N =1 Super Virasoro Tensor Categories.

Area of Science:

• Representation Theory
• Algebraic Quantum Field Theory
• Mathematical Physics

Background:

• The study of vertex operator superalgebras (VOSAs) is crucial in understanding conformal field theories and string theory.
• The N=1 super Virasoro VOSA, denoted S(c,0), is a fundamental object with applications in diverse areas of theoretical physics.
• Understanding the module categories of VOSAs, particularly their tensor category structures, is essential for classifying representations and their properties.

Purpose of the Study:

• To investigate the category of C1-cofinite modules for the universal N=1 super Virasoro VOSA S(c,0) at arbitrary central charge c.
• To establish the vertex algebraic braided tensor category structure for this module category.
• To analyze the properties (semisimplicity, rigidity, fusion rules) of this category for specific central charges.

Main Methods:

• Analysis of the category of C1-cofinite modules for the N=1 super Virasoro VOSA S(c,0).
• Application of the Huang-Lepowsky-Zhang framework for vertex algebraic braided tensor categories.
• Determination of fusion rules and investigation of properties like semisimplicity and rigidity for specific parameter values.

Main Results:

• The category of C1-cofinite modules for S(c,0) is shown to be locally finite and possess the Huang-Lepowsky-Zhang braided tensor category structure.
• For non-rational central charges c^ns(t) = 15/2 - 3(t + 1/t) with t not rational, the category is semisimple, rigid, and slightly degenerate, with determined fusion rules.
• For c^ns(1) = 3/2, the category is rigid and its simple modules share fusion rules with Rep(osp(1|2)). For other rational c^ns(t), the module S_{2,2} is rigid and self-dual, a key step towards proving category rigidity.

Conclusions:

• The C1-cofinite module category of the N=1 super Virasoro VOSA exhibits rich tensor category structures.
• Specific central charges lead to semisimple and rigid categories, with fusion rules determined.
• The rigidity of the module S_{2,2} is a crucial step in establishing the rigidity of the entire category for general rational central charges.