Questions tagged [gromov-witten-theory]

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48
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0answers
824 views

What is the current status of derived differential geometry?

I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many ...
27
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2answers
2k views

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 ...
24
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1answer
3k views

What are Gromov-Witten invariants in terms of physics?

What do Gromov-Witten invariants (of say a Calabi-Yau 3-fold) represent, or what are they supposed to represent, in terms of string theory? When I compute GW invariants, am I actually computing some ...
24
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2answers
2k views

Manifolds distinguished by Gromov-Witten invariants?

What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures ...
22
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3answers
2k views

How mirror of quintic was originally found?

In the 90-91 pager "A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY", Candelas, de la Ossa, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically ...
20
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3answers
2k views

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 ...
19
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7answers
2k views

Higher Dimensional Gromov-Witten Theories

So, a basic set-up in modern enumerative geometry is that you have some object $X$ (say, a "nice" stack, for whatever definition of "nice" you need) and then you want to "count" the curve of genus $g$ ...
18
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1answer
3k views

Hochschild (co)homology of Fukaya categories and (quantum) (co)homology

There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of ...
17
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3answers
2k views

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(...
17
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2answers
2k views

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 ...
16
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1answer
2k views

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 ...
16
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0answers
2k views

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 ...
15
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1answer
3k views

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 ...
15
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2answers
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 ...
14
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1answer
789 views

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 ...
14
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1answer
851 views

Curves on K3 and modular forms

The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
14
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2answers
2k views

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 "...
13
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4answers
2k views

Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...
13
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1answer
2k views

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 =...
13
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0answers
306 views

Log symplectic vortex equations in Hamiltonian log GW theory

Hamiltonian Gromov-Witten theory(see Mundet-Tian paper) corresponds to a new type of Symplectic vortex equations: Such type of models gives a connection to Hitchin-Kobayashi correspondence and Floer ...
12
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4answers
2k views

Why would I want to know (equivariant) quantum cohomology?

Let's say that I have a variety I think is interesting, and based on some papers I don't fully understand, I can compute quite explicitly its equivariant quantum cohomology in terms of explicit ...
12
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6answers
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Gromov-Witten theory and compactifications of the moduli of curves

Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather ...
12
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3answers
2k views

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 ...
12
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2answers
1k views

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 ...
12
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1answer
1k views

How to understand Givental's I- and J-functions?

I am learning about mirror symmetry and having trouble understanding Givental's I- and J-functions. For example the J-function for the quintic threefold $X$ is defined by the formula $$ J:=e^{(t_0+...
12
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1answer
506 views

Quantum cohomology of line bundles over $\mathbb P^N$

Let $n,N$ be two positive integers. Consider the total space of the line bundle $\mathcal O(-n)$ on $\mathbb C\mathbb P^N$. This is an algebraic variety with an action of $G=GL(N+1,\mathbb C)\times \...
11
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3answers
3k views

References for Donaldson-Thomas theory and Pandharipande-Thomas theory?

I'm looking for good introductory references for Donaldson-Thomas theory and Pandharipande-Thomas theory. I know a bit about Gromov-Witten theory, but almost nothing about Donaldson-Thomas and ...
11
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2answers
4k views

Picard-Fuchs equations

If I have the periods $$\pi_1(\lambda)=\int_0^1\frac{dx}{\sqrt{x(x-1)(x-\lambda)}}$$ and $\pi_2(\lambda)$ similarly defined of the cubic curve $$y^2z=z(x-z)(x-\lambda z)$$ Such functions will be ...
11
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2answers
1k views

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 $\...
11
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1answer
996 views

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 ...
11
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0answers
428 views

Does quantum cohomology have an $E_\infty$-ring structure?

Consdier the classical singular cohomology ring $H^*(X, \mathbb{Z})$ of a manifold $X$. The $H^n(X, \mathbb{Z})$'s are the cohomology groups of a chain complex and it turns out that the multiplication ...
11
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0answers
560 views

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 ...
10
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3answers
752 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 ...
10
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2answers
1k 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 ...
10
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1answer
1k views

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 $X$...
10
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1answer
580 views

Is there a tropical geometric proof for counting genus g curves in any n dimensional projective space?

Consider the following question: Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z})$ a fixed homology class. Let $\mu_1, \mu_2, \ldots, \mu_k$ denote certain generic ...
9
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2answers
2k views

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. ...
9
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0answers
363 views

The virtual fundamental class as derived intersection

Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
9
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0answers
689 views

Question on Ionel and Parker's paper: Relative Gromov Witten Invariants

In defining Gromov-Witten invariants using symplectic geometry, most of the trouble is to achieve transversality for moduli spaces of pseudo-holomorphic curves which are multiple covers of simple ones....
8
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1answer
665 views

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,...
8
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1answer
418 views

Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry

In section 0.3. of their paper "Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields," Barannikov and Kontsevich discuss the fact that Kontsevich's formality morphism (from his paper ...
8
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0answers
248 views

The closure of a locus in $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$

Consider the closure $K \subset \overline{\mathcal{M}}_{0,n}(\mathbb{P}^1, n)$ in the stable maps space of the locus $K_0$ of maps $(f : C \to \mathbb{P}^1, p_1, \ldots, p_n)$ where $C \cong \mathbb{P}...
8
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0answers
478 views

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 ...
7
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3answers
910 views

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, ...
7
votes
2answers
769 views

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 ...
7
votes
1answer
225 views

Gromov-Witten invariants and the mod 2 spectral flow

I'm reading Lee-Parker, “A structure theorem for the Gromov-Witten invariants of Kähler surfaces”, which studies Gromov-Witten invariants within symplectic geometry. Lee-Parker write (&...
7
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1answer
400 views

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}(\...
7
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1answer
523 views

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 ...
6
votes
2answers
2k views

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})}...
6
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1answer
546 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) ...