Abstract:

n this dissertation the connection between Frobenius algebras and topological
quantum field theories (TQFTs) is investigated. It is wellknown that each 2dimensional
TQFT (2TQFT) corresponds to a commutative Frobenius algebra and conversely, i.e., that
the category whose objects are 2TQFTs is equivalent to the category of commutative Frobe
nius algebras. Every 2TQFT is completely determined by the image of 1dimensional sphere
S1 and by its values on the generators of the category of 2dimensional oriented cobordisms.
Relations that hold for these cobordisms correspond precisely to the axioms of a commutative
Frobenius algebra.
Following the pattern of the Frobenius structure assigned to the sphere S1 in this way, we
examine the Frobenius structure of spheres in all other dimensions. For every d ≥ 2, the
sphere Sd−1 is a commutative Frobenius object in the category of ddimensional cobordisms.
We prove that there is no distinction between spheres Sd−1, for d ≥ 2, because they are all free
of additional equations formulated in the language of multiplication, unit, comultiplication
and counit. The only exception is the sphere S0 which is a symmetric Frobenius object but
not commutative.
The sphere S0 is mapped to a matrix Frobenius algebra by the Brauerian representation,
which is an example of a faithful 1TQFT functor. We obtain the faithfulness result for all
1TQFTs, mapping the 0dimensional manifold, consisting of one point to a vector space of
dimension at least 2.
Finally, we show that the commutative Frobenius algebra QZ5 ⊗ Z(QS3), defined as the ten
sor product of the group algebra and the centre of the group algebra, corresponds to the
faithful 2TQFT. It means that 2dimensional cobordisms are equivalent if and only if the
corresponding linear maps are equal. 