# Tag Info

15

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 ...

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

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 ...

9

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 ...

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

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 $N=4$ ...

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

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

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 ...

7

I'm not an expert in algebraic topology by the stretch of anyones imagination, but hopefully I can shed some light on this. The starting point is, as you mention, the Maxwell equations themselves. Cast into a geometric language the curvature 2-form $\bf{F}$, which you can think of as the Faraday tensor for $U(1)$-Maxwell theory (there are generalisations ...

7

There are several inequivalent definitions, used in different contexts, which is the reason for your confusion. The word "Chiral" originally refered to chirality, or handedness of spin along the direction of motion. This is still the most often used definition. The spinor representations of the Lorentz group in even dimensions have components with a ...

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

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 ...

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 ...

6

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 ...

6

There is nothing "wrong" with the Einstein field equations in $2+1$ as indicated by the comments, but they do have interesting, significantly restricted behavior in $2+1$ dimensions. For example, the Wikipedia page referred to by Olof in the comments says that in $2+1$, every vacuum solution is locally either $\mathbb R^{2,1}$, $\mathrm{AdS_3}$, or ...

6

Witten clearly writes the justification just on the line above the equation (3.2): the integral is independent because the integration measure is invariant under supersymmetry – the symmetry generated by $Q$. Just to be sure, $Q$ is the infinitesimal generator which is why $\exp(\varepsilon Q)$ is a finite transformation generated by this generator: ...

6

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 ...

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