About the notation OSP(8|4) etc.

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Discussion Overview

The discussion centers around the notation used in superalgebras, specifically OSP(N|4) and sl(m|n). Participants explore definitions, properties, and references related to these notations within the context of superconformal field theories and superspaces.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant inquires about the general definition of notations like OSP(N|4) and sl(m|n), noting their association with superalgebras.
  • Another participant explains that m|n indicates the number of bosonic and fermionic dimensions, with sl(m|n) representing the special linear algebra acting on a superspace with those dimensions.
  • A further contribution clarifies that OSP refers to ortho-symplectic, highlighting its relationship with orthogonal and symplectic groups and their respective metrics.
  • There is a request for references or books that provide more information on these topics.
  • A suggestion is made to refer to the book "Supermanifolds" by DeWitt as a potential resource.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and properties of the notations discussed, but there is no consensus on specific references or resources beyond the suggestion made.

Contextual Notes

The discussion does not resolve the broader implications or applications of these notations in various contexts, nor does it address potential limitations in understanding or definitions.

Who May Find This Useful

Readers interested in theoretical physics, particularly in the areas of superalgebras, superconformal field theories, and related mathematical frameworks.

isospin
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Are there any books or papers which explain these notations like OSP(N|4), sl(m|n)? It seems they are all considered as superalgebras, but how is this kind notation generally defined?
By the way, I know for a 3-d superconformal field theory, OSP(8|4) means a SO(8) R-symmetry and a Sp(4) conformal symmetry, which is isomorphic to SO(2, 3).
Thanks.
 
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m|n denotes the number of bosonic (ordinary, even) dimensions and the number of fermionic (odd) dimensions. Thus, sl(m|n) is the special linear algebra acting on a superspace with m even and n odd dimensions. OSP stands for ortho-symplectic, because it mixes aspects of the orthogonal and symplectic groups. Recall that the orthogoal group preserves the even metric, and the symplectic group preserves the odd symplectic metric; the OSP group preserves a metric which symmetry depends on Grassmann parity.
 
By any chance, do you know a good book or reference paper on these matters?
 
Try the book Supermanifolds by DeWitt
 

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