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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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{...
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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 ...
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What are these 'partial' reflective invariants and equivariants of a multiary function called?

In the univariable case we say that a function is even if $f(x)=f(-x)$ and odd if $-f(x)=f(-x)$ for all $x$ in some space of interest. In the multiary case we would similarly consider a function to be ...
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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 ...
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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))...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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....
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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 ...
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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 ...
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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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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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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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 ...
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6 votes
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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 ...
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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 ...
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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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8 votes
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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 ...
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22 votes
3 answers
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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 ...
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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 ...
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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 ...
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7 votes
2 answers
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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 ...
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12 votes
1 answer
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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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4 votes
1 answer
380 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 ...
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