Questions tagged [quantum-field-theory]
For questions about mathematical problems arising from quantum field theory, the branch of physics which describes subatomic particles and their interactions in terms of perturbations of the corresponding scalar, vector or tensor fields.
224
questions
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vote
0answers
20 views
Uniqueness of Witten-Dijkgraaf 2D TQFT at 0th dimension
If I understand correctly, Dijkgraaf-Witten TQFT in dimension 2 is the
following. Fix any finite group $G$, we define a field over a closed
2-manifold to be a principle $G$ bundle (it's automatically ...
7
votes
2answers
513 views
A toy model in 0-d QFT
Questions
For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a counting problem about Feynman diagrams.
Is there a tutorial for how ...
3
votes
0answers
77 views
Inverse semigroups and partial symmetries
I recently ran across the idea of inverse semi-groups in the context of partial symmetries, where the symmetry only acts on part of the system and not the entire system (e.g., in quasi-crystals).
My ...
8
votes
1answer
397 views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
17
votes
2answers
962 views
Formal mathematical definition of renormalization group flow
I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what ...
4
votes
0answers
122 views
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 ...
4
votes
0answers
198 views
Are vertex operator algebras ever conspiratorial?
I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
2
votes
0answers
39 views
Zero in the spectrum of an elliptic second order operator
This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...
2
votes
0answers
72 views
Spectrum of a linear elliptic operator
In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...
3
votes
1answer
377 views
Quantum Field Theory: completing the “A Bridge between Mathematicians and Physicists” series
I decided to read the series "A Bridge between Mathematicians and Physicists" written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series should be ...
17
votes
2answers
2k views
Quantum corrections to geometry
In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the ...
6
votes
0answers
122 views
Can we define topological quantum field theories on Calabi-Yau manifolds?
Calabi Yau manifolds are Kähler manifolds with vanishing first Chern class. According to the conjecture of E. Calabi , for a Kähler manifold M , if
$c_1 (M) = 0 $ , then M would admit a Ricci-flat ...
5
votes
1answer
196 views
Gauge invariance of Chern-Simons functional integral for a 3-manifold with boundary
I am trying to understand how the functional integral for Chern-Simons theory for a possibly non-compact 3-manifold with boundary is made gauge invariant.
For a compact 3-manifold, $M$, without ...
3
votes
0answers
80 views
Dimension of the skein module of a closed manifold?
I'm looking for a reference to Witten's conjecture that the free part of the (Kauffman bracket) skein module of a closed 3-manifold is finitely generated, i.e. the dimension of $K(M)$, where $M$ is a ...
4
votes
0answers
97 views
Conformal group and cobordism
In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations.
Namely, I like to know what has been done and explored in the past?
on ...
6
votes
0answers
79 views
Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)
My major question in this post here is that:
How can we relate the following two mod 2 indices:
$\eta$ invariant,
the number of the zero modes of the Dirac operator $N_0'$ mod 2,
associated to ...
8
votes
0answers
165 views
Lagrangian subgroups/submanifolds, 2d topological boundary and 3d “non-abelian” Chern–Simons theory
This post is meant to ask for proper references to fill a gap in the literature.
My short question is that are there known and precise ways to formulate 2d topological boundary conditions" for ...
17
votes
0answers
403 views
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 ...
9
votes
1answer
314 views
Does there exist a discrete gauge theory as a TQFT detecting the figure-8 knot?
My question: Does there exist a discrete gauge theory as TQFT detecting the figure-8 knot?
By detecting, I mean that computing the path integral (partition function with insertions of the knot/...
4
votes
0answers
121 views
$T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions
The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field
theories (CFT) by an operator that is quadratic in the stress-...
14
votes
0answers
276 views
How does quotienting by a finite subgroup act on the framed-cobordism class of a group manifold?
Let $G$ be a connected simple connected compact Lie group, and $\Gamma \subset G$ a finite subgroup. Then (the underlying manifold of) $G$ can be framed by right-invariant vector fields, and this ...
6
votes
1answer
311 views
One particle irreducible Feynman diagrams
In quantum field theory Feynman has invented a diagrammatic method to encode various terms in the Taylor decomposition of integrals of the following form below which I will write in a baby version as ...
10
votes
1answer
646 views
References on dualities on quantum field theory for mathematicians
Dualities on QFT–also called Quantum Field Theory dynamics–is a huge and fundamental research area. However, despite underpinning major mathematical breakthroughs such as the work of Kapustin and ...
3
votes
0answers
182 views
Yang-Mills theory v.s. Kaluza–Klein theory: Classical actions
In general Yang-Mills theory [1] seems to be different from the dimensional reduced Kaluza–Klein theory.
