Questions tagged [mirror-symmetry]

Use for questions about mirror symmetry in theoretical/mathematical physics. Associate with [tag:mathematical-physics] if necessary.

Filter by
Sorted by
Tagged with
2 votes
2 answers
81 views

Must all lines of symmetry for a shape pass through the same point?

Must all lines of symmetry for a shape pass through the same point? I'm unable to think of a shape that doesn't follow this rule but can't come up with rigorous proof for this. By shape I mean any ...
  • 29
1 vote
0 answers
19 views

Bondal-Orlov conjecture on Calabi-Yau varieties

Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties $\implies$ Equivalent derived categories. I have started reading the paper by ...
  • 518
0 votes
1 answer
126 views

How to construct the Green's function solution for the barotropic Rossby wave eq with Dirichlet b.c.?

The linear differential equation I have been working on is the barotropic Rossby wave equation: $$ L\psi=(\frac{\partial}{\partial t}\nabla^2+\beta\frac{\partial}{\partial x} )\psi=\delta(\vec{x}-\vec{...
0 votes
0 answers
30 views

S-curve Graph Mirror

I have an s-curve equation that uses cosine. Any trigonometric function is allowed but I can't use exponential or logarithmic equations as they don't perform to what is required. The cosine equation ...
  • 1
1 vote
1 answer
74 views

A variety is the moduli space of structure sheaves of points

In the last paragraph of the first page of this paper, it is mentioned that an $n$-dimensional Calabi-Yau manifold $X$ is the moduli space of structure sheaves of its points and I am not really sure ...
  • 152
1 vote
0 answers
121 views

Introductory papers in Mirror Symmetry for algebraic geometry students

Assuming that a student knows algebraic geometry at the level of Cox's (1) Ideals, Varieties and Algorithms and (2) Toric Varieties What are some good papers (meaning, suitable in light of (1) and (2))...
3 votes
0 answers
44 views

Moduli space of special Lagrangians

I'm currently reading Auroux's Mirror Symmetry and T-duality in the Complement of the Anticanonical Divisor and Special Lagrangian Fibrations, Wall-crossing, and Mirror Symmetry back and forth. I'm ...
  • 113
0 votes
1 answer
42 views

Convert from one segment to another in 2D-coordinate system

Given a 2D-coordinate system where we only look at the area between 0-1 on both, x and y axis. How can I divide that system into n segments and mirror a point at p(0.5,0.5), so that all points are ...
1 vote
0 answers
120 views

Mirror Symmetry References

For someone comfortable with Hartshorne's book chapters 2 and 3 which references would be good to learn mirror symmetry (specially the homological flavor), GW invariants and related problems? I intend ...
1 vote
0 answers
47 views

Unambiguous geometry terms for specific kinds of circular symmetry

I'm writing a paper on asymmetries in the human visual system and want to ensure that I am using correct/unambiguous terminology to describe the asymmetries in question. Unfortunately, I've had ...
  • 244
0 votes
1 answer
40 views

Pre-calc algebraic method for predicting symmetry

I am looking for an algorithm that can be used on any equation that contains polynomials containing x and y to determine if reflective or rotational symmetries exist. If it is possible, I would like a ...
  • 257
2 votes
0 answers
380 views

How graduate students get to work in homological mirror symmetry

My question is probably an odd one here but I would very much like to work in Homological Mirror Symmetry. An example of a course I'd like to be able to take and understand is https://faculty.math....
  • 35
1 vote
1 answer
1k views

Find Reflection of A point with respect to a line mirror in 3D

I need to find the reflection of point $P(1,2,3)$ w.r.t line mirror $(x-1)/2 =(y-1)/3 = (z+1)/1$ I know one method to do it i.e by first finding the foot of perpendicular of P on the line by using ...
0 votes
0 answers
52 views

Genus of a curve? (Mirror Symmetry book)

I'm reading Chapter 6 of Mirror Symmetry book. In Example 6.1.1 "A degree 3 polynomial f in $\mathbb P^2$ determines a curve of genus $g = \binom{3−1}{2} = 1$ that has the structure (induced from ...
  • 187
2 votes
0 answers
48 views

Is $(x,y)\rightarrow (-x,-y)$ an inversion transformation?

Does anyone know whether $(x,y)\rightarrow (-x,-y)$ is an inversion transformation or not? I know that the standard inversion (parity) transformation in two dimensions should be something like $(x,y)\...
  • 21
3 votes
1 answer
63 views

Defining asymmetries in Turing's reaction-diffusion paper

I'm reading Alan Turing's paper titled The Chemical Basis of Morphogenesis and there is a section in it with mathematical definitions that mystify me. I'm guessing that Turing tried to keep ...
5 votes
0 answers
110 views

Why is a DG-enhancement of the derived bounded category of coherent sheaves an enhancement?

