Frobenius Algebra Structures in Topological Quantum Field Theory

Home , Cohomology ring, Frobenius algebra, Quantum cohomology

Frob enius Algebra Structures

in Top ological Quantum Field Theory

and Quantum Cohomology

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

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

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

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 in and other lo cations

The distinguished element discussed in this work provides a dierent sec

tion of the tangent bundle of a Frob enius manifold Theorem shows that

this section also has a great deal to do with semisimplicity although the exact

connection with semisimplicity in the sense of Dubro vin is not yet clear

Another strong motivation for studying the distinguished element of the

FA structure of quantum cohomology is provided by the following

Prop ositions In classical cohomology the distinguished element

of the FA structure is the classical Euler class In quantum cohomology the

distinguished element the quantum Euler class is a deformation of the

classical Euler class

The bulk of chapter consists of an examination of the role of the distin

guished element in the classical and quantum cohomology rings of the nite

Grassmannian manifolds G These rings may be describ ed in a variety of

ways and each such approach sheds new light on the quantum Euler class

In particular viewing the rings as Jacobian algebras yields mo dulo technical

ities

Prop osition The quantum cohomology of G is semisimple

Another approach highlighting the tensor pro duct formula for Chern classes

leads to

Corollary There is a specic polynomial in the Chern classes of G

which is a lifting of both the classical and quantum Euler classes

The Schub ert calculus the last of the p oints of view considered provides

a canonical lifting the Giamb elli formula of elements of the classical co

homology of G to p olynomials in particular elements of the cohomology

This lifting b ehaves nicely in quantum cohomology and suggests an evo ca

tive conjecture regarding a lifting of the quantum Euler class Because of its

technical nature we leave the statement of this conjecture to section

Frob enius Algebras

Frob enius algebras FAs underwent a great deal of investigation circa

b ecause of their relevance to representation theory A prime player in

this work was Nakayama who in exp osed among many other results

the basic algebraic asp ects of Frob enius algebras which are summarized here

in prop osition The novel approach presented in this chapter to the

algebraic structure of Frob enius algebras yields new results which readily apply

to the material in chapters and A number of the results app earing here

ha ve been published in

Characterizations of Frob enius Algebras

This section sets down the basic approach to Frob enius algebras used in this

thesis The classical characterizations of Frob enius algebras are sup

plemented by a new characterization in terms of a sp ecial coalgebra

structure Additional prop erties of the comultiplication are given Sur

rounding these main results are useful p oints of interest regarding vector space

