Questions tagged [instantons]

An instanton (pseudoparticle) is a classical solution to the equations of motion, usually of Yang-Mills theory, with a finite, non-zero action, on a Euclidean spacetime. Instantons normally dominate the path integral of Y-M theory and determine its quantum tunneling behavior. May use for merons and other topological field-theoretic configurations.

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Questions about the large-instanton problem

The Problem. The issue that I'm talking about is the large-instanton problem of asymptotically-free non-abelian gauge theories. You can read about it in: [1.] Section 15 of 't Hooft's 1976 paper on ...
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About the Yang-Mills equation

Perhaps this is a sick question. How do we know that Yang-Mills equation does not have a non-trivial classical solution in Minkowski spacetime in vacuum? By non-trivial solution, I mean one that ...
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Saddle point approximation and finite action configurations forming a set of zero measure

In Coleman's "Aspects of Symmetry", chapter 7, section 3.2, he makes a claim that configurations of finite action form a set of zero measure and are therefore unimportant. Further, he goes on to prove ...
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Winding number of QCD vacuum

We know that QCD vacuum has instantons, which corresponds to tunneling process. Consider $SU(2)$ gauge theory without matter. We say that in the classical configuration of vacuum state $F^a_{\mu\nu}=0$...
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Regarding instantons in 2D Abelian gauge theory

I am trying to understand the analogues of instantons in a $U(1)$ gauge theory in 2D Euclidean spacetime. If we follow the same arguments as the 4D case and say that the the gauge field must tend to ...
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Consistency condition for Yang-Mills on a Torus

So I was recently studying 't Hooft's paper on self-dual solutions to Yang-Mills on $\mathbb{T}^4$. So the basic idea is that you consider a box with periodic boundary conditions and then you impose ...
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Bosonic fields in the presence of Instantons

If I understood it correctly the t'Hooft vertx induced by an instanton background in a non-abelian gauge theory comes from the zero modes of the Dirac operator (included in the classical action). ...
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Besides instantons and large-$N$ what are some other general non-perturbative methods for quantum field theory?

Besides large-$N$, instantons, lattice QFT, what are some other non-perturbative methods that help us better understand QFTs like the large distance dynamics of Yang Mills and QCD?
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Intuition about ADHM construction

I'm trying to understand reasons, why self-dual Yang-Mills equation can be reduced to algebraic equations. It's seem like a miracle. In article Construction of Instanton and Monopole Solutions and ...
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Why do we think that the $U(1)$ problem is solved by instantons?

It is usually thought that the $U(1)$ problem is solved when 't Hooft realized that instantons induce additional symmetry breaking of the $U(1)_A$ symmetry aside from the non-vanishing quark masses. ...
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ADHM construction of gravitational instantons

Is there a ADHM-like construction for gravitational instantons? Could somebody explain why there is not such construction?
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Instantons as classical solutions

Instantons are classical solutions of the Euclidean (i.e. imaginary time) classical equations of motion. The standard example from single particle 1D QM is taking a potential of the form $V(x)=(x^2-a^...
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Fermionic non-zero modes to standard $SU(2)$ 't Hooft instantons?

The fermionic zero modes to the "standard" $SU(2)$ 't Hooft instantons are known as follows (see page 42 in this article): For a 't Hooft instanton (self-dual field strength tensor) in the regular ...
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Topological number and gauge invariance

In QCD or other non-abelian gauge theories, we come across infinitely many vacua that are gauge equivalent but have different topological numbers. We then say that the instanton solution is tunnelling ...
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Understanding the prefactor $\frac{\theta g^2}{32\pi^2}$ of the $F\tilde{F}$ term in Yang-Mills theories

The most general Yang-Mills (YM) action consistent with Lorentz invariance, gauge invariance and renormalizability should contain a term $$\kappa F_{\mu\nu a}\tilde{F}^{\mu\nu a}\tag{1}$$ where $\...
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Are there winding-number vacua in Weinberg-Salam (Or are they a gauge artifact)?

