# 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.

**7**

votes

**1**answer

216 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

**0**answers

69 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

**0**answers

262 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 ...

**5**

votes

**1**answer

206 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

**1**answer

586 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

**0**answers

147 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

**0**answers

230 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-...

**10**

votes

**0**answers

104 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 ...

**4**

votes

**0**answers

172 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 ...

**7**

votes

**0**answers

285 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 ...

**4**

votes

**1**answer

130 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 ...

**7**

votes

**1**answer

260 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

**0**answers

126 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

**0**answers

142 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

**0**answers

123 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

**0**answers

203 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 ...

**8**

votes

**0**answers

87 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

**2**answers

426 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

**0**answers

262 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

**0**answers

94 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

**0**answers

192 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 $P\rightarrow M$ ...

**4**

votes

**0**answers

120 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

**0**answers

102 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

**0**answers

244 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

**1**answer

192 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

**0**answers

65 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

**2**answers

204 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

**1**answer

395 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 ...

**4**

votes

**1**answer

187 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 $...

**10**

votes

**1**answer

418 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 ...

**4**

votes

**1**answer

185 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

**4**answers

743 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 ...

**3**

votes

**1**answer

301 views

### Question on the integral $\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx$

From the wikipedia page on Gaussian integral https://en.wikipedia.org/wiki/Gaussian_integral the following formula holds:
$$\int_{-\infty}^{\infty} e^{a x^4+b x^3+c x^2+d x+f}\,dx =\frac12 e^f \ \...

**8**

votes

**0**answers

261 views

### Mysterious relationship between central charges of conformal field theories and the Beraha numbers

Background:
Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it)....

**6**

votes

**1**answer

619 views

### What is the relation between BRST quantization and gauge fixing quantization

To quantize gauge field, one usually use gauge-fixing procedure and then plus ghost field, my question is what the relation between BRST quantization and gauge fixing quantization is? Because it seems ...

**4**

votes

**0**answers

68 views

### Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$

It is known that contracting over the vector indices of two Pauli matrix can be simplified to a bunch of delta functions. This is done via Fierz formula
$$\delta_{ab}\sigma^a_{ij}\sigma^b_{kl}=\delta_{...

**2**

votes

**0**answers

179 views

### Gaussian integrals and Showing $ \int f({\vec {x}})e^{\left(-{\frac {1}{2}}\sum \limits _{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)}d^{n}x=e^{D}f|_{x=0}$

This is related to my other question on tackling a gaussian integral for $f(w,u)=\frac{1}{w-u}$.
Q1 Suggestions on evaluating gaussian integrals with "nice" functions (not necessarily polynomials)
...

**10**

votes

**2**answers

260 views

### Is there a 1-dimensional analogue of the correspondence between the Levin-Wen and Turaev-Viro models?

Given a spherical fusion category $\mathcal C$, the Levin-Wen model constructs a lattice field theory: to each oriented surface with a triangulation, it assigns a state space $\mathcal H$ and a ...

**7**

votes

**2**answers

330 views

### Rigorous definition of the commutator $[a(k_1), a^\ast(k_2)]$ of creation and annihilation operators in boson quantum field models

In their lecture notes "Boson Quantum Field Models" (in "Mathematics of Contemporary Physics", R.Streater (ed.)), Glimm and Jaffe define an annihilation operator $a(k), k \in \mathbb{R}$ on a certain ...

**4**

votes

**1**answer

351 views

### Some questions about correlation functions and amplitudes in quantum field theory

I have been trying to learn some quantum field theory recently and I have a few questions which should be easy to answer for experts. I understand the basics of quantum mechanics / statistical ...

**1**

vote

**0**answers

130 views

### Wightman axioms to Vertex algebra, the inspiration for the infinitesimal translation operator T?

In section 1.1, 1.2 of Kac's book Vertex Algebras for Beginners, he deduces the axioms of vertex algebras (or more precisely, right chiral algebras) from the Wightman axioms for $2$d CFT.
Denote $\...

**3**

votes

**1**answer

205 views

### Wave front set of vector-valued Dirac delta distribution

Context: I am reading a physics paper Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime which applies the notion of the wave front set to operator-valued ...

**44**

votes

**4**answers

4k views

### Why is Quantum Field Theory so topological?

I understand that my question suffers from my lack of knowledge about the field, but as a mathematician without much knowledge of physics I have been wondering much about the following and I always ...

**12**

votes

**1**answer

494 views

### Do all $\mathcal{N}=2$ Gauge Theories “Descend” from String Theory?

I asked this on PhysicsSE, but I think it also fits here as it's related to algebro-geometric connections to string and gauge theory.
I'm thinking about the beautiful story of "geometrical ...

**8**

votes

**2**answers

529 views

### Characters of irreducible unitary representations of the Poincaré group

Consider Poincare group $\mathrm{ISO}(1,d-1)$, given by $\mathbb R^{1,d-1}\rtimes SO(1,d-1)$ in signature $(1,d-1)$, for some odd $d \geq 3$.
Denote the universal cover of the component connected to ...

**4**

votes

**0**answers

144 views

### Seiberg-Witten theory in 4d is categorification of Seiberg-Witten in 3d

According to Gukov et al. in this 2017 paper Seiberg-Witten theory in 4d categorifies Seiberg-Witten theory in 3d. In what sense is this phrase mentioned? I know what the process of categorification ...

**5**

votes

**0**answers

293 views

### Mathematics of $\mathcal{N}=2$ Gauge Theory and Instantons

Someone may suggest I post this on PhysicsSE, but I would prefer to not have a physicist answer in jargon I cannot understand. In fact, the reason I'm asking this is that I'm sort of drowning in the ...

**7**

votes

**1**answer

469 views

### Instanton Moduli Space on ALE Spaces

I asked this on MathStackExchange and was instructed it would be better here.
I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather,...

**3**

votes

**0**answers

48 views

### Integration of Weyl operators multiplied by quasifree state over a symplectic space

I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...

**8**

votes

**0**answers

163 views

### Equivalence classes of Wilson lines in $SU(2)_k$ Chern-Simons theory

One basic aspect of the 3D TQFT/2D CFT correspondence that I'd like to understand better is the following. It is often said that the ground states of Chern-Simons theory on a (spatial) torus are in ...