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23

These are all good questions. Perhaps I can answer a few of them at once. The equation describing the violation of current conservation is $$\partial^\mu j_\mu=f(g)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$ where $f(g)$ is some function of the coupling constant $g$. It is not possible to write any other candidate answer by dimensional analysis ...

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The easiest way to see imaginary time used is in elementary quantum mechanics in one dimension. (This is the explanation cribbed from wikipedia). Suppose we're looking at a tunneling-through-a-barrier problem. We start with the Schrodinger equation: $$-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+V(x)\psi(x) = E\psi(x)$$ Make the ansatz $$\psi(x) ... 13 Let us look at the instantons of an ordinary pure Yang-Mills theory for gauge group G in four Euclidean dimensions: An instanton is a local minimum of the action$$ S_{YM}[A] = \int \mathrm{tr}(F \wedge \star F)$$which is, on \mathbb{R}^4, precisely given by the (anti-)self-dual solutions F = \pm \star F. For (anti-)self-dual solutions, \mathrm{tr}... 9 I will add to twistor59 answer. Hawking liked the concept of imaginary time \tau=\mathrm{i}t because it transforms a Lorentzian metric$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$into a four dimensional like Euclidean metric$$ds^2 = +c^2 d\tau^2 + dx^2 + dy^2 + dz^2$$Hawking and others believed that a quantum gravity theory could be developed in this ... 6 I) This is discussed around eq. (23.7.1) on p. 462 in Ref. 1. The task is to perform the path integral$$\tag{1} \int_{BC} [d\phi]e^{\frac{i}{\hbar}S[\phi]} ~=~\sum_{\nu}\int\! du \int_{BC_0} [d\phi_q]e^{\frac{i}{\hbar}S[\phi_{cl}+\phi_{\nu,u}+\phi_q]} $$over fields \phi with some (possible inhomogeneous) boundary conditions BC. This is done by ... 5 It's Stokes's theorem. Consider a field F = dA + A \wedge A such that A is pure gauge at infinity, that is, \lim_{x\to\infty} A(x) = \omega\, d \omega^{-1} for some \omega : S^3 \to SU(2) \sim S^3 where \omega is a function on the 3-sphere because the limit can depend on the direction out to infinity. In differential forms the first expression is ... 5 The reason why classical solutions add a "lot" to the path integral is that their action (phase) is stationary i.e. almost the same phase as the action (phase) in their reasonably large vicinity of the configuration space; one gets positive interference as a consequence. More generic paths cancel with the adjacent ones whose phases are different and random. ... 4 Fermionic zero modes on an instanton background are in one-to-one correspondence with solutions to the Dirac equation$$(i D_\mu \gamma^\mu - m)\Psi = 0$$where the partial derivative D_\mu contains the gauge field term with the gauge field defining the instanton solution substituted into it. We want to study these solutions in the Euclidean spacetime. ... 4 For the pure e.g. d=4 gauge theory instanton and gauge field perturbations around it, there is no negative mode – the counterpart of the bound state. It's the only one among the 3 classes that is absent here. One may see this absence by noticing that the gauge theory may be embedded into a supersymmetric theory with the same gauge-field degrees of freedom,... 4 A soliton is a localized, non-dispersive solution of a nonlinear theory in Euclidean space. It certainly is a real object: you have a famous story about a certain John Russell who observed soliton-like waves made by a boat on a river (wikipedia knows everything about it!) The so-called morning glory clouds in Australia (http://en.wikipedia.org/wiki/... 4 This is apparently too long for a comment, so it's going to be fleshed out into at least (hopefully) a partial answer. A big problem I see with this is determining which theories can live in the same universe. In this way, I think Liang Kong's Mathematical Theory of Anyon Condensation provides a way forward. To reconstruct your set up: Consider three ... 4 The sphaleron is kind of the opposite of the instanton, and kind of the same. Let's make that statement precise: An instanton is a local minimum of the action that mediates vacuum tunneling (link to an answer of mine how and why instantons do that). The sphaleron sits in-between the vacua, in a certain sense, it is the instanton "in the middle of tunneling":... 4 I) Here we will give an explanation at the physics level of rigor. Coleman et. al. in Ref. 1 are ultimately interested in the Minkowskian partition function/path integral$$\tag{A} Z^M~=~ \langle x_f | \exp\left[-\frac{iH \Delta t^M e^{-i\epsilon}}{\hbar} \right] | x_i \rangle ~=~N \int [dx] \exp\left[\frac{iS^M[x]}{\hbar} \right], $$with Minkowskian ... 4 1) Suppose that you have two configurations (here I've used Coulomb gauge with euclidean time \tau):$$ \tag 0 A_{i}(x) = \begin{cases} 0 = U^{(0)}\partial_{i}(U^{(0)})^{-1}, \quad \tau = -\infty \\ U^{(1)}\partial_{i}(U^{(1)})^{-1}, \quad \tau = \infty\end{cases} $$Such situation describes tunneling between vacua with topological charges 0 and 1. ... 