ORBIFOLDING FROBENIUS ALGEBRAS

RALPH M. KAUFMANN

UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGELES, USA^{∗}

Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define a geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super–graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi–homogeneous singularities and their symmetries.

Introduction

The subject of this exposition is the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures stemming from a geometry associated to a physical theory with a global finite gauge group [DW,DVVV,IV,V]. The most prominent example of this type in mathematics is the Gromov– Witten theory of orbifolds [CR], which are global quotients. The use of orbifold constructions, however, is much older and is the cornerstone of the original mirror construction [GP]. The orbifolds under study in that context are so–called Landau–Ginzburg orbifold theories, which have so far nor been studied mathematically. These correspond to the Frobenius manifolds coming from singularities and are studied in detail in the present paper.

A common aspect of the physical treatment of quotients by group actions is the appearance of so called twisted sectors. This roughly means that if one wishes to take the quantum version of quotient by a group action, one first has to construct an object for each element in the group together with a group action on this object and in a second step take invariants in all of these components. Whereas classically one considers only the G–invariants of the original object which constitutes the sector associated to the identity, the untwisted sector.

∗ Partially supported by NSF grant #0070681

The present work axiomatizes this procedure and points out that the construction of twisted sectors is not merely an auxiliary artifact, but is essential. This is clearly visible in the case of the Frobenius algebra associated to the singularity of type A_{n} together with a Z/nZ+1 action and the singularity of type A_{2n−3} with Z/2Z action, which are worked out in detail in the last paragraph. This particular example exhibits a version of Mirror symmetry in which it is self–dual. The twisted sectors,however, shed some light on this duality, since in reality the sum of twisted sectors is dual to the untwisted one.

In the present work, we develop the theory of orbifold Frobenius algebras along the now classical lines of Atiyah, Dubrovin, Dijkgraaf and Segal. I.e. we start by introducing the algebraic structures in an axiomatic fashion. There is a slight difference to the theory with trivial group in that there are two structures with slightly different G-action to be considered which are nevertheless present. These versions differ by a twists with a character. In the singularity version, on the non–twisted level, they correspond to the different G–modules on the cohomology and the Milnor ring [W]. In physical terms these two structures are related by spectral flow. To be even more precise, one structure carries the natural multiplication and the other the natural scalar product. The next step is a cobordism realization of the theory. There are ramifications of the present exposition with [FQ, T], if the above mentioned character is trivial. This is, however, not necessarily the case and in the most interesting examples worked out in the last paragraph, this is not the case. The key structure here is a non–commutative multiplication on the sum of all the twisted sectors, even before taking invariants.

To provide a larger basis of examples we introduce so–called special G–Frobenius algebras, which have as a subgroup Jacobian G–Frobenius algebras which in turn encompass the singularity examples or those of manifolds whose cohomology ring can be described as a quotient by a Jacobian ideal. Here it is important to note that everything can be done in a super (i.e. Z/2Z–graded) version. This introduces a new degree of freedom into the construction corresponding to the choice of parity for the twisted sectors. Lastly, we explicitely work out several examples including the transition from the singluarity A_{2n+3} to D_{n} via a quotient by Z/2Z.

In the case that the Frobenius algebra one starts out with comes from a semi–simple Frobenius manifold and the quotient of the twised sector is not trivial, then there has a unique extension to the level of Frobenius manifolds. This is the case in the above example of A_{2n+3} and D_{n}.

Acknowledgments

This work is the written version of talks presented at the WAGP2000 conference in October 2000 , which contained all agebraic constructions and exaples, and at the workshop on “Mathematical apspects of orbifold string theoy” in May 2001, which contained the geometric part of the third paragaph. I wish to thank the organizers of these conferences, which were immensely stimulating.

I would also like to thank the Max–Planck Institut fr̈ Mathematik in Bonn where much of this work was completed during my stays in 2000 and 2001. The research has profited from conversations with Yu. Manin, Y. Ruan, V. Turaev, C. Hertling and B. Penner whom it is a pleasure to thank.

Finally I would like to acknowledge the support from the NSF grant #0070681.

1. Frobenius Algebras and Cobordisms

In this paragraph, we recall the definition of a Frobenius algebra and its relation to the cobordism–category definition of a topological field theory [A,Du,Dij].

1.1. Frobenius algebras.

1.1.1. Definition. A Frobenius algebra (FA) over a field k of characteristic 0 is ⟨A, ∘, η, e⟩, where

A is a finite dim k–vector space,
∘ is a multiplication on A: ∘ : A ⊗ A → A,
η is a non-degenerate bilinear form on A, and
1 is a fixed element in A – the unit.

satisfying the following axioms:

a) Associativity (a ∘ b) ∘ c = a ∘ (b ∘ c)

b) Commutativity (a ∘ b) = b ∘ a

c) Unit: ∀ ∈ a : 1 ∘ a = a ∘ 1 = a

d) Invariance:

η(a, b ∘ c) = η(a ∘ b, c)

1.1.2. Remark. By using η to identify A and A^{∗} – the dual vector space of A– these objects define a one–form ϵ ∈ A^{∗} called the co–unit and a three–tensor μ ∈ A^{∗} ⊗ A^{∗} ⊗ A^{∗}

Using dualization and invariance these data are interchangeable with η and ∘ via the following formulas. Explicitly, after fixing a basis (Δ_{i})_{i∈I} of A, setting η_{ij} := η(Δ_{i}, Δ_{j}) and denoting the inverse metric by η^{ij}

ϵ(a) := η(a, 1),

μ(a, b, c) := η(a ∘ b, c) = ϵ(a ∘ b ∘ c),

a ∘ b = ∑_{ij} μ(a, b, Δ_{i})η^{ij}Δ_{j} and

η(a, b) = ϵ(a ∘ b).

1.1.3. Notation. We call ρ ∈ A the element dual to ϵ. This is the element which is Poincaré dual to 1.

1.1.4. Grading. A graded Frobenius algebra is a Frobenius algebra together with an group grading of the vector space A: A = ⨁_{i∈I} A_{i} together with the following compatibility equations: denote the I–degree of an element by deg.

1) 1 is homogeneous; 1 ∈ A_{d} for some d ∈ I

2) η is homogeneous of degree d + D.

I.e. for homogeneous elements a, b η(a, b) = 0 unless deg(a) + deg(b) = d + D.

This means that ϵ and ρ are of degree D.

3) ∘ is of degree d, this means that μ is of degree 2d+D, where again

this means that deg(a ∘ b) = deg(a) + deg(b) + d

1.1.5. Definition. An even derivation E ∈ Der(A, A) of a G twisted Frobenius algebra A is called an Euler field, if it is conformal and is natural w.r.t. the multiplication. i.e. for some d, D ∈ k it satisfies:

η(Ea, b) + η(a, Eb) = Dη(a, b)  (1.1)

and

E(ab) = Ea b + a Eb = d ab  (1.2)

Such derivation will define a grading on A by its set of eigenvalues.

1.1.6. Remark. For this type of grading, we will use the group Q. There are two more versions of grading: 1) a Z/2Z super-grading, which will be discussed in 1.1.13 and 2) a grading by a finite group G, which is the content section 2.1.

1.1.7. Definition. Given an I–graded Frobenius algebra A, we define its characteristic series as

χ_{A}(t) := ∑_{i∈I} dim(A_{i})t^{i}  (1.3)

We refer to the set {d, D; i : dim A_{i} ≠ 0} as the (I–)spectrum of A.

1.1.8. Scaling. If the group indexing the grading has the structure of a Λ-module, where Λ is a ring, we can scale the grading by an element λ ∈ Λ. We denote the scaled Frobenius algebra by λA := ⨁_{i∈I} A_{λi} where (λA)_{i} = A_{λi}

χ_{λA}(t) := ∑_{i∈I} dim(λA_{i})t^{i} = t^{λ}χ_{A}(t)  (1.4)

It is sometimes –but not always– convenient to normalize in such a way that deg(e) = e In the case that the grading is given by Q, this means deg_{Q}(1) = 0 and in the finite group case deg_{G}(1) = e.

1.1.9. Operations. There are two natural operations on Frobenius algebras, the direct sum and the tensor product. Both of these operation extend to the level of Frobenius manifold, while the generalization of the direct sum is straight forward, the generalization of the tensor product to the level of Frobenius manifolds is quite intricate [K1].

Consider two Frobenius algebras A′ = ⟨A′, ∘′, η′, e′⟩ and A′′ = ⟨A′′, ∘′′, η′′, e′′⟩ .

1.1.10. Direct sum. Set A′ ⊕ A′′ := ⟨A, ∘, η, e⟩ = ⟨A′ ⊕ A′′, ∘′ ⊕ ∘′′, η′ ⊕ η′′, e′ ⊕ e′′⟩. E.g. (a′, a′′) ∘ (b′, b′′) = (a′ ∘′ b′, a′′ ∘′′ b′′). The unit is 1 = 1′ ⊕ 1′′ and the co-unit is ϵ = (ϵ′, ϵ′′).

1.1.11. Lemma. If both Frobenius algebras are graded by the same I then their direct sum inherits a natural grading if and only if the gradings can be scaled s.t.

D′ + d′ = D′′ + d′′ := D + d  (1.5)

where

D = D′ = D′′  (1.6)

in this scaling.

Furthermore, the unit will have degree d′ = d′′ = d.

Proof

The equation 1.5 insures that the three tensor μ is homogeneous of degree D + d. The homogeneity of η yields the second condition: for η to be homogeneous, we must have that after scaling ρ′ and ρ′′ are homogeneous of the same degree D′ = D′′ = D. The two equations together imply the homogeneity of 1 = (1′, 1′′) of degree d = d′ = d′′.

1.1.12. Tensor product. Set A′⊗A′′ := ⟨A, ∘, η, e⟩ = ⟨A′⊗A′′, ∘′⊗ ∘′′, η′ ⊗ η′′, e′ ⊗ e′′⟩. E.g. (a′, a′′) ∘ (b′, b′′) = (a′ ∘′ b′, a′′ ∘′′ b′′). The unit is 1 = 1′ ⊗ 1′′ and the co-unit is ϵ = ϵ′ ⊗ ϵ′′.

There areno conditions for grading. I.e. if both Frobenius algebras are I–graded there is a natural induced I–grading on their tensor product. The unit is of degree d = d′ + d′′, the co–unit has degree D′ + D′′ and the multiplication is homogeneous of degree d = d′ + d′′.

1.1.13. Super-grading. For element a of a super vector space A = A_{0} ⊕ A_{1} denote by ã its Z/2Z degree, i.e. ã = 0 if it is even (a ∈ A_{0}) and ã = 1 if it is odd (a ∈ A_{1}).

1.1.14. Definition. A super Frobenius algebra a field k of characteristic 0 is ⟨A∘, η, e⟩, where

A is a finite dim k–super vector space,
∘ is a mutiplication on A: ∘ : A ⊗ A → A,

which preserves the Z/2Z–grading

η is a non-degenerate even bilinear form on A, and
1 is a fixed even element in A_{0} – the unit.

satisfying the following axioms:

a) Associativity (a ∘ b) ∘ c = a ∘ (b ∘ c)

b) Super–commutativity (a ∘ b) = (−1)^{ãb̃}b ∘ a

c) Unit: ∀ ∈ a : 1 ∘ a = a ∘ 1 = a

d) Invariance:

η(a, b ∘ c) = η(a ∘ b, c) The grading for super Frobenius algebras carries over verbatim.

1.2. Cobordisms.

1.2.1. Definition. Let COB be the category whose objects are one– dimensional closed oriented (topological) manifolds considered up to orientation preserving homeomorphism and whose morphisms are cobordisms of these objects.

I.e. Σ ∈ Hom(S_{1}, S_{2}) if Σ is an oriented surface with boundary ∂Σ ≡ −S_{1} ∐ S_{2}.

The composition of morphisms is given by glueing along boundaries with respect to orientation reversing homeomorphisms.

