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TASI Lectures on Solitons

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David Tong

High Energy Physics - Theory · Submitted 28 Sep 2005 · 263 references

Pith's verdict: A well-built review whose real contribution is the co-dimension ladder and the honest labelling of what is still conjecture; the only concrete slip I found is a normalization typo in eq. (2.78).

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CitationDavid Tong. "TASI Lectures on Solitons." arXiv:hep-th/0509216 (2005). https://pith.science/paper/4BJH7GGH
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The paper

The paper claims

Four solitons, one geometry: instantons down to kinks

The reading

These lectures argue that the four classical solitons of supersymmetric gauge theory — the instanton, the monopole, the vortex and the domain wall — are not four separate subjects but one subject seen at four co-dimensions. The organising object is the moduli space: the space of all solutions of a given topological charge, modulo gauge, equipped with a metric inherited from the field theory. That metric is what survives of the solution once the details are stripped away, and it is what governs the low-energy dynamics, the spectrum of BPS states, and the counting of zero modes. The lectures also press a second claim: the equations themselves are related by dimensional reduction, so that monopoles are instantons with one direction traded for a scalar expectation value, and domain walls are vortices with one direction traded for a fermion mass. If the picture is right, a problem about one soliton is a problem about all of them with different boundary data, and techniques built for one (the ADHM construction, the Nahm construction, brane engineering) transfer to the others.

What carries the argument

The moduli space metric $g_{alphabeta} = tfrac{1}{2e^2}int d^4x, mathrm{Tr}(delta_alpha A_mu, delta_beta A^mu)$, built from the overlap of zero modes of the linearised Bogomoln'yi equations with a background gauge-fixing condition $D_mu delta A^mu = 0$. Its power is that it converts an infinite-dimensional field problem into finite-dimensional geometry: geodesics on it are the low-energy scattering of solitons, harmonic forms on it are the BPS bound states, and its hyperKähler or Kähler character is guaranteed by the supersymmetry preserved on the soliton worldvolume. The auxiliary construction is the hyperKähler quotient: an ADHM-type matrix model with a $U(k)$ gauge symmetry, adjoint and fundamental multiplets, and a triplet of D-term constraints whose solution space, divided by $U(k)$, is the moduli space. The lectures stress that the metric obtained this way coincides with the field-theory metric for instantons, and that non-commutativity enters the same construction as a Fayet-Iliopoulos parameter that resolves the small-instanton singularity.

Extended reading notes

Core claim

The load-bearing claim is a web of identities linking the four solitons. The monopole equation $B_i = D_ivarphi$ is the self-dual instanton equation $F = star F$ with one coordinate replaced by a scalar and given a vacuum expectation value, so the two moduli-space metrics arise from the same linearised equations with different boundary conditions. The domain wall equation $D_3varphi = -tfrac{e^2}{2}(sum_i q_i q_i^dagger - v^2)$, $D_3 q_i = -(varphi - m_i)q_i$ is the vortex equation with one direction traded for a mass term, the masses playing the role the scalar vev played one step up. And the vortex moduli space sits inside the resolved instanton moduli space as the fixed locus of a rotation isometry, $V_{k,N} cong I_{k,N}|_{h=0}$. Alongside the identities, the lectures carry a method: read the auxiliary quantum mechanics or quiver theory on the soliton's own worldvolume (D-branes wrapping the soliton) and the soliton's moduli space appears as a Higgs branch, giving the ADHM construction for instantons and the Nahm construction for monopoles as brane-derived quotients rather than guesswork.

Load-bearing premise

The identification of the vortex moduli space with the Higgs branch of the auxiliary two-dimensional quiver rests on matching dimensions, matching isometries, and explicit checks at one vortex; the lectures state plainly that beyond that the metrics do not agree and that a direct field-theory proof remains to be found.

