Questions tagged [gromov-witten-theory]
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163 questions
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Cobordism of orbifolds?
Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...
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0
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Deformation theory with a view toward GW theory and DT theory
I am studying GW theory (and DT theory) in algebraic geometry. I now understand the heuristic "Aut, Def, Obs" argument written in Mirror Symmetry book (by Hori et al.), but it is too hard for me to ...
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votes
2
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957
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Explicit computation of Gromov-WItten invariants
After studying some foundation of Gromow-Witten invariants, I now would like to see an explicit computation. I heard that one should first take a look at the total space of $\mathcal{O}(-1)^{\oplus2}$ ...
- 241
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3
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Why are Gromov-Witten invariants of K3 surfaces trivial?
Why is GW invariants of K3 surfaces are trivial? My naive guess is that GW invariants are deformation invariant and you can always deform your K3 surface to non-projective one, which has no subcomplex ...
- 241
3
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2
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Conics in the quadric line complex
Hello, I apologize in advance if this question is misguided somehow, since my algebraic geometry is pretty shaky.
I am wondering if there is a way to understand all the conics in a generic quadric ...
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10
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0
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gromov witten donaldson thomas correspondence
Let $X$ be a nonsingular projective 3-fold. I am trying to understand the proof of the GW/DT correspondence as presented in Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds. I would ...
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0
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140
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Obstruction theories on non-smooth spaces with smooth fibres
Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by
$$[X, E^\bullet] = c_{top}\big((E^{-1})^...
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1
answer
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Is P^2 important in Kontsevich's recursion formula?
There is a famous recursion formula by Kontsevich to find the number of
genus zero degree $d$ curves in $\mathbb{CP}^2$ through $3d-1$ points.
My question is the following: Let $S$ be a complex ...
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0
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174
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Are there ways to make low degree checks for enumerative formulas except for curves in CP^2?
This is a concrete question in Enumerative geometry. Let $S$ be a compact
complex surface and $L\rightarrow S$ a holomorphic line bundle. Let
$$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$
...
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7
votes
1
answer
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Trivial obstructions and virtual fundamental classes
Suppose $X$ is a DM stack, and let $E^\bullet$ be a perfect obstruction theory of $X$ such that the $E^{-1}$ term admits a trivial quotient/sub-bundle. Is it true that the virtual fundamental class $[...
- 6,290
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Description of virtual fundamental class
For some concrete examples, is there an easy way to describe the virtual fundamental class (say, by capping off the moduli pace with an obstruction bundle ). Consider the moduli space of stable maps ...
- 147
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0
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247
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Transitive action on moduli space of holomorphic curves.
If $G$ is a complex semi simple Lie group and $P$ is a parabolic subgroup of it, the quotient $G/P$ is endowed with a Kahler structure and the action of $G$ on $G/P$ is holomorphic with respect to ...
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3
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2k
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Minimal genus, adjunction inequality
Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know ...
- 5,063
8
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1
answer
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To what extent does Poincare duality hold on moduli stacks?
Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the manifold,...
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7
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2
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892
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Intuition behind the age grading in quantum cohomology of orbifolds
Let $\mathscr{X}$ be a smooth DM-stack with projective coarse moduli space. I am interested in the orbifold cohomology ring $H^\mathrm{orb}(\mathscr{X})$, as defined by Chen-Ruan (for orbifolds) and ...
- 40.3k
16
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1
answer
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Donaldson-Thomas Invariants in Physics
First of all, I am sorry for there are a bunch of questions (though all related)and may not be well framed.
What are the DT invariants in physics. When one is computing DT invariants for a Calabi-Yau ...
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3
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1
answer
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Curve Splitting in the Degeneration Formula for Relative GW Invariants (MNOP II)
I am attempting to understand and use the degeneration formula for GW invariants as stated in 'Gromov-Witten theory and Donaldson-Thomas theory II' (Maulik, Nekrasov, Okounkov, Pandharipande). The ...
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The hypergeometric pullback conjecture
Here arXiv:math/0510287, Golishev proposed the following conjecture:
The hypergeometric pullback conjecture. Let $X$ be a Fano variety. Then,for any constituent $C$ of the quantum D-module $Q$ there ...
7
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1
answer
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Some questions on moduli of stable maps
Let $\overline{M}_{0,k}(\mathbb{P}^n,d)$
denote the moduli space of genus zero degree $d$ stable maps with $k$ marked points. This is an orbifold of expected dimension. Let $\overline{U}_{0,k}(\...
3
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0
answers
369
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genus one Gromov-Witten invariants of Calabi-Yau 3-folds
In
http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf
physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2).
Can any body explain to me (...
13
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3
answers
2k
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Computation of Gromov-Witten invariants for symplectic manifolds
According to references, Gromov-Witten invariants were first defined for symplectic manifolds and later for projective varieties algebraically, and they coincide on the overlap. Because I thought ...
- 1,398
3
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2
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661
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Quantum cohomology for open varieties
Hi, I know very little about the quantum cohomology (QC for short). I only got interested in the subject as the genus zero part may be relevant to a problem I'm working on. So I hope my question makes ...
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Gromov-Witten invariants of singular spaces
I wonder if there is any situation where one can talk about Gromov-Witten invariants
or quantum multiplication for singular varieties. Ideally, I would like have a situation
where for a singular ...
- 7,037
17
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3
answers
2k
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Is there a "motivic Gromov-Witten invariant"?
I recently attended an interesting seminar, where the concept of motivic Donaldson-Thomas invariants was explained (0909.5088).
Very roughly, the DT invariant is a generating function $\sum q^k e(...
