Crossed Systems and HQFT’s

NEHA GUPTA

SHIV NADAR UNIVERSITY, DADRI, UP

Presentation includes :

Introduction (TQFT’s and HQFT’s)

Concepts –

• Symmetric monoidal category

• Frobenius systems in a monoidal category

• Crossed systems in a symmetric monoidal category

• Cobordism Category

• Definition of an HQFT

Main Results

Introduction

Topological quantum field theories (TQFTs)

TQFTs produce topological invariants of manifolds using ideas suggested by quantum field theory; see [At], [Wi].

For d ≥ 0, a (d+1)-dimensional TQFT over a commutative ring K assigns to every closed oriented d-dimensional manifold M a projective K-module of finite type AM and assigns to every compact oriented (d+1)-dimensional cobordism (W; M0; M1 ) a K-homomorphism τ(W) : AM0

→ AM1.

These modules and homomorphisms should satisfy several axioms including tensor multiplicativity with respect to disjoint union and functoriality with respect to gluing of cobordisms.

The study of TQFTs has been especially successful in low dimensions d = 0, 1, 2, 3.

One-dimensional TQFTs (d = 0) bijectively correspond to projective K-modules of finite type.

Two-dimensional TQFTs (d = 1) are fully classified in terms of commutative Frobenius algebras, see [Kock].

Three-dimensional TQFTs (d = 2) are closely related to quantum groups and braided categories; see [RT], [Tu2]

Powerful four-dimensional TQFTs (d = 3) arise from the Heegaard–Floer homology of 3-manifolds due to P. Ozsváth and Z. Szabó

Algebraic structures underlying four-dimensional TQFTs are yet to be explored in detail.

TQFT’s contd.

Homotopy Quantum Field Theories (HQFT)

The general notion of a (d+1)-dimensional HQFT was introduced in 1999 by Turaev and independently by M. Brightwell and P. Turner [BT1] for d=1 and simply connected target spaces.

Turaev applied the idea of a TQFT to maps from manifolds to topological spaces.

So, a (d+1)-dimensional homotopy quantum field theory (HQFT) may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d+1)-dimensional cobordisms endowed with maps to a given space X.

Such an HQFT yields numerical homotopy invariants of maps from closed oriented (d+1)-dimensional manifolds to the topological space X.

A TQFT may be interpreted in this language as an HQFT with target space consisting of one point.

Presentation includes :

Introduction (TQFT’s and HQFT’s)

Concepts –

• Symmetric monoidal category

• Frobenius systems in a monoidal category

• Crossed systems in a symmetric monoidal category

• Cobordism Category

• Definition of an HQFT

Main Results

Symmetric monoidal Categories

Background : A symmetric monoidal category is a category with a product operation – (a monoidal category) for which the product is as commutative as possible.

Definition : A braided monoidal category, is a monoidal category equipped with a natural isomorphism (braiding)

τX,Y : X ⊗ Y → Y ⊗ X

which is compatible with the associator for the tensor product.

Definition : If the braiding squares to the identity, then the braided monoidal category is a  symmetric monoidal category.

Examples:

Symmetric Categories…S.No. Category Description Tensor Unit Symmetric

1 Set Category of sets and functions Cartesian product of sets

Singletons Yes

2

Cat Category of small categories  

Product category

Category with only one object and its

identity map

Yes

3

Ab Category of Abelian groups group homomorphisms

Quotient of free group on their direct sum by the tensor

relations

 

Z (group of integers)

Yes

4

R-Mod modules over comm ring R and ring homomorphisms

Tensor product of modules R

 

R

Yes

5

k-Vect Vector spaces over k and linear transformations

Tensor products of vector spaces

 

k

Yes

6

R-Bimod (R,R)-bimodules over comm ring R and ring homomorphisms

Tensor product of bimodules RR

 

R

No

Frobenius G-graded System

A Frobenius G-graded system in a monoidal category {C, ⊗, I} is a collection of objects {Ag}gϵG, equipped with following collection of maps

G-algebra : μ = {μg,h}g,hϵG where μg,h : Ag ⊗ Ah → Agh and,

η : I → Ae such that μ and η satisfy certain compatibility conditions.

G-coalgebra : ∆ = {∆g,h}g,hϵG where ∆g,h : Agh → Ag ⊗ Ah

ε : Ae → I , such that ∆ and ε also satisfy certain compatibility conditions.

