Questions tagged [string-theory]
A class of theories that attempt to explain all existing particles (including force carriers) as vibrational modes of extended objects, such as the 1-dimensional fundamental string.
141 questions
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Manifolds with negative dimension – Definition, References
Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?
My problem was born in ...
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2
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New series for $\pi$ from string theory
This is a direct followup to the post Possible new series for $\pi$ by Timothy Chow
and its numerous answers and comments.
Using another formula in the same string theory paper by Saha and Sinha one ...
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0
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modular properties of macmahon function?
How does the MacMahon function for counting plane partitions
$M(q) = \frac{1}{(1-q^n)^n}$
behave under modular transformations?
For instance for $q= e^{2 \pi i \tau}$ where $\tau \rightarrow -1/\tau$.
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answer
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Anomaly in QFT physics v.s. determinant line bundle
In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field ...
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answer
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What are "branes", and why do they form a category?
I've been trying to read Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program recently, as someone whose mathematical interests are in the Langlands program. I have some ...
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0
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Comparison between two volume forms on genus zero Teichmüller space
Consider a sphere with $n$ punctures. If you pick a holomorphic cotangent vector at each puncture, you can canonically construct a holomorphic top form in the corresponding moduli space. (The specific ...
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answer
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In M-theory, what can hypothesis H tell us that quantization in ordinary cohomology cannot?
In classical field theory, many fields and related objects are described as differential
forms. For example, in electromagnetism, the field $F := B - \mathrm dt\wedge E$ is a 2-form, and Maxwell's
...
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What is this correspondence between composition algebras over R and superstring theories?
In the page for superstring theory, Wikipedia states:
Another approach to the number of superstring theories refers to the mathematical structure called composition algebra. In the findings of ...
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how to derive this elliptic integral?
I am reading the article arXiv: 2207.09961, there are some interesting elliptic integrals, i.e. the formula (3.7) and (3.8). You can also see this image
where $p_0(z)=\sqrt{-Q_0(z)}$ and $Q_0(z)=-\...
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answers
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On determinants of Laplacians on Riemann surfaces
History of the formula: In their famous paper "On determinants of Laplacians on Riemann surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the ...
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Divisor cohomology through spectral sequences
I don't know if it belongs here but anyway, I need to compute arithmetic genus of divisors pulled back from a Fano base space to a bundle (which may or mayn't be trivial) defined through the ...
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Navier-Stokes fluid dynamics, Einstein gravity and holography
There was some activity a while ago, like 10 years ago, string theoreists try to relate
the fluid dynamics, for example, governed by Navier-Stokes equation,
to
the Einstein gravity, and its ...
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3
answers
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What exactly is the relation between string theory and conformal field theory?
Maybe it would be helpful for me to summarize the little bit I
think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and
an operator
$$A(X): {\cal H}^{\otimes n}\rightarrow {\cal H}^{\...
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Book on mathematical "rigorous" String Theory?
I've been looking high and low for a mathematical book on String Theory. The only book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and ...
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NSR superstring as a map of supermanifolds
On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
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Freeman Dyson's approach to string theory [closed]
Context:
In celebrating the centenary of Ramanujan's birth, Freeman Dyson presented the following career advice for talented young physicists [1]:
My dream is that I will live to see the day when our ...
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1
answer
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The Fuchsian monodromy problem
I want to understand the argument being made from equation 6.1 to 6.5 in this paper between pages 27-28
6.2, 6.4 and 6.5 are completely out-of-the-blue to me and I have no clue as to from where they ...
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Axiomatic string theory?
There have been many proposal of a mathematical definition of Quantum Field Theory, for instance through Wightman or Osterwalder-Schrader axioms. Were there any efforts toward doing the same for ...
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What is known about the cohomology of the U-duality group?
$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie
group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
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Enumerative geometry and restricted plane partitions
Donaldson-Thomas theory is an enumerative theory for virtual counts of ideal sheaves (with trivial determinant) of the structural sheaf $\mathcal{O}_{X}$ of some smooth projective manifold $X$.
There ...
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Physical Approach to Knot Categorification
Some recent work by Aganagic on knot categorification, Knot Categorification from Mirror Symmetry, Part II: Lagrangians, discusses two categorical approaches to categorification of quantum link ...
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L-infinity algebra of deformations of an L-infinity algebra?
