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Abstract:
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n this dissertation the connection between Frobenius algebras and topological
quantum field theories (TQFTs) is investigated. It is well-known that each 2-dimensional
TQFT (2-TQFT) corresponds to a commutative Frobenius algebra and conversely, i.e., that
the category whose objects are 2-TQFTs is equivalent to the category of commutative Frobe-
nius algebras. Every 2-TQFT is completely determined by the image of 1-dimensional sphere
S1 and by its values on the generators of the category of 2-dimensional 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 d-dimensional 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 1-TQFT functor. We obtain the faithfulness result for all
1-TQFTs, mapping the 0-dimensional 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 2-TQFT. It means that 2-dimensional cobordisms are equivalent if and only if the
corresponding linear maps are equal. |