QFT with defects



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Functorial quantum field theory



A (pre-)quantum field theory with defects is, roughly a field theory that assigns data not just to plain manifolds/cobordisms, but to spaces that may carry certain singularities and/or colorings. At the locus of such a singularity the bulk field theory may then undergo transitions.

Such defects are known by many names. In codimension 1 they are often called domain walls. If they are boundaries they are often called branes, the corresponding domain walls are then sometimes called bi-branes. Examples of Dimension-1 defects are Wilson lines and cosmic strings (at least in gauge theory) and dimension-0 defects are often called monopoles.



A plain nn-dimensional local FQFT (a bulk field theory) is a symmetric monoidal (∞,n)-functor from the (∞,n)-category of cobordisms.

Z:Bord n𝒞 . Z : Bord_n \to \mathcal{C}^\otimes \,.

If one replaces plain cobordisms here with cobordisms Bord n DefBord_n^{Def} “with singularities” including boundaries and corners but also partitions labeled in a certain index set DefDef, one calls a morphism

Z:Bord n Def𝒞 Z : Bord_n^{Def} \to \mathcal{C}

a TQFT with defects. A general formalization is in (Lurie, section 4.3), see at Cobordism theorem – For cobordisms with singuarities (boundaries/branes and defects/domain walls). See also (Davydov-Runkel-Kong and Carqueville-Runkel-Schaumann).

Such a morphism carries data as follows:

So one may think of the codimension kk colors as defects where the TQFT that one is looking at changes its nature.

In particular, when the QFT on one side of the defect is trivial, then the defect behaves like a boundary condition for the remaining QFT. Since at least for n=2n=2 QFT such boundary conditions are also called branes, defects are also called bi-branes.

The statement of the cobordism theorem with singularities (Lurie, theorem 4.3.11) is essentially the following:

Given a symmetric monoidal (∞,n)-category 𝒞 \mathcal{C}^\otimes, then for every choice of pasting diagram of k-morphisms for all kk, there is a type of manifolds with singularity DefDef, such that Bord n DefBord_n^Def is the free symmetric monoidal (,n)(\infty,n)-category on this data, hence such that TFTs with defects (Bord n Def) 𝒞 (Bord_n^Def)^\otimes \to \mathcal{C}^\otimes are equivalently given by realizing such a pasting diagram in 𝒞\mathcal{C}, where each of the given k-morphism appears as the value of a codimension (nk)(n-k)-defect. (See also Lurie, remark 4.3.14).

Topological defects from spontaneously broken symmetry

under construction

An old notion of defects in field theory – well preceeding the above general notion in the context of FQFT – is that of topological defects in the vacuum structure of gauge theories that exhibit spontaneous symmetry breaking (such as a Higgs mechanism).

A comprehensive review in in (Vilenkin-Shellard 94. Steps towards conceptually systematizing these broken-symmetry defects and their interaction are made in Preskill-Vilenkin 92. We now discuss this may be translated to and formalized in the general FQFT definition above along the lines of (Fiorenza-Valentino, FSS).

Let GG be a Lie group, to be thought of as the (local or global) gauge group of some gauge theory. Let HGH \hookrightarrow G a subgroup, to be thought of as the subgroup of global symmetries preserved by some vacuum configuration (which “spontaneously breaks” the symmetry from GG to HH, the archetypical example is the Higgs mechanism).

Then the space of such vacuum configurations is the coset space G/HG/H. So given a manifold XX (spacetime), vacuum configuratons are given by functions XG/HX \to G/H. Hence G/H 1G/H_1 is the “moduli space of vacua” or “vacuum space” in this context.

The functions XG/HX \to G/H are to be smooth functions in the bulk of spacetime. If they are allowed to be non-smooth or even non-continuous along given strata of XX, then these are called defects in the sense of broken gauge symmetry.

In particular (with counting adapted to dimX=4dim X = 4)

hm, need to fine-tune the technical conditions here, to make the following statement come out right…


Next consider a sequence of subgroups

H 2H 1H 0G H_2 \hookrightarrow H_1 \hookrightarrow H_0 \coloneqq G

to be thought of as coming from two consecutive steps of spontaneous symmetry breaking, the first one down to H 1H_1 at some energy-scale E 1E_1, and the second at some lower energy scale E 2<E 1E_2 \lt E_1.

Then we say that vacuum defects at energy E 2E_2 of codimension-kk which wind around an element π k(H 1/H 2)\pi_k(H_1/H_2) are metastable if they become unstable at energy E 1E_1, hence if their image in π k(H 0/H 2)\pi_k(H_0/H_2) is trivial.

So if we add to the singular cobordism category the kk-morphism which is the kk-dimenional unit cube with an open kk-ball removed, then the boundary field data for metastable codimension nkn-k-defects is

[0,1] kD k Π(H 1/H 2) [0,1] k Π(H 0/H 2) Π(H 0/H 1) \array{ [0,1]^k - D^k &\to & \Pi(H_1/H_2) \\ \downarrow &\swArrow& \downarrow \\ [0,1]^k &\to& \Pi(H_0/H_2) &\to& \Pi(H_0/H_1) }

we have a homotopy fiber sequence

Π(H 1/H 2)Π(H 0/H 2)Π(H 0/H 1). \Pi(H_1/H_2) \to \Pi(H_0/H_2) \to \Pi(H_0/H_1) \,.