1.2.2. Remark. The operation of disjoint union makes this category into monoidal category with unit ∅.

1.2.3. Remark. The objects can be chosen to be reprsented by disjoint unions of the circle with the natural orientation S^{1} and the circle with opposite orientation S̄^{1}. Thus a typical object looks like S = ∐_{i∈I} S^{1} ∐_{j∈J} S̄^{1}. Two standard morpisms are given by the cylinder, and thrice punctured sphere.

1.2.4. Definition. Let VECT_{k} be the monoidal category of finite dimensional k–vector spaces with linear morphisms with the tensor product providing a monoidal structure with unit k.

1.2.5. Theorem. (Atiyah, Dijkgraaf, Dubrovin) [A,Dij, Du] There is a 1–1 correspondence between Frobenius algebras over k and isomorphism classes of covariant functors of monoidal categories from COB to VECT_{k}, natural with respect to orientation preserving homeomorphisms of cobordisms and whose value on cylinders S ×I ∈ Hom(S, S) is the identity.

Under this identification, the Frobenius algebra A is the image of S^{1}, the multiplication or rather μ is the image of a thrice punctured sphere and the metric is the image of an annulus.

2. Orbifold Frobenius algebras

2.1. G–Frobenius algebras. We fix a finite group G and denote its neutral element by e.

2.1.1. Definition. A G–Frobenius algebra (FA) over a field k of characteristic 0 is < G, A, ∘, 1, η, φ, χ >, where

G finite group
A finite dim G-graded k–vector space

A = ⊕_{g∈G}A_{g} A_{e} is called the untwisted sector and the A_{g} for g ≠ e are called the twisted sectors.

∘ a multiplication on A which respects the grading:

∘ : A_{g} ⊗ A_{h} → A_{gh}

1 a fixed element in A_{e}–the unit
η non-degenerate bilinear form

which respects grading i.e. g|_{A_{g}⊗A_{h}} = 0 unless gh = e.

φ an action by algebra automorphisms of G on A,

φ ∈ Hom_{k−alg}(G, A), s.t. φ_{g}(A_{h}) ⊂ A_{ghg^{−1}}

χ a character χ ∈ Hom(G, k^{∗})

Satisfying the following axioms: Notation: We use a subscript on an element of A to signify that it has homogeneous group degree –e.g. a_{g} means a_{g} ∈ A_{g}– and we write φ_{g} := φ(g) and χ_{g} := χ(g).

a) Associativity (a_{g} ∘ a_{h}) ∘ a_{k} = a_{g} ∘ (a_{h} ∘ a_{k})

b) Twisted commutativity a_{g} ∘ a_{h} = φ_{g}(a_{h}) ∘ a_{g}

c) G Invariant Unit: 1 ∘ a_{g} = a_{g} ∘ 1 = a_{g} and φ_{g}(1) = 1

d) Invariance of the metric:

η(a_{g}, a_{h} ∘ a_{k}) = η(a_{g} ∘ a_{h}, a_{k})

i) Projective self–invariance of the twisted sectors φ_{g}|A_{g} = χ_{g}^{−1}id

ii) G–Invariance of the multiplication

φ_{k}(a_{g} ∘ a_{h}) = φ_{k}(a_{g}) ∘ φ_{k}(a_{h}) iii) Projective G–invariance of the metric

φ_{g}^{∗}(η) = χ_{g}^{−2}η iv) Projective trace axiom

∀c ∈ A_{[g,h]} and l_{c} left multiplication by c: χ_{h}Tr(l_{c}φ_{h}|_{A_{g}}) = χ_{g^{−1}}Tr(φ_{g^{−1}}l_{c}|_{A_{h}})

An alternate choice of data is given by a one–form ϵ the co–unit with ϵ ∈ A_{e}^{∗} and a three–tensor μ ∈ A^{∗} ⊗ A^{∗} ⊗ A^{∗} which is of group degree e, i.e. μ|_{A_{g}⊗A_{h}⊗A_{k}} = 0 unless ghk = e.

The relations between η, ∘ and ϵ, μ are analogous to those of 1.1.2.

Again, we denote by ρ ∈ A_{e} the element dual to ϵ ∈ A_{e}^{∗} and Poincaré dual to 1 ∈ A_{e}.

2.1.2. Remarks.

1) A_{e} is central by twisted commutativity and ⟨A_{e}, ∘, η|_{A_{e}⊗A_{e}}, 1⟩ is a Frobenius algebra.

2) All A_{g} are A_{e}-modules.

3) Notice that χ satisfies the following equation which completely determine it in terms of φ. Setting h = e, c = 1 in axiom iv)

dim A_{g} = χ_{g^{−1}}Tr(φ_{g}|_{A_{e}})  (2.1)

by axiom iii) the action of φ on ρ determines χ up to a sign

χ_{g}^{−2} = χ_{g}^{−2}η(ρ, 1) = η(φ_{g}(ρ), φ_{g}(1)) = η(φ_{g}(ρ), 1)  (2.2)

4) Axiom iv) forces the χ to be group homomorphisms, so it would be enough to assume in the data that they are just maps.

2.1.3. Proposition. The G invariants A^{G} of a G–Frobenius algebra A form an associative and commutative algebra with unit. This algebra with the induced bilinear form is a Frobenius algebra if and only if ∑_{g} χ_{g}^{−2} = |G|. If k = C and χ ∈ Hom(G, U(1)) this implies ∀g : χ_{g} = ±1

Proof. Due to axiom ii) the algebra is associative and commutative. And since 1 is G invariant, the algebra has a unit.

Now suppose ∑_{g} χ_{g}^{−2} = |G|. Then η|_{A^{G}⊗A^{G}} is non–degenerate: Let a ∈ A^{G} and choose b ∈ A s.t. η(a, b) ≠ 0. Set b̃ = \frac{1}{|G|} ∑_{g∈G} φ_{g}(b) ∈ A^{G}. Then

η(a, b̃) = ∑_{g∈G} η(a, φ_{g}(b)) = ∑_{g∈G} η(φ_{g}(a)φ_{g}(b)) = \frac{1}{|G|} ∑_{g∈G} χ_{g}^{−2}η(a, b)
= η(a, b) ≠ 0

On the other hand if η|_{A^{G}⊗A^{G}} is non–degenerate then let a, b ∈ A^{G} be such that η(a, b) = 1. It follows:

1 = η(a, b) = \frac{1}{|G|} ∑_{g∈G} η(φ_{g}(a), φ_{g}(b)) = \frac{1}{|G|} ∑_{g∈G} χ_{g}^{−2}η(a, b) = \frac{1}{|G|} ∑_{g∈G} χ_{g}^{−2}

so that ∑_{g} χ_{g}^{−2} = |G|.

The last statement follows from the simple fact that if |χ_{g}| = 1 and ∑_{g} χ_{g}^{−2} = |G| then χ_{g}^{−2} = 1 and hence χ_{g} ∈ {−1, 1}.

2.1.4. Definition. A G–Frobenius algebra is called an orbifold model if the data ⟨A^{G}, ∘, 1⟩ can be augmented by a compatible metric to yield a Frobenius algebra. In this case, we call the Frobenius algebra A^{G} a G–orbifold Frobenius algebra.

2.2. Super-grading. We can enlarge the framework by considering super-algebras rather than algebras. This will introduce the standard signs.

2.2.1. Definition. A G-twisted Frobenius super-algebra is over a field k of characteristic 0 is < G, A, ∘, 1, η, φ, χ >, where

G finite group
A finite dimensional Z/2Z × G-graded k–vector space

A = A_{0} ⊕ A_{1} = ⊕_{g∈G}(A_{g,0} ⊕ A_{g,1}) = ⊕_{g∈G}A_{g} A_{e} is called the untwisted sector and is even. The A_{g} for g ≠ e are called the twisted sectors.

∘ a multiplication on A which respects both gradings:

∘ : A_{g,i} ⊗ A_{h,j} → A_{gh,ij}

1 a fixed element in A_{e}–the unit
η non-degenerate even bilinear form

which respects grading i.e. g|_{A_{g}⊗A_{h}} = 0 unless gh = e.

φ an action by even algebra automorphisms of G on A,

φ ∈ Hom_{k−alg}(G, A), s.t. φ_{g}(A_{h}) ⊂ A_{ghg^{−1}}

χ a character χ ∈ Hom(G, k^{∗})

or if k = C, χ ∈ Hom(G, U(1))

Satisfying the axioms a)–d) and i)–iii) of a G–Frobenius algebra with the following alteration: b^{σ}) Twisted super–commutativity

a_{g} ∘ a_{h} = (−1)^{ã_{g}ã_{h}}φ_{g}(a_{h}) ∘ a_{g} iv^{σ}) Projective super–trace axiom

∀c ∈ A_{[g,h]} and l_{c} left multiplication by c:

χ_{h}STr(l_{c}φ_{h}|_{A_{g}}) = χ_{g^{−1}}STr(φ_{g^{−1}}l_{c}|_{A_{h}})

where STr is the super–trace.

2.2.2. Operations. Restriction

If H ⊂ G and A = ⊕_{g∈G}A_{g} then Ã := ⨁_{h∈H} A_{h} is naturally a H–Frobenius algebra.

Direct Sum Given a G–Frobenius algebra A and an H–Frobenius algebra B then A ⊕ B is naturally a G × H–Frobenius algebra with the graded pieces (A ⊕ B)_{(g,h)} = A_{g} ⊕ B_{h}.

We define the direct sum of two G–Frobenius algebras to be the G– Frobenius subalgebra corresponding to the diagonal Δ : G → G × G in A ⊕ A.

Tensor product Given a G–Frobenius algebra A and an H–Frobenius algebra B then ⨁_{(g,h)}(A_{g} ⊗B_{h}) is naturally a G×H–Frobenius algebra (A ⊗ B)_{(g,h)} = A_{g} ⊗ B_{h}.

We define the tensor product of two G–Frobenius algebras to be the G–Frobenius subalgebra corresponding to the diagonal G → G × G in A ⊕ A.

Braided Tensor product. If A and B are two G–Frobenius algebras with the same character χ, we can define a tensor on A ⊗ B by setting (A ⊗ B)_{g} := ⊕_{k∈G}A_{k} ⊗ B_{k^{−1}g}. For the multiplication we use the sequence

A_{k}⊗B_{k^{−1}g}⊗A_{l}⊗B_{l^{−1}h} ⟶^{(id⊗id⊗φ_{k^{−1}g}⊗id)∘τ_{2,3}} A_{k}⊗A_{k^{−1}glg^{−1}k}⊗B_{k^{−1}g}⊗B_{l^{−1}h}

→^{∘⊗∘} A_{glg^{−1}k} ⊗ B_{k^{−1}gl^{−1}h} (2.3)

and ⨁_{k}(φ_{k} ⊗ φ_{kh^{−1}}) for the action of h on ⨁_{k} A_{k} ⊗ B_{k^{−1}g}

Remark. If one thinks in terms of cohomology of spaces the the direct sum corresponds to the disjoint union and the tensor product corresponds to the Cartesian product. The origin of the braided tensor product, however, is not clear yet.

2.3. Geometric model – spectral flow. The axioms of the G– Frobenius algebra are well suited for taking the quotient, since the multiplication is G–invariant. However, this is not the right framework for a geometric interpretation. In order to accommodate a more natural coboundary description, we need the following definition:

2.3.1. Definition. A Ramond G–algebra over a field k of characteristic 0 is < G, V, ¯∘, v, η̄, φ̄, χ >

G finite group
V finite dim G-graded k–vector space

V = ⊕_{g∈G}V_{g}

V_{e} is called the untwisted sector and

the V_{g} for g ≠ e are called the twisted sectors.

¯∘ a multiplication on V which respects the grading:

¯∘ : V_{g} ⊗ V_{h} → V_{gh}

1 a fixed element in V_{e}–the unit

η̄ non-degenerate bilinear form

which respects grading i.e. η̄|_{V_{g}⊗V_{h}} = 0 unless gh = e.

