1. Crossed Systems and HQFT’s
Neha Gupta
Shiv Nadar University, Dadri, UP

2. 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

3. 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.

4. TQFT’s contd.
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.

5. 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.

6. 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

7. 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:

8. 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 3 Ab
Category of Abelian groups group homomorphisms
Quotient of free group on their direct sum by the tensor relations
Z (group of integers) 4
R-Mod
modules over comm ring R and ring homomorphisms
Tensor product of modules ⊗R R 5
k-Vect
Vector spaces over k and linear transformations
Tensor products of vector spaces k 6
R-Bimod
(R,R)-bimodules over comm ring R and ring homomorphisms
Tensor product of bimodules R⊗R No

9. 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)

10. 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.

11. 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

12. 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

13. Category X-Cobn over X in degree n :
Notions:
We do it in three stages.
Stage I: X−Cobn
0-morphism M = (M, fM, pM ), where M = n-dimensional closed manifold st each component is a pointed closed manifold, fM : M → X is continuous and pM = point on each component.
1-morphism from M=(M, fM, pM ) to K = (K, fK, pK ) is a triple A = (A, fA, αA ), where A= (n+1)- dimensional manifold, fA : A → X is continuous and αA : ∂A →(-M) ⊔ K is an X-homeomorphism.
Note : X−Cobn 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 boundary
Closed manifold = manifold without boundary
X = (X, x) is a pointed path-connected topological space.

14. 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, φ0, α 0) = A (ii) (A, φt, α t) are 1-morphisms (iii) γ : (A, φ1, α1) → B is an X -homeomorphism of 1-morphisms (which means γ preserves orientation/base points st fA =fB 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, pK) and K = (K, fK, pK) are isomorphic if there are 1- morphisms A = (A, fA, α A ) and B = (B, fB, 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: X−Cobn (i) Objects = 0-morphisms of X−Cobn
(ii) Morphisms = equivalence classes of 1-morphisms in X−Cobn

15. Stage III: Theorem [-] : X−Cobn is a category. Stage III: X−Cobn
Objects = Representatives {Xα} of isomorphism classes of connected 0-morphisms in X−Cobn
Let = < Xα> be the smallest full subcategory containing these objects such that it is closed under disjoint unions.
Then X−Cobn is a fully faithful monoidal subcategory of X−Cobn . Thus we have an inclusion functor G : X−Cobn ↪ X−Cobn
Theorem [-] : X−Cobn is a category.
Theorem [-]: The functor G : X−Cobn ↪ X−Cobn is a symmetric monoidal equivalence.

16. X = pointed K(G,1) space, for a multiplicative abelian group G.
We give description of X−Cob𝟏 in this case.
Objects/Circles : M = ⊔g( Mϵg, g ϵ G ) where Mϵg = (Sϵ1, g) is a circle S1 with standard +ve/-ve orientation ϵ ϵ { ±1} and homotopy class of the map into X 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 Dϵ which can be viewed as a cobordism between empty set and Sϵ1. The homotopy class of the map
fD : Dϵ → X is determined by the homotopy class g ϵ G
represented by the loop f|S1ϵ. See the figure.
S+1

17. (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|C0ϵ and f|s x [0,1] respectively.
Here the interval is oriented from 0 to 1. The loop f|C1μ represents (h-1 g-ϵ h)μ.
See the picture below:
C0ϵ = S1ϵ x 0
C1μ = S1μ x 1

18. (iii) a disc with two holes Pϵ, μ, γ(g1, g2, g3, g4)
Its also called a pant. Denote its boundary by Lϵ, M μ and Nγ 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, g2, g3, g4 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, g2 and (g4g2-ϵ g4-1 g3g1- μg3-1)γ respectively. See the picture below:

19. Structures-n-results related to X−Cob𝟏
A peep inside X−Cob𝟏

20. Circles in X−Cob𝟏 :
The morphisms discussed in (i), (ii) and (iii) are the basic morphisms that generate the whole of the category X−Cob𝟏
Circles : Let Ag = (S1- , g) and denote the collection (Ag)gϵG in X−Cob𝟏 as A. We call the collection A as circles in X−Cob𝟏 . 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 → I
ϕf : Af ⊗ Af-1 → I
ϕf : I → Af ⊗ Af-1

21. Theorem for Circles in X−Cob𝟏: [- , D. Rumynin]
With these notations and morphisms encapsulated with A, we have proved the following result :
Theorem : Circles in X−Cob𝟏 form a Frobenius G-graded system with X = (K(G, 1), x) space; multiplication = μf,g; unit η and pairing = ϕf as described above.

