scholarly journals Twin TQFTs and Frobenius Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Carmen Caprau
Keyword(s):  
Monoidal Category ◽  
Topological Quantum Field Theory ◽  
Algebra Homomorphism ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Generators And Relations ◽  
Symmetric Monoidal Category ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Algebraic Description

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

ALLOWABLE CLOSED SURFACE EVALUATIONS FROM TOPOLOGICAL QUANTUM FIELD THEORY

2014 ◽  
Vol 23 (01) ◽  
pp. 1450002
Author(s):  
PAUL DRUBE
Keyword(s):  
Field Theory ◽  
Closed Surface ◽  
Topological Quantum Field Theory ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Symmetric Polynomials ◽  
Base Field ◽  
Fundamental Part ◽  
Topological Quantum ◽  
Deep Relationship

Every 2D TQFT evaluates closed 2D surfaces to an element of the base field. These evaluations appear as a fundamental part of the skein relations induced by the TQFT, dating back to Bar-Natan. We definitively classify what sets of closed surface evaluations may arise from a 2D TQFT. Our answer reveals a deep relationship between achievable evaluations and the theory of symmetric polynomials. We then apply our results to precisely determine which 2D TQFTs have an associated Frobenius algebra that may be realized as the algebra of all surfaces with boundary S1, modulo surfaces that are "evaluated similarly" by the TQFT.

A NOTE ON DUALITY, FROBENIUS ALGEBRAS AND TQFTS

2004 ◽  
Vol 13 (08) ◽  
pp. 999-1006
Author(s):  
LUCIAN M. IONESCU
Keyword(s):  
Monoidal Category ◽  
The Self ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Self Duality ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Definition Of ◽  
Hermitian Structures

Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle). A self-dual definition of a Frobenius object without the use of a prefered dual is considered. The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors. Hermitian structures are modeled as a conjugation compatible with duality. It is the structure cobordism categories posses. A definition of generalized cobordism categories is proposed.

QFT and unification of knot theories

2014 ◽  
Vol 29 (24) ◽  
pp. 1430025
Author(s):  
Alexey Sleptsov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Knot Invariants ◽  
Topological Quantum

We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

TOWARDS A QUANTUM ALGORITHM FOR THE PERMANENT

2007 ◽  
Vol 05 (01n02) ◽  
pp. 223-228 ◽  
Author(s):  
ANNALISA MARZUOLI ◽  
MARIO RASETTI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Topological Quantum

We resort to considerations based on topological quantum field theory to outline the development of a possible quantum algorithm for the evaluation of the permanent of a 0 - 1 matrix. Such an algorithm might represent a breakthrough for quantum computation, since computing the permanent is considered a "universal problem", namely, one among the hardest problems that a quantum computer can efficiently handle.

Frobenius algebra with relaxed associativity constraint

2018 ◽  
Vol 27 (07) ◽  
pp. 1841008
Author(s):  
Zbigniew Oziewicz ◽  
William Stewart Page
Keyword(s):  
Associative Algebra ◽  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Frobenius Algebra ◽  
Frobenius Algebras ◽  
Frobenius Structures ◽  
Necessary And Sufficient

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra [Formula: see text] is said to be a Frobenius algebra if there exists a [Formula: see text]-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra [Formula: see text] is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided [Formula: see text]-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows [Formula: see text]-permuted opposite algebra to be extended to [Formula: see text]-permuted algebras.

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

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