Do Graphs Admit Topological Field...
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![Page 1: Do Graphs Admit Topological Field Theories?math.sharif.ir/faculties/uploads/daneshgar/Lectures/IUT... · 2019. 7. 14. · Einstein’s gravity!) I One of the main ideas:Gluing dynamics](https://reader035.fdocuments.us/reader035/viewer/2022071416/6113f8e3550b04285362b5c9/html5/thumbnails/1.jpg)
Do Graphs Admit Topological Field Theories?
Do Graphs Admit Topological Field Theories?
Amir Daneshgar
Department of Mathematical Sciences
Sharif University of Technology
February 23, 2011
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Do Graphs Admit Topological Field Theories?
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
![Page 3: Do Graphs Admit Topological Field Theories?math.sharif.ir/faculties/uploads/daneshgar/Lectures/IUT... · 2019. 7. 14. · Einstein’s gravity!) I One of the main ideas:Gluing dynamics](https://reader035.fdocuments.us/reader035/viewer/2022071416/6113f8e3550b04285362b5c9/html5/thumbnails/3.jpg)
Do Graphs Admit Topological Field Theories?
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
![Page 4: Do Graphs Admit Topological Field Theories?math.sharif.ir/faculties/uploads/daneshgar/Lectures/IUT... · 2019. 7. 14. · Einstein’s gravity!) I One of the main ideas:Gluing dynamics](https://reader035.fdocuments.us/reader035/viewer/2022071416/6113f8e3550b04285362b5c9/html5/thumbnails/4.jpg)
Do Graphs Admit Topological Field Theories?
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
![Page 5: Do Graphs Admit Topological Field Theories?math.sharif.ir/faculties/uploads/daneshgar/Lectures/IUT... · 2019. 7. 14. · Einstein’s gravity!) I One of the main ideas:Gluing dynamics](https://reader035.fdocuments.us/reader035/viewer/2022071416/6113f8e3550b04285362b5c9/html5/thumbnails/5.jpg)
Do Graphs Admit Topological Field Theories?
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!
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
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?
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
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.
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
The category Cob(d+1)
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.
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
The category Cob(d+1)
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!
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
The category Cob(d+1)
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?
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
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 → ∅.
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
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)?
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
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.
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
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.
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
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+)).
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
Invariants through surgery II
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).
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
Invariants through surgery II
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!
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
A cylinder
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
A sphere
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
A torus
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Do Graphs Admit Topological Field Theories?
What is a topological field theory?
Catch a fleeting glimpse of Atiyah’s TFT
Frobenius relation
I There is an equivalence of categories between the category of2-dimensional topological (quantum) field theories andcommutative Frobenius algebras.
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Do Graphs Admit Topological Field Theories?
On the category of graphs
The tangent space
I The tangent space at vertex v is the set of out-going vectorsfrom v!
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Do Graphs Admit Topological Field Theories?
On the category of graphs
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).
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Do Graphs Admit Topological Field Theories?
On the category of graphs
The homomorphism problem
A question
I Does there exist a homomorphism from the Petersen graph tothe triangle K3?
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Do Graphs Admit Topological Field Theories?
On the category of graphs
The homomorphism problem
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.
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Do Graphs Admit Topological Field Theories?
On the category of graphs
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.
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Do Graphs Admit Topological Field Theories?
On the category of 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?
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Do Graphs Admit Topological Field Theories?
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!
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Do Graphs Admit Topological Field Theories?
Graphical TFT’s at the basement level!
The cylindrical construction
Graphical cylinders
I Intuitively a cylinder is a graph M with boundary∂M = B− ∪B+.
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Do Graphs Admit Topological Field Theories?
Graphical TFT’s at the basement level!
The cylindrical construction
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.
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Do Graphs Admit Topological Field Theories?
Graphical TFT’s at the basement level!
The cylindrical construction
An example
I Hom(Cn�K2 ,K3) 6= ∅ ⇔ Hom(Cn , [K2�K2 ,K3 ]) 6= ∅.
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Do Graphs Admit Topological Field Theories?
Graphical TFT’s at the basement level!
The cylindrical construction
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
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Do Graphs Admit Topological Field Theories?
Graphical TFT’s at the basement level!
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.
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Do Graphs Admit Topological Field Theories?
The End
The End.
Thank You!