However, the historical account was that people tried to trace back the origin of non-Abelian ...
9
votes
0answers
357 views
Yang-Mills theory with non-compact gauge groups G
Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups).
However, it is not entirely clear the formulation of Yang-Mills theory with non-...
11
votes
0answers
132 views
Geometry of Affine Kac-Moody Algebras
I recently asked this question on phys.SE and it was suggested to me to ask it here.
One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
5
votes
0answers
216 views
Chern-Simons theory with non-compact gauge groups G
This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
8
votes
0answers
332 views
Witten zeta function v.s. Riemann zeta function
From a talk, we learned that
The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”:
where we sum over irreducible ...
5
votes
1answer
161 views
Nonlinear sigma models with non-compact groups / target spaces
A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...
8
votes
1answer
302 views
The Precise Meaning of the Moduli Space of Flat Connections?
Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists.
For 3d Chern-...
8
votes
0answers
183 views
Coleman–Mandula theorem and a mathematical proof
Coleman–Mandula theorem (by Sidney Coleman and Jeffrey Mandula) [1] is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a ...
4
votes
0answers
148 views
Complex projective algebraic variety, moduli space of flat connections, and instantons
In Looijenga's work below, if I understand correctly, it shows that
Statement 1: At an algebraic variety, the moduli space of SU($N$) flat
connections on a 2-torus $T^2$ is given by the space of ...
4
votes
0answers
162 views
Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer
Atiyah-Hitchin-Singer Ref 1 states that the number of
virtual dimensions of the instanton moduli space
for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
7
votes
0answers
208 views
The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
10
votes
1answer
204 views
Borel-Ecalle re-summation and resurgence: Criteria and Results
This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\...
16
votes
2answers
519 views
What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?
Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...
11
votes
0answers
292 views
CohFT: Witten vs. Kontsevich and Manin
Is there any connection to CohFTs as defined by Witten in his 1988 paper (via topological twist) and the CohFTs as defined by Kontsevich and Manin (in the context of Gromov-Witten theory of course).
...
1
vote
0answers
98 views
Determine all possible magnetic monopole of gauge theories
In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
9
votes
0answers
213 views
Twisted Chern-Simons, and Twisted Wess-Zumino Term
I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten.
Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
3
votes
0answers
140 views
Instanton configurations of self-dual and anti-self-dual instantons interplay
Yang-Mills gauge theory is given by the action
$$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$
whose Euler-Lagrange equations are the classical equations of motion. The classical ...
1
vote
0answers
147 views
Conventions / Normalizations of Yang-Mills Field Theories
Let the spacetime be 4-dimensional.
In the usual Maxwell theory of Abelian gauge fields $A$, where field strength $F=dA$ one considers the Maxwell action written as
$$
S_{Maxwell}\equiv\int
-\frac{...
7
votes
0answers
343 views
The Fock Space vs the Hilbert space in the context of Quantum Field Theory
Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ ...
7
votes
1answer
211 views
What is a formal definition of a Fermionic quantum field?
I could not locate a definition of Fermionic quantum field (like for an electron!) in even Kevin Costello's book, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&...
2
votes
0answers
66 views
About anti-commutation of gauge charged Fermionic quantum fields
[Please correct anything I might say wrong in what follows!]
For everything that follows I am thinking in the context of a supersymmetric QFT. Hence I guess everytime I say "spacetime" it needs to be ...
3
votes
2answers
407 views
$spin_{\mathbb{C}}$ Connection and Charge Parity
From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,
https://arxiv.org/abs/1602.04251
the authors states that on a curved manifold with a $spin_{\mathbb{C}}...
4
votes
1answer
497 views
Question on Witten’s paper “Supersymmetry and Morse theory”
EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)).
This well known article applies some tools developed by physicists (e.g. path integrals) to ...
3
votes
1answer
198 views
Spurious length-scale cutoff emerges in propagator defined in Costello's “Renormalization and EFT”
In page 9 of the introductory chaper of Renormalization and Effective Field Theory (the introductory chapter is available free here), Kevin Costello defines a propagator $P$ for the Laplace operator $...
12
votes
2answers
550 views
Mathematical/Physical uses of $SO(8)$ and Spin(8) triality
Triality is a relationship among three vector spaces. It describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation ...
5
votes
1answer
209 views
2TQFT and commutative Frobenius algebras
There is an equivalence between the category of commutative finite dimensional Frobenius algebras and 2 dimensional topological quantum field theories, see for example the book by Joachim Kock, which ...
9
votes
4answers
784 views
Geometric or conceptual way to understand supersymmetry algebra
Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ...
