17
votes
Accepted
What are "branes", and why do they form a category?
Let me start by putting your questions into a bit more context. Kapustin and Witten's story occurs within string theory, a theory of 1-dimensional extended objects. Strings may be "closed," ...
- 258
16
votes
Roadmap for Mirror Symmetry
Can't really say anything about the physics ... :-)
But for Kontsevich's HMS conjecture, my personal (very biased) all-time-favorite list is:
Two great survey papers to start with are: "A beginner's ...
- 1,981
13
votes
Geometric Langlands: From D-mod to Fukaya
To answer [a paraphrase of] your second question first: yes the Kapustin-Witten perspective on geometric Langlands has I think been taken very seriously by a segment of the math community. I find it ...
- 24k
10
votes
What is the mirror of symplectic field theory?
A partial answer is as follows.
In order to make use of the theory of holomorphic curves (with boundaries or asymptotics, whatever), we should restrict ourselves to symplectic manifolds with convex ...
- 3,187
10
votes
Accepted
B-model and Hochschild cohomology
The boundary conditions of the B-model, ie, the D-branes, are the objects in $\mathcal{D}^b(X)$. A little bit of playing with pictures gives that the space of closed string states must be in the ...
- 4,217
9
votes
Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture
Disclaimer: I am also not an expert.
According to Perutz (see 'Core homological mirror symmetry project'), it is expected that T-duality (SYZ, your 3) implies HMS (your 2), which should imply Hodge-...
- 2,808
8
votes
Accepted
Question on condition for a sheaf to be locally free in Orlov 2004
The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ ...
- 39.3k
8
votes
Geometric Langlands: From D-mod to Fukaya
One answer to your initial question is that the $D$-modules are supposed to actually do something - they're supposed to analogize to automorphic forms under the sheaf-functions dictionary. Therefore ...
- 148k
7
votes
Accepted
Multiple mirrors phenomenon from SYZ and HMS perspective
Some relation between these two ambiguities is indeed part of the expected but still largely conjectural big picture.
Let $X$ be a compact Calabi-Yau manifold, viewed in a symplectic way. Let $M_X$ ...
- 6,920
7
votes
Accepted
Lagrangian torus fibrations and Arnol'd-Liouville theorem
By the time I finished writing this answer someone has explained the whole idea in a comment, but I thought I'd post it anyway as there is some more detail here. I assume the version of Arnold-...
- 7,005
7
votes
Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?
I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.
In fact, ...
- 66.3k
6
votes
Wrapped Fukaya categories of Stein manifolds
By now we understand quite well generators of such categories. The co-core disks for any relative skeleton of the sector associated to the superpotential will do. (The argument for this is geometric,...
- 8,723
5
votes
Accepted
Mirror symmetry for singular Lagrangian torus fibrations
Yes, there is a concrete program on how to handle singular fibers in the SYZ fibration and several steps of this program are already completed.
You can watch the videos of Abouzaid lectures on the ...
- 6,239
5
votes
The mirror of the Landau--Ginzburg model given by elliptically fibered K3
In general, if $X$ is a compact smooth $n$-dimensional Calabi-Yau manifold, and $D\subset X$ is an ample (or numerically effective) divisor, then the mirror of $X$ is usually a degeneration of the ...
- 3,187
5
votes
Accepted
How to construct the mirror partner of a blowup?
You can find a lot of such examples in the paper of Abouzaid-Auroux-Katzarkov: https://link.springer.com/article/10.1007/s10240-016-0081-9.
Basically, they studied the case when $X$ is $(\mathbb{C}^\...
- 3,187
5
votes
Mirror symmetry for blowups of the projective plane
It depends on the positions of the points that you blow up. If you blow up respectively $p,q$ and $r$ points on the 3 irreducible components of the toric divisor $D\subset\mathbb{CP}^2$, with $p,q,r\...
- 3,187
5
votes
Geometric Langlands: From D-mod to Fukaya
More a comment than an answer:
I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed. In particular, it’s possible to immediately apply the dictionary ...
- 8,723
5
votes
Accepted
Higher homological mirror symmetry?
Yes. The full stable infinity category is the right answer. The reason is that both sides are topological quantum field theories. In two dimensions, which is the important case here, Costello showed ...
- 4,217
4
votes
Reference on rigorous formulation of mirror symmetry conjecture
http://www.claymath.org/publications/monographs/mirror-symmetry
would be the place to start. It has contributions by both mathematicians and physicists.
4
votes
Learning Quantum (Co)Homology and Landau Ginzburg Superpotential
You can find some very accessible discussion on quantum cohomology of toric (Fano) manifolds in On the quantum homology algebra of toric Fano manifolds by Ostrover & Tyomkin (and references ...
- 441
4
votes
Log Calabi-Yau surfaces without maximal boundaries
Firstly, I do not think that every maximal boundary has a smoothing. For example, if the linear system $|-K_X|$ as fixed part, then very likely it contains no smooth element.
On your question of the ...
- 1,164
4
votes
B-model and Hochschild cohomology
For correct attribution, one should at least mention the paper which the preprint of Moore and Segal itself quotes as the source for the particular case of the algebraic description of open-closed ...
- 149
4
votes
Accepted
Mirror partners of some Calabi-Yau threefolds
Your two examples are actually of very different characters. The first has Hodge numbers $h^{2,1} = 3$ and $h^{1,1} = 51$; the second is rigid. This means that in the first case you're looking for a ...
- 426
4
votes
Accepted
What does does the monodromy weight filtration represent?
Given a nilpotent endomorphism $N$ of a finite dimension vector space $V$, Jordan canonical form implies that we can decomponse $V$ into a sum of "blocks" on which we can find bases ...
- 35.2k
4
votes
Accepted
Kapustin-Witten branes and the derived moduli stack of Higgs bundles
The short answer is yes.
The sigma model to $M_G(X)$ is a low energy approximation to a 2d gauge theory with gauge group $Maps(X, G)$. Studying the sigma model to the stack $Higg_G(X)$ is basically ...
3
votes
Multiple mirrors phenomenon from SYZ and HMS perspective
Two algebraic varieties are called Fourier-Mukai partners if their bounded derived categories of
coherent sheaves are equivalent.
A "trivial" example of Fourier-Mukai partners is given by ...
- 314
3
votes
Relation between mirror symmetry, homological mirror symmetry, and SYZ conjecture
Another relation between 2) and 3) is that 2) is an algebraic statement while 3) is a geometric statement. So if one obtains an SYZ mirror by dualizing a Lagrangian torus fibration, then one obtains a ...
- 31
3
votes
Reference or explanation: Cup products, deformations of complex structure and Mirror Symmetry
Yes, this story is heavily expanded upon :-)
As far as I understand it, the genus-zero Gromov-Witten invariants of the A-side and the Hodge theory of the B-side can be arranged into a gadget called '...
- 1,981
Only top scored, non community-wiki answers of a minimum length are eligible