Editorial extensions

If this is right

  • Soliton spectra in different co-dimensions become one calculation with different boundary data, so index theorems and zero-mode counts transfer between instantons, monopoles, vortices and walls instead of being redone each time.
  • The instanton moduli space of $U(N)$ five-dimensional Yang-Mills is conjectured to carry exactly one normalisable harmonic form for each instanton number $k$; this is the statement that $k$ instantons form a threshold bound state, which is what the five-dimensional theory must show if it is secretly a six-dimensional $(2,0)$ theory on a circle.
  • The radius of the hidden circle in that six-dimensional lift is read off from the vortex-string worldsheet by T-duality, giving $e^2/8pi^2$ — large compared with the monopole width, unlike the naive $1/v$.
  • Vortex strings in four dimensions carry a chiral worldsheet: a four-dimensional Weyl fermion descends to a purely right-moving mode on the string, which is the mechanism behind the quantitative match between two-dimensional sigma models and four-dimensional gauge theories.
  • Reconnection of cosmic strings at low energy follows from the smoothness of the two-vortex moduli space: head-on collision is right-angle scattering, not rebound, because the single-valued coordinate is the square of the separation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the dimensional-reduction web is taken literally as a dictionary, one could try to derive the vortex moduli space metric from the instanton metric by restricting to the fixed locus of the rotation isometry, rather than by direct field-theory computation; the lectures note the two metrics do not agree beyond $k=1$, which suggests the restriction is not isometric and the dictionary is topological
  • The same logic suggests a systematic search: any pair of Bogomoln'yi systems related by trading a coordinate for a mass or a scalar vev should have moduli spaces related by a fixed-locus or quotient operation, which would predict relations not listed here, for instance between domain-wall moduli spaces and vortex moduli spaces at non-zero mass.
  • The fractional-instanton speculation offered at the end of the instanton lecture — that $4kN$ instanton moduli rearrange into $kN$ objects of mass $1/e^2N$ — is testable in principle on a compactified space, where the symmetric-product topology of the moduli space of instantons on $T^4$ or $K3$ is the evidence cited.
  • The claim that the vortex moduli space is a middle-dimensional complex submanifold of the resolved instanton moduli space invites a check at $k=2$, $N=2$, where both spaces are computable in principle and the embedding could be exhibited explicitly rather than inferred from dimensions and isometries.
Original abstract

These lectures cover aspects of solitons with focus on applications to the quantum dynamics of supersymmetric gauge theories and string theory. The lectures consist of four sections, each dealing with a different soliton. We start with instantons and work down in co-dimension to monopoles, vortices and, eventually, domain walls. Emphasis is placed on the moduli space of solitons and, in particular, on the web of connections that links solitons of different types. The D-brane realization of the ADHM and Nahm construction for instantons and monopoles is reviewed, together with related constructions for vortices and domain walls. Each lecture ends with a series of vignettes detailing the roles solitons play in the quantum dynamics of supersymmetric gauge theories in various dimensions. This includes applications to the AdS/CFT correspondence, little string theory, S-duality, cosmic strings, and the quantitative correspondence between 2d sigma models and 4d gauge theories.

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The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

These lectures cover aspects of solitons with focus on applications to the quantum dynamics of supersymmetric gauge theories and string theory. The lectures consist of four sections, each dealing with a different soliton. We start with instantons and work down in co-dimension to monopoles, vortices and, eventually, domain walls. Emphasis is placed on the moduli space of solitons and, in particular, on the web of connections that links solitons of different types. The D-brane realization of the ADHM and Nahm construction for instantons and monopoles is reviewed, together with related constructions for vortices and domain walls. Each lecture ends with a series of vignettes detailing the roles solitons play in the quantum dynamics of supersymmetric gauge theories in various dimensions. This includes applications to the AdS/CFT correspondence, little string theory, S-duality, cosmic strings, and the quantitative correspondence between 2d sigma models and 4d gauge theories.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 These lectures cover aspects of solitons with focus on applications to the quantum dynamics of supersymmetric gauge theories and string theory. contribution 0.72 0.30 0.08 asserted 0.99
s1 The lectures consist of four sections, each dealing with a different soliton. contribution 0.49 0.44 0.02 asserted 1.00
s2 We start with instantons and work down in co-dimension to monopoles, vortices and, eventually, domain walls. method 0.60 0.25 0.03 asserted 0.98
s3 Emphasis is placed on the moduli space of solitons and, in particular, on the web of connections that links solitons of different types. contribution 0.78 0.26 0.09 asserted 0.95
s4 The D-brane realization of the ADHM and Nahm construction for instantons and monopoles is reviewed, together with related constructions for vortices and domain walls. contribution 0.46 0.09 0.04 asserted 0.98
s5 Each lecture ends with a series of vignettes detailing the roles solitons play in the quantum dynamics of supersymmetric gauge theories in various dimensions. method 0.52 0.46 0.04 asserted 0.99
s6 This includes applications to the AdS/CFT correspondence, little string theory, S-duality, cosmic strings, and the quantitative correspondence between 2d sigma models and 4d gauge theories. contribution 0.86 0.12 0.08 asserted 0.96

