# Tag Info

28

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 ...

17

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 ... 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 (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 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 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 ... 8 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 ... 7 A quote from http://en.wikipedia.org/wiki/Symmetry_protected_topological_order : The SPT order (for both frermionic and bosonic systems) has the following defining properties: Distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry. However, they all can ... 7 A conformal transformation is one which alters the metric up to a factor, i.e.$$g_{\mu\nu}(x)\to\Omega^2(x)g_{\mu\nu}(x) A field theory described by a Lagrangian invariant up to a total derivative under a conformal transformation is said to be a conformal field theory. These transformations include Scaling or dilations $x^\mu \to \lambda x^\mu$ ...

7

There are two ways to quantize a gauge theory. 1) First quantize all degrees of freedom and then reduce the gauge freedom by imposing conditions on the quantum Hilbert space. 2) First reduce the gauge redundancy then quantize the reduced phase space. For realistic gauge theories in $4D$, there are problems in implementing either of these methods. The ...

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

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

Before going into the details, let me tell you that this type of actions describe the deepest connection between geometry and physics and generalizations of these types of theories are still under active research even today. The Lagrangian describes $N=1$ supersymmetric quantum mechanics on a Riemannian manifold The bosonic part of this Lagrangian is the ...

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

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

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

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 ...

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