From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
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Associating noncommutative geometries to 2D conformal field theories
I have recently been reading a bit about noncommutative geometry and string theory and it looked to be an open question (or at least this was open two decades ago) whether there are constructions ...
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Lifting in String Theory and QFT
I'm posting this here instead of Physics Stack as my question is on the precise mathematical meaning of a word which is often used in the physics literature.
In theoretical physics (especially string ...
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2
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On how to diagonalize a Casimir element
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}$I'm trying to read the physics paper Two Dimensional QCD as a String Theory. I'm struggling with my ignorance about ...
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Affine Kac-Moody algebra from quantum group exchange algebra
In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model.
...
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answers
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Mathematical predictions of AdS/CFT
What sorts of mathematical statements are predicted by the AdS/CFT correspondence?
My "understanding" (term used very loosely) is that this correspondence isn't a mathematically rigorous ...
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0
answers
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Holomorphic anomaly at genus 1
For computing instantons contributions from worldsheet torus to target torus, one can evaluate zero modes contribution of genus 1 partition function given by following expression:
$$Tr(-1)^FF_LF_Rq^{...
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Perverse sheaves and maximal genus Gopakumar-Vafa invariants
Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it ...
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Vafa-Witten invariants for mathematicians
As Richard Thomas has written (we paraphrase just slightly), mathematical physicists Vafa and Witten introduced new "invariants" of four-dimensional spaces in a paper:
A Strong Coupling Test of S-...
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1
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Supersymmetry charge $Q$ as anti-linear and anti-unitary operator
We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...
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References for superhomology
This question concerns topological string theory.
It was known sice its outset, that the BRST-cohomology ("observables") of the weakly coupled topological string B-model on a Calabi-Yau ...
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1
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Derive how the level quantization for 3d quantum Chern-Simons theory path integrals?
Let us consider abelian and non-abelian 3d quantum Chern-Simons theory path integrals:
abelian Chern-Simons theory on non-spin manifolds ---
$$
\int [DA]\exp(i \frac{k}{2\pi} \int_X (A \wedge dA ))
...
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1
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GKO (or coset) construction - all possible highest weights $h$
I am reading the famous paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive.
From a compact simple Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\...
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2
answers
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Spectral Flow Invariance for Calabi-Yau Sigma Models
I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.
If $X$ is a Calabi-Yau $d$-...
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References for quivers and derived categories of coherent sheaves for a string theory student
I'm a student mostly from physics knowledge hoping to learn about the math involved the string theory paper Topological Quiver Matrix Models and Quantum Foam.
Context: The topological string theory ...
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Calabi-Yau structures on dg-categories
A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here)
$$ A^! = A[-n]$$
Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...
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Anti-symmetric operators for the Dirac or Majorana spinors
In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...
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1
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Why is the inertia stack of a smooth Deligne-Mumford stacks called inertia?
Let $\mathcal{X}$ be a smooth Deligne-Mumford stack. Then there is an associated stack $I\mathcal{X}$, called the inertia stack of $\mathcal{X}$.
Why is the inertia stack called "inertia"?
We can ...
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Possible Context for this "Siegel-like" Modular Form Construction?
The following construction of something very nearly a Siegel modular form of degree 2 arose in my research. I'm outside the worlds of automorphic forms and number theory, so I'm wondering if it ...
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Bridgeland stability for restricted Kahler moduli?
Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical ...
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Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
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answers
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Mathematical uses of string theory
It is widely believed that correctness of string theory as a physical theory will not be decided in the near future. Regardless whether this will turn out to be correct or not, mathematical concepts ...
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what is the stringy Kähler moduli space?
I saw the stringy moduli space mentioned in a few papers but with little no explanation. I vaguely understand it is supposed to be the moduli space of complex structures on the mirror manifold.
Could ...
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Are there some known identities of elliptic polylogarithms similar to the Abel identity of polylogarithm?
Let
\begin{align}
Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2}.
\end{align}
This polylogarithm satisfies the following Abel identity:
\begin{align}
& Li_2(-x) + \log x \log y \\
& + Li_2(-...
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Degree-3 curves on the Calabi–Yau quintic
Robbert Dijkgraaf said,1
concerning the simplest
Calabi–Yau space, the quintic:
"A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. ...
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vote
0
answers
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GKO construction for (Super-)Virasoro algebras
I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...
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What are D-branes, really?
In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
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What is the definition of picture changing operation?
What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?
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The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
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