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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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/...
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$T\bar{T}$ 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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 $...
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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). ...
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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 ...
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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 $P\rightarrow M$ ...
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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 ...
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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{...
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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 $F_{-1}$ ...
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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&...
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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 ...
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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}}...
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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
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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 $...
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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 ...
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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 ...
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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 ...