duals the p ossibilities for Frob enius algebra structures and the

nature of maps of Frob enius algebras

Fix a eld K No assumption is made ab out K itmay b e nite or innite

dimensional algebraicallyclosed or not All algebras AK are assumed to b e

nite dimensional and commutative andtocontain a unit Multiplication

A A and A EndAwill denote the in A will be denoted by A

regular representation In other words ab a b The vector space

dual A has an Amo dule structure A A A given by a a

Prop osition The fol lowing conditions on A are equivalent

i There exists an Amodule isomorphism A A

ii There exists a linear form f A K whose kernel contains no nontrivial

ideals

iii There exists a nondegenerate linear form A A K which is asso

ciative ie ab c a bc

iv For al l ideals I A annannI I and

I K ann I K A K

Pro of A complete pro of is given in pages The pro of of the

equivalence of the rst three conditions rests on the following Given A

A satisfying condition i the linear form f satises condition ii

Given form f A K satisfying condition ii the linear form f

satises condition iii Given A A K the linear form f

satises condition ii and the linear map f satises condition i

An algebra A satisfying these conditions is called a Frob enius algebra

When A isaFA the maps f which are guaranteed to exist by conditions

i ii and iii will henceforth b e presumed to satisfy the relationships

f and f When it is useful to emphasize the FA structure of A

endowed by particular f and the algebra A will be denoted by A f

Prop osition shows how many such structures exist

Prop osition If A has a FA structure then so does A

Pro of If A f is a FA then by prop osition all elements of A are

of the form a f f aa for some a A The isomorphism A A

allows us to dene multiplication in A bya f b f ab f Dene A

K to b e evaluation at and note that aA aA f aA It

follows that if aA fg then a and hence a Prop osition

now shows that A is aFA

For the next result view A and A A as Amo dules via the usual mo dule

actions A A A and I A A A A A resp ectively In the

latter case we will denote the mo dule action by a b cab c

Theorem A nite dimensional commutative algebra A with unit is a

FA if and only if it has a cocommutative comultiplication A A A with

a counit which is a map of Amodules

Pro of Let A A A denote the multiplication in A let g K A

denote the unit map sending and let I A A denote the identity

K A

map

Assume A is a FA Wenow showhow the isomorphism of A with its dual A

structure Dene comultiplication A A A endows A with a coalgebra

to be the map

A A

It is clear from the denition of that A is coasso ciative and co commu

tative Note also that can be used to dene the multiplication in A since

a f b f a f b f ab f The commutativity of

f a

f af a

A K

guarantees the commutativity of

g I

A A K A

f I

A A K A

Since the top row is nothing other than I we see that the b ottom row is I

Thus f is the counit in A

To show that is an Amo dule map we must conrm commutativity of

A A

A A A A A

and Cho ose a basis e e and corresp onding tensor representations

f for f resp ectively Viewing as a map A A A the commutativity

e e

A A

ij j

k m

f e

f e e

ik k

ij kl l j

k m

f e e

A A

jl ik l j

follows from the commutativity and asso ciativity of and the fact that is

the adjoint of f By the denition of it is immediately evident that the

diagram

A A A

I I

A A A A

A A A A A A

commutes Thus commutativity of the diagram in question follows from the

commutativity of the outer edge of the following diagram

A A

A A A A A

I I

A A A A

Since the outer edge is nothing other than an expression of the asso ciativity

of it certainly commutes

NowassumethatA has a comultiplication satisfying the hyp otheses and

let f A K be the counit To show that A is actually a FA it suces to

show that the linear form f A A K satises the conditions

required by prop osition Asso ciativity of immediately follows from the

asso ciativity of It remains to show nondegeneracy

g K A A By assumption the following diagram Dene

commutes

A A A

g I I f

K A A A A A A K

By denition of g and f comp osition along the lower edge of this diagram

gives the identity Thus the top line shows that I I is the

identity map on A Cho osing a basis e e for A this comp osition maps

an arbitrary a A as follows

X X

a a u e a u e aa

K j j j j

j j

where the u are some elements in A In fact these u form a basis for A since

j j

they clearly span A and there are at most A K of them Taking a u

u e u so e u Assume that for some we see that u

j j i j i ij i

k e x for all x Plugging in x u we k k A we have

j j i n

see that k for all i In other words is nondegenerate and A is a FA

Given a basis e e for a FA A f let e e denote the dual

basis of A relative to In other words the e satisfy e e On

i ij

i j

the other hand let e e denote the basis of A satisfying e e

j ij

n i

Note that e e since

e f e e f e e e e

j j j ij

i i i i

to A f satis Prop osition The FA comultiplication corresponding es

i e e

A i

ae e for a A ii a

Pro of Using the notation of the pro of of theorem we have

u e

A K j j

The rst claim now follows by the co commutativity of and u e

The second claim is now an immediate consequence of theorem which

asserts that aa for all a A

Frob enius algebras and Hopf algebras are similar in spirit in that b oth have

acomultiplication satisfying some relationship with multiplication Neverthe

less the following prop osition shows that they have fundamentally dierent

algebraic structures

Prop osition The FA comultiplication A A A of A f is also

K and f id a Hopf algebra comultiplication if and only if A

Pro of We p ostp one the proofofthis prop osition until section

For an element a A dene the expression a to mean the map A

A A given by p ostmultiplication on the second term Sp ecically

a I a

Prop osition If A f is a FA then al l FAstructures on A are given

by A u f where u A may be any unit The comultiplication corresponding

to u f is u

Pro of If u A is a unit then for any a A such that u f ax

f uax for all x Aitmust b e that uaand thus a is By prop osition

u f is aFA form

Assume A g is another FA structure on A Now g A so we have

g u u f for some u A Since g is a FA form the map g

is an isomorphism A A as in the pro of of prop osition Thus there is

a v A such that f v v g vu f But f vu f vu

implies that vu since is an isomorphism so u is a unit

Now let and denote the appropriate maps of A u f By prop osition

u u

we know that u ue e Therefore b y the denition of we

have

u ue e

u i

e u e

u i u

and thus

u e u e

u u i

u u

FA map A f A f is a map of algebras which preserves the A

action of f ie f f

Prop osition All FA maps are injective In addition is also a map

of coalgebras if and only if it is an isomorphism

Pro of Injectivity of follows from the commutativity of

A A

This diagram commutes b ecause

f a f a f a f a

It also follows from this that is surjective Because

g g g g

A A

the map preserves the action of If is com ultiplicative then is multi

plicative hence an FA map and hence b oth injective and surjective If is an

isomorphism then must be multiplicative and thus is comultiplicative

Let FrobK denote the category of FAs A f andFAisomorphisms The

purp ose of restriction to isomorphisms will b ecome clear later

Decomp osition of Frob enius Algebras

The FA structure of A behaves well with resp ect to direct sums and

as well as tensor pro ducts and The former allows for a

classication of indecomp osable FAs Imp ortant asp ects of the F A

structure of such indecomp osable FAs relate to the so cle of A and the

choice of FA structure

Dene the direct sum A A A to b e adirect sum of algebras

Prop osition If A A then A f FrobK for some f if and

only if for each i there is an f such that A f FrobK Furthermore if

i i i

A f A f FrobK for al l i then f u f for some unit u A

i i i

Pro of Let e e be mutually orthogonal idemp otents suchthatA

n i

Ae for all i If A f FrobK then for each i dene f f j Assume

i i A i

that for some i there is an element a A Ae such that f a x for

i i i i i

all x A Then f a Af a A f a A It follows that a and

i i i i i i i

therefore A f is aFA for each i

i i

Conversely if A f FrobK for all i then dene the form f

i i

L P

f by f x x f x Assume that there is an element b

i n i i

i i

b b b e A such that f bx for all x A Then for all i

n i i

f b A f bA e showing that all b and hence b are Therefore

i i i i i i

A f is aFA

If A f A f FrobK for all ithenby prop osition anyFA form

i i

f A K must satisfy f u f for some unit u A Conversely for

each i there is a unit u A such that f u f j But u u u

i i i i A n

is a unit in A and f u f j for each i

i A

In light of this prop osition dene the direct sum in FrobK by

A f A f A A f f

Prop osition Comultiplication in A f A A f f respects

the direct sum structure

Pro of Let A A A By denition of direct sum for algebras multi

plication in A is amap

A A A A A A A A A A

A A A A A

Thus the comultiplication map as dened ab ove will satisfy this diagram

A A A A

A A

A A A A

A A

A A A A A A A A

The result easily follows

Let N N A denote the nilradical of A ie the sum of all nilp otent

ideals of A The smallest essential ideal of A is called the so cle of A When

A is indecomp osable N consists of all nonunits pp and the

ideal S annN is the so cle x

Prop osition S is a principal ideal any of whose elements is a gener

ator

Pro of Since N is an ideal it is a subspace of A Note that N A

since A hasanidentity element Wemay therefore cho ose a basis for A such

that all basis elements not in N are units Let U b e the nontrivial subspace

generated by the unit basis elements Note that any nonunit in U lies in N as

well so the only nonunit in U is It follows that N K U K A K

Assume nowthatA is a FA Since N is an ideal prop osition shows that

N K S K A K and thus S K U K

Denote the unit basis elements generating U by u u and let s be any

nonzero elementofS Assume that k su for some k k K

i i n

not all zero Now u k u is a unit so we have s u a

i i

contradiction Thus the elements su su of S are linearly indep endent

Since S K U K we see that S sU sA

If A is indecomp osable and N A then A contains only units and

so is just a eld extension of K If N A we will refer to A as an

annihilator algebra

Prop osition If A is a eld then any nonzero f A is a FA form If

A is an annihilator algebra then f A is a FA form if and only if f S

Pro of Assume that A is a eld and that f A is nonzero ie that there

is an x A such that f x Given any b A we have f bb x so

the kernel of f contains no nontrivial ideals By prop osition f is a FA

form

Now assume that A is an annihilator algebra with so cle S sA and that

f A is suchthatf sv for some sv S Because A is nite dimensional

It is clear asavector space there exists amaximum m such that N

by the maximality of m that S N In particular this shows that given

any a A there exists a b A such that ab su S for some unit u But

then f abu v f sv and prop osition again shows that f is a FA

form

The converse is trivial If f S then the kernel of f clearly contains a

nontrivial ideal

Combining the discussion ab ove with prop ositions and wehave

proven

Theorem Every FA A decomposes into a direct sum of elds and in

decomposable annihilator algebras and the FA form of A is determined by its

indecomposable constituents up to module action by a unit

In addition to a direct sum op eration the category FrobK also has a

tensor pro duct Dene the tensor pro duct A A A to b e a tensor pro duct

of K algebras

A then A f FrobK for some f if Prop osition If A and

only if for each i there is an f such that A f FrobK Furthermore if

i i i

A f A f FrobK for al l i then f u f for some unit u A

i i i

Pro of Supp ose A f FrobK Let a a a be an element

of A such that f a Extend fag to a basis for A such that f a

relative to that basis and for each i extend fa g to a basis for A anddene

i i

f a A K relative to each such basis Note that f f f

i i n

For each i if there is a b A suchthatf b A fgthenwemust also

i i i i i

have

f b A A

i n

f A f A f b A f A f A

i i i i n

Because A f isaFA it must b e that b ie that

b By prop osition this shows that A f is aFA

i i i

To prove the other direction assume that for each A there is an f such

i i

f f f A K If there is a that A f FrobK Dene

n i i

nonzero element b b b A such that

f bA f b A fg

i i i i

then there must be some i such that f b A But A f FrobK so

i i i i i

we must have b and thus b a contradiction It follows that A f

FrobK

The pro of of the last claim pro ceeds mutatis mutandis exactly as for the

corresp onding claim in prop osition

In light of this prop osition dene the tensor pro duct in FrobK by

A f A f A A f f

