Questions tagged [algebraic-geometry]

Use for questions about algebraic geometry as it applies to physics. Purely mathematical questions should NOT go here, instead, they belong on Math Stack Exchange.

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Does $\mathcal{N}=1$ SCFTs have geometric constructions?

There are class S theories which are 4d $\mathcal{N}=2$ SCFTs by compactifying 6d $\mathcal{N}=(2,0)$ SCFT on a Riemann surface. Does similar geometric constructions exist for 4d $\mathcal{N}=1$ SCFTs?...
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Positive geometry and log singularities

In order to define a positive geometry it is a requirement that has to be a logarithmic singularities on the boundaries, for example for an interval (endpoints $a$ and $b$) the canonical form is $$\...
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Is there any physical significance for the Cremona group / birrational transformations on the projective plane?

In algebraic geometry, the Cremona group is the group of birational transformations on the projective plane. I would like to know: Is there any physical interpretation for this group? Does this group ...
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Why do you need to count curves on Calabi-Yau manifolds in string theory?

One of the mathematical fields that string theory is said to have had a large bearing on is enumerative geometry which, roughly, deals with counting rational curves on hypersurfaces and its ...
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2answers
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Gaussian beam reflection derivation

In this PhD thesis I found a good treatment of refraction and reflection of Gaussian beams at a curved surface. My doubt is from where does it come the cube in the cosine at the denominator of $x^2$ ...
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1answer
279 views

References on mathematical stacks for a string theory student

This question was posted on mathoverflow (here) without too much success. I'm hoping to read the famous Kapustin-Witten Paper "Electric-magnetic duality and the geometric Langlands program" ...
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1answer
62 views

Application of algebraic geometry in studying geodesic and hypersurface

Hypersurface and geodesic can, in principle be written as function of spacetime co-ordinates but not all of them can be written as a polynomial therefore my question is: does algebraic geometry finds ...
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1answer
109 views

Four dimensional massless spectra of type IIA/B compactified on $\mathcal{M}_{4} \times {\rm CY}_3$

I am following “String Theory and M-Theory” by Becker, Becker, and Schwarz and I am currently studying chapter 9. I have a question - or better yet a point of confusion - regarding the derivation of ...
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Equivalence Between the Algebras of the Standard Model and Connes Non-Commutative Geometry Model

I've been watching some lectures on Connes non commutative geometry model and one of the things I don't understand is why the algebra he considers, $M_2(\mathbb{H}) \oplus M_4(\mathbb{C})$, is ...
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Why are worldsheets of strings _holomorphic_?

Disclaimer. I am a mathematician (algebraic geometer) who knows nothing about physics. Even worse, I might have major misconceptions about the objects I'll ask about. The level of the question is pop-...
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1answer
96 views

Doubt about mathematical construction underlying physical systems

Consider the first and second videos of this playlist $[1]$. It seems the professor tried to discuss some heuristic approach between number theory abstract algebra and physics; Classical Physics is ...
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For the torus to rotate 180 degrees around the East-West symmetry axis, what happens?

(Suppose to ignore the deformed friction and torus when rotating)
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Connectedness on Special Kaehler manifolds

I just wanted to make a short/concise question which is quite mathematical but the aim is physical so I would like to ask it here. Anyone knows if there is a general statement about connectedness on ...
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1answer
434 views

Algebraic geometry and topology for string theory [duplicate]

I am looking for a comprehensive book or notes in algebraic geometry and topology techniques used in string theory compactifications covering topics like orientifolds, orbiolds, Calabi Yau manifolds ...
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Geometric interpretation of dimensional reduction of $E_8 \times E_8$ string

Following on from this question. I'm trying to get a geometric picture of what happens under dimensional reduction. Let me focus on a single $E_8$ factor, in 10 dimensions - the Lagrangian will ...
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429 views

Relation between Topological String Theory and Physical String Theory?

I'm familiar with topological string theory from the mathematical perspective. In my narrow world, the topological string partition function is given by the Gromov-Witten partition function, which is ...
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Intuition for curves of self-intersection in F-Theory

I'm trying to study some papers on F-Theory where the standard language of communication involves ideas and terminology from algebraic geometry, specifically intersection theory. As a concrete ...
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445 views

Why is important to know the geometry of the moduli space of vacua in a SUSY gauge theory?