3 There is no better definition than what Wikipedia offers - in general, a topological excitation is a (field) state, i.e. a localized quantity since fields depend on spacetime, whose integral is a topological invariant. One prime example are Yang-Mills theories in 4D, where the integral \int \mathrm{Tr}(F\wedge F), as essentially the second Chern class of ... 3 Instanton calculations involve integrals over collective coordinates. One of these is the instanton size \rho. Reliable instanton calculations are those for which the integral over instanton sizes is dominated by small sizes so that (for asymptotically free theories) the coupling constant is small and higher order corrections in the semi-classical ... 3 Yes, you can recover the Seiberg-Witten prepotential from the Darboux coordinates X_\gamma (and indeed also from their "semiflat" versions X_\gamma^{sf}). The reason is the asymptotic property X_\gamma \sim exp(\pi R Z_\gamma / \zeta) as \zeta \rightarrow 0 (up to a \zeta-independent constant). Thus knowing X_\gamma is sufficient to recover ... 3 Okay, I cannot give you a full understanding of what is going on, but I can make the objects we are dealing with more precise: There are two spaces here: The moduli space M_\text{sh}(r,k) of framed torsion-free coherent sheaves of rank r and second Chern class k on the projective scheme \mathbb{P}^2 viewed as a complex analytic space with its ... 3 Instantons appear as classical solutions of Yang-Mills equation due to nontrivial topology of nonabelian gauge group. They may play some role in physics because of requirement of finiteness of vacuum energy. When we ask how explicitly instantons affect on physics, we must use quantum description: nontrivial homotopy group of nonabelian symmetry group and ... 2 TO have an instanton solution, you need to map the (euclideanized) "spacetime at infinity" to the group manifold. In the case of SU(2), both the spacetime at infinity and the group manifold are S^3 and instantons are characterized by the integers. I hope you understand that much, at least for SU(2). If you're interested in 4d instantons, they are ... 2 I think the path-integral is a complete red herring here! I'll try to convince you that Wick rotation yields completely equivalent way of writing the Lagrangian in classical field theory. Consider a classical action$$S[x] = \int L[x(t)] dt$$where x:\mathbb{R} \to \mathcal{M} for some target manifold \mathcal{M}. The Lagrangian is schematically given ... 2 I'm not sure why you are asking, because you seem to mention the answer already. This problem has been studied thoroughly in the late 70's by Belavin et al and 't Hooft. As far as I understand, the quantum vacuum is the lowest energy eigenstate of a Hamiltonian. It turns out that the classical solutions to the equations of motion (of a particle, or a field) ... 2 1) If \phi is non-singular, you can safely multiply both sides by \phi and get \square \phi = \phi*0 = 0 . If \phi is singular you can't do this because \phi * 0 is undefined. Equivalently, where 1/\phi=0 you can have \square\phi nonzero, as shown in (4.64). The singularities in \phi will map to zeroes in 1/\phi. 2) I believe only the ... 2 Where a string carves out a 2-dimensional world-sheet and a point particle carves out a 1-dimensional world-line of spacetime, the instanton carves out a 0-dimensional world-point. Counting only spatial dimensions, a string is 1-dimensional and a point particle is 0-dimensional. By logical extension, an instanton has dimension -1, if we only ... 2 First, the reasoning in the question about isomorphism classes of bundles is wrong, because the \check{H}^1(M,G) from the linked math.SE post is not the cohomology of M with coefficients in G, but actually the Čech cohomology of M for the sheaf \mathscr{G} : U\mapsto C^\infty(U,G). However, this indeed has a relation to the cohomology of M ... 2 You wrote$$ c_2 = \mathrm{Tr}(F\wedge F) which is a real-valued form, since $F$ is an $\mathrm{ad}$-valued form and the trace (which is supposed to be the adjoint trace in general) produces a number, so this is a real valued form. Elements of $H^{2n}(M,\mathbb{Z})$ are not forms. The deRham cohomology $H_\text{dR}^{2n}(M,\mathbb{R})$ of forms as the ...

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It depends on what you mean by vacuum. If you mean a field configuration that has $F=0$, then the gauge potential is locally pure gauge (cf. this answer by Qmechanic), so there can only be global obstructions to the gauge equivalence class of the $A$ corresponding to $F=0$ being $A=0$ everywhere. On $\mathbb{R}^{1,3}$, there is no such obstruction. If ...

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A vacuum is a field configuration that is pure gauge, i.e. $A = g^{-1}\mathrm{d}g$ for a gauge transformation $g$, and hence $F = \mathrm{d}_A A = 0$ (for $\mathrm{d}_A$ the gauge covariant derivative). An instanton is a local minimum of the action, which is given by an (anti-)self-dual configuration $F = \pm \star F$. It is not a vacuum for non-zero ...

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