Prop osition Comultiplication in A f A A f f respects

the tensor product structure

Pro of The pro of is essentially the same as that given for the direct sum

in prop osition except that now we have the following diagram

A A A A

A A

A A A A

A A

The direct sum and the tensor pro duct eachendow FrobK with a monoidal

structure In other words wehave asso ciative bifunctors FrobK FrobK

FrobK with identity K for the tensor pro duct and identity ie the

zerodimensional algebra for the direct sum For more details concerning

monoidal structures see Chapter Note that we app eal here and in the

sequel without explicit mention to MacLanes coherence theorem ibid in

order to assure asso ciativity Intuitively this theorem allows us to work with

natural equivalence in a monoid as if it were identity

The Distinguished Element

Every FA A f contains a canonical distinguished element

This distinguished element has a nice form for any choice of basis

and behaves well with resp ect to direct sums and tensor pro ducts

Most imp ortan tly for our purp oses is a unit if and only if A is

semisimple Particular prop erties of relative to the so cle of A and

Jacobian ideals will play an imp ortant role in chapter

e e where Prop osition The distinguished element of A f is

e runs over a basis for A If u A is a unit then u is the distinguished

element of A u f

Pro of The rst statement follows immediately from prop osition

and the second statement follows from prop osition

Prop osition The distinguished element respects direct sum and tensor

product structure Specical ly

A A

A A f f A f A f

and

A A

A A f f A f A f

Pro of By the case of direct sums follows from prop osition

and the case of tensor pro ducts follows from

The map A A A is dened to b e the comp osite

In a sense the s function merely as achange of basis Now is just

A A K viewed as an element of A A Considering the matrix

where the basis of A is tak en to b e B fe e g and that of A to

B n

be B fe e g we see that corresp onds to the

This latter matrix contains the co ecients of we have matrix

e e

i j B

Moreover we see that

e e

B i j

Before pro ceeding to the next theorem note that a eld extension is a

simple mo dule and that an annihilator algebra is not simple because it is

commutative and contains nilp otents p Thus any semisimple

commutative FAis a direct sum of elds

Theorem The distinguished element of a FA A is a unit if and only

if A is semisimple

Prop ositions and show that this theorem is indep endent of the

choice of FA form f for A

Pro of First we show that the distinguished element of an annihilator

algebra is nilp otent Let A be an annihilator algebra with so cle S sA and

ideal N consisting of all the nilp otents in A As in write A as a direct sum

UN of vector spaces where all nonzero elements of U are units Without loss

of generality construct U to include Cho ose a basis u u u for

A j

U Then s su su su is a basis for S Cho ose nilp otents n n

j k

such that n n ssu su is a basis for N Because every element

k j

of A divides s the linear form f s A K is a FA form See the pro of

Order the basis for A as follows su u n n su su

j k j

Then the matrix for is ablock matrix of the form

B C

j j j k j j

B C

k j k k k j

j j j k j j

Here denotes an l m matrix with arbitrary entries and denotes

lm lm

an l m zero matrix

Denote the j k j k square blo ck of by R and denote by S

the j j square blo ck in the upp er righthand corner of R

Since is invertible R is invertible as well Of course the lower righthand

blo ck of is R Because S is ab ove blo cks of s it must be that the

upp er lefthand j j square corner T of R satises TS We now show

that T

If detS then its columns are linearly dep endent This implies in

detR turn that the last j columns of R are dep endent and hence

contradicting the assumption that isinvertible Thus S must b e invertible

and TS implies T

In light of the remarks preceding this theorem this shows that con

tains no summands a b where b oth a and b are units Thus is

asum of nilp otents and hence is nilp otent

It is obviously true that if A is a eld extension then is a unit Since

the distinguished element resp ects direct sums by prop osition and the

direct sum of units is a unit we see that if A is semisimple then its distin

guished element is a unit If A is not semisimple then its decomp osition into

irreducibles contains an annihilator algebra This summand contributes a non

unit direct summand to and thus is not a unit

Prop osition In a FA A the ideal A is the socle of A

As was the case for theorem this result is indep endent of the choice

of FA form

The main construction of the following pro of is essentially taken from Sawin

although this result do es not explicitely app ear there

Pro of Because the so cle of a nitedimensional commutative algebra is

the direct sum of the so cles of its indecomp osable constituents x it suces

to prove this prop osition for the indecomp osable cases

If A is a eld extension then N A fg so S annN A But

theorem shows that is aunit so A A S

If A is an annihilator algebra dene a chain of ideals S S S

S A where each S is the preimage in A of the so cle of AS Cho ose

n k k

a basis for S Now starting with i iteratively take the basis for S and

extend it to a basis for S Denote the elements of the basis for S A

i n

by e e and let e e denote the corresp onding dual basis elements

Supp ose e S and that a A is any nilp otent element Then ae S

i k i k

and therefore can b e expressed as a linear combination of basis elements other

than e It follows that f ae e so e e N A Ker f But Ker f can

i i i

i i

contain no nontrivial ideals by so we must have e e N A ie

e e S This follows for each i so e e S By prop osition

i i

S A

For the remainder of this section assume that K has characteristic

Supp ose that an algebra A not necessarily a FA is nite dimensional as a

K x x R where vector space and is given by the presentation A

R f f is some nitelygenerated ideal in K x x Because A

p n

is nite dimensional we must have p n The Jacobian ideal J J R of

R is dened to b e the ideal generated by the determinants of the n n minors

of the matrix

f f

mo d R

x x

The ideal J is welldened since it is a Fitting ideal of the mo dule of

Kahler dierentials of A see x x

The following result of Scheja and Storch is rep orted in more generality

in ibid although for the denition of complete intersection we refer

the reader to

Prop osition J fg if and only if A is a complete intersection and J

generates the socle of A

element K x x R is a FA with distinguished Now assume that A

for some choice of FA form

Prop osition J fg if and only if J A If p n then J fg if

and only if

det mo d R u

for some unit u A

Pro of This prop osition follows immediately from and

Examples

Example Truncated polynomial algebras P K xx

Let w x Take the standard basis for P and let f w Since every

basis element divides w we see that w is a FA form In fact P contains

no idemp otents so is indecomp osable and is thus an annihilator algebra with

S wA By prop osition comultiplication is determined by

k nk n

x x and nx

Example Finite dimensional graded commutative connected Hopf al

gebras over K

This a generalization of the previous example Connected means here that

there are no elements of negative degree and that the mo dule generated by

elements of degree is onedimensional over K

Margolis shows that if A is such an algebra then it is a Poincare

algebra ie for each q there is a nondegenerate form A A K where

q nq

n A K As the discussion in section will show this implies the

A and A is an annihilator algebra with existence of isomorphisms A

nq q

so cle generated by the highest degree element

Example Finite dimensional local rings gradient algebras Qh

n n n

C R h where h R R is a smooth map h is the germ

of h at h is the ideal generated by the components of h and C R

n n

denotes the ring of germs at of smooth functions R R

Let J b e the Jacobian determinantofh and let J b e the residue class of J in

Qh It is shown in that any linear functional f Qh R such that

f J is a FA form By prop ostion this suces to show that Qh

has so cle S J Qh In fact Qh is lo cal and a complete intersection

so this example is just a sp ecial case of

Examples along these lines are provided by top ological LandauGinzburg

mo dels These arise in the study of the ring of states C x W where x

i i

denote chiral sup erelds and W x is a quasihomogenous sup erp otential

The algebraic structure of this ring is briey discussed in The app earance

of these LandauGinzburg mo dels in classical and quantum cohomology will

be discussed in more depth in chapter

Example Group algebras K H where H is a nite abelian group

These are actually just ungraded unconnected Hopf algebras

Let f Since each basis element has an inverse f is clearly a FA

form Moreover b ecause the basis elements form a group we have

h h and thus jH j If f is adjusted by a unit then more

interesting things can happ en For instance if h H is an element of order

d then the FA form jH j h f yields the comultiplication given by

h H

jH j hh h and h an element of order d

Example The character ring R H of representations of nite or com

pact groups H tensored with Q

Let V V denote the irreducible representations of H and let

n n

resp ectively denote their characters Assume is the trivial character The

bilinear form dened by h i dim Hom V V is nonsingular

i j H i j ij

It is asso ciative b ecause dim Hom V V equals the dimension of the space

H i j

of H invariant bilinear forms ie dimV V asso ciativity follows from

i j

the noncanonical isomorphism V V and the asso ciativity of the tensor

pro duct We see that denes a FA structure having FA form Since each

basis element of R H is selfdual relative to we have For

denote the number of memb ers of the conjugacy class of h in h H let ch

H The virtual character then has the following well known denition

jH jch if h and h are conjugates

otherwise

It follows that the following are equivalent

i Every element of H is conjugate to its inverse

ii All representations of H are real

iii is invertible

iv R H is a direct sum as algebras of eld extensions

Another example involving representations is that of fusion algebras which

are the representation rings of lo op groups See for details

Frob enius Ring Extensions

Applications in chapter require an understanding of Frob enius algebra struc

tures when the base ring is not a eld This section deals with particular

change of basering maps which preserve the distinguished element

and some of its nice prop erties The latter result has imp ortant appli

cation in chapter

Supp ose AR is a nitedimensional as a mo dule commutative ring ex

tension with identity and f A R is an R linear form We say that f is

nonsingular if the determinant of the asso ciated linear form A A R

taking a a f aa is a unit in R When f is nonsingular we refer to AR

as a Frob enius extension FE and call f a Frob enius form FE form

As with Frob enius algebras we will denote A with this choice of FE form

by A f In this case we also equivalently have the corresp onding mo dule

isomorphism A A and given any R basis fe e g for A we can

A n

form the dual basis fe e g relative to This immediately implies the

existence of a distinguished element as well Finally note that over a

general ring the notion of nonsingularity diers from the weaker notion of

nondegeneracy We say that f is nondegenerate if f aA fg a

Nondegeneracy do es not in this more general setting guarantee the existence

of mo dule isomorphisms as ab ove

Prop osition A FE A f over an integral domain R is torsion free

Pro of Supp ose that a A and ra for some r R Then we have

fg f raA rf aA

Since R is an integral domain wehave f aA fg But f is an FE form so

we have a

Supp ose R S is a surjective homomorphism of rings ie sending

Dene B A to be A S In this ring we have ra s

R S R

a r s for all r R s S and a A Let A B denote the ring

homomorphism a a Dene the linear form f B S by

f a ss f a

The form f is welldened since

f ra s s f ra s rf a s r f a f a r s

and f satises the diagram

A B

R S

Let e e denote a basis for A and let e e denote the corre

i n

sp onding dual basis relative to

Prop osition The form f endows B A with a FE structure and

Pro of It suces to show that the set f e e g is a basis for B

e to the form B B S determined by f is and that its dual basis relativ

f e e g The existence of a dual basis will show B is a FE The

particular form of the basis and dual basis together with the fact that is a

homomorphism will prove the claim ab out

We rst prove the orthogonality relations

e e f e e S e e f f

i i ij ij i

i i j

Toprove that wehave a basis as claimed note that the elements e e

