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29

They're variants, different kinds of quantum field theory, but they're not mutually exclusive. The different adjectives you mention separate quantum field theory to "pieces" in different ways. The different sorts of variants you mention are being used and studied by different people, the classification has different purposes, the degree of usefulness and ...


19

The absence of physical excitations in 3 dimensions has a simple reason: the Riemann tensor may be fully expressed via the Ricci tensor. Because the Ricci tensor vanishes in the vacuum due to Einstein's equations, the Riemann tensor vanishes (whenever the equations of motion are imposed), too: the vacuum has to be flat (no nontrivial Schwarzschild-like ...


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

As you say yourself, indeed every connection on a bundle is locally given by a Lie algebra valued 1-form and in general only locally. Let's say this more in detail: for $X$ any manifold, a $G$-principal connection on it is (in "Cech data"): a choice of good open cover $\{U_i \to X\}$; on each patch a 1-form $A_i \in \Omega^1(U_i)\otimes \mathfrak{g}$; on ...


11

We have thought a bit about the last paragraph of the above question and have some arguments as to what the answer should be. Since there have been no replies here so far, maybe I am allowed to hereby suggest an answer myself. Recall, the last part of the above question was: is there a nonabelian 7-dimensional Chern-Simons theory holographically related to ...


10

In a "fully defined" TQFT the spaces of states are necessarily finite dimensional. This follows simply from the fact that the correlators assigned to the cap and the cup cobordism (the "2-point functions") equip the space of states with the structure of a dualizable object in the corresponding monoidal category of vector spaces, which are precisely the ...


10

After stating the solution, I'll try to give some physical insights to the best of my knowledge and some more references. The dimension of the required state space is given by the Verlinde formula, having the following form for a general compact semisimple Lie group $G$ on a Riemann surface with genus $g$ corresponding to the level $k$: $$ \mathrm{dim} ...


9

First, the full paper is here: http://citeseer.ist.psu.edu/viewdoc/download;jsessionid=807BE383780883ACB4CAB8BD48E8C90B?doi=10.1.1.128.1806&rep=rep1&type=pdf Second, the paper has 150 citations. See all this information at INSPIRE (updated SPIRES): http://inspirebeta.net/record/278923?ln=en Third, the text between 3.4 and 3.5 looks ...


9

Luboš would know this already (he's acknowledged in this paper), but Neitzke and Vafa conjectured in 2004 that the mirror manifold of $CP^{3|4}$ is a quadric surface $Q$ in $CP^{3|3}$ x $CP^{3|3}$, and mirror symmetry is a type of T-duality. There were a few follow-ups, including a paper by Sinkovics and Verlinde which studies classical $N=4$ ...


9

The "topological" in "topological order" and the "topological" in "topological insulator" have different meanings. The 'topological' in topological order means 'robust against ANY local perturbations'. The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of ...


8

The Kapustin-Witten paper http://arxiv.org/abs/hep-th/0604151 says (on page 17) that two of the three twists are related to Donaldson theory: Two of the twisted theories, including one that was investigated in detail in [45: Vafa Witten], are closely analogous to Donaldson theory in the sense that they lead to instanton invariants which, like the ...


8

(sorry I don't have enough reputation to make a comment): This question is very broad/vague, as indeed algebraic/differential topology (symplectic geometry of course) is completely used in theoretical physics, in particular for Topological QFTs. From a physicist's perspective, start with Nakahara's Geometry, Topology, and Physics. Surgery, cobordism, and ...


8

It's not the making as opposed to verifying of topological superconductors that is difficult experimentally. One of the most useful techniques in identifying topological properties of a material is Angle-Resolved Photoemission Spectroscopy (ARPES). ARPES can independently image the bulk and surface modes of a 3-D solid with very good energy and momentum ...


8

How to obtain this braiding matrix from Non-Abelian Chern-Simon theory? To obtain braiding matrix $U^{ab}$ for particle $a$ and $b$, we first need to know the dimension of the matrix. However, the dimension of the matrix for Non-Abelian Chern-Simon theory is NOT determined by $a$ and $b$ alone. Say if we put four particles $a,b,c,d$ on a sphere, the ...


7

Irving Segal, the obnoxious and disliked genius mathematician of MIT, once went around asking physicists « ¿what is a quantum field? » As he tells the story, only Enrico Fermi gave him an answer (after pausing for a little thought). « The occupation number formalism.» What does this mean. What Fermi meant is that for a given particle, say an electron, ...


7

The relation is very deep and has a rich mathematical structure, so (unfortunately) most stuff will be written in a more formal, mathematical way. I can't say anything about Donaldson theory or Floer homology, but I'll mention some resources for Chern-Simons theory and its relation to the Jones Polynomial. There is first of all the original article by ...


7

The topological spin $h$ of an anyon (a quasi-hole in a FQH state) is the exponent in the Green function of the quasi-hole along the edge of the FQH state [see eq.(61) in my review paper http://arxiv.org/abs/1203.3268 ], which can be measured by the I-V curve: $I\propto V^{4h-1}$ in the tunnelling experiments between FQH edges.


7

Your term in the Lagrangian density is usually not given a special name; it is called the "F wedge F" term from the $p$-form notation $F\wedge F$ (the symbol is written as $\backslash{\rm wedge}$) and represents a tensor multiplication of antisymmetric tensor followed by a new antisymmetrization of all the indices (up to some normalization that depends on ...


7

This is explained in Section 3 of Witten's "Quantum Field Theory and the Jones Polynomial." The idea is to locally parametrize a three-manifold by $M\times \mathbf{R}$, where $M$ is some two-dimensional manifold and $\mathbf{R}$ is the time direction that we are quantizing along. Once we do this, we can fix temporal gauge, where the time component $A_0$ of ...


7

In the absents of any response, let me try to give a quick answer. I am a little bit confused about why you say the topological entanglement entropy (TEE) is usually calculated on surfaces with boundaries. You can, and I think this is whats usually done, calculate it on compact manifolds. A boundary will always be present since you need to do a bi-partition ...


7

A modern geometric-algebraic tool presently being under active research in the context of quantization of field theories with topological terms is the theory of gerbes. The main reference for this application (quantization) is the book by Brylinski : "Loop Spaces, Characteristic Classes, and Geometric Quantization". Gerbes are extensively used in string ...



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