in some checks of the AdS/CFT correspondence.Comment: 41 pages, LaTeX, 12 figures, minor changes, final version in NP
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