clearly span B since is surjective Supp ose that for some fs g S we have

s e Then for all j

i i

f f s e e s

i i j

It follows that e e are indep endent and thus form a basis The

as well orthogonality relations show that f e e g is a basis

In the next result let R K be any surjective K linear ring homo

morphism where K is a eld

Prop osition The element is either aunitinA or a zero divisor

i If is a unit in A then B A is semisimple

ii If is a zero divisor and ann Ker then A is not semisim

Af Af

ple

Pro of If is a unit then there exists a u A such that u

Af Af A

But then by

u u

Af A B

so is aunit as well

If is not a unit in A then by prop osition whose pro of carries

through verbatim for rings the map f is not a FE form In other words

there exists an a A suchthatf aA f aA fg But f is a FE

Af Af

form so it must b e that a If follows that a Since there

Af Af

exists some a ann such that a Ker we see that is

Af Af

a zero divisor as well Both statement i and ii now follow from theorem

Frob enius Extensions and Number Theory

The results in this section will not be used elsewhere in this thesis but place

the theory of Frob enius extensions nicely in the context of algebraic number

theory where the usual number theory trace provides a sp ecic choice of FE

and FA structures Of sp ecic interest are the following For Galois and

separable extensions the distinguished element is the identity element

The ring of integers B of a eld extension over a eld of fractions is a FE over

the basering if and only if the usual numb er theory dierent is all of B

Assume we have a tower of nitedimensional ring extensions

with a B linear FE form f C B and an Alinear FE form f B A

C B

Cho ose an Abasis b b for BA and a B basis c c for CB The

m n

elements fb c g are an Abasis for CAalso denote these basis elements by

i j ij

e b c Let and be the mo dule isomorphisms X X X B

ij i j B C X

C induced by FE forms f and f resp ectively and let and be or

B C B C

the distinguished elements of BA and CB resp ectively Denote dual basis

elements as follows c c and b b

i i

C B

Prop osition If f and f are FE forms then the Alinear form

C B

f f f C A

B C

is an FE form as wel l

Pro of Let C C C denote multiplication on C It suces to

exhibit abasis dual to fe g relative to f But we have

ij ij

f b b f f c c c c b b f f b b c f e b

jl ik jl B i B C j j i B C i ij

k l l k k l k

and we are done so b c e

k l kl

Let denote the mo dule isomorphism C C corresp onding to f and

let be the corresp onding distinguished element of CA

Prop osition Let B C be the inclusion map The fol lowing dia

gram commutes

C B

Q Q

C B

Qs Qs

Hom C B Hom C A Hom B A

B A A

Pro of The triangle on the left commutes b ecause f c f c

B C

f f cc c f c for all c C The square on the right commutes

B C

b ecause for b B

c f b f f cbf bf c f c f b

B C B C C B

Prop osition The fol lowing hold

i f C B

C B

ii b c f b c

C i j B i j

Pro of In lightof the pro of of we have

X X X

f f e e f b b c c b b C B

C C ij C i j i B

ij i j i

ij ij ij

pro ving i

Now because we have a mo dule isomorphism we also have a corre

sp onding comultiplication for which f serves as the counit In other

B B

words

f I f b b

B b B B B i B

Thus

X X

c c f b b c c

C j B i j

j j j

j ij

b c by the pro of of item ii now follows c Since b

i j

j i

Now let LK be a separable eld extension The usual number theory

trace f Tr satises the conditions of prop osition and thus endows

LK with a FA structure Let denote the corresp onding distinguished

element

Prop osition If LK is a Galois extension or a separable extension

then

Pro of Assume rst that LK is a Galois extension Let GalLK

f g denote the corresp onding Galois group By the normalbasis theorem i

p there is an element L suchthat is a K basis

for L We have

Tr Tr

i i i i LK ij LK

LK i

ie Thus It now follows that

i i i

X X

i i i

i i

showing that is invariant under action of the Galois group and must

therefore lie in K By FA considerations Tr L K Tr

LK LK LK

This proves the case when LK is Galois

If LK is a separable extension we may embed L in a Galois extension

L K The trace map

Tr Tr Tr

L K LK L L

is a FE form by Since we have Tr L L By

L K L L L K

prop osition this must be L L and thus also

LK LK

Let A be a Dedekind ring K its eld of fractions L a nite dimensional

separable eld extension of K and B the integral closure of A in L Let

e e be an Abasis for B Then e e is also a K basis for L

n n

Prop osition Any nonzero Alinear form f B A is nondegenerate

and has a unique extension to a FA form f L K

Pro of Assume there is a nonzero b B suchthatf bB Since there

ve is a nonzero a A such that ab B we ha

f bB f ab bB f aB af B

acontradiction Thus f is nondegenerate Since f is determined by its values

on e e there is clearly a unique nondegenerate extension f of f By

prop osition f is a FA form

Given aFA form f L K dene C to be the fractional ideal

fx L j x f B Ag

and dene D C Note that D is an ideal in B When f Tr D is

f f f

the usual number theory dierent D

Prop osition D xD

xf f

Pro of Let be the Lmo dule isomorphism L L determined by f

Note that

C Be Be

If we adjust the form to x f and denote the corresp onding isomorphism

map by we have e x e see the pro of of and therefore

x i

C x C The result now follows

xf f

Prop osition The ring extension BA is a FE if and only if D B

Pro of Assume BA isaFEwithFEformf and corresp onding B mo dule

isomorphism B B Extend f to an f L K as in and let

be the corresp onding Lmo dule isomorphism L L We have

fx L j x f j Hom B Ag fx L j x f B Ag C

B A f

Clearly B C f

Take x C and let b x f j Since b f x f j by prop osition

f B B

we have equality of their extensions b f x f Applying we see

that x b B and thus C B

Now f u Tr for some unit u B by so gives

C uC B implying that D C u B uB B

LK LK LK

Conversely if D B let f Tr and let be the mo dule homo

LK LK

morphism b b f Since f is nondegenerate must be injective

x y B x y xy b B f xbf yb f

Toshow surjectivityof extend f using to the FAformf L K

and dene as usual Take an arbitrary Hom B A Extending to an

element Hom L K let x C Now D D B

K f f

so C B It follows that x f is dened and

xx f x f j j

B B

Thus is surjective and hence an isomorphism

Top ological Quantum Field Theories

TwoDimensional Cob ordisms

This section consists of a precise denition of the category of twodimensional

cob ordisms and a description of that category in terms of generators and

relations

Let PreCob ord denote the twocategory dened as follows

Ob jects are disjoint unions of lab eled oriented compact one manifolds

Sp ecically dene B C by B t k e For each

k k

k orient the image of B in accordance with the parametrization and

lab el the image with the index k The ob jects are taken to be the

empty manifold and the disjoint unions n B for all

n N fg

Morphisms n m are oriented top ological surfaces not necessar

ily connected equipp ed with an orientation preserving homeomorphism

from the b oundary to the disjoint union n m Here n indicates

reversal of orientation In other words the orientation induced by on

the p ortion of corresp onding to n is the opp osite of the orientation

that p ortion inherits from n Each b oundary comp onentisgiven the la

b eling induced by its homeomorphic image Comp osition of morphisms

consists of gluing corresp ondinglylab eled b oundaries in an orientation

preserving manner

Twomorphisms are orientationpreserving homeomorphisms T

of morphisms such that the following diagram commutes

n m

T j

Note that T j must preserve lab eling

Dene Cob ord to be the category whose ob jects are those of Pre

Cob ord but whose morphisms are the equivalence classes of morphisms

induced by the twocategory structure of PreCob ord In other words

two morphisms are equivalent in Cob ord if there is a twomorphism

T in PreCob ord Because b oundarypreserving homeomorphisms

of surfaces are homotopy equivalences the morphisms of Cob ord are distin

guished only up to homotopy class

Cob ord has a monoidal structure induced by ordered disjoint union

whichwe will refer to as tensor pro duct Sp ecicallyif are morphisms

of Cob ord having b oundary comp onents lab eled k and k re

sp ectively then has b oundary comp onents lab eled k k where

lab els k corresp ond to the b oundary comp onents of with the same

lab els and k k k corresp ond to the boundary comp onents of

by the two lab eled k resp ectively The equivalence relation induced

morphisms guarantees welldenedness and hence asso ciativity of this tensor

pro duct

Prop osition The morphisms in Cob ord are generated by gluing copies

of the ve basic surfaces shown in gure subject to the ve sets of relations

shown in gure

Σ Σ Σ g f I

Σβ Σα

Figure The generators of Cob ord Incompatiblyoriented b oundaries are

shown to the left of each comp onent compatiblyoriented b oundaries are shown

to the right

b b (A)

c c

(I)

(C)

(U) (F)

Figure The relations of Cob ord Within each relation corresp ondingly ori

ented b oundaries are lab eled consistently from top to b ottom

m n

Figure Decomp osition of a generic morphism

Pro of According to the classication theorem for twodimensional ori

ented surfaces with oriented b oundaryeach connected morphism Cob ord

is determined up to homotopy class by a triple m g n where g is genus m

is the number of incompatiblyoriented b oundaries and n is the number of

compatiblyoriented b oundaries Thus each such with m g n may be

canonically decomp osed as shown in gure If this has m n

then the left right p ortion of the shown decomp osition will b e replaced with

If g then the central p ortion will be deleted It is clear that

g f

the ve basic shap es of gure generate all whether connected or not via

comp osition and tensor pro duct

Since none of the relations feature morphisms with nonzero genus their

validityisveried by counting b oundary comp onents Toshow the suciency

of the relations we will show that any decomp osition of a morphism

Cob ord can be transformed into the canonical decomp osition for using

only the given relations Here dene the canonical decomp osition of a dis

be the tensor pro duct of connected morphisms each connected morphism to

of which viewed in isolation is in canonical decomp osition

Supp ose we are given a decomp osition If j we

j j

(A) (C) (A)

(I)

Figure Example of application of relations C

are done Otherwise by the relations I no generality is lost in assuming that

up to relab eling of b oundary comp onents each is of the form

i X

where X I f or g For ease of notation we will refer to

such padded morphisms simply as and assume that the padding is

always exactly what is necessary for valid comp osition

Note rst that is necessarily already in canonical form The pro of of the

general case pro ceeds by induction Assume that can

i i

be transformed into canonical form using the given relations We will show

that can also be transformed into canonical form

i i i

Let n denote the quantity of outgoing b oundary comp onents in If

n then is trivially in canonical form Assume then that n If

then relations I C and A guarantee that is still in canonical

i I i

form Figure shows an example of how this works

If or then relations U and A resp ectively show that

i f i

the only change is an adjustment of the quantity of b oundary comp onents

Zero or more applications of relation A yield the canonical decomp osition

for

Assume that n and There are two cases to consider The

ingoing b oundaries of either glue to the same comp onentof or glue to

i i

the comp onentof dierent comp onents In the rst case denote by to

which the glues it suces to show that can b e transformed using

(A) move (i):

move (ii): (F)

Figure Moves used when gluing to a single comp onent

the given relations to canonical form Let n denote the quantity of outgoing

b oundaries in If n then is already in canonical form If n

then note that there are at least n copies of in not used in the

genus section of the canonical decomp osition In accordance with diagram

we refer to these l copies of as b eing to the left of the newly added

in Pro ceeding by induction on l the n case serves as the base

case assume that for l copies of to the left of a transformation to

canonical form exists Apply relation A rep eatedly as for example in move

i of gure until an application of F is p ossible as shown in move ii of

gure Now apply the inductive hyp othesis

glue to dierent comp o Supp ose now that ingoing b oundaries of

nents of together having a total of l generators Assume that if

there were fewer than l generators a transformation to canonical form would

be p ossible Pro ceeding by induction consider the cases as outlined in gure

(I) (F) * * *

(I) (A) ** *

(U) (I) *

(I)

