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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
38 views

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 $SU(2)$ ...
user avatar
  • 1,123
3 votes
0 answers
88 views

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 (...
user avatar
3 votes
1 answer
63 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\...
user avatar
  • 2,881
2 votes
0 answers
75 views

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 ...
user avatar
  • 820
4 votes
1 answer
304 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 ...
user avatar
  • 43
6 votes
0 answers
348 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 ...
user avatar
2 votes
0 answers
118 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 ...
user avatar
  • 131
1 vote
0 answers
69 views

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....
user avatar
  • 197
4 votes
2 answers
536 views

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\...
user avatar
  • 1,936
3 votes
1 answer
137 views

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}...
user avatar
3 votes
2 answers
248 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 ...
user avatar
  • 7,340
7 votes
0 answers
99 views

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 ...
user avatar