Frobenius structures over Hilbert C*-modules

Chris Heunen Manny Reyes

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Naive quantum field theory

“Categorical quantum mechanics needs to come toterms with infinite-dimensionality. This paper is justdipping its toe into those cold waters, working withsituations where all infinite-dimensionality is concentratedin the commutative part. But the theorems are sufficientlysimple and beautiful that they will stand the test of time.”

– Anonymous referee

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Classical limit

~0

physicaltheories

qu

antu

m

qu

antu

m

clas

sica

l

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Theories varying over time

tbig

bang

theo

ry

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Continuing higher categories

U U∗ =

Colours = classical outcomes

finite:infinite:

continuous:

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Base space

Let X be locally compact Hausdorff space.C0(X) = {f : X → C cts | ∀ε > 0∃K ⊆ X cpt : f(X \K) < ε}

C

Xε

f

K

Cb(X) = {f : X → C cts | ∃‖f‖ <∞∀t ∈ X : |f(t)| ≤ ‖f‖}

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Hilbert spaces

C-module H with complete inner product valued in C

tensor product over C monoidal categorytensor unit C tensor unit Icomplex numbers C scalars I → Ifinite dimension dual objectsadjoints daggerorthonormal basis commutative dagger Frobenius structureC*-algebra dagger Frobenius structure

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Hilbert modules

C0(X)-module with complete inner product valued in C0(X)

tensor product over C0(X) monoidal categorytensor unit C0(X) tensor unit ICb(X) scalars I → I? dual objectsadjointable morphisms dagger? dagger Frobenius structure

“Scalars are not numbers”

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Beware

�

I Bounded map may not be adjointable

I Closed subspace may not be complemented

I Subobject of dual object may not be dual

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Bundles of Hilbert spaces

Bundle E � X, each fibre Hilbert space, operations continuous

, with

E

Xt

Et

Hilbert C0(X)-modules ' bundles of Hilbert spaces over Xsections vanishing at infinity ←[ E � X

E 7→ localisation

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Bundles of Hilbert spaces

Bundle E � X, each fibre Hilbert space, operations continuous, with

E

Xt

Et

Hilbert C0(X)-modules ' bundles of Hilbert spaces over Xsections vanishing at infinity ←[ E � X

E 7→ localisation

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Bundles of Hilbert spaces

Bundle E � X, each fibre Hilbert space, operations continuous, with

E

Xt

Et

Hilbert C0(X)-modules ' bundles of Hilbert spaces over Xsections vanishing at infinity ←[ E � X

E 7→ localisation

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Dual objects

if X compact

E has dual object when : I → E∗ ⊗ E and

: E ⊗ E∗ → I satisfy = and =

⇐⇒

finite Hilbert bundle:⇐⇒

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Dual objects

if X paracompact

E has dual object when : I → E∗ ⊗ E and

: E ⊗ E∗ → I satisfy = and =

⇐⇒

finite Hilbert bundle:supt∈X dim(Et) <∞

⇐⇒

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Dual objects

if X paracompact

E has dual object when : I → E∗ ⊗ E and

: E ⊗ E∗ → I satisfy = and =

⇐⇒

finite Hilbert bundle:supt∈X dim(Et) <∞

⇐⇒

finitely presented projective Hilbert module:

E C0(X)ni

i†

id

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Dual objects

if X compact

E has dual object when : I → E∗ ⊗ E and

: E ⊗ E∗ → I satisfy = and =

⇐⇒

finite Hilbert bundle:∀t ∈ X : dim(Et) <∞

⇐⇒

finitely generated projective Hilbert module:

spanC0(X)(x1, . . . , xn) = E

F

G

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Frobenius structures

E has special dagger Frobenius structure : E ⊗ E → E:

= = =

⇐⇒

E is a finite bundle of C*-algebras:each fibre is C*-algebra, operations continuous, sup dim(Et) <∞

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Commutative Frobenius structures

p : Y � X finite covering: continuous, each t ∈ X has openneighbourhood whose preimage is union of disjoint open sets

homeomorphic to it, supt∈X |p−1(t)| <∞

pp

z

z2

C0(Y ) special dagger Frobenius: 〈f | g〉(t) =∑

p(y)=t f(y)∗g(y)

comultiplication comes from Y ×X Y = {(a, b) ∈ S1 × S1 | a2 = b2}

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Commutative Frobenius structures

p : Y � X finite covering: continuous, each t ∈ X has openneighbourhood whose preimage is union of disjoint open sets

homeomorphic to it, supt∈X |p−1(t)| <∞

pp

z

z2

C0(Y ) special dagger Frobenius: 〈f | g〉(t) =∑

p(y)=t f(y)∗g(y)

comultiplication comes from Y ×X Y = {(a, b) ∈ S1 × S1 | a2 = b2}

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Commutative Frobenius structures

p : Y � X finite covering: continuous, each t ∈ X has openneighbourhood whose preimage is union of disjoint open sets

homeomorphic to it, supt∈X |p−1(t)| <∞

pp

z

z2

C0(Y ) special dagger Frobenius: 〈f | g〉(t) =∑

p(y)=t f(y)∗g(y)

comultiplication comes from Y ×X Y = {(a, b) ∈ S1 × S1 | a2 = b2}

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Caveats

�

I not determined by orthonormal basis of sectionsmight be no copyable states at all!(even though category is well-pointed)

I unital Frobenius algebra ⇐⇒ finite Hilbert bundle

p

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Caveats

�

I not determined by orthonormal basis of sectionsmight be no copyable states at all!(even though category is well-pointed)

I unital Frobenius algebra ⇐⇒ finite Hilbert bundle

p

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Nontrivial central Frobenius structure

D = {z ∈ C | |z| ≤ 1}S1 = {z ∈ C | |z| = 1}

X = S2 = {t ∈ R3 | ‖t‖ = 1}

{x ∈ C0(D,Mn) : x(z) =

(z1·1

)x(1)

(z1·1

)if z ∈ S1

}is special dagger Frobenius structure: 〈x | y〉(t) = tr(x(t)∗y(t))

central: Z(E) = C0(X)

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Nontrivial central Frobenius structure

D = {z ∈ C | |z| ≤ 1}S1 = {z ∈ C | |z| = 1}

X = S2 = {t ∈ R3 | ‖t‖ = 1}

{x ∈ C0(D,Mn) : x(z) =

(z1·1

)x(1)

(z1·1

)if z ∈ S1

}is special dagger Frobenius structure: 〈x | y〉(t) = tr(x(t)∗y(t))

central: Z(E) = C0(X)

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Transitivity

E is special dagger Frobenius structure in HilbC0(X)

⇐⇒

E is special dagger Frobenius structure in HilbZ(E)

and

Z(E) is specialisable dagger Frobenius structure in HilbC0(X)

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Additive structure

With adjointable morphisms:

I Direct sums provide dagger biproducts

I FHilbC0(X) has dagger kernels

I HilbC0(X) has dagger kernels =⇒ X totally disconnected

I clopen subsets of X ↔ dagger subobjects of tensor unit

Conjecture: if compact dagger category has dagger biproducts,equalisers, and isometries are kernels, it embeds into FHilbC0(X)

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This is just the beginning

Message:

I New category for arsenal

I Scalars are not numbers

I Frobenius structures may have no copyable states

Next talk:

I Abstract spatial structure to any monoidal category

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