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14

The Atiyah-Segal axioms and generally the axioms of FQFT formalize the Schrödinger picture of quantum physics: to a codimension-1 slice $M_{d-1}$ of space one assigns a vector space $Z(M_{d-1})$ -- the (Hilbert) space of quantum states over $M_{d-1}$; to a spacetime manifold $M$ with boundaries $\partial M$ one assigns the quantum propagator which is the ...


12

Algebraic geometry as such appears because it happens to capture important aspects of the geometry of strings. For instance the partition functions of superstrings are elliptic genera and the best way to understand this is to regard a torus-shaped string worlsheet as an elliptic curve, regard the moduli space of possible worldsheet tori as the moduli stack ...


9

Recently, it is realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states. The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher ...


7

Here is a motivation for the general notion of sheaf and sheaf cohomology: motivation for sheaves, cohomology and higher stacks A general introduction to differential geometry as needed in physics in terms of sheaves is at geometry of physics in the section on smooth spaces. More along these lines is in section 1.2 of arXiv:1310.7930, which ...


7

EDIT #3: My other answer gives a more detailed and structured account (I hope). (I would leave this as a comment, but I don't have enough reputation so…) You should check out Atiyah's paper itself. He makes attempts to explain at least some of these things. Unfortunately, I need to get going at the moment (but I'll come back and edit this with a more ...


5

I decided to include this as a separate answer, rather than mess with the above. Sorry in advance for the length. I still wholeheartedly suggest that you check out: (1) Quantum Quandaries, by Baez; (2) Frobenius Algebras and 2D Topological Quantum Field Theories, by Koch (a portion of it is here, and there is a "short version" here); (3) An ...


5

Here is a belated reply. (I come across this question only now, by chance. This was posted right when our daughter was born, which was kind of distracting for me...) The quick answer to the question is the following somewhat remarkable statement Identity types in the new foundations of mathematics in homotopy type theory correspond in physics to spaces of ...


4

My short answer is No, they're not too useful, but let me discuss some details, including positive ones. Categories, especially derived categories, have been appropriate to describe D-brane charges - and not only charges - beyond the level accessible by homology and K-theory. See e.g. http://arxiv.org/abs/hep-th/0104200 However, I feel it is correct ...


3

Nigel, A few comments. Firstly I think we should separate the questions for Category Theory and Quantum Logic here, as I think they are rather different in a sense I will explain below. Category Theory: My view is that most physicists look for insights and theories from a Geometric viewpoint. This somewhat unites Relativity Theorists and String Theorists. ...


3

Category theory has some potential for physics. Quantum logic I am less sanguine about. It has always struck me as a way of expressing something we understand in set theoretic language. It has always struck me as a formalistic study that brings little additional content. Philip Goyal demonstrated how the above summation over intermediate points is ...


2

To respond to Marek's comment: as a reader familiar with physics and categories, I can say why I am unable to even approach a response. The question cites some papers, but is not readable on its own (not "self-contained"). We only learn is that there is some nonlocal theory out there which obeys some consistency conditions. I am sure that this would ...


1

Look at this and see if it helps http://arxiv.org/pdf/0905.3010v2


1

I cannot comment on the category theory part, but the ideas regarding 'numbers' to a 3-manifold and vector spaces to Riemann surfaces comes about naturally in Gromov-Witten theory (see here: http://www.math.harvard.edu/~jbland/ma273x_notes.pdf for a nice introduction). The heuristic recipe (physical explanation) is as follows: Take a closed symplectic (as ...


1

I expect this is somewhat of an axiomatic statement, since the TQFT axioms include the association of a Riemann surface $\Sigma$ to a vector space (or a module) $Z(\Sigma)$, and an element $Z(M)\in Z(\partial M)$ to a manifold $M$. They do not include direct reference to categories. It sounds like the categories are natural extensions to the lower dimensions ...


1

There are maybe three different stages to be distinguished and to be understood here: first: maybe part of the question is why an $n$-dimensional QFT should assign numbers to closed $n$-dimensional manifolds, and vector spaces to closed $(n-1)$-dimensional manifolds. That is what I had replied to in that other discussion linked to above: the vector spaces ...



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