In pure SU(2) Yang-Mills the vacua van be grouped in homotopy classes labeled by their winding number. Instantons connect these giving rise to the theta-vacuum. I’m studying the SU(2) sphaleron in ...
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Why do quarks flip chirality when exchanging an instanton?

Quarks are elementary particles, part of the SM. There are a lot of different questions and answers on this site about chirality, I did not find a really good description, so I will use this wiki ...
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What exactly do the zero-modes of the instanton mean?

I am studying instantons in quantum mechanics. My question is regarding the the zero mode of the fluctuation determinant that we get because the solution for the instanton breaks time translation ...
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Equilibrium points of bounce/instanton solution after Wick's rotation

In Coleman's paper Fate of the false vacuum: Semiclassical theory while working out the exponential coefficient for tunneling probability through a potential barrier, he studies the problem with Wick'...
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Supersymmetric Quantum Mechanics and the Localization Theorem

I am working through Tachikawa's review on instanton counting arXiv:1412/7121, and in his treatment of Atiyah's localization theorem (see section 3.1.3), he mentions the following equations: $$\text{...
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Why does the Holst term not affect gravitational dynamics?

The general first-order Palatini action in four dimensions is given by $$S[e,\omega]=\frac{1}{2\kappa}\int_{\mathcal{M}} F_{IJ}[\omega]\wedge\left(\star+\frac{1}{\gamma}\right)\left(e^I\wedge e^J\...
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Implications of Instanton Corrections (to Degenerate Vacuua) for Spontaneous Symmetry Breaking

We consider that if the classical vacuua of a theory are degenerate then each of them can be non-invariant under one or more of the symmetries of the Lagrangian. We can choose one of the vacuua and ...
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Instantons, renormalization, and the Schwinger Model

Instantons in QCD contribute to the up, down, and strange quark masses (see, e.g., Georgi and McArthur (1981) or Kaplan and Manohar (1986)). Some papers claim that this contribution is equivalent to ...
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Winding number in 4D & $SU(2)$ group

In the book Quantum field theory by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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A question on 't Hooft's paper “Computation of the quantum effects due to a four-dimensional pseudo-particle”

In the avove paper 't Hooft considers the following (rotation invarinat) operators on the Euclidean space $\mathbb{R}^4$: \begin{eqnarray} \mathcal{M}_0=-\left(\frac{\partial}{\partial r}\right)^2-\...
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Quantum Fluctuation Contribution in the Path Integral of a Meta-stable Potential

In Wen XiaoGang's QFT of Many-Body Systems Sect 2.4.2, He studied the decay of a Meta-stable state via path integral method. The real-time potential is A state at $x=-a$ decays to $x=\infty$. The ...
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Are theta vacua topologically protected?

In discussions of Yang-Mills instantons it is often stated that one should sum in the path integral over all contributions of fluctuations around all the topologically distinct vacua labelled by ...
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Are Instantons Massless?

That is, are the only field configurations which give a non-zero winding number ones in which the Fourier transform includes a factor like $\theta(k^0)\hat{D}\delta(k^2)$, where $\hat{D}$ is some ...
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How does the instanton break the $U(1)_A$ symmetry in QCD?

The $U(1)_A$ symmetry in QCD is anomalous. Its supposed to be broken by the instantons. Can anyone physically describe how does that happen?
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Reference on instantons in gauge theories

Is there a reasonably detailed and systematic exposition of the theory of instantons in non-abelian gauge theories?
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What is the difference between real and complex instantons (mathemtically, and their physical significance), and connection to Wick rotation

I am struggling to understand the difference and physical significance between real and complex instantons- I think these are also sometimes called ghost instantons? There are also anti-instantons. ...
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What is the 't Hooft determinant?

The 't Hooft vertex/determinant is somehow generated by instantons and is responsible for the generation of mass gap in pseudo-Goldstone bosons, such as an axion. For example, the complex Peccei-...
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Is there any connection between instantons and surface-interacting polymers?

