All Questions
20 questions
2
votes
0
answers
143
views
Enumerative geometry and restricted plane partitions
Donaldson-Thomas theory is an enumerative theory for virtual counts of ideal sheaves (with trivial determinant) of the structural sheaf $\mathcal{O}_{X}$ of some smooth projective manifold $X$.
There ...
- 502
11
votes
1
answer
2k
views
Vafa-Witten invariants for mathematicians
As Richard Thomas has written (we paraphrase just slightly), mathematical physicists Vafa and Witten introduced new "invariants" of four-dimensional spaces in a paper:
A Strong Coupling Test of S-...
- 10.5k
6
votes
1
answer
726
views
Degree-3 curves on the Calabi–Yau quintic
Robbert Dijkgraaf said,1
concerning the simplest
Calabi–Yau space, the quintic:
"A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. ...
- 151k
11
votes
3
answers
1k
views
In Gromov-Witten theory, why is the string coupling constant weighted by $2g-2$?
Let $X$ be a Calabi-Yau threefold and let us fix a homology class $\beta\in H_2(X,\mathbb Z)$, just for simplicity. The generating series of Gromov-Witten invariants of $X$ in class $\beta$, $$\mathsf ...
- 1,534
29
votes
3
answers
5k
views
Topologically distinct Calabi-Yau threefolds
In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...
- 23.5k
14
votes
2
answers
3k
views
what is the stringy Kähler moduli space?
I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold.
Could ...
- 1,889
18
votes
0
answers
549
views
Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
- 1,701
11
votes
0
answers
600
views
The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
- 1,298
4
votes
0
answers
211
views
Bridgeland stability for restricted Kahler moduli?
Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical ...
- 1,701
8
votes
1
answer
805
views
How to construct the mirror partner of a blowup?
Question: Let's assume we have a pair $(X,\check{X})$ that are mirror dual to each other in the sense of Homological mirror symmetry (EDIT: this does not have to be CY n-folds, but can also be a Fano ...
- 1,981
10
votes
2
answers
2k
views
Gromov-Witten and integrability.
The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are ...
- 1,343
3
votes
0
answers
343
views
Localization of the pushforward in equivariant cohomology
I am reading Nekrasov's paper and in page 2 he considers the $G \times T^2$ equivariant cohomology of the (compactified) moduli space $\tilde{M_k}$ of $U(N)$ instantons on $\mathbb{C}^2$. Here $G$ ...
- 587
2
votes
0
answers
349
views
SYZ conjecture for varieties of general type or Fano
Let $X$ and $Y$ are Calabi-Yau varieties and mirror to each other. Then from HMS the Fukaya Floer category of Lagrangian intersections in $X$, is equivalent to bounded derived category of coherent ...
user21574
15
votes
2
answers
2k
views
Higher genus closed string B-model
The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which ...
- 21k
7
votes
0
answers
225
views
Relation between Donaldson invariants and GW invariants
What is known about the relation of Donaldson invariants on a complex surface $\Sigma$ and GW invariants (or equivalent) of local Calabi-Yau 3folds such as the canonical bundle of $\Sigma$? (if any of ...
- 661
14
votes
0
answers
577
views
State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds
I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...
- 509
8
votes
1
answer
843
views
Virasoro constraints for the generating function of Hurwitz numbers.
Generating function of the simple Hurwitz numbers is known to be connected with Gromov-Witten potential of the point (Kontsevich $\tau$-function) (see e.g. Ian Goulden, David Jackson and Ravi Vakil). ...
- 1,343
8
votes
1
answer
566
views
Multiple Hodge integrals and integrability
It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function (see Kazarian). ...
- 1,343
6
votes
1
answer
577
views
Gromov-Witten and integrability 2.
This is a followup of my previous question Gromov-Witten and integrability. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) ...
- 1,343
8
votes
0
answers
1k
views
triangulated/derived categories in Physics and algebraic geometry
Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...