## Project Details

### Description

PROJECT SUMMARY

DAVID AYALA AND JOHN FRANCIS

Proposed Activities

This project involves the development of the theory of factorization homology with coe�cients

in (1; n)-categories. The initial proposed use of this theory is to prove the cobordism hypothesis of

Baez-Dolan and Lurie, which asserts that topological quantum �eld theories valued in a symmetric

monoidal (1; n)-category are in bijection with fully dualizable object of that category { in particular,

that a �eld theory is determined by its value on a point. A second application is an unexpected

re�nement of the cobordism hypothesis using a formulation of the state-operator correspondence in

terms of factorization homology. This re�ned cobordism hypothesis produces numerical invariants

of manifolds under a strictly weaker condition than the full dualizability proposed by Baez-Dolan

and Lurie. These invariants satisfy only a weaker version of locality, which does not allow them

to be computed from handlebody presentations. Ayala and Francis further propose to relate their

construction to sigma models by way of a conjectural analogue of Tannakian duality for factorization

homology, and to employ these techniques to study what physical duality results from a form of

Poincar�e duality for factorization homology.

Intellectual Merit

The cobordism hypothesis of Baez-Dolan and Lurie is the structural crux in the current math-

ematical view of topological quantum �eld theory. It informs the work of every mathematician in

this �eld, but a complete proof has not, to this point, been written. The present proposal o�ers

both a complete proof of the cobordism hypothesis as well as a conceptually natural construction

of the correspondence therein. It also o�ers a new basis for locality in topological quantum �eld

theory centered around the observables of a physical theory rather than the state spaces. This is

based on the theory of strati�cations rather than on Morse theory and handlebody presentations,

which have formed the basis for locality since Atiyah's axioms. This modi�cation potentially allows

for new manifold invariants and new examples from physics which did not conform to the original

cobordism hypothesis.

The manifold topology required for the foundations of their theory is signi�cant: the existence

of factorization homology necessitates that the collection of disk-strati�cations of manifold can be

assembled into an 1-category { the requisite lifting condition uses a coherently functorial con-

struction of resolutions of singularities. Likewise, to establish even the most basic property of

factorization homology involves showing that the classifying space of disk-strati�cations of mani-

fold is contractible { this is an analogue of Whitehead's theorem, that smooth manifolds have an

essentially unique piecewise linear structure. Similarly, the essential local-to-global principle for

factorization homology requires an excision result for di�eomorphism groups which is new and of

independent interest.

Broader Impact

Ayala and Francis have a number of graduate students, and the scope and design of their agenda

is well-suited toward incorporating young researchers, certainly including graduate students, into

the work. They have previously collaborated with and mentored graduate students.

1

DAVID AYALA AND JOHN FRANCIS

Proposed Activities

This project involves the development of the theory of factorization homology with coe�cients

in (1; n)-categories. The initial proposed use of this theory is to prove the cobordism hypothesis of

Baez-Dolan and Lurie, which asserts that topological quantum �eld theories valued in a symmetric

monoidal (1; n)-category are in bijection with fully dualizable object of that category { in particular,

that a �eld theory is determined by its value on a point. A second application is an unexpected

re�nement of the cobordism hypothesis using a formulation of the state-operator correspondence in

terms of factorization homology. This re�ned cobordism hypothesis produces numerical invariants

of manifolds under a strictly weaker condition than the full dualizability proposed by Baez-Dolan

and Lurie. These invariants satisfy only a weaker version of locality, which does not allow them

to be computed from handlebody presentations. Ayala and Francis further propose to relate their

construction to sigma models by way of a conjectural analogue of Tannakian duality for factorization

homology, and to employ these techniques to study what physical duality results from a form of

Poincar�e duality for factorization homology.

Intellectual Merit

The cobordism hypothesis of Baez-Dolan and Lurie is the structural crux in the current math-

ematical view of topological quantum �eld theory. It informs the work of every mathematician in

this �eld, but a complete proof has not, to this point, been written. The present proposal o�ers

both a complete proof of the cobordism hypothesis as well as a conceptually natural construction

of the correspondence therein. It also o�ers a new basis for locality in topological quantum �eld

theory centered around the observables of a physical theory rather than the state spaces. This is

based on the theory of strati�cations rather than on Morse theory and handlebody presentations,

which have formed the basis for locality since Atiyah's axioms. This modi�cation potentially allows

for new manifold invariants and new examples from physics which did not conform to the original

cobordism hypothesis.

The manifold topology required for the foundations of their theory is signi�cant: the existence

of factorization homology necessitates that the collection of disk-strati�cations of manifold can be

assembled into an 1-category { the requisite lifting condition uses a coherently functorial con-

struction of resolutions of singularities. Likewise, to establish even the most basic property of

factorization homology involves showing that the classifying space of disk-strati�cations of mani-

fold is contractible { this is an analogue of Whitehead's theorem, that smooth manifolds have an

essentially unique piecewise linear structure. Similarly, the essential local-to-global principle for

factorization homology requires an excision result for di�eomorphism groups which is new and of

independent interest.

Broader Impact

Ayala and Francis have a number of graduate students, and the scope and design of their agenda

is well-suited toward incorporating young researchers, certainly including graduate students, into

the work. They have previously collaborated with and mentored graduate students.

1

Status | Finished |
---|---|

Effective start/end date | 6/1/15 → 5/31/19 |

### Funding

- National Science Foundation (DMS‐1508040-001)

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.