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Questions tagged [mirror-symmetry]

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

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Quantization Laguerre equation

I saw this interesting article https://www.mdpi.com/2073-8994/14/4/741 about how the quantization of the eigenvalues of the Legendre equation is a consequence of parity symmetry and imposing that the ...
Marc Navarro's user avatar
4 votes
0 answers
93 views

What is the global complex moduli space for dimensions higher than 1?

I am trying to read the book 'Mirror Symmetry and Algebraic Geometry' by D. Cox and S. Katz. In the book it claims that 'the space of all complex structures on a given manifold $V$ is a well known ...
Hyunbok Wi's user avatar
2 votes
2 answers
56 views

C3 symmetry, chirality and polytopes [closed]

Is there some analogy for the phenomenon of chirality related to enantiomorph polyhedrons, but where the family of enantiomorphs is more than two polyhedra, in three or more dimensions, and related ...
Esteban R Q's user avatar
4 votes
0 answers
167 views

What was the difficulty in enumerative geometry problems before physics?

I have read the book 'Enumerative Geometry and String Theory' by Katz, and it left me with some questions. It is outlined in the text how ideas from String theory and TQFT has enriched enumerative ...
Hyunbok Wi's user avatar
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0 answers
35 views

Why/when does a kaleidoscope not generate 'continuous' images?

I'm building a kaleidoscope in code. It seems to work, but I'm getting unexpected results. Either my code is wrong or my expectations were wrong. The code is mimicking a number of mirrors, ...
commonpike's user avatar
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0 answers
37 views

Studying Chapter 10 of Weibel's Introduction to Homological Algebra

In Weibel's text on Homological Algebra, Chapter 10 deals with the Derived Category. Is it necessary to study all 9 chapters before it to make Chapter 10 accessible? (Assuming a background of Groups, ...
BeefStew's user avatar
2 votes
0 answers
45 views

Non-straight lines of symmetry

This is a less concrete question I was just curious about. We always talk about straight lines of symmetry and reflections but can non-straight lines of symmetry exist when reflecting something, and ...
d0uble_a_b4ttery's user avatar
2 votes
2 answers
117 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 ...
SiD's user avatar
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1 vote
0 answers
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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 ...
Rio's user avatar
  • 562
1 vote
1 answer
190 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{...
YZ-GDOU's user avatar
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0 answers
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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 ...
landt5's user avatar
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1 vote
1 answer
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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 ...
P. Usada's user avatar
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1 vote
0 answers
207 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))...
Salazar_3854708's user avatar
3 votes
0 answers
49 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 ...
shaine's user avatar
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1 answer
85 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 ...
Marvin Krüger's user avatar
1 vote
0 answers
150 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 ...
Fernando Martins's user avatar
1 vote
0 answers
68 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 ...
nben's user avatar
  • 294
0 votes
1 answer
49 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 ...
acacia's user avatar
  • 247
2 votes
0 answers
508 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....
MaryM's user avatar
  • 35
1 vote
1 answer
2k 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 ...
Ashu tosh's user avatar
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59 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 ...
KoKo's user avatar
  • 197
2 votes
0 answers
49 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)\...
Yang's user avatar
  • 21
3 votes
1 answer
79 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 ...
Bertram Wooster's user avatar
5 votes
0 answers
147 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 ...
C.D.'s user avatar
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3 votes
0 answers
118 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 ...
zeta space's user avatar
2 votes
0 answers
33 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 ...
vanmeri's user avatar
  • 153
2 votes
1 answer
651 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 ...
Or Kedar's user avatar
  • 921
3 votes
1 answer
375 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 ...
Benighted's user avatar
  • 2,563
2 votes
1 answer
188 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
332 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 ...
Meer Ashwinkumar's user avatar
6 votes
1 answer
1k 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 ...
Daniel Mejia's user avatar
6 votes
1 answer
485 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 ...
Benighted's user avatar
  • 2,563
2 votes
0 answers
47 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 \...
phy_math's user avatar
  • 6,480
2 votes
0 answers
29 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 ...
phy_math's user avatar
  • 6,480
8 votes
0 answers
405 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 ...
H. Shindoh's user avatar
  • 2,070
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
267 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 ...
Euna's user avatar
  • 535
7 votes
1 answer
936 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
2k 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 ...
jj_p's user avatar
  • 2,390
11 votes
1 answer
3k 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
434 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 ...
hao's user avatar
  • 211