φ̄ an action by algebra automorphisms of G on V,

φ̄ ∈ Hom_{k−alg}(G, V), s.t. φ̄_{g}(V_{h}) ⊂ V_{ghg^{−1}}

χ a character χ ∈ Hom(G, k^{∗})

Satisfying the following axioms: Notation: We use a subscript on an element of V to signify that it has homogeneous group degree –e.g. v_{g} means v_{g} ∈ V_{g}– and we write φ̄_{g} := φ̄(g) and χ_{g} := χ(g).

a) Associativity (v_{g}¯∘v_{h})¯∘v_{k} = v_{g}¯∘(v_{h}¯∘v_{k})

b’) Projective twisted commutativity

v_{g}¯∘v_{h} = χ_{g}φ̄_{g}(v_{h})¯∘v_{g} = φ̄_{g}(v_{h}¯∘v_{g}) c’) Projectively invariant unit:

v¯∘v_{g} = v_{g}¯∘v = v_{g} and φ̄_{g}(v_{0}) = 1χ_{g}v_{0}

d) Invariance of the mtric:

η(v_{g}, v_{h}¯∘v_{k}) = η(v_{g}¯∘v_{h}, v_{k}) 1’) Self–invariance of the twisted sectors

φ̄_{g}|V_{g} = id 2’) Projective G–invariance of multiplication

φ̄_{k}(v_{g}¯∘v_{h}) = χ_{k}φ̄_{k}(v_{g})¯∘φ̄_{k}(v_{h}) 3’) G–Invariance of metric

φ̄_{g}^{∗}(ḡ) = g 4’) Trace axiom

∀c ∈ V_{[g,h]} and l_{c} left multiplication by c: Tr(l_{c}¯∘φ̄_{g}|_{V_{h}}) = Tr(φ̄_{h^{−1}} ¯∘l_{c}|_{V_{g}})

2.3.2. Definition. A state space for a G–Frobenius algebra A is a triple ⟨V, v, η̄, φ̄⟩, where

V is a G-graded free rank one A–module: V = ⨁_{g∈G} V_{g},
v is a fixed generator of V – called the vacuum,
η̄ is non–degenreate bilinear form on V
φ̄ is a linear G-action on V fixing v projectively

i.e. φ̄(g)(span(v)) ⊂ span(v) such that these stuctures are compatible with those of A (we denote

φ̄_{g} := φ̄(g)

a) The action of A respects the grading: A_{g}V_{h} ⊂ V_{gh}.

b) V_{h} is a rank one free A_{h}–module and V_{h} = A_{h}v.

c) φ̄_{g}(av) = φ_{g}(a)φ̄(v) : ∀a ∈ A, v ∈ V d) For a, b ∈ A : η̄(av, bv) = η(a, b)

e) ∀gβ ∈ π, c ∈ A_{[g,h]} ∀c ∈ V_{[g,h]} and l_{c} left multiplication by c: Tr(l_{c}¯∘φ̄_{g}|_{V_{h}}) = Tr(φ̄_{h^{−1}} ¯∘l_{c}|_{V_{g}})

2.3.3. Remark. A state space inherits an associative multiplication ¯∘ with unit from A via

∀a, b ∈ A : (av)¯∘(bv) := (a ∘ b)v  (2.4)

2.3.4. Definition. We call two state spaces isomorphic, if there is a A–module isomorphism between the two.

Since state spaces are free rank one A–modules It is clear that all automorphisms are rescalings of v.

2.4. Proposition. Given a G–twisted Frobenius algebra A then there is at most a unique state space up to isomorphism and the form η̄ is G–invariant (i.e. φ̄_{g}^{∗}(η̄) = η̄).

Proof. We start with a free rank one A–module V and reconstruct all other data. The G–grading on V is uniquely determined from that of A by axiom b) and this grading satisfies axiom a). Up to isomorphism, we may assume a generator v ∈ V is fixed, then axiom d) determines η̄ from η. We denote the Eigenvalue of φ̄_{g} on v by λ_{g}: φ̄_{g}v = λ_{g}v Notice that due to c) φ̄ is determined by λ_{g}. Using axioms b), c) and e), we find that λ_{g} = χ_{g}, thus fixing the G–action φ̄.

Namely with c = 1 and h = e in e):

Tr(φ̄_{g}|_{V_{e}}) = λ_{g}Tr(φ_{g}|_{A_{e}}) = λ_{g}χ_{g}^{−1}Tr(φ_{e}|_{A_{g}}) = λ_{g}χ_{g}^{−1}Tr(φ_{e}|_{V_{g}})

The equality φ̄_{g} = φ_{g}λ_{g} implies that η̄ is φ̄ invariant:

η̄(φ̄_{g}(av), φ̄_{g}(bv)) = λ_{g}^{2}η(φ_{g}(a), φ_{g}(b)) = λ_{g}^{2}χ_{g}^{−2}η(a, b) = η̄(av, bv)

In general:

Tr(l_{c}φ̄_{g}|_{V_{h}}) = λ_{g}Tr(l_{c}φ_{g}|_{A_{h}})
= λ_{g}χ_{g}^{−1}χ_{h^{−1}}Tr(φ_{h^{−1}} ∘ l_{c}|_{A_{g}})
= λ_{g}χ_{g}^{−1}χ_{h^{−1}}λ_{h^{−1}}^{−1}Tr(φ̄_{h^{−1}} ¯∘l_{c}|_{V_{g}})
= Tr(φ̄_{h^{−1}} ¯∘l_{c}|_{V_{g}})

so that with this choice of φ̄ axiom e) is satisfied.

This fact leads us to the following definitions:

2.4.1. Definition. The Ramond space of a G–Frobenius algebra A is the G–graded vector–space

V := ⊕_{g}V_{g} := ⨁_{g} A_{g} ⊗ k

together with the G-action φ̄ := φ ⊗ χ, the induced metric ḡ and the induce multiplication ¯∘, Replacing A_{g} by V_{a} this changes the axioms as follows:

2.4.2. Theorem. There is a one–to–one correspondence between isomorphism classes of G–Ramond algebras and G-Frobenius algebras

Proof. The association of a Ramond space to a G–Frobenius algebra provides the correspondence. The inverse being the obvious reverse twist by χ.

2.4.3. Remark. In the theory of singularities, the untwisted Ramond sector corresponds to the forms H^{n−1}(V_{ϵ}, C) while the Frobenius algebra corresponds to the Milnor ring [W]. These are naturally isomorphic, but have different G–module structures. In that situation, one takes the invariants of the Ramond sector, while we will be interested in invariants of the G–Frobenius algebra and not only the untwisted sector.

3. Bundle cobordisms, finite gauge groups, orbifolding and G–Ramond algebras

In this section, we introduce two cobordism categories which correspond to orbifold Frobenius algebras and G–Ramond respectively. G–Frobenius algebras. Again G is a fixed finite group.

3.1. Bundle cobordisms. Gluing along boundaries induces the composition and the disjoint union provides a monoidal structure.

3.1.1. Definition. Let GBCOB be the category whose objects are principal G–bundles over one–dimensional closed oriented (topological) manifolds whose morphisms cobordisms of these objects (i.e. principal G–bundles over oriented surfaces with boundary)

More precisely, B_{Σ} ∈ Hom(B_{1}, B_{2}) if Σ is an oriented surface with boundary ∂Σ = −S_{1} ∐ S_{2} and B_{Σ} is a bundle on Σ which restricts to B_{1} and B_{2} on the boundary.

The composition of morphisms is given by gluing along boundaries with respect to orientation reversing homeomorphisms on the base and covering bundle isomorphisms.

3.1.2. Remark. The operation of disjoint union makes this category into monoidal category with unit ∅ formally regarded as a principal G bundle over ∅.

3.1.3. Remark. Typical objects are bundles B over S = ∐_{i∈I} S^{1} ∐_{j∈J} S̄^{1}

3.1.4. Structure Lemma. The space Bun(S^{1}, G) of G bundles on (S^{1}, ν) can be described as follows:

Bun(S^{1}, G) = (G × F)/G

where F is a generic fiber regarded as a principal G-space and G acts on itself by conjugation. and the quotient is taken by the diagonal action. Furthermore the space Bun′(S^{1}, G) of pointed G bundles on S^{1} can be described as follows:

Bun(S^{1}, G) = G × F ∼ G × G.

3.1.5. Remark. Usually one uses the identification Bun(M, G) = Hom(π_{1}(M), G)/G. Which we also use in the proof. However, for certain aspects of the theory – more precisely to include non–trivial characters– it is vital to include a point in the bundle respectively a trivialization rather then just a point in the base.

Proof of the structure Lemma. Given a pointed principal G bundle (B, S^{1}, π, G, G), b ∈ B we set ν = π(b) ∈ S^{1}. The choice of b ∈ F = B_{ν} gives and identification of F with G, i.e. we let β : G ↦ F be the admissible map that satisfies β(e) = b. We set g ∈ G to be the element corresponding to the monodromy around the generator of π_{1}(S^{1}). Notice, that since we fixed an admissible map everything is rigid – there are no automorphisms– and the monodromy is given by an element, not a conjugacy class. Thus we associate to (B, b) the tuple (g, b).

Vice versa, given (g, b) we start with the pointed (S^{1}, ν) and construct the bundle with fiber G, monodromy g and marked point b = β(e) ∈ B_{ν}.

The bijectiveness of this construction follows by the classical results quoted above [St].

The choice in this construction corresponds to are a choice of a point b ∈ B. Changing b amounts to changing β and the monodromy g. Moreover, Moving the point ν = ϕ(b) and moving b inside the fiber by parallel transport keeps everything fixed. Moving b inside the fixed fiber by translation (once ν is fixed) corresponds to translation by the group action in the fiber i.e. the translation action of G on G and simultaneous conjugation of the monodromy. Hence, the first claim follows.

This observation leads us to the following definition:

3.1.6. Definition. We call a bundle over a closed one–dimensional manifold rigidified if its compomnents are labelled and the bundle is pointed in each component. or equivalently if a point and a trivialization around that point are fixed in each component. We denote such a bundle (B, b) where b ⊂ B : b = (b_{0}, …, b_{n}) is the set of base–points for each component. Furthemore, if the surface has genus zero we realize it in the plane as a pointed disc with all boundaries being S^{1}. We label the outside circle by 0. If π is the bundle projection, we set x_{i} := π(b_{i}) we call b_{0} the base–point, the B_{x_{i}} the special fibers and B_{x_{0}} the initial fiber.

3.2. Rigidified bundle cobordisms and G–Frobenius algebras.

3.2.1. Definition. Let GBCOB^{∗} be the category whose objects are rigidified bundles principal G–bundles over one–dimensional closed oriented (topological) manifolds considered up to pull–back under orientation preserving homeomorphism of the base respecting the markings and whose morphisms are cobordisms of these objects (i.e. principal G–bundles over oriented surfaces with boundary together with rigidification on the boundary, a choice of base–point x_{0} ∈ ∂Σ compatible with the rigidification and a choice of curves Γ_{i} from x_{0} to x_{i}.

I.e. objects are bundles B over S = ∐_{i∈I} S^{1} ∐_{j∈J} S̄^{1} with base– points on the various components {(b_{i} ∈ B|_{S_{i}^{1}} : i ∈ I}{b_{j} ∈ B|S̄_{j}^{1} : j ∈ J}). and B_{Σ} ∈ Hom((B_{1}, b_{1}), (B_{2}, b_{2})) if Σ is an oriented surface with boundary considered up to orientation preserving homeomorphism with boundary ∂Σ = −S_{1} ∐ S_{2} – again up to homeomorphism – and B_{Σ} is a bundle on Σ which restricts to B_{1} and B_{2} on the boundary together with rigidification data for the boundary i.e.(b_{1}, b_{2}) with b_{1} ⊂ B_{1} and b_{2} ⊂ B_{2}).