22. Cylinders and X –Cylinders in X−Cob𝒏 :
Consider a morphism A : M → N in X−Cob𝒏 . Then A is called a cylinder if A ≅ M x I as topological manifolds.
And A is an X−cylinder of M if A is X -homeomorphic to M = (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 X -cylinder is a cylinder, but a cylinder may not necessarily be an X -cylinder. For example, consider the handle ∅ → S1⊔ S1 in X−Cob𝟏 .
The concept of cylinders and X -cylinders have two directions to go about.
One is considering the mapping class group of M containing all the equivalent classes of the X -cylinders of M.
And the other direction is to enrich the collection of circles in X−Cob𝟏 with crossed system using X -cylinders of circles.

23. Theorem for Cylinders in X−Cob𝟏 : [- , D. Rumynin]
Theorem : Cylinders in X−Cob𝟏 define a G-crossed system on circles (with X = (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 C− + (g,h−1) Ahgh-1

24. Presentation Slides : Introduction (TQFT’s and HQFT’s) Concepts -
Frobenius systems in a monoidal category
Crossed systems in a symmetric monoidal category
Cobordism Category X−Cob𝒏 and results in X−Cob
Definition of an HQFT
Main Results

25. (n+1)-dimensional X -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 X using a version of Atiyah's axioms for a TQFT.
Our mechanism shall regard an (n+1)-dimensional HQFT as a monoidal functor Z : X−Cob𝟏 → C for any monoidal category C.
Definition : An (n + 1)-dimensional X -HQFT Z assigns to any
- { n-dimensional X -manifold (M, gM ) }  {some object Z M in C.}
- { X -homeomorphism of n-dimensional X-manifolds f : M ---> N}  { an isomorphism f# : ZM  Z𝑁}
- { (n+1)-dimensional X -cobordism (W, M0, M 1) }  { a morphism Z(W) : ZM0  ZM1in C}.
These objects and morphisms satisfy the following eight axioms.

26. Axioms of an HQFT
(i) Z respects compositions of X –homeomorphisms between X – manifolds. That is, for two composable X –homeomorphisms, we have (f’f)# = f’# f#
(ii) Z respects monoidal structure on objects naturally : ∃ a natural isomorphism ZM⨆N ≅ ZM ⊗ ZN
(iii) Z takes identity to id : Z∅ ≅ IC
(iv) Z respects X –homeomorphism between X –morphisms For F : (W, M0, M1 , g) (W’, M’0, M’1 , g’), we have Z(W’) ◦ (F|M0 )# = (F|M1 )# ◦ 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”) : ZM0  ZM1
(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.

27. 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 X – cobordism with empty boundaries. And thus we get the corresponding endomorphism of A∅ = IC
Turaev does not say explicitly about symmetric structure. There exists an X-homeomorphism f: M⊔N  N⊔M for any two n-dimensional X –manifolds M and N. Correspondingly, ∃ an isomorphism f# : ZM⨆N  ZN⨆M in C. Then Axiom (ii) gives the isomorphism ZM ⊗ ZN ≅ ZN ⊗ ZM
This forces a symmetric structure on the objects ZM in C. Suppose C is already symmetric, then to avoid confusion between the two symmetric structures in C, we introduce axiom (ix) : The original braiding of the category C agrees with the forced symmetric structure on the objects ZM
Rigidity on the objects ZM in C is enforced automatically by X -HQFT by setting (ZM)* = Z(-M)

28. Results : [N. Gupta]
Theorem : Suppose X is a K(G; 1) space. Then any (1+1)-dimensional X -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.

29. 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.

30. Future Interest/work :
TQFT structure on Symplectic manifolds.
HQFT structure on Loop spaces and if possible obtain some invariants of these spaces.

31. References: [Kock] http://mat.uab.es/~kock/TQFT.html
[Tu1] /abs/math/
[Tu2]
[RT]
[kock]
[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.,
[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.

32. Thanks 