Questions tagged [mirror-symmetry]

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

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between Calabi–Yau manifolds and refers to a situation where two Calabi–Yau manifolds look very different geometrically but are actually equivalent when employed as extra dimensions of string theory. Mathematically, it describes a correspondence between complex geometry and symplectic geometry.

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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 geometry and conversely. I am interested in knowing a…
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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 help of techniques in Physics. My current…
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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 sufficient condition) especially in the context of mirror…
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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 appreciated.
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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 Taimanov? My background is from theoretical and…
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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 seen in regular courses at my university. So, I…
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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 I've been thinking about this, is that I'm doing…
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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 cohomology of this degree we can integrate by…
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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 of an affine Cech cover. This is the perspective…
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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 manifold?
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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 having some difficulties with the moduli space of…
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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 mathematical jargon out of the definitions to perhaps…
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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 symplectic spaces but I'd like to gain a deeper…
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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 of the smooth, non-compact resolutions of the…
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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 one show this rigorously? Solution from physics: In…
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