do graphs admit topological field...

Do Graphs Admit Topological Field Theories?


Amir Daneshgar

Department of Mathematical Sciences

Sharif University of Technology

February 23, 2011


Outline

What is a topological field theory?The category Cob(d+1)Catch a fleeting glimpse of Atiyah’s TFT

On the category of graphsThe homomorphism problemSome computational facts

Graphical TFT’s at the basement level!The cylindrical constructionConcluding remarks


What is a topological field theory?

Basic ideas and motivations

I Basic physical motivation: Is it possible to construct a theoryof everything? (in a physical sense!)

I This general unknown theory is usually referred to as quantumgravity.

I A breakthrough: Feynman’s approach through actionfunctionals and Feynman’s integrals. (This does not coverEinstein’s gravity!)

I One of the main ideas: Gluing dynamics on space-timestogether in a compatible way.

I Surprise: This gives rise to very interesting mathematicalconcepts and results too!



Atiyah’s formulation

I In its simplest form, a (d+ 1)-dimensional TFT,

Z : Cob(d+ 1)→ FVecK ,

is a monoidal functor from the category Cob(d+ 1) to thecategory of finite dimensional vector spaces (over a fixed fieldK).

What does this mean?



The category Cob(d+1)

Objects and 1-maps of Cob(d + 1)

I The objects of Cob(d+ 1) are d-dimensional manifoldswithout boundary.

I Let B− and B+ be two such objects. Then a 1-map (or anordinary map) is a diffeomorphism f : B− → B+. This spaceis naturally equipped with the identity map and thecomposition of such maps.




Cobordisms and 2-maps in Cob(d + 1)

I A cobordism in Cob(d+ 1) is an equivalence class of (smooth,compact and oriented) manifolds, as [M ] : B− → B+, ofdimension (d+ 1) with boundary ∂M = (B−)op ∪B+.

I Two such cobordisms, M and N are equivalent if there is adiffeomorphism f : M → N that restricts to identity map ontheir common boundary.




Cobordisms and 2-maps in Cob(d + 1)

I A cobordism is a model of a space-time.

I Composition of cobordisms are defined by gluing!

I 2-morphisms between cobordisms are defined naturally!




Cob(d + 1) and FVecK

as monoidal categories

I The category Cob(d+ 1) has a monoidal structure withrespect to disjoint union of cobordisms.

What does this mean?



Catch a fleeting glimpse of Atiyah’s TFT

Invariants through TFT’s

I Identify V ∗ ' Hom(V,K).

I Identify a (d+ 1)-manifold M with boundary ∂M with acobordism ∅ → ∂M .

I Identify a (d+ 1)-manifold Mop with boundary ∂Mop with acobordism ∂Mop → ∅.




Invariants through TFT’s

I Then applying Z we have:

Z(M) ≡ (f : Z(∅) = K → Z(∂M)) ≡ (z(M) def= f(1) ∈ Z(∂M)).

I Note that z(Mop) can be identified with a map

zop(M) def= (Z(∂M)→ Z(∅) = K) as an element of Z(∂M)∗.

How one may compute z(M)?




Invariants through surgery I

I Denote the duality in FVecK by 〈., .〉 : V ∗ ⊗ V → K.

I Let a (d+ 1)-manifold M be cut through an embeddedd-manifold B into two parts M1 and M2 , i.e.M = M1 ∪B M2 with ∂M1 = B and ∂M2 = Bop.




Invariants through surgery I

I Thinking of M1 and M2 as cobordisms we haveZ(M) = Z(M1) ◦ Z(M2) and consequently an elementz(M) = 〈zop(M2), z(M1)〉.

I Note that when ∂M = ∅ then through identification we havethe numerical invariant zop(M) ≡ z(M) ∈ K.




Invariants through surgery II

I More generally if ∂M = (B−)op ∪B+ then

Z(∂M) = Z((B−)op ∪B+) ' Z(B−)∗ ⊗ Z(B+) '

' Hom(Z(B−), Z(B+)).





I Hence for a gluing M = M1 ∪B M2 with ∂M1 = A ∪B and∂M2 = Bop ∪ C, we have,

Z(M) = Z(M2) ◦ Z(M1).





I Idea: One may obtain invariants of closed manifolds andcategorize them if one can construct all such manifoldsthough amalgams of a finite number of basic atomics ones.

Let us see a couple of examples in dimension 1 + 1!




A cylinder




A sphere




A torus




Frobenius relation

I There is an equivalence of categories between the category of2-dimensional topological (quantum) field theories andcommutative Frobenius algebras.


On the category of graphs

The tangent space

I The tangent space at vertex v is the set of out-going vectorsfrom v!



The homomorphism problem

A graph homomorphism

I A graph homomorphism σ from a graph G to a graph H is amap σ : V (G) −→ V (H) such that uv ∈ E(G) impliesσ(u)σ(v) ∈ E(H).




A question

I Does there exist a homomorphism from the Petersen graph tothe triangle K3?




Graph coloring

I Homomorphisms to Kn is equivalent to coloring the verticesof the graph by n colors such that the terminal ends of eachedge have different colors.



Some computational facts

The homomorphism problem I

I Fix a graph H and ask: Given a graph G, is it true thatHom(G,H) 6= ∅.

I It is known that this decision problem is NP-complete in mostsymmetric cases of H, as for simple graphs andvertex-transitive directed graphs.



Some computational facts

The homomorphism problem II

I It is known that any CSP reduced to a directedhomomorphism problem.

I The dichotomy conjecture for CSP’s imply thatthe homomorphism problem of graphs is either in P or isNP-complete.

How can we tackle this conjecture?


Graphical TFT’s at the basement level!

Graphical TFT’s

I Edges are counterparts of vector spaces (tangent spaces) ingraphs.

I Hence, in order to mimic the scenario of a TFT in graphtheory we ought to ask about a nice category of graphicalcobordisms (i.e. cylinders)!

In what follows we investigate the existence of such a category ofcylinders!



The cylindrical construction

Graphical cylinders

I Intuitively a cylinder is a graph M with boundary∂M = B− ∪B+.




Cylindrical and exponential constructions

I Given a cylinder C, there exist a cylindrical constructionG� C and an exponential construction [C,H], which arefunctorial and moreover,

a) Hom(G� C,H) 6= ∅ ⇔ Hom(G, [C,H]) 6= ∅.b) There exist a retraction

rG,H

: Hom(G� C,H)→ Hom(G, [C,H])

and a section

sG,H

: Hom(G, [C,H])→ Hom(G� C,H)

such that rG,H◦ s

G,H= 1, where 1 is the identity mapping.




An example

I Hom(Cn�K2 ,K3) 6= ∅ ⇔ Hom(Cn , [K2�K2 ,K3 ]) 6= ∅.




Some examples

A directed cylindrical construction Show(C12)Cartesian product as a cylindrical construction Show(C3�P2)The Petersen graph ShowSubdivision and powers ShowThe exponential graph [K2�K2 ,K3 ] Show



Concluding remarks

Epilogue

I The problem of defining a well-behaved category of graphicalcylinders (cobordisms) and corresponding TFT’s is not onlyimportant from a mathematical point of view but also isclosely linked to fundamental problems in computer science as

P?= NP .

I It seems that such constructions may lead to situations whichare quite similar (but maybe weaker than) Tensor-Hom dualitywhich may be of independent interest in graph theory.


The End

The End.

Thank You!

do graphs admit topological field...

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