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.85.

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Figures
Figure 1. Dp-branes as instantons. where it’s useful to consider Yang-Mills instan￾tons embedded as solitons in a p + 1 dimensional theory with p ≥ 3. With this in mind, let’s consider a configuration of N Dp-branes, with k D(p − 4)-branes in type II string theory (Type IIB for p odd; type IIA for p even). A typical configuration is drawn in figure 1. We place all N Dp-branes on top of each other so that, at low-energies, their worldvolume dynamics is de￾scribed by d = p + 1 U(N) Super Yang-Mills with 16 Supercharges For example, if p = 3 we have the familiar N = 4 theory in d = 3 + 1 dimensi
Figure 2. F-strings give rise to hypermultiplets. transforms in the k of the U(k) gauge symmetry, and the N¯ of a SU(N) flavor symmetry. In contrast ψ˜ transforms in the (k¯, N) of U(k)×SU(N). (One may wonder about the difference between a gauge and fla￾vor symmetry in zero dimensions; again the reader is invited to lift the configuration to higher dimensions where such nasty questions evaporate. But the basic point will be that we treat configurations related by U(k) transformations as physically equivalent). These hypermultiplets can be thought of as the dimensional reduction of N = 2 hype
Figure 3. by studying the winding of the scalar field φ around S 2 ∞. How￾ever, since the winding is related to the magnetic charge, and the latter is a characteristic of the long range behavior of the monopole, it’s somewhat easier to neglect the non-abelian struc￾ture completely and study just the U(1) fields. The equivalence between the two methods is reflected in the equality between first and second homotopy groups in (2.6). For this purpose, it is notationally simpler to work in unitary, or singular, gauge in which the vev hφi = φ~ · H~ is fixed to be constant at infinity. This necessar
Figure 4. class of solutions since it implies that monopoles can be placed at arbitrary separation and feel no force. But this doesn’t happen for electrons! Any objects carrying the same charge, whether electric or magnetic, repel. So what’s special about monopoles? The point is that monopoles also experience a sec￾ond long range force due to the massless components of the scalar field φ. This gives rise to an attraction between the monopoles that precisely cancels the electromagnetic repulsion [82]. Such cancellation of forces only occurs when there is no potential for φ as in (2.3). The…
Figure 6. The D-brane set-up for monopoles of charge ~g = P a na~αa. We start with N = 4 U(N) super Yang-Mills, realized on the worldvolume of D3- branes. To reflect the vev hφi = diag(φ1 . . ., φN ), we separate the D3-branes in a transverse direction, say the x 6 direction. The a th D3-brane is placed at position x6 = φa. As is well known, the W-bosons correspond to fundamental strings stretched between the D3-branes. The monopoles are their magnetic duals, the D-strings. At this point our notation for the magnetic charge vector ~g = P a na~αa becomes more visual. This monopole in secto
Figure 7. The D3-branes give rise to impurities on the worldvolume of the D1-branes. The theory on the D-strings is a d = 1 + 1 U(n) gauge theory with 16 supercharges (known as N = (8, 8)). Each D3-brane impurity donates a hypermultiplet to the theory, breaking supersymmetry by half to N = (4, 4). As in lecture 1, we write the hypermultiplets as ωa =
Figure 9. Separating the Dp-branes gives rise to a mass for the hypermultiplets How can we see this behavior from the perspective of the worldvolume theory? We can return to the D-brane set-up, now with the Dp-branes separated in one direction, say x6, to mimic the vev hφi. Each Dp-D(p − 4) string is now stretched by a different amount, reflecting the fact that each hypermultiplet has a different mass. The potential on the worldvolume theory of the D-instantons is now V = 1 g 2 X 10 m,n=5 [Xm, Xn] 2 + X m,µ [Xm, Xµ] 2 + X N a=1 ψ a† (Xm − φ m a ) 2ψa + ψ˜a (Xm − φ m a )