The extra structure of curves and base–point allows us to identify the special fibers with the initial fiber via parallel transport. Thus we can describe the rigidification data in terms of b_{0}, and the group elements g_{i} defined via Γ_{i}(b_{0}) = g_{i}b_{i} .

The composition of morphisms is given by gluing along boundaries with respect to orientation reversing homeomorphisms of the base and covering bundle isomorphisms identifying the base–points.

3.2.2. Remark. The operation of disjoint union makes this category into monoidal category with unit G regarded as a bundle over ∅ with base point e.

3.2.3. Remark. By the structure Lemma 3.1.4 we can identify the monoidal generators of objects of RGBCOB with G × G.

A generating object can thus be depicted by an oriented circle with labels (g, h) where (g, h) ∈ G×G. We use the notation as (g, h) for positively oriented circles and \overline{(g, h)} for negatively oriented circles. Due to the involutive property we have that \overline{(g, h)} ≃ (g^{−1}, h)^{∗}. General objects are then just disjoint unions of these, i.e. tuples (g_{i}, h_{i}). homomorphisms on the generators are given by the trivial bundle cylinder with different trivializations on both ends represented the natural diagonal action of G by conjugation and translation described in 3.1.4, so that this diagonal G–action is realized in terms of cobordisms.

The natural G × G action, however, cannot be realized by these cobordisms and we would like to enrich our situation to this case by adding morphisms corresponding to the G action on the trivializations.

3.2.4. Definition. Let RGBCOB be the category obtained GBCOB∗ by adding the following morphisms. For any n–tuple (k_{1}, …, k_{n}) : k_{i} ∈ G and any object (g_{i}.h_{i}) : i = 1, … n we set

τ(k_{1}, …, k_{n})(g_{i}.h_{i}) : (g_{i}, k_{i}h_{i})_{i∈{1,...n}}

we call these morphisms of type II and the morphisms coming from GBCOB∗ of type I.

3.2.5. Remark. There is a natural forgetful functor from RGBCOB to GBCOB.

Given a character χ ∈ Hom(G, k^{∗}) we can form the fiber–product G ×_{χ} k this gives a functor k[G]_{χ} from the category VECT_{k} to k[G] − MOD, the category of k[G]–modules.

3.2.6. Definition. A G_{χ}–orbifold theory is a monoidal functor V from RGBCOB to k[G] − MOD satisfying the following conditions:

i) The image of V lies in the image of k[G]_{χ} ii) Objects of RGBCOB which differ by morphisms of type II are

mapped to the same object in k[G] − MOD. iii) The value on morphisms of type I does not depend on the choice

of connecting curves or base–point. iv) The morphisms of type II are mapped to the G–action by χ v) The functor is natural with respect to morphisms of the type

τ(k, …, k). I.e. V (τ(k, …, k)∘Σ) = V (τ_{out}(k^{−1}, …, k^{−1}))∘V (Σ)∘ V_{in}(τ(k, …, k))

vi) V associates id to cylinders B × I, b ∈ B × 0, b ∈ B × 1 considered

as cobordisms from (B, b) to itself. vii) V is involutory: V (S̄) = V^{∗}(S) where ∗ denotes the dual vec-

tor space with induced k[G]–module structure. In accordance, the morphism of type II commute with involution, i.e. they are mapped to the G–action by χ^{−1}.

vii) The functor is natural with respect to homeomorphisms of the underlying surface of a cobordism and pull–back of the bundle.

3.2.7. Corollary. G_{χ}–orbifold theories are homotopy invariant. Proof. By standard arguments using naturality and vi) a homotopy

of an object S induces an identity on its image. More precisely given a homotopy of objects f_{t} : S ↦ S , it induces a map F : S × I ↦ S × I and we have the commutative diagram:

V (S) →^{V (S×I)=id} V (S)
V (f_{0}) ↓  ↓ V (f_{1})
V (S) →^{V (S×I)=id} V (S)

In particular, if V (f_{0}) = id then V (f_{1}) = id

3.2.8. Remark. Given a choice of connecting curves, we can identify all fibers over special points. Therefore after fixing one identification of a fiber with G, we can identify the other marked points as translations of points of parallel transport and label them by group elements, which we will do.

The action of τ(k, …, k) corresponds to a change of identification for one point and simultaneous re-gauging of all other points via this translation, i.e. a diagonal gauging. Therefore given a cobordism, we can fix an identification of all fibers with G.

3.2.9. Definition and Notation. We will fix some standard bundle cobordisms pictured below.

I: The standard disc bundle is the disc with a trivial bundle and positively oriented boundary considered as a cobordism between ∅ and (e, e) it will be denoted by D.

II: The standard g–cylinder bundle is the cylinder S^{1} × I with the bundle having monodromy g around ((S^{1}, 0)) considered as a cobordism between (g, e) ∐ \overline{(g, e)} and ∅ it will be denoted by C_{g}.

III: The (g, k)^{h}–cylinder bundle is the cylinder with a bundle having monodromy g around (S^{1}, 0) considered as a cobordism between (g, e) and (hgh^{−1}, h) it will be denoted by C_{g,h}.

IV: Thestandard (g, h)–trinion bundle is the trinion with the bundle having monodromies g around S_{1}^{1} and h around S^{2} and translations e for τ_{01}, τ_{02} considered as a cobordism between (g, e), (h, e) and (gh, e) it will be denoted by T_{g,h}.

V: The (g, h)–torus bundle is the once–punctured torus with the principal G–bundle having monodromies g and h around the two standard cycles considered as a cobordism between ([g, h], e) and ∅ it will be denoted by E_{g,h}^{k}.

3.2.10. Lemma. The bundles over a cylinder that form cobordisms between (g, e) and (h, k) are parameterized by G; it is necessary that g = khk^{−1}, these cobordisms are given by the C_{(g,h)}.

Proof. Given such a bundle over the cylinder Σ_{0} := S^{1} × I the translation from B_{ν,0} to B_{ν,1} along γ(t) := (ν, t) is a complete invariant. We fix this element k ∈ G Since in π_{1}(Σ_{0}, (ν, 0) C_{1} = γC_{2}γ^{−1} we must have g = khk^{−1} we call these bundles Φ_{k,g}

3.2.11. Proposition. To fix a G_{χ}–orbifold theory on the objects of the type (g, e) and to fix a G_{χ}–orbifold theory on the morphisms it suffices fix it on bundles over the standard cylinder bundle C, the (g, h)–cylinder bundles C_{g,h}, and on the standard trinion T.

Proof. The first claim follows from the condition ii) and the monoidal structure. For the second claim first notice that due to the homotopy Lemma 3.2.7 V is fixed on D. Also, any bundle over the cylinder is trivial, furthermore by v) we may regard the cylinder as a cobordism from B|_{S^{1}} to itself and after applying a morphism of type II we can assume that the boundary objects ore of the type (g, e), (h, k). Therefore we know the functor V on all bundles over cylinders. Dualizing and gluing on cylinders, we find that once the functor is defined on the standard trinion, it is defined on all bundles over all trinions. Lastly, given any surface, we can choose a decomposition by a marking into discs, cylinders and trinions, then gluing determines the value of V on this surface.

3.2.12. Proposition and Definition. For any G_{χ}–orbifold theory V set

\begin{aligned}V ((g, e)) & = & V_{g} \\ V (T_{g,h}) & = & ¯∘ : V_{g} ⊗ V_{h} → V_{gh} \\ V (D)(1_{k}) & = & v_{0} \\ C_{g} & = & η̄|_{V_{g}⊗V_{g^{−1}}} \\ V (C_{g,h}) & = & φ̄_{h} : V_{g} ↦ V_{hgh^{−1}} \end{aligned}

then this data together with g and χ form a Ramond–G algebra ⟨G, V, ¯∘, v, η̄, φ̄, χ⟩ which we call the associated G–Ramond algebra to V.

Proof. It is clear by 3.2.11 that given a G_{χ} orbifold theory it is completely fixed by its associated G–Ramond algebra. The converse is also true:

The axiom a) follows from the standard gluing procedures of TFT. I.e. the usual gluing of a disc with 3 holes from two discs with two holes in two different ways.

For axiom b’) we regard the following commutative diagrams

(g, e) ∐(h, e)⟶^{τ_{12}}(h, e) ∐(g, e) (h, e) ∐(g, e) (g, e) ∐(h, e)
↓ T_{g,h}^{e} ⇒  ↓ II  ⇒ ↓ III ⇒  ↓ IV
(gh, e) ⟶^{id} (gh, e)  (gh, g)  (gh, g)
⇓ V
V_{g} ⊗ V_{h} ⟶^{τ_{12}} V_{h} ⊗ V_{g} ⟶^{id} V_{h} ⊗ V_{g} ⟶^{id} V_{h} ⊗ V_{g}
↓ ¯∘  ↓ φ̄_{g}¯∘
V_{gh} ⟶^{id} V_{gh} ⟶^{id} V_{gh} ⟶^{id} V_{gh}

where we have used axiom vii) for the first move and axiom iii) for the second and gluing for the last one.

The unit of axiom c) is given by D̄^{e}. The projective invariance follows from

∅ ⟶^{D} (e, e)
↓ ∥  ↓ ∥
∅ →^{D}(e, e) →^{C_{(e,e)}^{k}}(e, k) ⟶^{II_{k^{−1}}}(e, e)
↓ ↓ ↓ ↓
k →^{1_{e}} V_{e} →^{φ̄_{k}} V_{e} →^{χ̄_{k^{−1}}} V_{e}
↓ ∥  ↓ ∥
k ⟶^{1^{e}} V_{e}

where in the third line 1_{k} ↦ v_{0} ↦ χ_{k}v_{0} ↦ v_{0} The axiom d) of the invariance of the metric is again a standard

gluing argument.

I.e. using ϵ̄ and associativity:

η̄(v_{g}¯∘v_{h}, v_{(gh)^{−1}}) = ϵ̄((v_{g}¯∘v_{h})¯∘v_{(gh)^{−1}}) = ϵ̄(v_{g}¯∘(v_{h}¯∘v_{(gh)^{−1}})) = η̄(v_{g}, v_{h}¯∘v_{(gh)^{−1}})

For axiom 1’) We use the following diagram:

(g, e)→^{id}(g, e) (g, e)
↓ id ⇒ ↓ C̃ ⇒ ↓ C
(g, e)→^{id}(g, e) (g, g)
⇓ V
V_{g} →^{id} V_{g} →^{id} V_{g}
↓ id  ↓ φ̄_{g}
V_{g} →^{id} V_{g} →^{id} V_{g}

where we used axiom vii) for the first move and axiom iii) for the second.

