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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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)\...
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1answer
22 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 ...
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0answers
43 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 ...
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0answers
89 views

Describe the process of fibrating the following symplectic spaces and studying their complex mirror spaces

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 ...
2
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0answers
28 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 ...
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241 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 ...
3
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1answer
242 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
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1answer
115 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 ...
3
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1answer
191 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 ...
4
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1answer
593 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
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1answer
271 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
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39 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 \...
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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 ...
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228 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 ...
18
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3answers
1k 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 ...
4
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1answer
189 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 ...
7
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1answer
600 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 ...
7
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2answers
922 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 ...
10
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1answer
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 ...
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1answer
257 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 ...