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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 ...


14

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 ...


11

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 ...


10

I don't really understand your question, but since you link to my paper on higher Yang-Mills theory (which I never tried publish because it has problems, even though everything stated in it is true to the best of my knowledge), it sounds like maybe you're interested in approaches that treat gravity using ideas from higher gauge theory. For this, I urge you ...


8

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 ...


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

Here is how higher category theory (homotopy theory) arises in gravity: First of all, the precise version of the statement that "gravity is a gauge theory" is that gravity in first order formulation is "Cartan geometry" for Minkowski spacetime regarded as the quotient of the Poincare group by the Lorentz spin-group. This statement generalizes to super-...


4

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

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 ...


3

I have been writing something in this direction in section 1 of the book Differential cohomology in a Cohesive topos (pdf). Have a look, just focus on section 1 and ignore the remaining sections on first reading. The survey-part is presently also appearing as a series on PhysicsForums. See at Higher prequantum geometry I, II, III, IV, V and Examples of ...


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 ...


2

John Baez, a well-known mathematical physicist presently (2014) in the Department of Mathemetics at the University of California, Riverside is trying to find more and more ways that category theory connects with physics, chemistry, electrical engineering, network theory, etc. You can read more about his efforts: At his UCR homepage here; and In particular, ...


2

A brane is more than just a manifold, it is a physical object. You can think of it as a higher dimensional version of a particle. It can carry a charge, it can couple to gauge fields, it can decay, etc. We tend to study a brane in the same way we study a string. We study strings in terms of their worldsheet, we study branes in terms of their worldvolume. The ...


1

One less well-known but great reference are the classical field theory notes by Deligne and Freed in the '99 IAS lectures. Some good things about them Very elegant treatment written for mathematicians Begins with a nice discussion of ordinary classical mechanics using principal bundles and connections Useful comments on supersymmetric gauge theories ...


1

An anyonic superselection sector is defined to be an equivalence class of local excitations under local operations. A non-trivial anyon (i.e. a non-trivial superselection sector) is a "local excitation which cannot be created locally". Here is a precise statement of this notion which works for systems with zero correlation length, where anyons can be ...


1

Mathematicians like to write tensor product, since in many cases (or maybe in all cases) anyon types (simple objects) are indeed irreducible representations of some algebraic object (e.g. Hopf algebra, quantum groups), and irreducible representations of finite groups provide a large family of examples for fusion categories, where $\otimes$ and $\oplus$ ...


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

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 ...



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