The axiom 2’) follows from the diagrams

(g, e) ∐(h, e) ⟶^{(C_{(g,e)}^{k},C_{(h,e)}^{k})}(kgk^{−1}, k) ∐(khk^{−1}, k) ⟶^{T_{kgk^{−1},khk^{−1}}^{k}}(kghk^{−1}, k) ⟶^{C_{(kghk^{−1},k)}^{k^{−1}}}(gh, e)
↓  ↓  ↓  ↓
V_{g} ⊗ V_{h}  ⟶^{φ̄_{k}⊗φ̄_{k}}  V_{kgk^{−1}} ⊗ V_{khk^{−1}}  ⟶^{∘^{k}}  V_{kghk^{−1}}  ⟶^{φ̄_{k^{−1}}} V_{gh}

(g, e) ∐(h, e)  ⟶^{T_{gh}^{e}}  (gh, e)
↓ ∥  ↓ ∥
(g, e) ∐(h, e) ⟶^{(II_{k},II_{k})}(g, k) ∐(h, k)⟶^{T_{g,h}^{k}}(gh, k) ⟶^{II_{k^{−1}}}(gh, e)
↓  ↓  ↓  ↓
V_{g} ⊗ V_{h}  ⟶^{χ̄_{k}⊗χ̄_{k}} V_{g} ⊗ V_{h} ⟶^{∘^{k}} V_{gh} ⟶^{χ̄_{k^{−1}}} V_{gh}
↓ ∥  ↓ ∥
V_{g} ⊗ V_{h} ⟶^{¯∘} V_{gh}

The axiom 3’) – G–invariance of the metric – follows from the following diagrams

(g, e) ∐ \overline{(g, e)}  ⟶^{A^{e}}  K
↓ ∥  ↓ ∥
(g, e) ∐ \overline{(g, e)} →^{(C,C̄)} (kgk^{−1}, k) ∐ \overline{(kgk^{−1}, k)} →^{A^{k}} K
↓ (id,¯ )  ↓ (id,¯ )  ↓ (id,¯ )
(g, e) ∐(g^{−1}, e) →^{(C,C)} (kgk^{−1}, k) ∐(kg^{−1}k^{−1}, k)→^{A^{k}} K
↓  ↓  ↓
V_{g} ⊗ V_{g^{−1}}  →^{φ̄_{k}⊗φ̄_{k}} V_{kgk^{−1}} ⊗ V_{kg^{−1}k^{−1}}  →^{η^{k}} K

and

(g, e) ∐ \overline{(g, e)} ⟶^{A^{e}} K
↓ ∥  ↓ ∥
(g, e) ∐ \overline{(g, e)} →^{(II_{k},II_{k})} (g, , k) ∐ \overline{(g, k)} →^{A^{k}} K
↓ (id,¯ )  ↓ (id,¯ )  ↓ (id,¯ )
(g, e) ∐ (g^{−1}, e) →^{(II_{k},II_{k}} (g, , k) ∐ (g^{1}, k)→^{A^{k}} K
↓  ↓  ↓
V_{g} ⊗ V_{g^{−1}}  →^{χ̄_{k}⊗χ̄_{k^{−1}}} V_{g} ⊗ V_{g^{−1}k} →^{η^{k}} K
↓ ∥  ↓ ∥
V_{g} ⊗ V_{g^{−1}} ⟶^{η} K

Lastly axiom 4’) comes from gluing a once punctured torus in two different ways.

3.2.13. Proposition. Given a G–Ramond algebra V there is a unique G_{χ} orbifold theory V s.t. V is its associated G–Ramond algebra.

Proof.

We need to show that the data is sufficient to reconstruct the functor. For the objects this is clear, due to the monoidal structure. For discs, annuli and trinions the functor is defined by its basic ingredients, glueing annuli C_{g,h} to the basic trinion, and gauging with morphisms of type II. For other morphisms of type I we notice that we can always decompose a surface into trinions, annuli and discs. For each decomposition there are three choices. The choice of a marking for the decomposition, a choice of orientation and a choice of pairs of points on over the marked curves. The second and the third choice can be seen to be irrelevant by insering two annuli C_{e,k} and A_{k,e} with suitable orientation in a normal neighborhood of the curve in question. The first choice is unique up to two operations [HT] which are correspond to associativity and the trace axiom, and is thus also irrelevant.

Combining the Propositions 3.2.11 and 3.2.13 with 2.4.2 we obtain:

3.2.14. Theorem. There is a 1–1 correspondence between G-twisted FA and isomorphism classes of G_{χ}–orbifold theories as χ runs though the characters of G.

Proof. In the standard way, we make the G–twisted Frobenius algebras and the G_{χ}–orbifold theories into categories by introducing the following morphisms. For G–twisted Frobenius algebras we use algebra homomorphism respecting all the additional structures and for G_{χ}–orbifold theories we use natrural transformations among functors. The map of associating a G-twisted FA to a G_{χ}–orbifold theory then turns into a full and faithful funcor which is by reconstruction surjective on the objects.

An immediate consequence of this is:

3.2.15. Corollary. There is a 1–1 correspondence between G-orbifold FA and isomorphism classes of monoidal functors which are identity on cylinders and have the involutary property from GBCOB to VECT which lift to RGBCOB.

3.3. Spectral flow. In the previous paragraph, we chose the geometric version to correspond to the Ramond picture, as is suggested by physics, since we are considering the vacuum states in their Hilbert– spaces at the punctures. By spectral flow, however, the vacuum states should bijective correspond to the chiral algebra. In the current setting the difference only manifests itself in a change of G–action given by a twist resulting from the character χ. We can directly produce this G–action and thereby the G–Frobenius algebra by considering the G–action not by C_{e,k}^{e}, but by II_{k^{−1}} ∘ C_{e,k}^{e}. This statement is proven by re-inspection the commutative diagrams in the proof of 2.4.2

4. Special G–Frobenius algebras

4.1. Definition. A special G–Frobenius algebra is a G–twisted Frobenius algebra whose components A_{g} are cyclic A_{e}–modules, together with two collections of maps (r_{g}), (i_{g}) indexed by G where

r_{g} : A_{e} → A_{g} is the map of A_{e}–modules induced by multiplication, we write a_{g} := r_{g}(a) in particular if one sets 1_{g} := r_{g}(1) we obtain r_{g}(a) = a1_{g}. Notice that it is equivalent to specify the map r_{g} or generators 1_{g} of A_{g} as cyclic A_{e}–modules. i_{g} : A_{g} ↦ A_{e} are collections of maps s.t. each i_{g} is an injection which splits the exact sequence of A_{e}-modules.

0 ⟶ I_{g} ⟶ A ⟶^{r_{g}} ; ⟵_{i_{g}} A_{g} ⟶ 0  (4.1)

Here I_{g} ⊂ A_{e} is the annihilator of 1_{g} and thus of A_{g}.

Furthermore the following should hold:

∀g ∈ G : φ_{g}(1_{h}) = φ_{g,h}1_{ghg^{−1}}

with φ_{g,h} ∈ K and φ_{g,1} = 1.

We denote the concatenation of i_{g} and r_{g} by π_{g}

π_{g} = r_{g} ∘ i_{g} : A_{e} → A_{e}

Notice that multiplication by 1_{g} acts as projection: a1_{g} = π_{g}(a)1_{g}.

4.1.1. Special super G–Frobenius algebra. The super version of special G–Frobenius algebras is straightforward. Notice that since each A_{g} is a cyclic A_{e}–algebra its parity is fixed to be (−1)g̃ := ˜1_{g} times that of A_{e}. I.e. a_{g} = i_{g}(a_{g})1_{g} and thus ã_{g} = { \atop i_{g}(a_{g})} ˜1_{g} . In particular if A_{e} is purely even A_{g} is purely of degree g̃.

4.2. Proposition. Let η_{g} : i_{g}(A_{g}) ⊗ i_{g}(A_{g}) → K : be given by the following formula.

η_{g}(a, b) = η(r_{g^{−1}}(a), r_{g}(b))

Then i_{g}(A_{g}) ⊂ A_{e} together with η_{g} and the induced multiplication a∘_{g} b = i_{g}(a∘r_{g}(b)) = π_{g}(a∘b) is a Frobenius sub–algebra. Furthermore η_{g} establishes an isomorphism of A_{e}–sub–modules between i_{g}(A_{g}) and i_{g^{−1}}(A_{g^{−1}}).

Proof. That the multiplication is commutative and associative can be seen as follows

a ∘_{g} b = i_{g}(ab1_{g})) = i_{g}(ba1_{g}) = b ∘_{g} c

likewise

(a∘_{g}b)∘_{g}c = i_{g}((ab)c1_{g})) = i_{g}(a(bc)1_{g}) = a∘_{g}(b∘_{g}c) η_{g} is an invariant non–degenerate form.

The non–degeneracy follows directly from the non–degeneracy of η: fix a ∈ i_{g}(A_{g}) and let a_{g} ∈ A_{g} and b_{g^{−1}} ∈ A_{g^{−1}} s.t. i_{g}(a_{g}) = a and η(a_{g}, b_{g^{−1}}) ≠ 0. Set b := i_{g^{−1}}(b_{g^{−1}}) and b̃ = π_{g}(b)

Then

η_{g}(a, b̃) = η(a1_{g}, b̃1_{g^{−1}}) = η(b̃a1_{g}, 1_{g^{−1}}) = η(ba1_{g}, 1_{g^{−1}})
= η(a1_{g}, b1_{g^{−1}}) = η(a_{g}, b_{g^{−1}}) ≠ 0

The invariance follows from

η_{g}(ab, c) = η(r_{g^{−1}}(ab), r_{g}(c)) = η(ab1_{g^{−1}}, c1_{g}) = η(a1_{g^{−1}}, bc1_{g}) = η_{g}(a, bc)

With these two properties, we get a map i_{g}(A_{g}) → i_{g}(A_{g})^{∗} and η gives an isomorphism i_{g}(A_{g})^{∗} → i_{g^{−1}}(A_{g^{−1}}) via (a, b) ∈ A_{g} ⊗ A_{g^{−1}} ↦ η(a1_{g}, b1_{g^{−1}}).

4.3. Proposition. The following equality holds π_{g^{−1}} ∘π_{g} = π_{g} showing that π_{g^{−1}}|_{i_{g}(A_{g})} = id identifying i_{g}(A_{g}) and i_{g^{−1}}(A_{g^{−1}}).

Proof. We have the following:

η(π_{g^{−1}} ∘ π_{g}(a)1_{g}, b_{g^{−1}}) = η(1_{g}, π_{g^{−1}}(π_{g}(a)i_{g^{−1}}(b_{g^{−1}}))1_{g^{−1}})
= η(π_{g}(a)η(1_{g}, i_{g^{−1}}(b_{g^{−1}})1_{g^{−1}})
= η(π_{g}(a)1_{g}, b_{g^{−1}})

which proves the statement due to the non–degeneracy of η.

4.4. Definition. Let A be special G–twisted Frobenius algebra A. We define a G graded 2–cocycle with values in A_{e} to be a map γ : G × G → A_{e} which satisfies

γ_{g,h} := γ(g, h) ∈ i_{gh}(A_{gh})  (4.2)

and

π_{ghk}(γ_{g,h}γ_{gh,k}) = π_{ghk}(γ_{h,k}γ_{g,hk})  (4.3)

A non–abelian G 2–cocycle with values in K^{∗} is a map φ : G×G → K which satisfies:

φ_{gh,k} = φ_{g,hkh^{−1}}φ_{h,k}  (4.4)

where φ_{g,h} := φ(g, h) and

φ_{e,g} = φ_{g,e} = 1

Notice that in the case of a commutative group G this says that the φ_{g,h} form a two cocycle with values in K^{∗}.

Furthermore setting g = h^{−1}, we find:

φ_{g^{−1},ghg^{−1}} = φ_{g,h}^{−1}

4.5. Proposition. A special G–twisted Frobenius algebra gives rise A to a g 2–cocycle γ with values in A and to an non–abelian G 2–cocycle φ with values in K^{∗}, which satisfy the compatibility equations

φ_{g,h}γ_{ghg^{−1},g} = γ_{g,h}  (4.5)

and

φ_{k,g}φ_{k,h}γ_{kgk^{−1},khk^{−1}} = φ_{k}(γ_{g,h})φ_{k,gh}  (4.6)

Proof.

Given a special G twisted Frobenius structure on A, we define γ_{g,h} ∈ i_{gh}(A_{gh}) by

1_{g}1_{h} = γ_{g,h}1_{gh}

and φ_{g,h} ∈ K^{∗} by:

φ_{g}(1_{h}) = φ_{g,h}1_{ghg^{−1}}.

By associativity of the multiplication, we find that the γ_{g,h} define a graded 2–cocycle with values in A.

γ_{g,β}γ_{gh,k}1_{ghk} = γ_{g,h}1_{gh}1_{k} = (1_{g}1_{h})1_{k} = 1_{g}(1_{h}1_{k}) = 1_{g}γ_{h,k}1_{hk} = γ_{h,k}γ_{g,hk}1_{ghk} (4.7)

So that

π_{ghk}(γ_{g,h}γ_{gh,k}) = π_{ghk}(γ_{h,k}γ_{g,hk}).