Figure Moves used when gluing to two comp onents Boundaries corresp onding

to dierent comp onents are shaded dierently The new s are marked with

asterisks

In each case the new is moved to the left of one generator in

and thus the inductive hyp othesis applies to the p ortion of the transformed

including and everything to its left The remaining p ortion

to the right of the new if any is now being glued to a connected comp o

nent which may be presumed to be in canonical form and the argument of

the previous paragraph applies The diagrams in gure are in of themselves

base cases for this induction so we are done

Top ological Quantum Field Theories

The main ob jects of interest in this chapter top ological quantum eld theo

ries TQFTs form a category with two monoidal structures This section

describ es the denitions and prop erties of morphisms of TQFTs as well as

tensor pro ducts and direct sums of TQFTs The behavior of direct sums

requires some additional attention Because it is necessary for under

standing morphism of TQFTs we rst include whichmay b e considered

the rst half of

Let VectK denote the category consisting of nite dimensional K vector

spaces and K linear maps with the monoidal structures given by direct sums

and tensor pro ducts A top ological quan tum eld theory is a monoidal

functor Z Cob ord VectK taking K and n V for some V

VectK As in the functor Z is normalized so that Z id

I Z

Prop osition Each TQFT Z induces a FA structure on Z

Note The following pro of uses the functoriality of Z rep eatedly and im

plicitely

Pro of Denote Z by V The op erator Z V V V

denes a K linear multiplication on V Relation A from gure guarantees

asso ciativity of and C shows that is commutative Relation U shows

that the op erator g Z K V provides a multiplicative identity

Similarly the relations guarantee that the op erators Z V V V

and f Z V K dene a co commutative coalgebra structure with f

counit on V Relation F implies that wehavecommutativity of the diagram

V V

V V V V V

In words is a map of mo dules It follows from theorem that Z induces

aFA structure on V

A map Z Z of TQFTs is a monoidal natural transformation

Explicitly consists of a collection of linear maps A

A where A A A K id and A A are Z Z

resp ectively such that the follo wing diagram commutes for all n and any

n m Cob ord

n n

A A

Z Z

m m

A A

Note that satises all three of the following commutative diagrams

A A

K A A A

It follows from prop osition that is aFA isomorphism

Dene TQFTK to b e the category whose ob jects are TQFTs and whose

morphisms are the maps dened ab ove TQFTK has two dierent monoidal

structures corresp onding to the two monoidal structures on VectK

Given algebras A A letT A A A A b e the canonical twist map

n n n

sending a b b a For any n dene T A A A A n

A A A A A A

R R

A A A A A A

Figure Aschematic diagram illustrating T

to be the comp osite

n n n n n

I T I I T I I T I

where I generically denotes the identity map Allow T and T to denote the

corresp onding maps for any algebras A A Figure illustrates an example

of T

Now given Z Z TQFTK dene the tensor pro duct Z Z on

ob jects by Z Z n Z Z Given a morphism m

n Cob ord dene Z Z T Z Z T Letting

A Z and A Z we have

m n

A A A A

Z Z

T T

m n

Z Z

m m n n

A A A A

Note that this denition of tensor pro duct ensures that the FA structure

induced by Z Z on Z Z as p er is the structure of a tensor

pro duct of FAs

Dene the direct sum Z Z on ob jects by

Z Z nZ Z

On the generating morphisms of Cob ord dene Z Z to act comp onent

wise

Z Z Z Z

f f f

Z Z Z Z

These requirement automatically guarantee that Z Z b ecause it is

the multiplicative unit map and Z Z by prop osition also act

comp onentwise For the latter this means that the image of the map contains

no cross terms Of course the map Z Z which is assumed to b e the

identity map acts comp onentwise All this guarantees that the FA structure

induced on Z Z is that of a direct sum of FAs

The following prop osition describ es how Z Z behaves on general mor

phisms

Prop osition The image under a TQFT Z Z Z of any c onnected

morphism of Cob ord acts componentwise If consists of more than one

component then Z wil l general ly not act componentwise

Pro of A connected morphism m g n has a decomp osition

R M L

corresp onding to the right middle and left p ortions of the canonical decom

p osition as shown in gure Thus Z is a comp osition of op erators

be Z n Z g Z m Let A

Rn Mg Lm

the images under Z of resp ectively Supp ose a a a is

an element of A By relations A C

a a a a

Lm m Lm

for any p ermutation of the indices fmg Moreover

a a a a a

Lm m Lm

Because was dened to act comp onentwise crossterms contributed by

a a will have no eect on the value of a Iterating this pro cedure

we see that no cross terms due to terms a a haveany eect on a

i i

Because is an arbitrary p ermutation we see that acts comp onentwise

ignoring all crossterms

It is clear that acts comp onentwise since

is simply a comp osition of comp onentwise maps

The map readily follows suit First acts on an element of A

Rn Rn

which by denition has no crossterms Second acts by various applica

tions of the map which never pro duces crossterms by prop osition

We see that ignores all crossterms of its op erand

Rn Mg Lm

and never pro duces crossterms in short acts comp onentwise

To show that disconnected morphisms need not yield comp onentwise maps

it suces to provide the example f f which is the image of Let

f f

a a A and b b A all be elements not in the kernel of the resp ective

maps f f Then in general

f f a b a b f a f b f a f b

f a f a f b f b

The Corresp ondence Between Two Dimensional

TQFTs and Frob enius Algebras

This section presents the fundamental theorem of this chapter A rig

orous diagramoriented pro of of then serves as an illustrative application

Theorem The functor F TQFTK FrobK which maps objects

by Z Z Z and morphisms by is an equivalence of cate

gories which respects tensor products and direct sums

Pro of Prop osition shows that F is welldened on ob jects and the

discussion in section shows that F is in fact a welldened monoidal functor

for both the tensor pro duct and direct sum structures Given an arbitrary

A f in FrobKitisnecessary to construct a corresp onding TQFT Dene

A and Z TQFTK by n

f g

Since f the unit g and I already satisfy the conditions of coasso cia

tivity cocommutatitivity counit and identity we need only check the re

lation in A corresp onding to relation F In other words we must conrm

commutativity of

A A

A A A A A

However this has already been shown in theorem

Given a morphism of FrobK there is a unique morphism of TQFTK

having since by denition

Σ Σ

H ω

Figure The surfaces corresp onding to the handle op erator and the distin

guished element

Assume that Z A A f f FrobK Denote by Z Z

the functors sending to A f A f resp ectively Using the maps T

discussed in we can extend to Z Z Z Similarly if Z Z

A A f f then we can canonically dene Z Z to havevalues on the

Z Z generators of Cob ord such that Z

The equivalence of the categories FrobK and TQFTK shows that the

orem may be viewed as a decomp osition theorem for TQFTs

Any TQFT Z sends the morphism depicted in gure to

a map H A A called a handle op erator Here we use A to

denote Z That this H is in fact a mo dule homomorphism follows from the

utativity of comm

A A A A A A A

A a

A A

Commutativity of the left hand square follows from theorem The

right hand square simply expresses asso ciativity of Thus H a for

some a A Since H it follows that H

A A

Note that Z sends the morphism shown in gure to A

Using the corresp ondence between FrobK and TQFTK we can now

provide a clean pro of of prop osition

Pro of Assume that FA A f has aFA comultiplication A A A

which is also a Hopf algebra comultiplication Then satises both of the

following diagrams

A A

A A A A A

A A

Hopf

A A A A A A

I T I

A A A A

Combining these diagrams along their shared comp osition gives the outer

edge of the following diagram the arrow across the middle is trivially valid

A A

A A A A A A A

I I

I T I

A A

Commutativity of the top triangle and the outer edge implies commutativ

ity of the lower triangle as well Using theorem let Z be the TQFT

corresp onding to A f The lower triangle now translates into the relation

Here we use to denote that Z Z Pre and post

comp osing with the same morphisms preserves this Z equivalence so we

have

But this shows that the map A A given by a f a is the identity

It follows that f id and thus A is onedimensional

Examples of TQFTs and Invariants of Surfaces

Example Unitary topological eld theories

Durhuus and Jonsson classify twodimensional TQFTs with KC in terms

of the sp ectrum of the handle op erator Z mentioned ab ove The dier

ences between that classication and the classication results given here arise

from issues regarding choice of base eld duals of algebras and orientation of

morphisms in Cob ord

Figure An example of the orientation relations

If is a morphism in Cob ord let denote with orientation reversed

Because the twomorphisms in Cob ord were dened using only orientation

preserving maps there is no a priori reason to assume that Z Z as

is assumed in Of course there is necessarily a relationship b etween Z

and Z b ecause any such can be decomp osed as

m n

I I

for some mn See gure for an example of such a decomp osition

of For a given choice of FA A f and corresp onding TQFT Z these

relationships determine a unique Amo dule isomorphism A A namely

the adjointofZ This isomorphism determines the eect in VectK

of the reversal of orientation in Cob ord

In Atiyah sp ecically leaves the p ossible axiom Z Z where

the latter use of indicates the vector space dual as an op en issue It is

imp ortant to note that this axiom is assumed in The two assumptions

tity Z Z which forces and the made there yield the strong iden

handle op erator to be simultaneously diagonalizable Durhuus and Jonsson

use the phrase unitary top ological eld theory UTFT to indicate these

particular assumptions They show that if Z and Z are UTFTs then

the following are equivalent

i Z Z for all morphisms without boundary

ii The op erators Z and Z have the same sp ectrum

H H

iii There exists a unitary natural transformation Z Z

The remainder of this section deals with the notion of a TQFT as a means

of distinguishing morphisms of Cob ord