Figure 10. have string-like objects which can terminate on D-brane ob￾jects, where their end is electrically charged. Yet all this is within the context of a gauge theory, with no reference to string theory or gravity. Let’s remind ourselves about some further properties of D-branes in string theory to see if the analogy can be pushed further. For example, there are two methods to understand the dynamics of D-branes in string theory, using either closed or open strings. The first method — the closed string description — uses the supergravity solution for D-branes to com￾pute their scattering.…
Figure 11. cial, and far from the only theory admitting vor￾tex solutions. Indeed, the vortex zoo is well pop￾ulated with different objects, many exhibiting cu￾rious properties. Particularly interesting examples include Alice strings [148, 149], and vortices in Chern￾Simons theories [150]. In this lecture we shall stick with the vortices arising from (3.1) since, as we shall see, they are closely related to the instantons and monopoles described in the previous lectures. To my knowledge, the properties of non-abelian vortices in this model were studied only quite recently in [151] (a related…
Figure 12. A sketch of the vortex profile. lecture, and embed the abelian vortex — let’s denote it q ⋆ and A ⋆ z — in the N × N matrices of the non-abelian theory. We have Az =   A⋆ z 0 . . . 0   , q =   q ⋆ v . . . v   (3.11) where the columns of the q matrix carry the color charge, while the rows carry the flavor charge. We have chosen the embedding above to lie in the upper left-hand corner but this isn’t unique. We can rotate into other embeddings by acting with the SU(N)diag symmetry preserved in the vacuum. Dividing by the stabilizer,
Figure 13. Right-angle scattering from the moduli space of two vortices. Even without the exact form of the metric, we learn something very important about vortices. Consider two vortices colliding head on. This corresponds to the trajectory in moduli space that goes up and over the tip of the cone, as shown in the figure. What does this correspond to in real space? One might think that the vortices collide and rebound. But that’s wrong: it would correspond to the trajectory going to the tip of the cone, and returning down the same side. Instead, the trajectory corresponds to vortices…
Figure 14. consisting of D-branes suspended between a pair of NS5- branes. We work in type IIA string theory, and build the d = 3+1, U(N) gauge theory6 with N = 2 supersymmetry. The D-brane set-up is shown in figure 14, and consists of N D4-branes with worldvolume 01236, stretched between two NS5-branes, each with worldvolume 012345, and separated in the x 6 direction. The gauge coupling e 2 is determined by the separation between the NS5-branes, 1 e 2 = ∆x 6 ls 2gs (3.24) where ls is the string length, and gs the string coupling. The D4-branes may slide up and down between the NS5-branes in…
Figure 15. is achieved by taking the right-hand NS5-brane and pulling it out of the page in the x 9 direction. In order to remain in the ground state, the D4-branes are not allowed to tilt into the x 9 direction: this would break supersymmetry and increase their length, reflecting a corresponding increase in the ground state energy of the theory. Instead, they must split on the D6-branes. Something known as the S-rule [91, 162] tells us that only one D4-brane can end on a given D6-brane while preserving supersymmetry, ensuring that we need at least N D6-branes to find a zero-energy ground…
Figure 16. We expect them to appear as other D-branes in the config￾uration. There is a unique BPS D-brane with the correct mass: it is a D2-brane, lying coincident with the D6-branes, with worldvolume 039, as shown in figure 16 [163]. The x 3 direction here is the direction of the vortex string. The problem is: what is the worldvolume theory on the D2-branes. It’s hard to read off the theory directly because of the boundary conditions where the D2-branes end on the D4-branes. But, already by inspection, we might expect that it’s related to the Dp-D(p − 4) system described in Lecture 1 in the