These maps are such that they give rise to non-degenerate pairings ζ = {ζg}gϵG :

ζg : Ag ⊗ Ag-1 → I (ζg = ε. μg, g-1)

Example and results for Frobenius System:

Equivalent definitions : A G-algebra A = {{Ag}gϵG, μ, η} in a monoidal category C forms a Frobenius G-system if any of the following holds

If A is equipped with a non-degenerate set of pairings ζ = {ζg}gϵG, then A has a co-algebra structure.

the associated induction and coinduction functors from C to A-Mod are naturally isomorphic;

Ind(M)g = Ag ⊗ M and CoInd(M)g = A*g-1 ⊗ M

which is same as A being isomorphic to its dual A * = { {A*g-1}gϵG, {μ*h-1,g-1}, η* } in A-Mod

Example:

When G={e}, it is simply a Frobenius algebra in non-graded case!

In Vectk, a Frobenius systm is a graded Frobenius algebra with fixed non-degenerate pairing.

G-Crossed Systems

Given a symmetric monoidal category C, and a group G, we define a Turaev G-crossed system in C as an ordered quintuple T = (A, μ, η, ϕ, φ) where (A, μ, η, ϕ) has a Frobenius G-graded structure with a non degenerate pairing ϕ and a set of morphisms ϕ = { ϕg,h : Ag → Ahgh-1 | g,hϵ G} in C , such that they satisfy the following set of axioms

Compatibility axiom of φ and μ

Compatibility axiom of φ and ϕ

φg,g-1 = Id

Compatibility axiom of φ and τ (the braiding in the category C )

The trace of the following two compositions are equal

b : Ah ⊗ Af → Ah ⊗ Agfg-1 → Af where b = μ ◦ (1⊗ ϕf,g

) and

c : Ah ⊗ Ag → Ahg → Ag where c = ϕhg,f-1

◦ μ

Example of G-Crossed systems:

Trivial eg : Crossed algebra over a trivial group G = {e} in the category of Vector spaces over a field k are nothing but but commutative Frobenius algebras over k.

Conversely, given such a Frobenius algebra A in this category, it determines a G-crossed system for any group G. The structure is determined by Ag=Ag for gϵG. Multiplications are (αg) (βf) = (αβ) (gf) where α,βϵA and f,gϵG. Pairing is determined by multiplication of scalors. And the set of morphisms ϕ are simply permutation of the copies of A.

G-Crossed system from 2-cocyles : Let θ = {θ(f,g) ϵ K* | f,g ϵ G} be a normalized 2-cocycle of G. Then,

A = { Af = Kf : f,g ϵ G }

multiplication (f,g) θ(f,g) fg

Pairing (f,g) θ(f,g)

Φf,g : f θ(f,g) θ(fgf -1, f)-1 gfg-1

Category X-Cobn over X in degree n :

Notions:

We do it in three stages.

Stage I:

0-morphism M = (M, fM, p

M ), where M = n-dimensional closed manifold st each component is a

pointed closed manifold, fM : M → X is continuous and p

M = point on each component.

1-morphism from M=(M, fM, p

M ) to K = (K, f

K, p

K ) is a triple A = (A, f

A, α

A ), where A= (n+1)-

dimensional manifold, fA : A → X is continuous and α

A : ∂A →(-M) K is an X-homeomorphism.

Note : is a weak 2-category in two senses.

(i) composition is not associative (ii) associativity and identity defined up to 2-isomorphism

manifold = compact oriented topological manifold with boundaryClosed manifold = manifold without boundaryX = (X, x) is a pointed path-connected topological space.

Stage II:

2-morphisms : A 2-morphism is defined as homotopies up to an isotopy on the boundary. Explicitly, ϕ : A → B is a triple (φ, α, γ) where φ : A x [0,1] → X and α : ∂A x [0,1] → (-K) M are continuous maps such that (i) (A, φ

0, α

0) = A (ii) (A, φ

t, α

t) are 1-morphisms (iii) γ : (A, φ

1,

α1) → B is an -homeomorphism of 1-morphisms (which means γ preserves

orientation/base points st fA =f

B γ and also preserves the bdry st …similar on both the

boundaries)

Equivalent 1-morphisms (A ~ B) : Two 1-morphisms A, B : K → M are equivalent if there exists a 2-morphism from A to B.

Isomorphic 0-morphisms : M = (M, fK, p

K) and K = (K, f

K, p

K) are isomorphic if there are 1-

morphisms A = (A, fA, α

A ) and B = (B, f

B, α

B) from M to K and K to M respectively, such

that IK ~ A ◦B and I ~ B ◦ A. In this case we say B is an inverse of A.

Stage II: (i) Objects = 0-morphisms of

(ii) Morphisms = equivalence classes of 1-morphisms in

Stage III:

Stage III:

Objects = Representatives {Xα} of isomorphism classes of connected 0-morphisms in

Let = < Xα> be the smallest full subcategory containing these objects such that it is closed

under disjoint unions.

Then is a fully faithful monoidal subcategory of . Thus we have an inclusion functor G :

Theorem [-] : is a category.