Excluded volume polymers interacting with a penetrable hypersurface of variable dimension is a very interesting system to study critical behavior via perturbative renormalization. Since a penetrable ...
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Checking modularity-like transformation property

Assume $M$ is a 4 manifold. Let $Z_v$ be partition function of fixed magnetic flux $v$ with all instanton configuration summed over where $v\in H^2(M,Z/nZ)$. $\tau$ denotes complex parameter on upper ...
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Isn't there a unique vacuum of the Yang-Mills quantum theory?

The theta vacua$^1$ of a Yang-Mills quantum theory are given by $$|\theta\rangle=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}|n\rangle.$$ In Srednicki's Quantum Field Theory, he claims that the ...
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What does $B+L$ anomaly have to do with a phase redefinition of the left-handed quark field?

According to this answer, the reason why $SU(2)_L$ weak theory does not have a theta vacuum is because any theta term can be reabsorbed with a phase redefinition of the left-handed quark field. ...
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How many connected components are there in the vacuum manifold of $\phi^4$ theory?

Consider the following theory in $1+1$ dimensions $$\mathcal L = \frac12(\partial\phi)^2 - \frac\lambda4 (\phi^2 - v^2)^2 \,,$$ which exhibits a $\mathbb Z_2 = \{0,1\}$ symmetry, $\phi \to -\phi$, ...
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Does Coleman-de Luccia instanton approach a Hawking-Moss instanton?

Suppose a Coleman-de Luccia instanton terminates in the basin with the true vacuum between the top of the potential barrier and the field value with potential energy equal to the potential energy of ...
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Multi-instanton contribution to path integral

Briefly, I would like to have a reference to a clear detailed exposition of the computation of the multi-instanton contribution to the path integral while computing the energy levels splitting of the ...
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What is the relationship between the JNR instantons and the BPST instanton?

The JNR instantons are related to the t'Hooft ansatz, and take the form \begin{equation} A=\sigma_{\mu\nu}\frac{\partial_\nu \rho}{\rho}dx^\mu, \end{equation} where $\rho$ takes the form \begin{...
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Why not regard all large gauge transformations as genuine ones?

A large gauge transformation is a gauge transformation that is not connected to the identity. When quantizing a gauge theory, we must take configurations related by ordinary gauge transformations to ...
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What justifies compactifying space and spacetime, in the context of instantons?

When studying Yang-Mills instantons, there are two instances where one compactifies a space. When classifying vacuum states, one demands $A_\mu(\mathbf{x})$ to become a constant as $\mathbf{x} \to \...
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Vacuum Stability

I am studying one of the paper of Sidney Coleman, "Fate of the False Vacuum. II. First quantum corrections". Just before eq. (2.18) he says "Because of time translation invariance, this equation ...
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Simple explanation of the QCD VEV in terms of instantons

I've heard that instantons in QCD generate quark bilinear condensate $\langle \bar{q}_{L}q_{R}\rangle$ which is responsible for spontaneous symmetry breaking. Is there any clear and simple way to ...
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Comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons

I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of: Chern class (1st, 2nd), and ...
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Does the ABJ anomaly for the Abelian gauge field have a topological argument?

We known that the ABJ anomaly for non-abelian gauge fields with gauge group containing $SU(2)$ as a subgroup has a topological argument from the Euclidean path integral. Through studying the Euclidean ...
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Validity of the thin wall approximation

Inspired by the question How can I understand the tunneling problem by Euclidean path integral where the quadratic fluctuation has a negative eigenvalue?, I decided to come back to the first paper by ...
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Instanton Counting (Tachikawa's Review) and “Hilbert Space”

I'm reading this review on instanton counting. I cannot understand in section 3 how supersymmetric Hilbert spaces come about. For instance in section 3.1.1, covering the supersymmetric particle on $\...
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Instanton contributions in quantum gravity

Suppose a low-energetic System, i.e. a System, where the presence of "classical" gravitational fields can be assumed to be Zero. Classically we would have e.g. the ordinary Minkowski metric or more ...
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Which vacuum is the Universe really in?

There ate two types of vacuum of the Standard model-the vacuum of the Higgs potential and that of the vacuum of the Yang-Mills fields labelled by the Chern-Simons number. See the figure 5 here. The ...