φ is a group homomorphism so that

φ_{gh,k}1_{ghkh^{−1}g^{−1}} = φ_{gh}(1_{k}) = φ_{g}(φ_{h}(1_{k})) =
φ_{g}(φ_{h,d}1_{hkh^{−1}}) = φ_{g,hkh^{−1}}φ_{h,k}1_{ghkh^{−1}g^{−1}}

which yields

φ_{gh,k} = φ_{g,hkh^{−1}}φ_{h,k}  (4.8)

By G–twisted commutativity

γ_{g,h}1_{gh} = 1_{g}1_{h} = φ_{g}(1_{h})1_{g} = φ_{g,h}1_{ghg^{−1}}1_{g} = φ_{g,h}γ_{ghg^{−1},g}1_{gh}

So γ_{g,h} and the φ_{g,h} satisfy

φ_{g,h}π_{gh}(γ_{ghg^{−1},g}) = π_{gh}γ_{g,h}  (4.9)

φ is also an algebra automorphism:

φ_{k}(1_{g})φ_{k}(1_{h}) = φ_{k}(1_{g}1_{h})

Expressed in the φ’s and γ’s:

φ_{k,g}φ_{k,h}γ_{kgk^{−1},khk^{−1}} = φ_{k}(γ_{g,h})φ_{k,gh}  (4.10)

which gives a formula for the action of φ on the γ’s

4.5.1. Corollary. A special G-Frobenius algebra gives rise to collection of Frobenius–algebras (A_{g}, ∘_{g}, 1_{g}, η_{g})_{g∈G} together with a G–action on A_{e}. a A graded G 2–cocycle with values in A_{e} and a compatible non–abelian G 2–cocycle with values in k^{∗}. Furthermore:

i) η_{e}(φ_{g}(a), φ_{g}(b)) = χ_{g}^{−2}η_{e}(a, b)

ii) η_{g}(a_{g}, b_{g}) = η(i_{g}(a_{g})i_{g}(b_{g}))γ_{gg^{−1}}, 1) iii) The projective trace axiom ∀c ∈ A_{[g,h]} and l_{c} left multiplication

by c:

χ_{h}Tr(l_{c}φ_{h}|_{A_{g}}) = χ_{g^{−1}}Tr(φ_{g^{−1}}l_{c}|_{A_{h}})  (4.11)

4.6. Proposition. (Reconstruction) Given a Frobenius algebra (A_{e}, η_{e}) and a G-action on A:φ : G × A → A together with the following data

- Frobenius algebras (A_{g}, η_{g}|g ∈ G \ {1}).

- Injective algebra homomorphisms i_{g} : A_{g} → A_{e} s.t. i_{g}(A_{g}) = i_{g^{−1}}(A_{g^{−1}}).

- A graded G 2–cocycle γ with values in A_{e} and a compatible non– abelian G 2–cocycle φ with values in K^{∗}

- A group homomorphism χ : G → K^{∗} such that i)-iii) of 4.5.1 hold, then there is a unique extension of this set of data to a special G–twisted Frobenius algebra, i.e. there is a unique special G–twisted Frobenius algebra with these underlying data.

Proof.

Denote the unit of A_{e} by 1 and the unit of A_{g} by 1_{g}.

Define the algebra epimorphism r_{g} : A_{g} ↦ A by r_{g}(1) := 1_{g} Denote by π_{g} : A_{e} → i_{g}(A_{g}) the projection to A_{g}. Let the A_{e} module structure on A_{g} be given by

ab_{g} := i_{g}^{−1}(π_{g}(ai_{g}(b_{g}))

it is clear that A_{g} is a cyclic A_{e}–module generated by

1_{g}. The algebra structure is then determined by:

a_{g}b_{h} := i_{g}(a_{g})i_{h}(b_{h})γ_{g,h}1_{gh}

This multiplication is associative as can be seen by reading 4.7 backwards.

The φ’s determine the G-action by: φ_{g}(b_{h}) := φ_{g}(i_{h}(b_{h}))φ_{g,h}1_{ghg^{−1}}. The compatibility and 4.6 guarantee that this is a representation and

the G action is indeed an action by algebra automorphisms.

The G–twisted commutativity a_{g}b_{h} = φ_{g}(b_{h})a_{g} also follows from the compatibility.

The form η is defined the following way:

η(a_{g}, h_{g^{−1}}) := η_{e}(i_{g}(a_{g})i_{g^{−1}}(b_{a^{−1}})γ_{gg^{−1}}, 1) = η_{g}(i_{g}(a_{g}), i_{g^{−1}}(b_{a^{−1}}))

and

η(a_{g}, h_{h}) := 0 if gh ≠ 1

This form is non–degenerate by assumption and it is invariant:

η(a_{g}b_{h}, d_{(gh)^{−1}}) = η_{e}(i_{g}(a_{g})i_{h}(b_{h})i_{h^{−1}g^{−1}}(d_{h^{−1}g^{−1}})γ_{gh}γ_{gh,h^{−1}g^{−1}}, 1)
= η_{e}(i_{g}(a_{g})i_{h}(b_{h})i_{h^{−1}g^{−1}}(d_{h^{−1}g^{−1}})γ_{h,h^{−1}g^{−1}}γ_{g,g^{−1}}, 1)
= η(g_{g}, b_{h}d_{h^{−1}g^{−1}})

4.6.1. Remark. By straightforward calculation we have that the projective trace axiom is equivalent to

∀g, h ∈ G; c1_{ghg^{−1}h^{−1}} ∈ A_{[g,h]}, c ∈ i_{hgg^{−1}h^{−1}}(A_{[gh]})

χ_{h}φ_{h,g}Tr(l_{γ_{ghg^{−1}h^{−1},hgh^{−1}}c}φ_{h}|_{i_{g}(A_{g})}) =

χ_{g^{−1}}φ_{g^{−1},ghg^{−1}}Tr(φ_{g^{−1}}l_{γ_{hg^{−1}h^{−1}g,g^{−1}hg})c}|_{i_{h}(A_{h})}) (4.12)

Notice that in the graded case (see below) this condition only needs to be checked for deg(γ_{ghg^{−1}h^{−1},hgh^{−1}}c) ≠ 0 and deg(γ_{hg^{−1}h^{−1}g,g^{−1}hg})c) ≠ 0.

Furthermore if [g, h] = e then γ_{ghg^{−1}h^{−1},hgh^{−1}} = γ_{hg^{−1}h^{−1}g,g^{−1}hg}) = 1_{e} and φ_{g^{−1},ghg^{−1}} = φ_{g,h}^{−1} = φ_{g^{−1},h}.

4.7. Remarks.

1) If A_{g}A_{h} ≠ 0 the compatibility condition (4.5) already determines the φ_{g,h} ∈ K^{∗}.

2) In particular: γ_{g,g} = 0 unless χ_{g} = 1 and if [g, h] = e we have φ_{g,h}φ_{hg} = 1 or γ_{g,h} = γ_{h,g} = 0.

3) If also A_{g}A_{h}A_{k} ≠ 0 the elements defined by (4.5) automatically satisfy the conditions of non–abelian 2–cocycles and the condition (4.6) is automatically satisfied.

Proof Without loss of generality, we may assume that 1_{g}1_{h}1_{k} ≠ 0 then due to the condition 4.5

1_{g}1_{h}1_{k} = (1_{g}1_{h})1_{k} = φ_{gh}(1_{k})(1_{g}1_{h}) = (φ_{gh}(1_{k})1_{g})1_{h} = φ_{gh,k}1_{ghkh^{−1}g^{−1}}1_{g}1_{h}

and using associativity, we similarly obtain

1_{g}1_{h}1_{k} = 1_{g}(1_{h}1_{k}) = 1_{g}(φ_{h}(1_{k})1_{h} = φ_{g}(φ_{b,k}1_{hkh^{−1}})1_{g}1_{h} = φ_{g,hkh^{−1}}φ_{h,k}1_{ghkh^{−1}g^{−1}}1_{g}1_{h}

For the first statement in 2) one just needs to plug in g = h into (4.5) and for the second one has to apply the formula twice. For 3) notice that 1_{k}1_{g}1_{h} = 1_{k}(1_{g}1_{h}) = φ_{k}(1_{g}1_{h})1_{k} and on the other hand 1_{k}1_{g}1_{h} = φ_{k}(1_{g})1_{k}1_{h}) = φ_{k}(1_{g})φ_{k}(1_{h})1_{k}.

4.7.1. Graded special G–Frobenius algebras. Consider a set of graded Frobenius algebras satisfying the reconstruction data: {(A_{g}, η_{g}) : g ∈ G} with degrees d_{g} := deg(η_{g}) s.t. A_{g} ≃ A_{g^{−1}}. E.g. in the cohomology of fixed point sets d_{g} is given by the dimension for the Jacobian Frobenius manifolds d_{g} fixed by the degree of Hess(f_{g}) = ρ_{g}. Furthermore, the reconstructed {η|_{(A_{g}⊗A_{g^{−1}}}, g ∈ G} have degree d_{g} = d_{g^{−1}}.

For a G–twisted FA the degrees all need to be equal to d := d_{e}. To achieve this, one can shift the grading in each A_{g} by s_{g} this amounts to assigning degree s_{g} to 1_{g}. This is the only freedom, since the multiplication should be degree–preserving and all A_{g} are cyclic..

Set s_{g}^{+} := s_{g} + s_{g^{−1}} s_{g}^{−} := s_{g} − s_{g^{−1}}

Then s_{g}^{+} := d − d_{g} for grading reasons, but the shift s^{−} is more elusive.

4.7.2. Definition. The standard shift for a Jacobian Frobenius algebra is given by

s_{g}^{+} := d − d_{g}

and

s_{g}^{−} := \frac{1}{2πi}tr(log(g)) − tr(log(g^{−1})) := \frac{1}{2πi}(∑_{i} λ_{i}(g) − ∑_{i} λ_{i}(g^{−1}))

= ∑_{i:λ_{i}≠0} 2(\frac{1}{2πi}λ_{i}(g) − 1)

where the λ_{i}(g) are the logarithms of the eigenvalues of g using the branch with values in [0, 2π) i.e. cut along the positive real axis.

In this case we obtain:

s_{g} = \frac{1}{2}(s_{g}^{+} + s_{g}^{−}) = \frac{1}{2}(d − d_{g}) + ∑_{i}(\frac{1}{2πi}λ_{i}(g) − \frac{1}{2})

4.7.3. Remark. The shift s_{g}^{−} is canonical in the case of quasi–homogeneous singularities upon replacing the classical monodromy operator J by Jg. This will be discussed in the elsewhere [K2].

The degree of γ_{g,g^{1}} is s^{+} from comparing degrees in the equation 1_{g}1_{g^{−1}} = γ_{g,g^{1}}1_{e}

4.7.4. Reconstruction for graded special G–Frobenius algebras. In the Reconstruction program-m the presence of a non–trivial grading can greatly simplify the check of the trace axiom. E.g. if A_{[g,h]} has no element of degree 0 then both sides of this requirement are 0 and if [g, h] = e in venereal one needs only to look at the special choice c with deg(c) = 0 which most often is just c = 1 the identity.

4.7.5. Ramond–grading. The grading in the Ramond–sector is by the following definition

deg(v_{0}) := −\frac{d}{2}

This yields

deg(η̄) = 0 and deg(¯∘) = \frac{d}{2}

5. Jacobian Frobenius algebras

5.1. Definition. A Frobenius algebra A is called Jacobian

if it can be represented as the Milnor ring of a function f. I.e. If there is a function f ∈ O_{A_{K}^{n}} s.t. A = O_{A_{K}^{n}}/J_{f} where J_{f} is the Jacobian ideal of f. And the bilinear form is given by the residue pairing. This is the form given by the the Hessian of ρ = Hess_{f}.