Theorem [-]: The functor G : is a symmetric monoidal equivalence.

X = pointed K(G,1) space, for a multiplicative abelian group G.

We give description of in this case.

Objects/Circles : M = g( M

ϵ

g, g ϵ G ) where Mϵ

g = (Sϵ

1, g) is a circle S1 with standard +ve/-ve

orientation ϵ ϵ { 1} and homotopy class of the map into given by g ϵ G.

Duals : Dual of (S+

1, g) is (S+

1, g-1) and for (S-1, g) is (S-

1, g-1)

Morphisms : Any 2-d compact oriented X –manifold can be constructed using three basic structures which are (i) a disc D

ϵ which can be viewed as a cobordism between empty set and

Sϵ

1. The homotopy class of the map

fD : Dϵ → X is determined by the homotopy class g ϵ G

represented by the loop f|S

1ϵ. See the figure.

S+

1

(ii) a disc with one hole Cϵ, μ

(g,h)

Its an annulus that can be viewed as a cylinder C = S1 x [0,1] with

as in-boundary of C as out-boundary of C

Provide C0 and C1 with base points : s x 0 and s x 1 respectively where s ϵ S1.

The homotopy class of the map fC : Cϵ, μ→ X is determined by the homotopy classes g, h

ϵ G represented by the loops f|C

0ϵ and f|

s x [0,1] respectively.

Here the interval is oriented from 0 to 1. The loop f|C

1μ represents (h-1 g-ϵ h)μ.

See the picture below:

C0ϵ = S1

ϵ

x 0 C1μ = S1

μ

x 1

(iii) a disc with two holes Pϵ, μ, γ

(g1,

g2, g

3, g

4)

Its also called a pant. Denote its boundary by Lϵ, M

μ and N

γ with base points l, m, n resp.

The homotopy classes of the map fC : Pϵ, μ, γ → X is determined by the homotopy classes

g1, g

2, g

3, g

4 ϵ G represented by the loops f|

Lϵ

, f|Mμ

, f|nl , f|

nm respectively. Here the

mutually disjoint embedded arcs nl and nm are in P.

Note : This establishes a bijective correspondence between set of homotopy classes of maps f and G4.

The loops f|Lϵ

, f|Mμ

, f|Nγ

represent the classes g1, g

2 and (g

4g

2

-ϵ g4

-1 g3g

1

- μg3

-1)γ respectively.

See the picture below:

Structures-n-results related to

A peep inside

Circles in :

The morphisms discussed in (i), (ii) and (iii) are the basic morphisms that generate the whole of the category

Circles : Let Ag = (S1

-

, g) and denote the collection (Ag)gϵG in as A. We call the

collection A as circles in . Note that A has following structures:

(i) The Disc D+ gives the unit structure on A

(ii) The Pant P- -+

(f,g,1,1) between L M and N where L= (S1

- , f), M = (S1

- , g) and N = (S1

+

, fg) gives the following structure on A

(iii) The Disc D- gives the counit

(iv) Composition of P- -+

(f,f-1,1,1) and D- gives pairing

(iv) Composition of D+ and P- ++

(1,f,1,1) gives co pairing

η : I → A1

μf,g: Af Ag → Afg

ϵ : A1 → Iϕf : Af

Af-1 → I

: I → Af A

f-1

Theorem for Circles in : [- , D. Rumynin]

With these notations and morphisms encapsulated with A, we have proved the following result :

Theorem : Circles in form a Frobenius G-graded system with = (K(G, 1), x) space; multiplication = μ

f,g; unit η and pairing = ϕf as described above.

Cylinders and –Cylinders in :

Consider a morphism : → in . Then is called a if A M x I as topological manifolds.

And is an of if A is -homeomorphic to = (M x I, fM , α ), with α given as

identity on one-end and out-boundary map of A at the other end.

Clearly, an -cylinder is a cylinder, but a cylinder may not necessarily be an -cylinder. For example, consider the handle → S1 S1 in .

The concept of cylinders and -cylinders have two directions to go about.

One is considering the mapping class group of M containing all the equivalent classes of the -cylinders of M.

And the other direction is to enrich the collection of circles in with crossed system using -cylinders of circles.

Theorem for Cylinders in : [- , D. Rumynin]

Theorem : Cylinders in define a G-crossed system on circles (with = (K(G, 1), x) space; and morphisms as described before.

The information of G-action on circles is carried by the cylinder C- +

(g,h-1)

ϕg,h

: Ag Ahgh-1

Presentation Slides :

Introduction (TQFT’s and HQFT’s)

Concepts -

• Frobenius systems in a monoidal category

• Crossed systems in a symmetric monoidal category

• Cobordism Category and results in

• Definition of an HQFT

Main Results

(n+1)-dimensional -HQFT

HQFTs were introduced by Turaev in 1999, and they are essentially TQFTs in a background space X, up to homotopy.

