Frob enius Algebra Structures in Top ological Quantum Field Theory and Quantum Cohomology by Lowell Abrams A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Do ctor of Philosophy Baltimore Maryland Abstract We prove that a commutative nitedimensional algebra A is a Frob enius algebra if and only if it has a co commutative comultiplication with counit Based on this we prove the onetoone corresp ondence between top ological quantum eld theories and Frob enius algebras formulated as an equivalence of monoidal categories For each Frob enius algebra A we dene a canonical characteristic class and show that this characteristic class is a unit if and only if A is semisimple tum cohomology the Frob enius algebra characteristic class the In quan quantum Euler class is a deformation of the classical Euler class whichisthe Frob enius algebra class of classical cohomology Weshow that in the case of the classical and quantum cohomology rings of the nite Grassmannian manifolds the quantum Euler class can be viewed as the determinant of a Hessian giving it a geometric interpretation We also discuss a particular p olynomial P which provides a lifting of b oth the classical and quantum Euler classes and conjecture the validity of a sp ecic expression for P which strengthens its connection to Frob enius algebra structure ii How great are Your works God Your thoughts are exceedingly deep Psalms iii Acknowledgments Great thanks are due to my advisor Jack Morava for his guidance and supp ort throughout my development as a research mathematician and par ticularly the pro cess of pro ducing this thesis Steven Zucker has also given generously of his time over the past six years and I am grateful for his help I have also b enetted from discussions and corresp ondence with Aaron Bertram Michael Boardman and Steve Sawin as well as others The camaraderie of Mohammad Ghomi Zoran Petrovic and many of my other p eers in the department of mathematics at Johns Hopkins provided me vironment conducive to research without which I would certainly with an en have made less progress Mohammad Ghomi was also kind enough to allow me at critical points of physical and mental exhaustion to nap on the futon in his oce My deep est debt is no doubt to my family My parents made myadvance ment in mathematics p ossible and have always supp orted all my decisions Without the understanding and constant supp ort of my wife Annick this thesis and the work it represents would never hav e come to fruition I will be forever grateful iv Contents Intro duction Frob enius Algebras Characterizations of Frob enius Algebras Decomp osition of Frob enius Algebras The Distinguished Element Examples Frob enius Ring Extensions Frob enius Extensions and Number Theory Top ological Quantum Field Theories TwoDimensional Cob ordisms Top ological Quantum Field Theories The Corresp ondence Between Two Dimensional TQFTs and Frob enius Algebras Examples of TQFTs and Invariants of Surfaces Quantum Cohomology The Frob enius Algebra Structure of Classical and Quantum Co homology Quantum Cohomology of Grassmannian Manifolds Quantum Cohomology of Hyp erplanes v Intro duction Frob enius algebras have b een ob jects of study since the earlier half of this century but they are nowenjoying renewed attention b ecause of their connec tions with currentinterests in mathematical physics This thesis deals with the structure of Frob enius algebras motivated by applications to twodimensional top ological quantum eld theory TQFT and quantum cohomology Top ological Quantum Field Theory Top ological quantum eld theories were rst describ ed axiomatically by Atiyah in Lo osely sp eaking a ddimensional TQFT is a representation in a sense analogous to representation of groups of the category of ddimensional cob ordisms of d dimensional compact manifolds In such a categorythe d dimensional compact manifolds are the ob jects and the cob ordisms themselves are the morphisms Comp osition of morphisms o ccurs by gluing cob ordisms along homeomorphic b oundaries Details of these denitions are discussed in sections and Interest sp ecically in twodimensional TQFTs go es back to such sources as Segals presentation in of twodimensional conformal eld theories and Wittens work in relating the same to results in higher dimensions In Voronov presents a folk theorem asserting that a twodimensional TQFT is hes a pro of Similar material also equivalenttoaFrob enius algebra and sketc app ears earlier in Intuitively the folk theorem states that the category of twocob ordisms is the universal Frob enius algebra ie that there is a onetoone corresp ondence between Frob enius algebras and representations of the category of twocob ordisms Chapter centers around a rigorous pro of of this theorem which we have strengthened to a statement ab out categories Theorem There is an equivalence of categories between the category of Frobenius algebras and the category of TQFTs which respects their monoidal structures The pro of of this theorem rests heavily on a new characterization of Frob enius algebras Theorem An algebra A is a Frobenius algebra if and only if it has a comultiplication which is a map of Amodules Section shows that FA structures behave well with resp ect to direct sum and tensor pro duct and this provides a means for dening direct sums and tensor pro ducts of TQFTs In light of the following decomp osition theorem for Frob enius algebras provides a decomp osition theorem for direct sums of TQFTs as well Theorem Every FA A decomposes into a direct sum of elds and inde composable annihilator algebras and the FA structure of A is more or less determined by the FA structures of its indecomposable constituents of the imp ortant morphisms in the category of twocob ordisms is the One torus with two punctures Gluing to one boundary of this morphism H increases genus by one without changing connectivity and without increasing the number of b oundaries Any TQFT Z sends to an op erator A A H where A is the Frob enius algebra corresp onding to Z This op erator is called the handle op erator and it acts simply bymultiplying by a certain distin guished element of A see section for details This canonically dened element is a kind of characteristic class for Frob enius algebras section explains its close relationship to the Euler class in the cohomology ring of an oriented manifold The distinguished element satises a notion of naturality and has a particularly nice algebraic prop erty Theorem A Frobenius algebra is semisimple if and only if its distin guished element is a unit The drive to nd algebraic invariants for top ological ob jects has a long history ie the study of algebraic top ology In recent times knots have b een ob jects of intense study in this regard and new top ological invariants have been found in terms of various algebraic structures such as von Neumann algebras and Hopf algebras In a similar vein TQFTs provide a new source of invariants for higher dimensional top ological ob jects Chapter closes with an example of several twodimensional TQFTs based on a single FA which collectively provide a source of invariants p erhaps rich enough to distinguish all twodimensional cob ordisms This requires further investiga tion Quantum Cohomology The quantum cohomology ring of a manifold M is additively the same as the classical cohomology ring of M but p ossesses a multiplication which is a deformation of the classical cup pro duct Section sketches the basic notions and denitions of quantum cohomology rings and how they generalize classical cohomology and explains how b oth these rings are FAs Although quantum cohomology is a ring extension and not an algebra section provides a means to deal with this discrep ency The manner in which quantum cohomology grows out of physicsoriented considerations is discussed in and In the latter source Dubrovin denes a Frob enius manifold M the tangent bundle TM has a FA to be a manifold such that each b er of structure which varies nicely from b er to b er This context allows for a close investigation of the nature of the quantum deformations of classical co homologywhich is generally realized as T M Moreover the fact that M is a Frob enius manifold is equivalent to the existence of a GromovWitten p oten tial on M satisfying various dierential equations including the WDVV equations ibid p Sp ecial manifolds which Dubrovin calls massive Frob enius manifoldshave the additional prop erty that for a generic p oint t M the FA T M is semisimple In this case a variety of additional results t relating to the classication of Frob enius manifolds hold ibid Lecture Kontsevich and Manin discuss asp ects of Frob enius manifolds in but deal with a dierent notion of semisimplicity Working with a manifold M which is essentially the cohomology ring of some space they dene a partic ular section K M TM and at each point M the linear op erator B T M T M which is multiplication by K They also dene a particular extension TM of TM and show that if over a sub domain of M the op erator B is semisimple ie has distinct eigenvalues then TM exhibits some sp ecial prop erties The notion of semisimplicity of B is referred to as semisimplicity in the sense of Dubrovin