I'm studying the moduli space of vacua for some supersymmetric gauge theory and I want to know specifically why it is important to know the geometry of this space. I know everything about the division ...
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Proving a geometric relation in 2D Minkowski spacetime [closed]

The triangle on the picture is the (future) causal domain of dependence of the interval of length $A$. The intervals of length $B$ and $C$ are relatively boosted to each other. How do I prove that $AD=...
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ADE Gauge Theory and String Theory

I should preface this my saying that I'm an algebraic geometer, and am not terribly knowledgeable about physics, nor the physics literature. I want to consider the singular surfaces $\mathbb{C}^{2}/...
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1answer
121 views

Connection between homotopic maps from $X\to Y$ and homotopic paths in $Y$ in the context of SU(2) Yang-Mills instantons

EDIT: I was reading little bit of homotopy theory in trying to understand the difference between homotopic maps from $X\to Y$ and homotopic paths in $Y$, and their significance in the context of SU(2) ...
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409 views

Relation between second cohomology and central extensions

In Blumenhagen's text on conformal field theory, after deriving the central extension of the Witt algebra, namely the Virasoro algebra, $$[L_m,L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$...
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Relation between QFT and algebraic geometry

So, I've read this article who mentioned that physicists in LHC, while calculating the Feynman diagrams in one of their experiments, noticed a strange pattern: the numbers emerging from Feynman's ...
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1answer
174 views

How should I physically understand the slope stability of vector bundles on a manifold X?

Very basic question here as I try to come to grips with the geometry associated with string theory. I can (almost) understand how a manifold $X$ can admit or not admit a particular vector bundle on ...
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339 views

Applications of group theory in astronomy [closed]

I am familiar with applications of representation/group theory in terms of symmetries in quantum mechanics and general relativity, but what are some of their applications in astronomy? More ...
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2answers
207 views

Flavor symmetry fixes the Higgs branch in any 4D ${\cal N}=2$ QFT

Let us consider two different quantum field theories in 4 dimensional Minkowski spacetime, call them theory A and theory B, with 8 supercharges. (i.e. 4D $\mathcal{N}=2$ theories). Let $G_A$ be the ...
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Metric transformation, polygons and gravitons

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
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1answer
106 views

Simple question about holonomy and the orbifold $C^2/Z_{2}$

This query is from Gubser's TASI lectures on Special Holonomy in string theory and M-theory. There is a detailed description in the first lecture, of the orbifold $C^2/Z_2$ and I think I understand ...
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151 views

Reference for orbifolds in string- and M-theory

A number of orbifold constructions have been studied heavily in string- and M-theory over the years, establishing various dualities between different theories. Can someone point me to a slightly more ...
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1answer
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Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
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1answer
941 views

How can I understand instantons as sheaves?

In specific, instantons are considered or interpeted as torsion free coherent sheaves. Why is that the case? Is there a nice way to understand this relation and of course also understand how the two ...
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1answer
283 views

What algebraic structure does the collection of all physical quantities form?

What algebraic structure -- by which I'm referring to abstract algebra theoretic ones such as ring, field, module, etc. -- does the collection of all physical quantities form? An related and/or ...
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211 views

Tropical Geometry and Quantization

Recently I saw this question posted on Math Overflow asking about the motivations behind tropical geometry. The OP mentions that tropical geometry can be viewed as the classical limit of regular ...
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992 views

TQFTs and Feynman motives

Questions Is a topological quantum field theory metrizable? Or else a TQFT coming from a subfactor? For a given metric, are there always renormalization and Feynman diagrams? Is there always a Feynman ...
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231 views

Estimating volume of moduli space of genus-g Riemann surface with n marked points

I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus $g$ and having $n$ marked points. I am reading some old string theory papers which discuss divergences ...
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1answer
1k views

Which is the role of Algebraic Geometry in String Theory? [closed]

Could someone sketch me what algebraic geometry has to do with string theory? Are there other mathematical disciplines that are interwoven with string theory? I'm aware of a similar question on math....
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103 views

Is it possible to build up holography in a closed manifold, i.e., in a manifold with a mathematical boundary?

I was wondering about the AdS/CFT correspondence basics. It is constructed on the idea of conformal compactification, in which a open manifold $M$ is homeomorphic related to a closed one $N$ through a ...
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1answer
604 views

Etale bundles and sheaves

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
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1answer
3k views

How algebraic geometry and motives appears in physics?

First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high ...
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98 views

Spinors on algebraic plane curves

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, ...
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2answers
834 views

Is there a physical motivation to study finite fields?

Clearly finite groups are of immense value in physics and these are also substructures of fields. However I never came across any computations involving finite fields at university and so I never ...
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1answer
334 views

transformations with commutators and anticommutators that generate displacements

is well known that composition of point reflections generate pure displacements. This implies that the commutator of two point reflections will be a pure displacement. Are there similar elemental ...
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765 views

What is Motivic mathematics and how is it used in physics?

In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being ...
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What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...
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Crash course on algebraic geometry with view to applications in physics

Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an ...