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[CM ICTP - Trieste]

JOINT ICTP/SISSA STATISTICAL PHYSICS SEMINAR



Tuesday, 13 January    -    12:30 hrs.





Seminar Room   - ICTP Leonardo Building - 1st floor





S. CARACCIOLO  ( Università degli Studi di Milano )



"Algebraic properties of Grassmann and Clifford subalgebras related  
to spanning hyperforests"





Abstract

A supersymmetric theory with osp(1|2)-invariance can be used to  
describe spanning (hyper-)forests on a given graph. In a sigma-model,  
the bosonic part of the superfield can be integrated out and the  
symmetry is realized

non-linearly in the remaining Grassmann degrees of freedom.

We investigate the invariant osp(1|2) Grassmann and Clifford  
subalgebras. We discover that, given an arbitrary vertex ordering, a  
basis of independent elements is given by terms labeled by non- 
crossing partitions, respectively of the vertices of the graph, and  
of a doubling of this set.

The invariant Clifford subalgebra is isomorphic to a Temperley--Lieb  
Algebra, in a peculiar even/odd generalization.

A relation involving four complex fermions plays a central role in  
the construction, and disentangles the only 4-set crossing partition  
as a linear combination of the other partitions with up to two  
blocks. This relation appears in different flavours, and equivalence  
of the formulations is shown.