I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have tried to find answers in many books on topological defects, gauge field theories, etc. but most of these books start with some Lagrangian and start talking about kinks and stuff like that. I have not been able to get a clear picture of what a topological object is physically? Is it just a mathematical construct or has some deep physical meaning (I am sure it has one) ? What is the difference between a topological object and a non-topological one. I understand that, in topology, we study properties under continuous deformations, stretching, twisting, etc.....so there is a context here but I do not understand its significance in physics. I need a very clear physical picture of this...I wouldn't mind some math though.
I think the broadest definition of a topological object would go as any object whose relevant characteristics do not depend on any metric. Instead it can depend on topological properties such as dimensionality, boundaries, compactedness, connectedness (connected or disjoint components), Connectivity (simply connected vs multiply connected), orientability (e.g. Mobius band vs sphere) and so on. That is a quite abstract definition and definitely would imply different interpretations depending on the context. The term topological object can refer either to a mathematical term entering the theory at a fundamental level (e.g. the topological theta term) or to physical object which is the "output" of a theory.
In the context of field theory (which I believe is the one you are interested in) topological objects are finite (energy density) solutions to non-linear (classical) field equations whose stability is guaranteed by some conserved topological charge or invariant. They are extended solutions, in the sense they are localized in a finite region of space and apart from that they behave like particles.
In this sense, topological objects are not a mere mathematical construction. They arise naturally in some theories, for instance non-linear and vacuum degenerate theories, and play important roles in areas such as field theory, condensed matter physics and astrophysics/cosmology.
The difference between a topological solution (object) and a non-topological one regards the mechanism that gives stability to each solution. For example, there are solutions to non-linear field equations (such as the KdV equation) that are also stable extended solutions however they are non-topological. Their stability comes from Noether Theorem and is not related to any topological property of the field.
Now it comes to the point where we should understand the relation between topological invariants and fields. In other words, how can the topology of the fields (continuous deformations) infer about its stability?
To have a grasp of the answer of the above question, consider the simplest topological solution in relativistic field theory, a kink in the $\phi^4$ theory in $1+1$ spacetime. The figure below shows the potential $V(\varphi)$, the static field configuration $\varphi(x)$ and the energy density $\epsilon(x)$ of a topological solution on that theory
If this solution were unstable then it would decay into the vacuum, that is the field configuration would be continuously deformed (topology showing up) into a straight line either either on $+v$ or on $-v$. But since to move any infinitesimal piece of the field from $-v$ to $+v$ one has to go through a finite energy barrier (the bump in the potential) and the energy cost for doing this to the entire field configuration is infinite. No non-singular transformation can deform this topological solution into the vacuum so it is stable.
How can we translate physical argument into the so-called topological invariants? Topological solutions in field theory are normally classified according to the homotopy classes of the vacuum manifold (the set of field expectation values that vanishes the scalar potential). In the case of kinks one is interested in $π_0(\mathcal M)$, the set set of path components of $\mathcal M$. This object simply says whether the vacuum manifold is connected or not, or in another words if the vacuum is made of disjoint parts. Note from our example that this is exactly what is going to be relevant for the stability of the solution. The vacuum manifold has two disjoint pieces, $\pm v$, and the fact that the scalar field asymptotically interpolates these two points is what is preventing it to decay into the trivial solution.
Other topological solutions have a similar description. For vortices (flux tubes or cosmic strings) we are interested in the fundamental group $\pi_1(\mathcal M)$ which classifies $\mathcal M$ accordingly how closed paths can or cannot be continuously shrank to a point. For monopoles (or hedgehogs) the relevant topological feature is how closed surfaces can be shrank to a point so the homotopy group characterizing these solutions is the second homotopy group $\pi_2(\mathcal M)$. The fact that for each type of solution there is a respective homotopy group classifying it is associated to the spacetime dimension and how the scalar field (Higgs) provides a map between spatial infinity and the vacuum manifold.
In other areas, there might be some differences when referring to a "topological object". For instance I hear condensed matter people saying that term when referring to some material itself, such as a topological insulator. However, topology enters in this case also in a similar way to the one described above. One studies how certain maps (from the first Brillouin zone to the some energy manifold) are continuously deformed.
A topological defect is a solution of your equation of motion (i.e. an eigenstate of your Hamiltonian say) which is stable under perturbation theory. The precise definition nevertheless requires a specific condition, and there is no global mathematical definition as far as I know (see the review by Mermin 1979 who tries to give some). For instance, a domain wall is a configuration of the electron spin which is stable under application of a small magnetic field, or small deformation of the lattice if you put your electrons on a lattice. The difficulty is usually in defining what small means.
Also, note a topological object (as the ones defined in quantum field theories) is usually a classical object. I like the presentation given by Rubakov in Classical theory of gauge field, which shows clearly what a zero-energy mode is in term of perturbation. As for their deep physical meaning, well, they are solution of your equation of motion (say the Schrödinger equation if you wish), so they are usual modes to that respect, but they live in between different vacuum, i.e. they link the different degenerate vacua (also called ground state). That's why they require ground state degeneracy
A simple example (at least for me who works on superconductivity) is the Ginzburg-Landau model. When uncharged, it exhibits the phase-slip solitons as an example of topological object (see also Coleman book Aspect of symmetry for a nice introduction + on the subject), see e.g. an answer of mine : physics.stackexchange.com/a/112206/16689 . When the condensate is charged (i.e. you have a covariant derivative) there is also vortex solutions which are stable (see Rubakov's book cited above)