If we write O_{A_{K}^{n}} = K[x_{1} … x_{n}], J_{f} is the ideal spanned by the \frac{∂f}{∂x_{i}}.

A realization of a Jacobian Frobenius algebra is a pair (A, f) of a Jacobian Frobenius algebra and a function f on some affine K space A_{K}^{n}, i.e. f ∈ O_{A_{K}^{n}} = K[x_{1} … x_{n}] s.t. A = K[x_{1} … x_{n}] and ρ := det(\frac{kel^{2}f}{kelx_{i}∂x_{j}}).

A small realization of a Jacobian Frobenius algebra is a realization of minimal dimension, i.e. of minimal n.

5.2. Definition. A natural G action on a realization of a Jacobian Frobenius algebra (A_{e}, f) is a linear G action on A_{K}^{n} which leaves f invariant.

Given a natural G action on a realization of a Jacobian Frobenius algebra (A, f) set for each g ∈ G, O_{g} := O_{Fix_{g}(A_{K}^{n})} this is the ring of functions of the fixed point set of g for the G action on A_{K}^{n}. These are the functions fixed by g: O_{g} = K[x_{1}, …, x_{n}]^{g}.

Denote by J_{g} := J_{f|_{Fixg(A^{n}_{K})}} the Jacobian ideal of f restricted to the fixed point set of g.

Define

A_{g} := O_{g}/J_{g}  (5.1)

The A_{g} will be called twisted sectors for g ≠ 1. Notice that each A_{g} is a Jacobian Frobenius algebra with the natural realization given by (A_{g}, f|_{Fix_{g}}). In particular, it comes equipped with an invariant bilinear form η_{g} defined by the element Hess(f|_{Fix_{g}})

For g = 1 the definition of A_{e} is just the realization of the original Frobenius algebra, which we also call the untwisted sector.

Notice there is a restriction morphism r_{g} : A_{e} → A_{g} given by a ↦ a mod J_{g}.

Denote r_{g}(1) by 1_{g}. This is a non–zero element of A_{g} since the action was linear. Furthermore it generates A_{g} as a cyclic A_{e} module.

The set Fix_{g}A_{K}^{n} is a linear subspace. Let I_{g} be the vanishing ideal of this space.

We obtain a sequence

0 → I_{g} → A_{e} →^{r_{g}} A_{g} → 0

Let i_{a} be any splitting of this sequence induced by the inclusion: î_{g} : O_{g} → O_{e} which descends due to the invariance of f.

In coordinates, we have the following description. Let Fix_{g}A_{K}^{n} given by equations x_{i} = 0 : i ∈ N_{g} for some index set N_{g}.

Choosing complementary generators x_{j} : j ∈ T_{g} we have O_{g} = K[x_{j} : j ∈ T_{g}] and O_{e} = K[x_{j}, x_{i} : j ∈ T_{g}, i ∈ N_{g}]. Then I_{g} = (x_{i} : i ∈ N_{g})_{′_{e}} the ideal in O_{e} generated by the x_{i} then O_{e} = I_{g} ⊕ i_{g}(A_{g}) using the splitting i_{g} coming from the natural inclusion î_{g} : K[x_{j} : j ∈ T_{g}] → K[x_{j}, x_{i} : j ∈ T_{g}, i ∈ N_{g}] We also define the projections

π_{g} : A_{1} → A_{g}; π_{g} = i_{g} ∘ r_{g}

which in coordinates is given by f ↦ f|_{x_{j}=0:j∈N_{g}} Let

A := ⨁_{g∈G} A_{g}

where the sum is a sum of A_{e} modules.

5.3. Theorem. Given a natural G action on a realization of a Jacobian Frobenius algebra (A_{e}, f) with a quasi–homogeneous function f of degree d and type q = (q_{1}, …, q_{n}) together with the natural choice of splittings i_{g} the possible structures of special G twisted Frobenius algebra on the A_{e} module A := ⨁_{g∈G} A_{g} are in 1–1 correspondence with elements of Z̄^{2}(G, A^{G}). the Z̄^{2} are G graded cocycles and a choice of parity which corresponds to a character σ ∈ Hom(G, Z/2Z), s.t. for all elements of order 2 the character is trivial.

Proof By the reconstruction Theorem the structure of special G twisted Frobenius algebra on the given data A_{g}, η_{g}, i_{g} is determined by:

- A graded G 2–cocycle with values in A_{e} and a compatible non– abelian G 2–cocycle with values in K^{∗}

- A group homomorphism χ : G → K^{∗}

subject to the conditions:

i) η_{e}(φ_{g}(a), φ_{g}(b)) = χ_{g}^{−2}η_{e}(a, b)

ii) η_{g}(a_{g}, b_{g}) = η(i_{g}(a_{g}b_{g}))γ_{gg^{−1}}, 1) iii) For any c ∈ A_{[gh]}:

χ_{h}STr(l_{c}φ_{h}|_{A_{g}}) = χ_{g^{−1}}STr(φ_{g^{−1}}l_{c}|_{A_{h}})

where we can restrict to the c with deg(c) = 0. The χ_{g} are fixed by the equation

(−1)g̃ dim(A_{g}) = STr(id|_{A_{g^{−1}}}) = χ_{g}STr(φ_{g}|_{A_{e}}) = χ_{g}Tr(φ_{g}|_{A_{e}})

To calculate the trace on the RHS, we use the character function for a morphism g of degree 0 on graded module V = ⊕_{n}V_{n}:

χ_{V_{n}}(g, z) := ∑_{n,μ} μ dim(V_{μ,n})z^{n}

where V_{μ,n} is the Eigenspace of Eigenvalue μ on the space V_{n}. We will use the grading induced by the quasi–homogeneity. I.e. let N be such that q_{i} = Q_{i}/N with Q_{i} ∈ N and N s.t. |G||N. Then a monomial has degree n if its quasi homogeneous degree is n/N. This is the natural grading for the quasi–homogeneous map grad(f). Notice that since g commutes with f it preserves the grading. It is clear that this character behaves multiplicatively under concatenations of quasi–homogeneous functions. Therefore by applying Arnold’s method, we can pass to a cover of K^{n} with the projection map T : T (x_{1}, …, x_{n}) = (x^{q_{1}}, …, x^{q_{n}}) and calculate the character for T and for grad(f) ∘ T. Then repeating the argument in a simultaneous Eigenbasis of g and the grading of [A], we obtain:

χ_{A_{e}}(g, z) = ∏_{i=1}^{n} \frac{(μ̃_{i})z^{N−Q_{i}} − 1}{(μ̃_{i}z)^{Q_{i}} − 1}

where the μ̃_{i} are the Eigenvalues of some lift of the action of g i.e. μ̃_{i}^{Q_{i}} = μ_{i}. Notice that since |G||N μ^{N} = 1, so that in the limit of z → 1, we obtain:

Tr(φ_{g}|_{A_{e}}) = ∏_{i:μ_{i}≠1} −μ_{i}^{−1} ∏_{i:μ_{i}=1} \frac{1}{q_{i}} − 1 = (−1)^{|N_{g}|}det(g)^{−1} dim(A_{g})

so that

χ_{g} = (−1)g̃(−1)^{|N_{g}|}det(g)

We set

σ(g) := g̃ + |N_{g}| mod 2

and call it the sign of g. Notice that σ ∈ Hom(G, Z/2Z) since both det and χ are characters. Also notice that a choice of sign corresponds to a choice of parity and vice–versa. Thus we obtain

χ_{g} = (−1)^{σ(g)} det(g)

This ensures condition i). Since for ρ = Hess(f):

φ_{g}(ρ) = det(g)^{−2}ρ

denote the element defining η_{g} by ρ̃_{g} and by ρ_{g} := i_{g}(ρ̃_{g}) condition ii) implies

γ_{g,g^{−1}}ρ_{g} = ρ

also we need

γ_{g,g^{−1}} = χ_{g}γ_{g^{−1},g}(−1)^{g̃g̃^{−1}}

Here we have a freedom of scaling, which we choose to distribute equally. More precisely, let Jac(f) the Jacobian matrix of f in the above basis be given by the block form

Jac(f) = (\begin{matrix}A & B \\ C & D \end{matrix})

then ρ_{g} = det(D) and this equation is solved by

γ(g, g^{−1}) = ((−1)g̃χ_{g})^{1/2}det(A − BD^{−1}C)

where we choose to cut the plane along the negative real axis. This uniquely defines a square root unless (−1)g̃χ_{g} = −1 in the case that g^{2} ≠ e we can choose roots i and −i for g respectively g^{−1}, the only case that has no solution is the case of g^{2} = e and (−1)g̃χ_{g} = −1, since there is no non–degenerate invariant form on an odd cyclic algebra. This means that if g^{2} = e either χ_{g} = −1 and g̃ = 1 or χ_{g} = −1 and g̃ = 1 or in other words we need (−1)^{|}N_{g}| = det(g).

Finally, we need to check the validity of iii). Notice that since the multiplication is graded the traces are 0 unless deg(γc) = 0 so that we can assume that γc = 1. In this case, we have to show:

χ_{h}STr(φ_{h}|_{A_{g}}) = χ_{g^{−1}}STr(φ_{g^{−1}}|_{A_{h}})

Let x_{i} be a basis of A_{e} in which g is diagonal. Then we have to compute the trace of the action of h on the sub-algebra generated by the x_{i} with eigenvalue 1 under the action of g. This is just the truncated version of the calculation above, so diagonalizing h on K[x_{i} : i ∈ T_{g}] we find using the same characteristic functions and rationale as before:

χ_{h}STr(φ_{h}|_{A_{g}}) = χ_{h}φ_{h,g}(−1)g̃Tr(φ_{h}|_{i_{g}(A_{g})})
= χ_{h}φ_{h,g}(−1)g̃ ∏_{j:ν_{j}≠1} −ν_{j}^{−1} ∏_{j:ν_{j}=1} \frac{1}{q_{j}} − 1
= χ_{h}φ_{h,g}(−1)g̃(−1)^{|T_{g}|}(−1)^{|T_{g}∩T_{h}|} det(h|_{T_{g}})^{−1} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h}))
= φ_{h,g}(−1)^{σ(g)}(−1)^{σ(h)}det(h) det(h|_{T_{g}})^{−1}(−1)^{N}(−1)^{|T_{g}∩T_{h}|} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h}))
= φ_{h,g}(−1)^{σ(hg)} det(h|_{N_{g}})(−1)^{|T_{g}∩T_{h}|+N} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h}))
= ϵ(h, g)T (h, g)

where ν_{j} are the Eigenvalues of h on i_{g}(A_{g}) and

ϵ(g, h) = φ_{g,h}(−1)^{σ(g)σ(h)} det(g|_{N_{h}})

and

T (h, g) = (−1)^{σ(g)σ(h)}(−1)^{σ(g)+σ(h)}(−1)^{|T_{g}∩T_{h}|+N} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h}))

we have that

T (h, g) = T (g, h) = T (g^{−1}, h)

Notice also that if [g, h] = e

ϵ(gh, k) = ϵ(g, k)ϵ(h, k)

and

T (gh, h) = T (hg, h) = T (g, h)

On the other hand

STr(φ_{g^{−1}}|_{A_{h}}) = ϵ(g^{−1}, h)(−1)^{σ(g^{−}1h)}(−1)^{|T_{g}∩T_{h}|+N} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h})

= φ_{h,g}(−1)^{σ(g^{−1}h} det(g^{−1}|_{N_{h}})(−1)^{g̃h̃}(−1)^{|T_{g}∩T_{h}|} dim(i_{g}(A_{g}) ∩ i_{h}(A_{h})
= ϵ(g^{−1}, h)T (g^{−1}, h)

where μ_{j} are the Eigenvalues of g^{−1} on i_{h}(A_{h}).