In order to make mirror symmetry more compatible with homological machinery, I understand it is common to give the derived bounded category on a variety a "DG-enhancement" by keeping around the data ...
  • 1,424
3 votes
0 answers
115 views

Fibrating $X=\Bbb R^2 / \{0\}$ by breaking up the space with hyperbola?

Attending graduate school this Fall and need to understand fibrations better. I will be taking geometry and algebra. I've read a neat article Quanta Magazine Article on the topic of mirror worlds and ...
  • 208
2 votes
0 answers
32 views

Why Lagrangians of two (real) torus are lines?

I am reading an article on Mirror Symmetry, where an example is given : the two (real) dimensional torus. My question is a basic one : taking the symplectic form (if ones focuses on the symplectic ...
  • 153
2 votes
1 answer
527 views

Book References about Complex Geometry

I took an introductory course in differential geometry, and now I take an advanced course about mirror symmetry and Calabi-Yau manifolds. I know this is way out of my league but I just want to have a ...
  • 861
3 votes
1 answer
339 views

Mirror Symmetry of Calabi-Yau Surfaces?

This isn't a terribly refined question, but more broad-brush: are there nice results on explicit mirror pairs of certain Calabi-Yau surfaces? In particular, I'm curious if we know the mirror partners ...
  • 2,493
2 votes
1 answer
167 views

What is mirror of symplectic $\mathbb{CP}^{2}$?

As far as I understand, mirror symmetry is an involution on the set of Calabi-Yau manifolds which acts at Hodge numbers by $h^{p,q} \leftrightarrow h^{q,p}$. Kontsevich in 1994 conjectured an ...
user avatar
3 votes
1 answer
275 views

First Chern class of toric manifolds

I have been reading a Mirror Symmetry monograph, and its physical arguments seem to imply that all toric manifolds have semipositive-definite first Chern class. Is this true, and if yes, how does ...
6 votes
1 answer
949 views

Prerequisites for book "mirror symmetry and algebraic geometry" by Cox and Katz

As the title suggest, I am trying to read the book mentioned, but I find that it uses a lot of material that I don't know yet. For example, it uses toric geometry and polytopes, topics that I've never ...
6 votes
1 answer
442 views

Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
  • 2,493
2 votes
0 answers
46 views

Mathematical terminology about Holomorphic vector bundle over Grassmanian.

This question is relevant to Mirror symmetry and moduli space. The linear sigma model in $U(k)$ with $Nk$ chiral fields vacua equation can be reduced as \begin{align} \sum_{i,j=1}^k \left(\sum_s^N \...
  • 6,176
2 votes
0 answers
24 views

Moduli space about $CP^{N-1}$ and $T^* CP^{N-1}$.

For complex $\phi$ in $U(1)$ gauge theory, we have \begin{align} |\phi_1|^2 + |\phi_2|^2 +\cdots |\phi_N|^2 =r \end{align} This equation $|\phi|^2=r$, describes sphere $S^{2N-1}$. Dividing the space ...
  • 6,176
8 votes
0 answers
330 views

The Kähler form and the anticanonical line bundle

Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive). On the other hand, I sometimes see the following definition (or ...
  • 2,000
22 votes
3 answers
3k views

Areas of contemporary Mathematical Physics

I have often heard that some developments in Physics such as Gauge Theory, String Theory, Twistor Theory, Loop Quantum Gravity etc. have had a significant impact on pure mathematics especially ...
5 votes
1 answer
246 views

Construction of virtual class at Homological Mirror Symmetry

In Homological Mirror Symmetry it is necessary to integrate cohomology class at stable moduli. For this, we can define virtual dimension that stable moduli space should have, and at moduli defined at ...
  • 535
7 votes
1 answer
860 views

Reference for Fukaya Categories and Homological Mirror Symmetry

What references are there for learning Fukaya categories (specifically, good references for self-study)? In addition, any references with an eye toward homological mirror symmetry would be greatly ...
user avatar
7 votes
2 answers
1k views

Reference request: toric geometry

What is a good book on algebraic geometry, with focus on toric varieties, similar both in the philosophy and in the prestige of the authors to Modern Geometric Structures and Fields by Novikov and ...
  • 2,294
11 votes
1 answer
2k views

Mathematics and Physics prerequisites for mirror symmetry

I am a physics undergrad interested in Mathematical Physics. I am more interested in the mathematical side of things, and interested to solve problems in mathematics inspired by physics maybe with the ...
user avatar
4 votes
1 answer
410 views

About Homological Mirror Symmetry

Why in homological mirror symmetry, we restrict us to a projective variety (Calabi-Yau manifold)? Because in physics we don't need this condition. What's the general picture for general Calabi-Yau ...
  • 211