Example General TQFTs

In general anyTQFTZ will distinguish b etween arbitrary morphisms

Cob ord if their resp ectivequantities of ingoing andor outgoing b oundaries

dier This is simply b ecause any nonzero linear op erator in the image of

Z carries with it all necessary information regarding the dimensions of its

domain and range

Example Annihilator TQFTs

If A f is an annihilator algebra or a direct sum of annihilator algebras

then its distinguished element satises Letting Z be the TQFT

corresp onding to A f it follows that Z In other words a

H H

TQFT based on A will send all connected morphisms having genus g

to the map Nevertheless it is interesting to note that Z do es distinguish

andm n m n between such morphisms as m m n n

the former morphism is necessarily sent to whereas the latter need not be

The following class of examples demonstrate much greater distinguishing

power

Example Group algebras A K G where K has characteristic and

G is nite

Let e e e denote the elements of G and cho ose them as a basis

G n

of A Relative to this basis cho ose the FAformA K to b e e Then for all

i e e and thus n e If Z denotes the TQFT corresp onding

i i

to A e we see that Z n e If is a morphism of

genus g then Z n In other words Z detects genus

For well chosen G the TQFT Z also easily distinguishes b etween morphisms

having the same quantities of ingoing and outgoing b oundaries but diering

in terms of the distribution of these b oundaries on the various connected com

p onents of the morphisms

ZZ is given as a multiplicative group by the For example supp ose G

elements e e where e e Again cho ose the FA form to be e

P N

m n m g n Cob ord let m Given a morphism

i i i i

i i

m n

n and Z A A For each of the injective maps

f g f mg dene a A to be the usual basis element having

all other p ositions Now apply e in the and p ositions and e in

n n

f to a for each if f a then the ingoing b oundaries

of lab eled and resp ectively belong to the same comp onent of

If f a then these b oundaries belong to dierent comp onents

Because chose all p ossible pairs of ingoing b oundaries we have completely

determined the distribution of the ingoing b oundaries

To determine the distribution of the outgoing b oundaries dene f A

n n

A to be the usual basis element of A having e in the j th p osition

and e in all other p ositions Similarly dene a A to be the usual basis

element having e in all p ositions except the ith where it has e For each

pair i j fmgfng evaluate f a Because e is the

j i

counit in A e this evaluates to if and only if the j th outgoing b oundary

do es not belong to the same comp onent as the ith ingoing b oundary By

varying i j over all pairs wehave completely determined the distribution of

outgoing b oundaries

The algebra K ZZ can also be used to distinguish morphisms on the

basis of distribution of genus Cho ose the FA form to be e e and de

note the corresp onding TQFT by Z We have e e Using a

diagonalization of the regular matrix representation it is easy to show

that

k k k

e e

and therefore

k k

g Cob ord having total genus g and no Given a morphism

b oundary the element Z Z g K has the following

i i

prop erties The highest power of which divides is If the power

of in the denominator is then g s is the quantity of comp onents of

not having genus The highest p ower of dividing is where t is the

quantity of comp onents having even genus including genus

It seems that these invariants contain even more information Each par

tition g g g of g determines a unique as ab ove and thus a sp ecic

value in K At least in some cases these values will be unique for example

if g we have

The partition Yields

Quantum Cohomology

The Frob enius Algebra Structure of Classical and

Quantum Cohomology

In the case of classical cohomology the approachtoFAs presented in chapter

eg the coalgebra structure and the distinguished element manifests itself in

terms of wellknown maps and elements These generalize to quantum

cohomology and it is a straightforward result that the quantum cohomology

ring is aFE

The exp ository material in this section on classical cohomology is based on

Ch VI

Let X denote a connected K orientable ndimensional compact manifold

where n is even Throughout this chapter except where noted otherwise

homology and cohomology groups will use co ecients in a eld K of charac

teristic Denote by X H X the fundamental orientation class of X

and let h i H X H X K denote the Kronecker index Poincare

duality tells us that for each k the map

H X H X

is an isomorphism By the canonical homomorphism of a nitedimensional

the isomorphism for each k vector space with its double dual we have

H X HomHomH X KK

nk nk

X h X i

Since and are adjoint op erations we have the isomorphisms

Hom HomH X KK HomH X K

h X i h X i

Comp osing all these isomorphisms shows that the linear form

f H X K h X i

denes a mo dule isomorphism

H X HomH X K

f h X i h X i

of H X mo dules We see that f is a FA form In fact it is where

denotes the generator of H X satisfying h X i The corresp onding

map H X H X K is classically referred to as the intersection

form Up to the canonical isomorphism of a vector space and its double dual

the isomorphism is nothing other than the Poincare duality map Henceforth

we will not distinguish between H X and Hom H X K

Multiplication in H X ie the cup pro duct is essentially given bythe

map

H X H X H X X H X

induced by the diagonal map X X X The dual map in the sense of

vector spaces is simply

X H X H H X H X X

Thus the FA comultiplication on H X is simply

the transfer map H X H X H X The Thom class of the

tangent bundle of X is D X where D H X X

X X X X

H X X is the inverse of the Poincare duality map In our notation

X But X so The Euler class

H X

of the tangent bundle of X which we will simply refer to as the Euler class

eX of X is dened to be In other words eX and we

have shown

Prop osition The distinguished element of the FA H X is the

Euler class eX

This proves the well known formula

eX e e fe g is a basis

i i

In previous chapters weonlyworked in the commutative setting since this

is necessary for a FA to corresp ond to a TQFT Nevertheless the algebraic

results on FAs still hold even if A H X is not commutative Sp ecify the

Amo dule action on A by a g g a where denotes multiplication

L L

on the left To see what happ ens take a basis element a H X of odd

degree and let b be the basis element such that ab Then a b and

b a Assuming a b recall that K is presumed to b e of characteristic

these basis elements contribute a bb a to the distinguished

Of course elements of even degree will contribute copies of with element

p ositive co ecients

The Euler class is never a unit b ecause degree considerations demand

that all units of H X lie in H X Theorem therefore shows that

H X is not semisimple However this argument is gratuitous any nite

dimensional algebra with a plain grading ie not cyclic in any manner

necessarily has nontrivial nilp otent elements purely for degree reasons In

fact H X is indecomp osable since its idemp otents must lie in H X again

for degree reasons and are therefore units The so cle S of H X is simply

eX H X H X

For a FA A f dene the map A A A K by h h h

h h h Here h h h denotes the multiplication on A describ ed in the

pro of of In the case A H X the multiplication on A is called

the homology intersection pro duct and is the triple intersection form

Note that the multiplication on any A f is determined by for a b A

w e have

ab abe e

where e ranges over a basis of A

For a symplectic manifold X the quantum cohomology algebra QH X is

dened via a collection of deformations of the triple intersection form For

more details regarding the denition of quantum cohomology than are given

here see

Let J b e an almostcomplex structure on n real dimensional X For an

MB J to b e the space of J element B H X Z dene the mo duli space

S holomorphic curves u C P X which represent B Recall that C P

so this denition makes sense The evaluation map MB J C P X

sending u z z z uz uz uz behaves well under action by the

reparametrization group G PSL C ie for G

u z z z u z z z

so we may pass to the quotient W B J MB J C P In

practice the denition of W B J generally involves a compactication of

MB J see x for details This space has dimension n c B

where c is the rst Chern class of the J tangent bundle TXJ of X

Given homology classes s s s H X Z of degrees d d d resp ectively

cho ose resp ective representatives s s s in X which are in general p osition

Naively the deformed triple intersection forms are dened as follows If

d d d n c B then

s s s fu z z z WB J j u z s g

B i

otherwise s s s When dened rigorously is seen to

B B

satisfy the skewsymmetry relation

d d d

s s s s s s

B B

For dimensional reasons the maps called the GromovWitten invariants

are welldened Under certain conditions these invariants are indep endent of

J Notice that when B wehave the triple intersection form the choice of

dened ab ove

Let denote the torsionfree part of the group algebra K H X ZTaking

B B

q B B to denote a basis of H X Z may b e expressed as K q

where q is a formal variable and the addition of exp onents is the group op er

ation of H X This is essentially an algebraic version of the Novikov ring

see x Note that is an integral domain since it is a lo calization of

an in tegral domain pp Dene the quantum cohomology algebra

QH X to have the same additive structure as H X but to have the

quantum multiplication

QH X QH X QH X

dened on elements a b H X QH X by

a b ab q e

B i i

Here e ranges over a basis of H X and the H X are such that

i i

he i In general we view H X as a subring of QH X When

i j ij

X QH X Notice that necessary we will use the injection map H

the q term of the pro duct is exactly the classical cohomology pro duct a b

The multiplication easily extends by linearity to all of QH X That it is

asso ciative is nontrivial see

Extend H X K by linearit y over to a form on QH X

Prop osition The form f endows QH X with a FE structure

Pro of Order a basis fe g for H X so that the corresp onding sequence of

degrees is nondecreasing The matrix having i j th entry equal to e e

i j

is the matrix for the classical Poincare pairing and is a blo ck matrix of the

form

B C

denotes a square blo ck with unsp ecied entries The matrix Here

