8
votes
Accepted
Relation between QFT and algebraic geometry
The Quanta magazine article you link references "Knots and Numbers in $\varphi^4$ Theory to 7 Loops and Beyond" by Broadhurst and Kreimer from 1995 which does not even mention the word "period" ...
- 129k
6
votes
Accepted
References on mathematical stacks for a string theory student
For the "physics part" of my question: I have found the papers String Orbifolds and Quotient Stacks
, D-branes, orbifolds, and Ext groups and Stacks and D-Brane Bundles very useful, readable ...
5
votes
Accepted
Why do you need to count curves on Calabi-Yau manifolds in string theory?
When we compactify string theory, we are interested in the effective theory in the remaining dimensions, and perform a generalization of Kaluza-Klein reduction. Now, where in flat 10 dimensions the ...
- 129k
4
votes
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Relation between Topological String Theory and Physical String Theory?
The crucial idea to understand why topological string theory cannot be in itself a fundamental description of nature is because, by construction, all the operators of the theory (including the energy-...
- 3,892
4
votes
How should I physically understand the slope stability of vector bundles on a manifold X?
The physical interpretation of slope stability of vector bundles is revealed once one thinks of the vector bundles as being the "Chan-Paton gauge fields" on D-branes. Then then rank of the vector ...
- 13.9k
3
votes
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Is the Godel universe Wick rotatable?
Yes, it can be Wick-rotated to a negative-definite Riemannian metric on the (Berger sphere)xR. See the my more detailed answer at https://mathoverflow.net/questions/462704/is-the-g%c3%b6del-universe-...
3
votes
Connection between homotopic maps from $X\to Y$ and homotopic paths in $Y$ in the context of SU(2) Yang-Mills instantons
A definition might clarify the confusion. The $n$-th homotopy group $\pi_n(Y)$ of a space $Y$ is constructed using the set of maps $S^n\to Y$. If we identify two maps when they're homotopic and add ...
- 4,733
2
votes
Accepted
Relation between second cohomology and central extensions
That the central extensions of a Lie algebra $\mathfrak{g}$ by $\mathbb{R}$ are in bijection to the cohomology classes
$$ H^2(\mathfrak{g},\mathbb{C}) := \{\theta : \mathfrak{g}\times\mathfrak{g}\to\...
- 129k
2
votes
Accepted
Why are worldsheets of strings _holomorphic_?
I suggest learning the basics of conformal field theory before diving into string theory. To illustrate where holomorphicity comes in, let's consider the free bosonic string (the simplest example of a ...
- 196
2
votes
Positive geometry and log singularities
I'm not sure I understood exactly what you want to know.
Anyhow, a positive geometry is an oriented geometry that has a canonical form. A canonical form is a differential form with dlog divergences, ...
- 31
2
votes
Why do you need to count curves on Calabi-Yau manifolds in string theory?
@NiharKarve Although the question is old, I think the paper for the "famous calculation of the quintic" that you are looking for is the paper of Candelas et al. "A Pair of Calabi-Yau ...
- 41
2
votes
ADE Gauge Theory and String Theory
In the literature, we refer to an ALE space as being a solution to the field equations which is a blow up of $\mathbb C^2 / \Gamma$ for a finite subgroup $\Gamma \hookrightarrow \mathrm{SU}(2)$.
A ...
- 19.5k
2
votes
Decomposition of vector bundle in $M$-theory
Surprised that no expert tried to answer this. It all probably has a simple explanation... For now, the best I can suggest is section 2 of https://arxiv.org/abs/1212.1467. It starts with an even ...
- 13.4k
1
vote
Accepted
How can I find the unitaries for the Bloch-Messiah decomposition?
You can write the transformation of in the Eq. 10 from Braunstein as
\begin{align}
\begin{pmatrix} b_1 \\ b_2 \\b_1^\dagger \\ b_2^\dagger \end{pmatrix} = \mathcal{S} \begin{pmatrix} a_1 \\ a_2 \\a_1^...
- 527
1
vote
Shankar Monopole belonging to the class $-1$ of $\pi_3(\mathbb{R}P^3)$
The answer might be a little late, but I'm also going through Nakahara at the moment:)
So I think, you might have missed a point of defining the negative elements of $\mathbb{Z}$; see Example 4.13 ...
1
vote
Application of algebraic geometry in studying geodesic and hypersurface
This answer was supposed to be a comment, but got too long. So it is not a full answer by any means.
I don't know enough about this subject to give a definite answer (and right now I don't have the ...
- 11.6k
1
vote
Accepted
ADE Gauge Theory and String Theory
M-theory compactified on a ADE singularity space is rather nice to describe qualitatively: Let $\Gamma\subset\mathrm{SU}(2)$ be an ADE group and consider the compactification of M-theory on $\mathbb{C}...
- 129k
1
vote
Accepted
Connectedness on Special Kaehler manifolds
I think that this question of yours is interesting and it was correctly posted in physics (not math) stack exchange. :)
I am not aware of any evidence or counterexample to the statement of ...
- 3,892
1
vote
Algebraic geometry and topology for string theory
Lecture notes by Candelas:- lectures on Complex manifolds [https://docs.google.com/viewer?a=v&pid=sites&srcid=Y29sb3JhZG8uZWR1fHRhc2ktMjAxNy13aWtpfGd4OjEyMzQ4M2MyZDNmOWMyNmM]
This notes ...
1
vote
Accepted
Flavor symmetry fixes the Higgs branch in any 4D ${\cal N}=2$ QFT
1.) NO
2.) Consider the lagrangian theories with gauge group G= USP(2N), four fundamental hypers and one antisymmetric, all these models have flavor symmetry SU(2) x SO(8), but the higgs branch is ...
- 34
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