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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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Is the Godel universe Wick rotatable?

Take Wick Rotatability being as the way defined in the article by Helleland: Wick rotations and real GIT Is the Gödel universe Wick rotatable according to this definition?
Bastam Tajik's user avatar
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4 votes
1 answer
230 views

Decomposition of vector bundle in $M$-theory

I was studying this paper where the authors construct some field theory solutions by wrapping M5-branes on holomorphic curves on Calabi-Yau. I have some questions about their construction. What they ...
Davide Morgante's user avatar
1 vote
0 answers
67 views

Resource recommendation for geometrical viewpoint of physics [closed]

I am interested in seeing the geometrical viewpoint of the formulation of quantum field theory and related fields. I have been studying The Geometry of Quantum States and find it a very good approach ...
1 vote
1 answer
144 views

Shankar Monopole belonging to the class $-1$ of $\pi_3(\mathbb{R}P^3)$

I am having a lot of trouble trying to understand how the classes of homotopy groups relate to point-defects in physics (and how they can be used/represent in general). This is a problem from Nakahara'...
MathZilla's user avatar
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1 vote
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Are there examples of harmonic differential forms in non-singular projective algebraic varieties that appear in general relativity or quantum physics?

I would like to study some real physical examples of harmonic differential forms in non-singular projective algebraic varieties. As Calabi - Yau and Kahler manifolds are used in supersymmetry theories ...
Rajaram Venkataramani's user avatar
1 vote
0 answers
43 views

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?...
Nugi's user avatar
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2 votes
1 answer
89 views

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 $$\...
MZperX's user avatar
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3 votes
0 answers
43 views

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 ...
Byron Andrade's user avatar
6 votes
2 answers
462 views

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 ...
Nihar Karve's user avatar
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0 votes
2 answers
45 views

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 $x^2$ ...
Massimo Valerio Preite's user avatar
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1 answer
485 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" ...
0 votes
1 answer
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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 ...
aitfel's user avatar
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2 votes
1 answer
152 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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3 votes
0 answers
105 views

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 ...
Insight's user avatar
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4 votes
1 answer
187 views

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-...
azaha89's user avatar
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2 votes
1 answer
107 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 ...
M.N.Raia's user avatar
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0 votes
0 answers
26 views

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)
Peter Parker's user avatar
2 votes
1 answer
157 views

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 ...
Martin's user avatar
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3 votes
1 answer
568 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 ...
2 votes
0 answers
55 views

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 ...
nonreligious's user avatar
10 votes
1 answer
846 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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0 answers
153 views

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 ...
leastaction's user avatar
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3 votes
0 answers
701 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 ...
Alessandro Mininno's user avatar
1 vote
0 answers
108 views

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=...
user avatar
3 votes
2 answers
540 views

ADE Gauge Theory and String Theory

I should preface this by 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}/\...
Benighted's user avatar
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2 votes
1 answer
144 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) ...
SRS's user avatar
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7 votes
1 answer
674 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}$...
JamalS's user avatar
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5 votes
1 answer
2k views

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 ...
embedded_dev's user avatar
3 votes
1 answer
255 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 ...
nonreligious's user avatar
1 vote
0 answers
541 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 ...
Arlenne González de la Rosa's user avatar
7 votes
2 answers
257 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 ...
Federico Carta's user avatar
8 votes
0 answers
215 views

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^{...
user1201349's user avatar
2 votes
1 answer
156 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 ...
leastaction's user avatar
  • 2,095
3 votes
0 answers
191 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 ...
20 votes
1 answer
3k views

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 ...
Nahc's user avatar
  • 2,071
10 votes
1 answer
1k 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 ...
Marion's user avatar
  • 2,188
5 votes
1 answer
326 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 ...
user78219's user avatar
2 votes
0 answers
285 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 ...
23 votes
0 answers
1k 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 ...
Sebastien Palcoux's user avatar
3 votes
0 answers
263 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 ...
user29126's user avatar
  • 131
5 votes
1 answer
2k 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....
Anne O'Nyme's user avatar
  • 3,872
3 votes
0 answers
109 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 ...
Mike's user avatar
  • 76
7 votes
1 answer
728 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 ...
14 votes
1 answer
4k 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 ...
user40276's user avatar
  • 1,043
3 votes
0 answers
104 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, ...
hailekofi's user avatar
  • 131
6 votes
2 answers
1k 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 ...
Nikolaj-K's user avatar
  • 8,513
1 vote
1 answer
359 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 ...
Meex3300's user avatar
15 votes
1 answer
845 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 ...
twistor59's user avatar
  • 16.9k
7 votes
0 answers
110 views

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 ...
user avatar
32 votes
8 answers
12k views

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 ...