Questions tagged [algebraic-topology]

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.

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What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?

I posted the original question on Math SE but it seems like a more appropriate question for Physics SE: https://math.stackexchange.com/q/4417225/ In calculating the instanton solutions for $SU(2)$ ...
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Utility of Topological Data Analysis in Theoretical Physics

I audited a lecture on Topological Data Analysis, and I found it really interesting, primarily because of the connection to algebraic topology. I asked the professor if there are any connections to (...
3 votes
1 answer
69 views

What is the topology of non-entangled states region for a 2 qubit Bloch hypersphere?

Preamble A two qubit/spin-1/2 system can be represented as $$|\psi\rangle=\alpha|\uparrow\uparrow\rangle+\beta|\uparrow\downarrow\rangle+\gamma|\downarrow\uparrow\rangle+\delta|\downarrow\downarrow\...
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Reidemeister Torsion

Can somebody in a layman's language explain what is a Reidemeister Torsion? This seems to play an important role in path integral of 2+1-gravity as demonstrated here in arXiv:gr-qc/9406006. This is ...
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4 votes
1 answer
336 views

Physical meaning of the Yang-Baxter equation

I'm a graduate student in mathematics, and I have lately been interested in the relation between knot theory and statistical mechanics. As I understood, the Yang-Baxter equation (shown below) is the ...
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6 votes
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395 views

Classification of higher Symmetry Protected Topological (SPT) phases

Suppose that we have a $d$ dimensional bosonic SPT phase, protected by some $p$-form symmetry, $G^{[p]}$. Suppose also that it is classified within group cohomology, so that we don't have to run into ...
2 votes
0 answers
120 views

String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops ...
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1 vote
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Which geometry does not allow the existence of matter?

I have seen these lectures by Fredric Schuller that discuss the obstruction theory and the role of global geometric properties in admitting a spin structure. See the video at 01:27:52 https://youtu....
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4 votes
2 answers
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Simple explanation for what a torsor is

I am studying Chris Elliott's notes on Line and Surface Operators in Gauge Theories (available here). In the notes, there's a mention of the fact that (for $G = U(1)$), $$W_{\gamma, n}(A) = e^{in\...
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3 votes
1 answer
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Large gauge transformation and intersection form

I am reading this paper and on pp.19-20 it states the following relation between large gauge transformation and intersection form: for the action on a 4-manifold $M^4$ $$S[A,B] = \int_{M^4}{\sum_{I=1}...
3 votes
2 answers
254 views

Physical application of Postnikov tower, String$(n)$ and Fivebrane$(n)$

We know that the Spin group is quite a useful concept in physics. For example, Spin$(3)=SU(2)$ (and Spin$(6)=SU(4)$) that describe gauge groups in the Standard model and the isospin symmetry in the ...
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What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...
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