Figure 17. Moving the D6-branes
Figure 19. The D4-branes contribute hypermultiplets (ψa, ψ˜ a) with a = 1, . . ., N. These hypermultiplets get a mass only when the D2-branes and D6-branes are separated in the X4 and X5 directions. This means we have a coupling like X N a=1 ψ † a {σ † , σ} ψa + ψ˜ a {σ † , σ} ψ˜† a (3.28) But there is no such coupling between the hypermultiplets and Z. The coupling (3.28) breaks supersymmetry to N = (2, 2). So we now understand the D2-brane theory of figure 19. However, the D2-brane theory that we’re really interested in, shown in figure 18, differs from this in two ways • The
Figure 20. in three dimensions. In particular, it’s possible to tinker with the mirror theories so that they actually coincide at all length scales, rather than simply at low-energies [189]. Mirror pairs for non-abelian gauge theories are known, but are somewhat more complicated to due to presence of instan￾ton corrections (which, recall, are monopoles in three dimen￾sions) [190, 191, 192, 193, 194]. Finally, one can find mirror pairs with less supersymmetry [195, 196], including mirrors for interesting Chern-Simons theories [197, 198, 199]. Finite quantum correction to the vortex mass in N =…
Figure 21. The reconnection cosmic strings. Slicing vertically, one sees a vortex-anti-vortex pair annihilate. Slicing horizontally, one sees two vortices scattering at right angles. Hence, reconnection of cosmic strings requires both the annihilation of vortex-anti￾vortex pairs and the right-angle scattering of two vortices. The former is expected (at least for suitably slow collisions). And we saw in Section 3.3.2 that the latter occurs for abelian vortices in the moduli space approximation. We conclude that abelian cosmic strings do reconnect at low energies. Numerical simulations reveal…
Figure 22. The three layer structure of the domain wall when e 2v 2 ≪ m2 . x 3 direction. The latter is also easy to see: it arises from acting on the domain wall with the U(1)F flavor symmetry of the theory [210]: U(1)F : q1 → e iθq1 , q2 → e −iθq2 (4.13) In each vacuum, this coincides with the U(1) gauge symmetry. However, in the interior of the domain wall, it acts non-trivially, giving rise to a phase collective coordinate θ for the solution. It can be shown that X and θ remain the only two collective coordinates of the domain wall when we return to finite e 2 [213]. 4.2.2 Classifica
Figure 23. The relative moduli space of two domain walls is a cigar. separation between domain walls is denoted by R. The tip of the cigar, R = 0, corre￾sponds to the two domain walls sitting on top of each other. At this point the relative phase of the two domain walls degenerates, resulting in a smooth manifold. The metric on this space has been computed in the e 2 → ∞ limit, although it’s not particularly illuminating [212] and gives a good approximation to the metric at large finite e 2 [218]. Asymptotically, it deviates from the flat metric on the cylinder by exponentially sup￾pressed…
Figure 24. the surviving U(1)Nf −1 flavor symmetry, an Nf - vector with i th component given by Si = i(qiD0q † i − (D0qi)q † i ) (4.26) Maximizing with respect to α results in the Bo￾gomoln’yi bound for dyonic domain walls, H ≥ q v 4 (~m · ~g) 2 + (~m · S~) 2 (4.27) This square-root form is familiar from the spectrum of dyonic monopoles that we saw in Lecture 2. More on this soon. For now, some further comments, highlighting the some similarities between dyonic domain walls and monopoles. – 101 –
Figure 25. Two interacting domain walls cannot pass through each other. The ~α1 domain wall is always to the left of the ~α2 domain wall. An Example: Nc = 2, Nf = 4 Consider the domain walls in the U(2) theory with Nf = 4 flavors. We’ll start at left infinity in the vacuum Ξ− = {1, 2} and end at right infinity in the vacuum Ξ+ = {3, 4}. There are two different possibilities for the intermediate vacua. They are: Ξ− = {1, 2} −→ {1, 3} −→ {1, 4} −→ {2, 4} −→ {3, 4} = Ξ+ Ξ− = {1, 2} −→ {1, 3} −→ {2, 3} −→ {2, 4} −→ {3, 4} = Ξ+ In terms of domain walls, these two ordering become, ~α2 −→ ~α3 −→ ~α1…