Finally we see that the φ are determined by the ϵ(g, h) which have to satisfy

ϵ(g, h) = ϵ(h^{−1}, g) ϵ(g, g) = 1 ϵ(g_{1}g_{2}, h) = ϵ(g_{1}, h)ϵ(g_{2}, h)

which means that the choices of the projective factors for G–action correspond in a 1–1 fashion to a choice of discrete torsion.

5.3.1. Remarks.

1) There is always one choice of cocycle γ given by

γ_{g,h} = \{ \begin{matrix}χ_{g}^{− \frac{1}{2}} π_{g}(ρ) & h = g^{−1} \\ 0 & else \end{matrix}

2) For any other solution γ by 4.7, we must have γ_{g,h} = 0 if [g, h] = e unless ϕ_{g,h} = ϕ_{h,g} or in other words

det(g|_{N_{h}}) det(h)|_{N_{g}}) = 1

since ϵ(g, h) = ϵ(h^{−1}, g) = eps(h, g)^{−1}

6. Mirror construction for special G–Frobenius algebras

6.1. Double grading. We consider Frobenius algebras with grading in some abelian group I.

A = ⨁_{i∈I} A_{i}

This grading can be trivially extended to a double grading with values in I × I in two ways

A^{cc} = ⨁_{i∈I} A_{i,i}

and

A^{ac} = ⨁_{i∈I} A_{i,−i}

corresponding to the diagonal Δ : I → I × I and (id, −) ∘ Δ : I → I × I. We call bi–graded Frobenius algebras of this form of (c, c)– type and of (a, c)–type respectively. In the language of Euler fields we consider the field (E, Ē) = (E, E) or (E, Ē) = (E, −E)

These gradings become interesting for special G–Frobenius algebras, since in that case the shifts will produce a possible non–diagonal grading.

6.2. Definition. Given a graded special G–Frobenius algebra we assign the following bi–degrees to 1_{g}

(E, Ē)(1_{g}) := (s_{g}, s_{g}^{−1})

It is clear that A_{e} is of (c, c)–type. A is however only of (c, c) type if s_{g} = s_{g}^{−1}.

Furthermore for the Ramond–space of A we assign the following bi– degree to v_{0}

(E, Ē)(v_{0}) := (−\frac{d}{2}, −\frac{d}{2})

6.3. Euler–twist (spectral flow). In this section, we consider a graded special G–Frobenius algebra and construct a new vector–space from it. We denote the grading operator by E:

6.3.1. Definition. The twist–operator j for an Euler–field E is

j := exp(2πiE)

We denote the group generated by j by J.

we call a special G–Frobenius algebra Euler if there is a special J×G– Frobenius algebra Ã of which A is a subalgebra.

6.3.2. Definition. The dual Ǎ to an Euler special G–Frobenius algebra of (c, c)–type is the vector space

Ǎ := ⨁_{g∈G} Ǎ_{g} := ⨁_{g∈G} V_{gj^{−1}}

with the A_{e} and G–module structure determined by τ_{j}^{g} : Ǎ_{g} ≃ V_{gj^{−1}}. together with the bi–grading

(E, Ē)(1′_{g}) := (s_{gj^{−1}} − d, s_{gj^{1}})

where ˇ1_{g} denote the generator of Ǎ_{g} as A_{e}–module and the bi–linear from

η̌:= τ_{j}^{∗} = (η̄)

where τ_{j} := ⊕_{g∈G}τ_{j}^{g}.

7. Explicit Examples

7.1. Self duality of A_{n}. We consider the example of the Jacobian Frobenius Algebra associated to the function z^{n+1}

A_{n} := C[z]/(z^{n})

together with the induced multiplication and the Grothendieck residue. Explicitly:

z^{i}z^{j} = \{ \begin{matrix}z^{i+j} & if i + j ≤ n \\ 0 & else \end{matrix}

the form

η(z^{i}, z^{j}) = δ_{i,n−1−j}

and the grading:

E(z^{i}) := \frac{i}{n + 1}

which means ρ = z^{n−1} and d = \frac{n−1}{n+1}.

We consider just the group J ≃ Z/nZ+1 with the generator j acting on z by multiplication with ζ_{n+1} := exp 2πin + 1. We have

F ix_{j^{i}} \{ \begin{matrix}C & if i = 0 \\ 0 & else \end{matrix}

and thus

A_{j^{i}} \{ \begin{matrix}A_{n} & if i = 0 \\ 1_{j^{i}} & else \end{matrix}

Furthermore we have the following grading;

(E, Ē)(1_{j^{i}}) = \{ \begin{matrix}(0, 0) & for i = 0 \\ (\frac{k−1}{n+1}, \frac{n−k}{n+1} & else \end{matrix}

which means ρ_{j^{i}} = 1_{j^{i}} and d_{j^{i}} = 0.

Using the reconstruction Theorem we have to find a cocycle γ and a compatible action φ. There is no problem for the metric since always |N_{g}| = 1 and if n + 1 is even det(ζ^{\frac{n+1}{2}} ) = −1. Since the group J is cyclic there is no freedom of choice for ϵ and just two choices of parity are possible corresponding to j ↦ ±1.

From the general considerations we know γ_{j^{i},j^{n−1−i}} ∈ A_{e} and deg(γ_{j^{i},j^{n−1−i}}) = d − d_{j^{i}} = \frac{n−1}{n+1} which yields

γ_{j^{i},j^{n−1−i}} = ((−1)^{j̃i}ζ^{i})^{1/2}ρ = ((−1)j̃ζ)^{i/2}z^{n−1}

for the other γ notice that deg(1_{j^{i}})+deg(1_{j^{k}}) = \frac{i+k−2}{n+1} while deg(1_{j^{i+j}}) = \frac{i+k−1}{n+1} if i + k ≠ n + 1, but there is no element of degree \frac{1}{n+1} in A_{j^{i+k}} for i + k ≠ n + 1.

Hence

γ_{j^{i},j^{k}} = \{ \begin{matrix}(−1)j̃ζ)^{i/2}z^{n−1} & for i + j = n + 1 \\ 0 & else \end{matrix}

this means that

φ_{j^{i},j^{k}} = (−1)^{j̃^{i}j̃^{k}}ζ^{−i}

Therefore the G–invariants A^{G} = A_{1} generated by the identity 1 The Ramond grading of this algebra is

(E, Ē)(1_{j^{i}}v_{0}) = \{ \begin{matrix}(0, 0) & for i = 0 \\ (\frac{k}{n+1} − \frac{1}{2}, −\frac{k}{n+1} + \frac{1}{2}) & else \end{matrix}

Since j ∈ G the special G–Frobenius algebra is Euler and the dual is defined, moreover G = J so that the vector–space structures of A and Ǎ coincide. The grading is given by:

(E, Ē)(1_{j^{i}}v_{0}) = \{ \begin{matrix}(0, 0) & for i = 0 \\ −\frac{k−(n+1)}{n+1} − \frac{1}{2}, \frac{n+1−k}{n+1} + \frac{1}{2}) & else \end{matrix}

The G–action is given by

φ̌_{j^{i},j^{k}} = \{ \begin{matrix}(−1)^{j̃^{i}j̃^{k}}ζ^{i} & for k = 1 \\ (−1)^{j̃^{i}j̃^{k}} & for k = 1 \end{matrix}

This G–action leaves all even sectors Ǎ_{j^{i}} invariant except for i = 1. Thus with the choice of all even sectors we have as G–modules

Ǎ ≃ A_{n}

where more explicitly

ˇ1_{j^{i}} ↦ z^{n+1−k} : k = 2, … n and ˇ1_{0} ↦ 1.

Notice that this A_{n} is of (a, c)–type however. Also since J = G the form η̄ pulls back and gives a non–degenerate form on Ǎ^{G}, which is the usual form on A_{n} Furthermore the usual multiplication on A_{n} is compatible with everything so that we can say that A_{n} is self–dual under this operation.

7.2. D_{n} from a special Z/2Z–Frobenius algebra based on A_{2n−3}. In this section, we show how to get D_{n} from a special Z/2Z–Frobenius algebra based on A_{2n−3}. The function for the Frobenius algebra A_{2n−3} is z^{2n−2}. Since this is an even function Z/2Z acting via z ↦ −z is a symmetry. There are two sectors the untwisted and the twisted sector containing the element 1_{−1} with degree 0 The multiplication is fixed by deg(γ_{−1,−1}) = \frac{2n−4}{2n−2} thus

γ_{−1,−1} = z^{2(n−2)}

again the group is cyclic so the G–action only depends on the choice of parity of the −1–sector, we have to be careful since (−1)^{2} = 1, but since |N_{−1}| = 1 we have that (−1)^{˜−1}χ_{−1} = 1 to either σ(−1) = 1 or σ(−1) = 0.

In the untwisted sector we have A_{e}^{Z/2Z} = ⟨1, z^{2}, …, z^{2(n−1)}⟩ ≃ A_{n−1} and the action of −1 on 1_{−1} is given by

φ_{−1,−1} = (−1)^{˜−1 ˜−1} + 1)

so that if σ(−1) = 1

A_{2n−3,Z/2Z}^{Z/2Z} ≃ D_{n}

and if σ(−1) = 0 we just obtain the invariants of the untwisted sector which are isomorphic to A_{n−1}.

The untwisted sector is given by the singularity A_{n−1} as expected upon the transformation u = z^{2}, Notice, that the invariants of the Ramond sector yield the singularity A_{n−2} as expected from [W]. These are of the form u^{i}duor z^{2i+1}dz with i = 0, …, n − 3.

7.3. Point mod G. In the theory of Jacobian Frobenius algebras there is the notion of a point is played by a Morse singularity z_{1}^{2} + ⋯ + z_{n}^{2}. Any finite subgroup group G ⊂ O(n, k) leaves this point invariant.

The G–twisted algebra after possibly stabilizing is the following.

A = ⨁_{g∈G} k1_{g}

And the grading is deg(1_{g}) = (\frac{1}{2}s_{g}^{−}, \frac{1}{2}s_{g}^{−}), since d = d_{g} = 1 We have several different choices for the cocycle by solving the equa-

tion of the previous paragraph with choices for ϵ and the super–sign and the Ansatz:

γ_{g,h} = ψ_{g,h}1_{e}

an in the graded case with the degree restriction: \frac{1}{2}s_{g}^{−} + \frac{1}{2}s_{h}^{−} = \frac{1}{2}s_{gh}^{−}. Special case. If we assume that G ⊂ SO(n, C) and that s_{g}^{−} = 0 (i.e. ∑_{i:λ_{i}≠0} \frac{1}{2πi}λ_{i} =

\frac{|N_{g}|}{2} ∈ N, then we can choose the cocycle

γ_{g,h} = 1_{e}

which makes A into the group algebra.

7.3.1. Point mod Z/nZ. By the above analysis we realize Z/nZ as the sub–group of rotations of order n in C. We have that s_{g}^{−} = 0 and thus we can choose the full cocycle making A into A_{n}, multiplicatively, with trivial grading and trivial G–action one chooses all even sectors.

The metric, however will not be consistent. The identity pairs with itself for instance. Dualizing A,we obtain the following space

Ǎ = ⊕_{j^{i}}A_{j^{i}−1}

with again a trivial (E, Ē) grading. Choosing the generator j̃ := j^{−1} for J the metric reads

η̃(˜1j̃^{i}, ˜1j̃^{k}) = \{ \begin{matrix}1 & if i + j = n − 1 \\ 1 & i = j = n \\ 0 & else \end{matrix}

This metric is compatible with the following multiplication:

˜1j̃^{i} ∘ ˜1j̃^{k} = \{ \begin{matrix}˜1j̃^{i+k} & if i + j ≤ n − 1 \\ ˜1j̃^{n−1} & i = j = n \\ 0 & else \end{matrix}

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E-mail address: kaufmann@math.usc.edu

University of Southern California, Los Angeles, USA