whose i j th entry is e e is the matrix for the inner pro duct

q i j

determined by f and It is a blo ck matrix of the form

B C

The upp er triangle of blo cks is for dimensional reasons Supp ose je j

l e where fl g Then the denition of je j n and e e

k k k j i j

guarantees that je j n for all k such that l so no term of e e is

k k i j

detected by f It follows that the i j th entry of is

Similarly the blo cks on the crossdiagonal of are exactly the corre

sp onding blo cks on the crossdiagonal of If je j je j n then

i j

h i

f e e f e e q e

i j B i j k k

q e e X

B i j

e e X

i j

since we must have

n je j je j jX j n c B

i j

It is a simple exercise in the denition of determinanttoshow that the de

terminant of a crosstriangular blo ck matrix is the pro duct of the determinants

of the crossdiagonal blo cks Because and share the same crossdiagonal

and the determinant of is a nonzero element of K in fact it is we

see that the inner pro duct dened by is nonsingular By the discussion

in section QH is a FE

When referring to H X QH X as a FA FE we will always mean

H X QH X Note that although H X and QH X share

the same basis fe g and essentially the same FA form the resp ective dual

bases are not necessarily equal In other words the fact that the element e is

the dual in H X to e do es not necessarily imply that e is dual to e

i i

in QH X However it do es hold the q term of e is in fact e It

follows that the q term of the distinguished element of QH X is eX

In other words we have

Prop osition The distinguished element is a deformation of the clas

sical Euler class

Because of this we refer to as the quantum Euler classanddenote

it by e X Unlike eX the quantum Euler class may very well be a unit

Quantum Cohomology of Grassmannian Manifolds

Let G denote the Grassmann manifold of complex k planes in C The

quantum cohomology algebra QH G and its distinguished element can b e

describ ed in several ways The most basic is in terms of generators and

relations The LandauGinzburg p otential provides a means for H G

G to be viewed as Jacobian algebras and thus allows for the and QH

application of This leads to a pro of that lo osely said QH G is

always semisimple Using the tensorpro duct formula for Chern

classes it is p ossible to describ e liftings of the quantum Euler class

The Schub ert calculus behaves well in quantum cohomology and this

leads to a new conjecture regarding liftings of the quantum Euler class

In discussing each approach we return to particular examples for illustrative

purp oses

As ab ove K denotes a eld

Generators and Relations

Let E and F denote the canonical and normal vector bundles over

kn kn

G Sp ecically E has total space

kn kn

E E fv V j V G v V g C G

kn kn kn

and pro jection map v V V G Similarly F has total space

kn kn

E F fv V j v V E E g

kn kn

and pro jection map v V V Here V denotes the n k dimensional

subspace of C which is orthogonal to V

Denote by x x H G the Chern classes of G ie the Chern

k kn kn

classes of E and by y y H G the normal Chern classes of

kn n kn

G ie the Chern classes of F Note that jx j jy j i for all i Dene

kn kn i i

x for i k and y for i n By the Whitney pro duct theorem

i i

and the fact that E F is a trivial bundle we see that these two

kn kn

collections of classes satisfy the following wellknown relationships

x y for j n

i j i

Notice that for i n k these relationships determine the y s as p oly

nomials in the x s The classical cohomology of the Grassmannians is ibid

x y y H G K x

n nk n kn

The quantum cohomology algebra of G has b een shown to be

QH G K q q x x y y q

kn n nk n

Because H G Z Z we simply view q as an additional invertible

variable taking exp onents in Z Notice that q behaves like an element of

degree jy j n

Example If k then X G is simply complex pro jective space

n n n

and K x x C P We have H C P

n n

K q q x x q A QH C P

n n

In this case eX e X nx the FE form b eing x The element

q xe X Recall that K is presumed e X is obviously a unit

q q

to be of characteristic Take K R and for each nonzero r R dene

R q q R to b e the map taking q r By prop osition A

r r P

is semisimple In fact if n is even and r is p ositivethen A decomp oses

r P

Otherwise it is aeld extension

Example Take k and n Then X G has quantum

cohomology

A QH X K q q x x x x x x x x q

The set fx x x x x x x g is a vectorspace basis for A and has the

following multiplication table

x x x x x x x

x x x x x x x qx

x x x x x x x qx qx

x x x x x x x q qx qx

x x x x qx qx qx qx x

x qx qx x qx qx x qx x

Taking the FE form x x it is straightforward to calculate that e X

x q and e X x x q x q The ring extension A has

orthogonal idemp otents e q x x and e The ring eA is isomorphic to

K q q w w q under the map sending ex w and the ring eA

is isomorphic to K q q v v q under the map sending ex v

In other words

K q q v v q K q q w w q A

Take K R and dene as ab ove If r is p ositive then the lefthand

summand of A decomp oses in A If r is negative then the right

G r G

hand summand decomp oses

LandauGinzburg Potential

Dene the Chern p olynomial of X G to b e

k k

Y X

t x t c G

i i t kn

i i

where t is a formal variable and the x s are the Chern classes The are

i i

referred to as the Chern ro ots of G but they are not ro ots of c Ob

kn t

viously x is the ith elementary symmetric p olynomial in the

i i k

Chern ro ots Dene

and

n k

q W

i q k

W qx

The function W is called the LandauGinzburg p otential of G Because

q kn

W and W are symmetric functions in the they may also be viewed as

q i

W W

functions of x x Dene dW to be the ideal and dene

x x

dW similarly Then

K x x dW H G

k kn

and

K q q x x dW QH G

k q kn

Denote by H and H the determinants of the Hessians

W W

H H

x x x x

i j i j

of W and W resp ectively

Prop osition For each X G thereisa K such that H eX

and H e X

q q

Pro of Because H and eX are the q terms of H and e X resp ectively

q q

it suces to prove the prop osition for the quantum case

The p olynomial W is homogenous of degree n x In other

words each summand of W has degree n in QH G where q is

q kn

taken to have degree n Also jx j i for each i Thus for xed i j wehave

jW j jx j jx j jW j i j

q i j q

x x

i j

Wenow showby induction that H is homogenous of degree k n k Each

s s minor M of H is a matrix with entries m H where i and j run

ij ij

over elements of some ordered subsets I J fkg resp ectively where

the minor of M which do es not include I J s Dene M i j to be

the entry m We have already shown that when s the single entry of

each M is homogenous of degree jW ji j Assume that for all minors M

of H of size less than s s the determinant of M is homogenous

of degree

X X

sjW j i j

iI j J

Now consider any s s minor M of H with index sets I J Take

any i I Then

detM sgn i j m detM i j

i j

j J

where sgn is the appropriate function I J f g By the induction

hyp othesis

i j j jm j jM m det M i j

i j i j

X X

A A

j jW ji j sjW j i

q q

j J nfj g iI nfi g

X X

s jW j i j

iI j J

But this is indep endent of the choice of j so detM is homogenous of degree

P P

s jW j i j In particular we can take M H and

iI j J

thus H is homogenous of degree

k k

X X

k jW j i s j k n k k k n k

i j

Of e e course e is also homogenous of degree k n k since e

i q q

where e runs over a basis for H X and since je j k n k je j

i i

Prop ositions and show that H ve X for some unit

q q

v QH X Write v v q where each v H X is homogenous of

ij ij

degree i and jv j jv j if and only if i i and j j For a xed i j we

ij i j

have

j j

v q e X jv j jq j je X j i j nk n k

ij q ij q

k n k either i j n or Since jH j

j j

v q e X v q e X

ij q i j q

i j I J

for some index sets I J But the latter is imp ossible by degree restrictions

due to the denition of v Thus jv q j for all i j In other words v is

ij ij

an element K

Take K R or C and let denote a homomorphism K q q K

as ab ove In the following paragraph any reference to QH X or any ele

ment a therein should be interpreted as referring to QH X and a

resp ectively

In this context the relationship b etween the distinguished element and the

Hessian provides e X with a nontrivial geometric interpretation Denote the

critical p oints of W by z z and note that H maybeviewed as a function

q j

K K asmay all the elements of QH X It is well known that for each

j H z if and only if the critical p oint z is degenerate Because the

j j

elements of QH X viewed as functions are completely determined by their

values on the critical p oints of W we see that H and hence e X is a unit

q q

in QH X if and only if the critical p oints of W are all nondegenerate

The relationship between e X and H also yields a new approach to the

following known result

Prop osition For al l G and al l nonzero r R the algebra QH G

kn r kn

is semisimple

The pro of is based on calculations app earing in

Pro of The Jacobian matrix V x asso ciated to the elemen

i j

tary symmetric functions x is a Vandermonde matrix and has determinant

Let r denote the gradientvector op erator with resp ect to

i j x

x x and let r denote the gradient op erator with resp ect to

k k

Viewing the gradien t op erators as row vectors we have r W V r W

x q q

Let r W denote the ith entry of r W and let V denote the ith row

x q i x q i

of V Then the Hessian of W with resp ect to the s is

T T

r r W r r W V

q x q

T T T

W V r W r V V r r

q x q i i x

Evaluating at the critical points of W ie assuming r W and ex

q x q

pressing everything in terms of the s we see that

r W det V H det r

i j

Now b ecause V is invertible the relation r W isequivalenttor W

x q q

n k

In other words for each i we have q at the critical points of i

Figure From left to right n even and r no critical points n even and

r n odd and r

W This implies that

k k

Y Y

n n k k k

x q

i i

Since q the numerator of H and thus H itself is nonzero at the critical

p oint of W It follows that H as an element of QH G has an inverse

q kn

and therefore by so do es e G By prop osition QH G

q kn r kn

is semisimple

n n

x and the Landau Example For X C P we have W

Ginzburg p otential function is W W qx Take K R and dene as

q r

ab ove In this case H is simply the second derivative For W the second

derivative at a critical point of W is always If n is even then the only

critical p oint is an inection p oint and if n is o dd then the only critical p oint