Figure 26. The ordering of many domain walls. The horizontal direction is their position, while the vertical denotes the type of domain wall. Notice that the ~α1 domain wall is trapped between the two ~α2 domain walls. These in turn are trapped between the three ~α3 domain walls. However, the relative positions of the ~α1 and middle ~α3 domain walls are not fixed: these objects can pass through each other. 4.6 What Became Of...... Now let’s play our favorite game and ask what happened to the other solitons now that we’ve turned on the masses. We start with..... 4.6.1 Vortices The vortices desc
Figure 27. The kink on the vortex string. up magnetic field B = diag(0, 1) and spit out magnetic field B = diag(1, 0). In other words, it is a source for the magnetic field B = diag(1, −1). This is precisely the magnetic field sourced by an SU(2) ’t Hooft-Polyakov monopole. What’s happening here? We are dealing with a theory with a mass gap, so any magnetic monopole that lives in the bulk can’t emit a long-range radial magnetic field since the photon can’t propagate. We’re witnessing the Meissner effect in a non-abelian superconductor. The monopole is confined, its magnetic field departing in…
Figure 28. The confined magnetic monopole. In summary, Kink on the Vortex String = Confined Magnetic Monopole The BPS confined monopole was first described in [233], but the idea that kinks on string should be interpreted as confined monopoles arose previously in [234] in the context of ZN flux tubes. More recently, confined monopoles have been explored in several different theories [235, 236, 237, 238]. We’ll devote Section 4.7 to more discussion on this topic. 4.6.3 Instantons We now ask what became of instantons. At first glance, it doesn’t look promising for the instanton! In the bulk, the
Figure 29. The D-brane configuration for an elementary ~g = ~α1 domain wall when Nc = 1 and Nf = 3. dynamics from D-brane set-up [246, 247], although some qualitative features can be seen. However, we can make progress by studying this system in the limit e 2 → ∞, so that the two NS5-branes and the Nf D6-branes lie coincident in the x 6 direction [248]. The portions of the D4-branes stretched in x 6 vanish, and we’re left with D4- branes with worldvolume 01249, trapped in squares in the 49 directions where they are sandwiched between the NS5 and D6-branes. Returning to the system of domain…
Figure 30. The D-brane configuration in the e 2 → ∞ limit. We can now read off the gauge theory living on the D4-branes. One might expect that it is of the form Q i U(ni). This is essentially correct. The NS5-branes project out the A9 component of the gauge field, however the A4 component survives and each U(na) gauge theory lives in the interval mi ≤ x4 ≤ mi+1. In each segment, we have A4 – 113 –
Figure 31. Plot of a field theoretic D-brane configuration [229]. • In each vacuum there are Nc different vortex strings. Not all of them can end on the bordering domain walls. There exist selection rules describing which vortex string can end on a given wall. For the ~αi domain wall, the string associated to qi can end from the left, while the string associated to qi+1 can end from the right [214]. • For finite e 2 , there is a negative binding energy when the string attaches itself to the domain wall, arising from the monopole central charge in (4.62). Known as a boojum, it was studied in th

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  • We can study the dynamics of $K$ instantons on the D$p$ branes via the correspondence between the ADHM construction [ref] and D$(p-4)$ branes within D$p$ branes [ref] .

    Instanton counting in Class $\mathcal{S}_k$ · Introduction · 0.86

  • One simple, but heuristic, way to see that the local vortex count might be [equation] follows by considering symmetry arguments based on colour-flavour locking, as used in [ref] to understand the internal moduli of vortices.

    Vortex counting and the quantum Hall effect · A heuristic argument for the local vortex count · 0.95

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