Turaev gives the axiomatic definition of an HQFT with target using a version of Atiyah's axioms for a TQFT.

Our mechanism shall regard an (n+1)-dimensional HQFT as a monoidal functor Z : → for any monoidal category .

Definition : An (n + 1)-dimensional -HQFT Z assigns to any

- { n-dimensional -manifold (M, g) } {some object Z in .}

- {-homeomorphism of n-dimensional X-manifolds f : M ---> N} { an isomorphism f : Z Z}

- { (n+1)-dimensional -cobordism (W, M0, M

1) } { a morphism Z(W) : Z Zin C}.

These objects and morphisms satisfy the following eight axioms.

Axioms of an HQFT

(i) Z respects compositions of –homeomorphisms between – manifolds. That is, for two composable –homeomorphisms, we have (f’f)# = f’# f#

(ii) Z respects monoidal structure on objects naturally : a natural isomorphism ZMN ZM ZN

(iii) Z takes identity to id : Z I

(iv) Z respects –homeomorphism between –morphisms For F : (W, M0, M1 , g) (W’, M’0, M’1 , g’), we have Z(W’) ◦ (F|M

0 )# = (F|M

1 )# ◦ Z(W’)

(v) Z respects monoidal structure on morphisms : Z(W) = Z(W’) Z(W”) where W is a disjoint union of W’ and W”

(vi) Z respects gluing of cobordisms : If (W, M0, M1) is obtained by gluing of (W’, M0, N) and (W”, N’, M1 ) along an X-homeomorphism f : N N’ then Z(W) = Z(W’) ◦ f# ◦ Z(W”) : ZM

0 ZM

1

(vii) Z(M x [0,1], M x 0, M x 1, g) = id : ZM ZM

(viii) Z is preserved under any homotopy of g relative to ∂W.

Remarks on HQFT :

The axioms (i)-(vii) constitute definition of a TQFT.

In case a (n+1)-dim HQFT has a trivial target space then it is simply a TQFT.

Axiom (viii) implies that Z(W) is a homotopy invariant of g. A closed oriented (n + 1)-dimensional manifold W endowed with a map g : W X can be considered as an – cobordism with empty boundaries. And thus we get the corresponding endomorphism of A I

Turaev does not say explicitly about symmetric structure. There exists an X-homeomorphism f: MN NM for any two n-dimensional –manifolds M and N. Correspondingly, an isomorphism f# : ZMN ZNM in . Then Axiom (ii) gives the isomorphism ZM ZN ZN ZM

This forces a symmetric structure on the objects ZM in . Suppose is already symmetric, then to avoid confusion between the two symmetric structures in , we introduce axiom (ix) : The original braiding of the category agrees with the forced symmetric structure on the objects ZM

Rigidity on the objects ZM in is enforced automatically by -HQFT by setting (ZM)* = Z(-M)

Results : [N. Gupta]

Theorem : Suppose is a K(G; 1) space. Then any (1+1)-dimensional -HQFT with values in a monoidal category C defines a G-crossed system in C.

Theorem : A G-crossed system (Ag, μ, ∆, ϵ) in C

defines (up to isomorphism) a (1+1)-dimensional symmetric HQFT over K(G,1) space with values in C.

Result : [N.Gupta]

Theorem : Given a monoidal category C, if X = (X, x) is a K(G,1) space, then the functor F : Z2(X; C) T(G, C) is an equivalence of categories.

Note :

Z2(X; C) = category of (1+1)-dimensional X -HQFTs with values in C.

T(G, C),= category whose objects are Turaev crossed G-systems in a symmetric monoidal category C.

Future Interest/work :

TQFT structure on Symplectic manifolds.

HQFT structure on Loop spaces and if possible obtain some invariants of these spaces.

References:

[Kock] http://mat.uab.es/~kock/TQFT.html

[Tu1] http://arxiv. /abs/math/9910010

[Tu2] http://arxiv.org/pdf/1202.6292.pdf

[RT] http://arxiv.org/pdf/math/0105018.pdf

[kock] http://mat.uab.es/~kock/TQFT.html

[BT1] M. Brightwell, P. Turner and S. Willerton, Homotopy quantum field theories and related ideas, Proceedings of the Tenth Oporto Meeting on Geometry, Topology and Physics (2001). Int. J. Modern Phys. A, 18 (2003), October, suppl., 115--122.

[At] Michael Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math.68(1989), 175–186.

[Wi] Edward Witten, Topological quantum field theory , Comm. Math. Phys. 117 (1988), 353–386.

http://warwick.ac.uk/38110/

Thanks