is at the origin where for n the second derivative is anyway For the

quantum case the fact that H is nonzero simply says that all critical p oints

of W are either maxima or minima Figure shows generic examples

of what will happ en for various choices of n and r These cases corresp ond

directly to the cases for the decomp osition of A

Example For X G solving for x x in W yields

W x x x x x and W W qx The partial derivatives of W are

q q

x x x x q

and

x x x

Although these are not the same p olynomials as the relations obtained ab ove

they do generate the same ideal as exp ected We now apply for some

r R The critical curves for W x are given by x andy x These

are indep endent of r On the other hand the critical curves for W x do

dep end on r They are given by

q q

p p

x y y r and x y y r

Figures and show generic examples of what happ ens for various choices

of r The surface shown is given by z W x y In each case there

r q

are only two critical points ie where b oth partial derivatives are This is

b ecause the application of to A is equivalent to taking the quotientbythe

r G

ideal q r If r then the critical points lie along the line x x

and the idemp otent e x x r evaluated at the critical points is If

r then the critical p oints are x y r r so the idemp otent

e takes the value Thus in each case the summand which do es not

decomp ose is in fact annihilated

Lifting the Euler Class

Recall that E and F denote the canonical and normal vector

kn kn

is dened to have total bundles over G resp ectively The dual bundle E

kn kn

Figure The classical case r The rst o ctantisshown in the far upp er

right and the critical curves intersect at the origin which is a saddlep oint

space

E E fv V j V G v HomV C g

and the obvious pro jection map

Prop osition The tangent bund le T of G is isomorphic to the ten

kn kn

sor product E F

Pro of Sketch

Let L be a pointinG ie a k plane through the origin in C Points in

G near to L can be viewed as elemen ts HomL L since every such

element can be identied with its graph in L L C If e e form a

i k

basis for L then for each i denote e by v For small enough in R

i i

if t then the elements e v t are indep endent so dene a p oint in

i i

v te v t can be viewed as a curve in G through G Thus e

k k kn kn

L which determines a unique tangent vector to L

Figure The rst o ctant is shown in the far upp er right the intersection of the

critical curves for W x marks the origin Ab ove r p ositive Below r negative

On the other hand if a tangent vector to L is determined by the curve

T t T tT t in G with T L then we can dene

k kn

HomL L by e T e

i i i

It follows that for all L that the b er T of T ab ove L is isomorphic

kn L kn

to Hom L L HomE F Since the Hom functor b eha ves well

kn L kn L

with resp ect to bundles the result follows

As discussed ab ove the Chern classes x x arising from the bundle

E are the elementary symmetric p olynomials in the Chern ro ots

kn k

An analogous situation holds for the normal classes which arise from the

bundle F Dene to be the Chern ro ots corresp onding to the

kn nk

formal p olynomial

y t

Then for all i we have y In fact the s and s are

i i nk i i

the rst Chern classes of the line bundles in the splitting of E and F

kn kn

resp ectively x Together with this fact allows us to write the char

acteristic classes c T in terms of the x s and y s The Chern p olynomial

i kn i i

for T is

k nk

Y X

t c T t

j i i kn

ij i

j in the pro duct range over p ossible indices ibid In other where i and

words for each i we have c T f g This shows that each

i kn i j i ij

c T is symmetric in the s and the s and can therefore b e written in

i kn i j

terms of the x s and y s

i j

In particular the Euler class eG can be lifted to a p olynomial

P K x x y y

k nk

or using the relations between the x s and y s to a p olynomial

i j

P K x x

P is referred to as the Euler p olynomial

Bertram has proven the following

Prop osition For each k n the Euler polynomial P is a lifting of

H QH G

q kn

is a lifting of e Corollary For each k n P G and we have

q kn

n n

eG H and e G H

kn q kn q

Pro of Prop osition shows that e e G H for some

q q kn q

K Let QH G H G denote the mo dule homomorphism

kn kn

we sending q By denition P is a lifting of e eG so by

have

K x x

H G QH G q

kn kn

where the unlab eled arrows are the canonical pro jection maps Now e e

by denition of the quantum multiplication so prop osition shows that

n n n

e e H e

q q

and thus This proves the second claim By the rst claim

now follows as well

Example For k the top Chern class of T lifts to

n n

K x y y

ni n n

This is simply x y As elements of A we have y

ni P ni

ni n n

x and thus P maps down to e C P nx

Example For k n we have

After multiplying this out and solving for the x s and y s we see that

i i

x y y y y x y x y x x y x P x

Applying the relations y x and y x x yields P x x

x x K x x To see what this maps to in A use the multiplication

table given ab ove to rewrite this p olynomial in terms of the given basis We

have x x x q and x x x so P maps to e G x x q

Schub ert Calculus

The classical cohomology H G can be describ ed in a very convenient

combinatorial manner For a more detailed intro duction than given here see

A Schub ert symbol for G is an ordered k tuple a a satisfying

kn k

a a n k Each such Schub ert symbol represents an

element of H G of degree a and the set of all p ossible Schub ert

kn i

symbols forms a free additive basis called the Schub ert basis for H G

In particular we have

x and y i

i i

for all i such that the Schub ert symb ols make sense The multiplicative struc

ture of H G is realized with this basis via the following two formulas kn

Pieri formula

b b a a y

k k j

a b a

i i i

P P

b j a

i i

Here we formally dene a and a n k

Giambel li formula

a a det y

k a ij

Here we formally dene y for j and j n k

By rep eatedly alternating applications of these two formulas it is p ossible

to compute any pro duct of elements expressed in the Schub ert basis

Of course the Schub ert basis also serves as a basis for QH G but in

addition to this the ab ove formulae generalize nicely to the quantum case

Let denote the multiplication map QH G QH G QH G

kn kn kn

Extending the Giamb elli formula by linearityallows us to view it as providing

Bertram has proven that the a lifting G H G K y y

k kn kn n

following diagram commutes

K q q y y K q q y y

nk nk

G G

kn kn

QH QH QH

H H

QH QH

In other words extending G by linearity to QH G provides a mo dule

kn kn

homomorphic lifting QH G K q q y y which we will also

kn nk

denote by G

Bertram has also develop ed a quantum Pieri formula but wewillnothave

use for that here

Finally note that if a a is a Schub ert symbol representing an ele

ment a H G then the symb ol corresp onding to a is n k a n

kn k

k a

The map G G which sends an n k plane L C to the

nkn kn

k plane L orthogonal to L is a homeomorphism whic h provides a canonical

isomorphism H G H G Under this isomorphism we may

kn nkn

view the Chern classes of G as the normal classes of G and viceversa

kn nkn

Dene G H G K x x to be the comp osite G Of

kn k nkn

course this map will also extend by linearity to QH G Finally via the

obvious inclusion maps we may view both G and G as b eing liftings as

depicted in the following diagram

K q q x x y y

k nk

K q q y y K q q x x

nk k

G G

kn nkn

QH G QH G

kn nkn

Conjecture The lifting P G e G e where e and e

G i kn i

i i

range over a basis and the corresponding dual basis of H G is exactly the

polynomial P K x x y y

k nk

That P has no q terms follows from the fact that all elements lifted b elong

to H G It is clear from the prop erties of G that P is in fact a lifting

kn kn G

of eG However the surprising implication of this conjecture is that P is

kn G

also a lifting of e G

q kn

Example The situation for G C P is quite satisfactory

Prop osition The conjecture holds for C P

Pro of The Schub ert symb ols for G are n and the symb ols

for G are For each i sends

x y i

G sends i y and G sends

n i nn

C B

det y

C B

Here we use x and y to denote the Chern classes and normal classes of

i i

G resp ectively Now i n i so

nkn

X X X

i i i

P G i G n i y x x y

G kn ni

kn ni

i i i

which is P

In order to calculate more complicated examples it is convenient to use

the Young diagram representation of the Schub ert symb ols Up to sign the

image under of the symb ol corresp onding to such a diagram can b e read o

Figure The symbol a H G is shown in white and

a is shaded Up to sign a and a

by reading columns rather than rows We reiterate that the sign change arises

b ecause H G H G maps

kn nkn

i i i

x y i H G

i nkn

Note that is determined by its values on the x s since it is an isomorphism of

rings Figure shows an example of a Young diagram and the corresp onding

dual diagram

Example

For G we have the following data note the implicit use of the isomor

phisms x y and y x

i i

a a a G a G a

x y y

x x y

x y y

x x y

From the last two columns of the table it is easy to read o

P y x y y x y x y y x x x y x

which is exactly P

A pro of of the ab ove conjecture will no doubt require a complex combi

natorial analysis of the relationship between the tensor pro duct formula for

belli formula However vector bundles and the Giam

Computer calculation has veried the conjecture for al l G with n

Quantum Cohomology of Hyp erplanes

For the sake of contrast with the Grassmannians this section provides another

class of examples of a quantum cohomology ring and determines which of

these are semisimple In Tian and Xu discuss a more general class of

examples along these lines from the point of view of semisimplicity in the

sense of Dubrovin as dened in the intro duction

intersection of degree d d Let X C P be a smo oth complete

and dimension n satisfying n d Let denote the hyp erplane i

class generating H X Z By the primitive cohomology H X of X we

n n n

mean H X ifn is o dd and the subspace of H X orthogonal to if n is

even Beauville shows in although he unnecessarily presumes q that

QH X is the algebra over K q q generated by andH X sub ject to

the relations

n d d

d d q

and for all a b H X

d n d

a and ab ha bi q d d

Here h i denotes the classical intersection form a b f a b where

Assume now that X is a hyp erplane of degree d Denote H X K

by R and cho ose a basis e e for H X Together with the elements

this provides a full vectorspace basis for QH X Thus the

distinguished element of QH X fis

P P

R n

i ni

e e

i i

n n d d

n d q

Notice that if d then is divisible by so e for example Since

e Ker it is a basis element for an co ecients prop osition shows

that QH X is not semisimple for any choice of r K

If d then we have

Rq n

and thus nq Order the basis for QH X as follows

e e Then the matrix corresp onding to under the regular rep

resentation is

Rq n

B C

n q

B C

n q

B C

n q

B C

Since the determinant of this matrix is a unit in K q q we see that is

a unit in QH X by this showsthat QH X is semisimple for any

choice of r

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Vita

The author was born and raised in Washington DC He received his BA

in mathematics from Yeshiva University in and his MA in mathematics

from The Johns Hopkins University in The author is married and has

two young children