- 6,133
13
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1
answer
2k
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Who streamlined Kontsevich's count of rational curves?
Let $N_d$ denote the number of rational curves in $\mathbf P^2$ passing through $3d-1$ points in general position. Maxim Kontsevich discovered a famous recursion for these numbers:
$$ N_d = \sum_{k+l =...
- 40.3k
5
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0
answers
280
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Gromov-Witten theory of equivariant local projective plane
Can I find written explicitly in the literature a formula for the genus zero equivariant Gromov-Witten theory of local $\mathbb{P}^2$?
I understand that the method of Givental will give the answer, ...
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14
votes
1
answer
875
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When can Witten-esque moduli spaces be used to define invariants of geometric structures?
I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.
It is striking ...
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1
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Gromov-Witten invariants counting curves passing through two points
Let us say that a closed symplectic manifold $X$ is $GW_g$-connected if there is a nonvanishing Gromov-Witten invariant of the form
$GW_{g,n}^{X,A}(\beta,point, point,\alpha_3,\ldots,\alpha_n)$ --in ...
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Relative Gromov-Witten Invariants
A central issue in defining relative GW-invariants on a symplectic manifold is the possibility that a sequence of relative pseudoholomorphic curves can degenerate in such a manner, that components lie ...
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2
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Gromov-Witten classes (as opposed to invariants)?
Let $\overline{M}_{g,n}(X,\beta)$ be the moduli of stable maps into $X$ of class $\beta \in H_2(X)$. We have the evaluation maps $\operatorname{ev}_i : \overline{M}\_{g,n}(X,\beta) \to X$. Given $\...
- 21k
17
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2
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Quantum cohomology of partial flag manifolds
Is there a place in the literature where the quantum differential
equation (or even just quantum cohomology algebra)
of partial flag manifolds $G/P$ is computed for arbitrary semi-simple $G$ and
...
- 7,037
6
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1
answer
577
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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) ...
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2
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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 ...
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2
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Deformation quantization and quantum cohomology (or Fukaya category) -- are they related?
Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of "...
- 21k
1
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1
answer
500
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Does any one understand the details of M Kazarian's work in enumerative geometry of $\mathbb{C}\mathbb{P}^2$ ?
I wanted to know if anyone understood the details of the paper
"Multisingularities, cobordisms, and enumerative geometry" available at the site
http://www.mi.ras.ru/~kazarian/.
In particular does ...
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5
votes
2
answers
647
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Is anything known about the enumeration of degree d, genus g curves in CP^2 where g >1 ?
I wanted to know if there is something analogous to Kontsevich's recursion formula for
enumeration of genus zero curves in $\mathbb{C}\mathbb{P}^2$, for higher genus curves.
There is a
similar ...
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votes
3
answers
1k
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Twisted curves, admissible covers, and an algebraic analogue of a specific monodromy computation
This problem arose when trying to understand the stack of twisted stable maps into a stack (specifically BG), as introduced by Dan Abramovich, Angelo Vistoli and several co-authors (Olsson, Graber, ...
- 40.3k
2
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2
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874
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Why are people interested in defining GW invariant in algebraic geometry category
Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.
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4
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1
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Do the virtual fundamental classes satisfy functorial properties?
In Gromov–Witten theory, if the symplectic virtual fundamental classes constructed by B.Siebert satisfy functorial properties, i.e., if $f\colon X\to Y$ is an appropriate map between symplectic ...
- 1,499
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0
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761
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a question on Costello's theorem
Costello's theorem,"open TCFT=Calabi Yau A-infinity category",he also mentions when applied to Fukaya category we can recover Gromov-Witten theory, but I see that it needs some assumption,also even if ...
- 1,499
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1
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385
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What's the chain level Gromov-Witten theory
I think I heard there is such a theory, but I just can't find reference.So I am asking if there really has such a theory and reference if yes. Thanks firstly!
- 1,499
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1
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What is the Gromov-Witten potential associated to String Topology?
Kevin Costello's article on the Gromov-Witten potential associated to a TCFT constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary ...
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2
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597
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About topological B model
I was heard (by an expert) that, in mirror symmetry, we have constructed a Quantum Master Equation associated to topological B model, and a solution to it. But I can't find any material about this. Is ...
- 1,499
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1
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What is the geometry behind psi classes in Gromov-Witten theory?
Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma_1, \ldots, \gamma_n$ corresponding to geometric cycles $Z_i$ on a target space $X$...
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1
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Computing 3 points Gromov-Witten invariants of the Grassmannian
This is from an exercise in Koch, Vainsencher - An invitation to quamtum cohomology.
Background
The exercise asks to compute the 3-points Gromov-Witten invariants of the Grassmannian $G = \mathop{Gr}...
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0
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MNOP conjecture
Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).
To define Gromov-Witten invariants, we consider moduli spaces of stable ...
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2
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What classes am I missing in the Picard lattice of a Kummer K3 surface?
Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not ...
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Different definitions of Novikov ring?
Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \in H_2(X;\mathbb{Z})}...
- 21k
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Are Fukaya categories Calabi-Yau categories?
Let X be a compact symplectic manifold. There is an idea, I think probably originally due to Kontsevich, that we should be able to get Gromov-Witten invariants of X out of the Fukaya category of X. ...
- 21k
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votes
1
answer
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Where does the Givental reconstruction formula come from?
In (for example) Semisimple Frobenius structures at higher genus (section 1.2) and Gromov-Witten invariants and quantization of quadratic Hamiltonians (section 6.8), Givental gives a conjectural ...
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