This induces a long exact sequence of homotopy groups

π k+1(H 0/H 1)π k(H 1/H 2)π k(H 0/H 2)π k(H 0/H 1)π k1(H 1/H 2). \cdots \to \pi_{k+1}(H_0/H_1) \to \pi_k(H_1/H_2) \to \pi_k(H_0/H_2) \to \pi_k(H_0/H_1) \to \pi_{k-1}(H_1/H_2) \to \cdots \,.

So for every metastable defect of codimension nkn-k given by cker(π k(H 1/H 2)π k(H 0/H 2))c \in ker(\pi_k(H_1/H_2) \to \pi_k(H_0/H_2)) there is an element in π k+1(H 0/H 1)\pi_{k+1}(H_0/H_1) of one codimension higher. One says (Preskill-Vilenkin 92) that the codimenion (nk)(n-k)-defect may end on that codimension (nk1)(n-k-1)-defect.


In order to formalize this we introduce, following the cobordism theorem with singularities, cells in Span n(H)Span_n(\mathbf{H}) which label spontaneous-symmetriy-breaking defects as well as their defects-of-defects which exhibit their decay by higher codimension defects.

Consider a span of the form

[Π(S k1),Π(H 1/H 2)] [Π(S k1),Π(H 1/H 2)] *. \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\rightarrow& \ast } \,.

Comparing this to the span that comes from the “cap” S k1D kS^{k-1} \to D^{k} \leftarrow \emptyset in the theory at energy level E 2E_2, which is just

[Π(S k1),Π(H 1/H 2)] [Π(D k),Π(H 1/H 2)] * \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast }

shows that the former models a kk-disk which is not filled with spacetime, but nevertheless closes the disk. This is the defect given as a removal of a piece of spacetime.

In order to formalize how these defects may decay at higher energy, consider next a span of field configurations of the form

[Π(S k1),Π(H 1/H 2)] [Π(S k),Π(H 0/H 1)] [*,Π(H 0/H 2)]. \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k}), \Pi(H_0/H_1)] &\rightarrow& [\ast, \Pi(H_0/H_2)] } \,.

Comparing again to the span that comes from the “cap” S k1D kS^{k-1} \to D^{k} \leftarrow \emptyset in the theory at energy level E 2E_2, which is just

[Π(S k1),Π(H 1/H 2)] [Π(D k),Π(H 1/H 2)] * \array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast }

shows that the former models a kk-disk whose center point carries a singularity: the fields at the bounding S k1S^{k-1} take values in the moduli space of the ambient theory Π(H 1/H 2)\Pi(H_1/H_2), but then at the tip of the “cap” there is a “field insertion” of a field with values in Π(H 0/H 2)\Pi(H_0/H_2). Hence this labels a defect of codimension kk.

To construct such a decay-process span that captures the above story from Preskill-Vilenkin 92, consider the following diagram:

[Π(S k),Π(H 0/H 1)] [Π(S k1),ΩΠ(H 0/H 1)] [Π(S k1),Π(H 1/H 2)] (pb) [Π(D k),Π(H 0/H 2)] [*,Π(H 0/H 2)] [Π(S k1),Π(H 0/H 2)] \array{ && [\Pi(S^{k}), \Pi(H_0/H_1)] \\ && \downarrow \\ && [\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)] \\ & \swarrow && \searrow \\ [\Pi(S^{k-1}), \Pi(H_1/H_2)] && (pb) && [\Pi(D^k), \Pi(H_0/H_2)] & \simeq & [\ast, \Pi(H_0/H_2)] \\ & \searrow & & \swarrow \\ && [\Pi(S^{k-1}), \Pi(H_0/H_2)] }

This may be read as follows:

  1. on the far left [Π(S k),Π(H 1/H 2)][\Pi(S^k), \Pi(H_1/H_2)] is the space of fields of the ambient theory at energy scale E 2E_2 around the defect;

  2. the bottom left map is the “fluctuation” map that sends these fields to fields at the higher energy scale E 1E_1;

  3. the bottom right map exhibt the possible “decays”: a lift through this map takes a field configuration that winds around a kk-ball and contracts it through that kk-ball, hence going forth and back through the bottom two maps corresponds to carrying a defect over the energy barrier from E 2E_2 to E 1E_1 and there having it decay away.

  4. these decaying configurations are therefore given by the homotopy fiber product of the bottom two functions, which is [Π(S k1),ΩΠ(H 0/H 1)][\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)], as indicated. But this space is really given by field configurations at energy scale E 1E_1 that wind around a (k+1)(k+1)-ball, as shown at the very top.

Hence the top part of this diagram is a span that exhibits a defect-of-defects which tells just the story that (Preskill-Vilenkin 92) is telling: a codimension-kk defect of the low energy theory decays at higher energy, and the decay is witnessed by the appearance of a codimension (k+1)(k+1)-defect of the high energy theory.

highenergycodim(k+1)defects lowenergycodimkdefectswiththeirdecayprocesses lowenergycodimkdefects (pb) highenergydecayprocesses tunnel apply codimkdefectsraisedtohigherenergy \array{ && {high\;energy \atop codim-(k+1)\;defects} \\ && \downarrow \\ && {low\;energy\;codim-k\;defects \atop with\;their\;decay\;processes} \\ & \swarrow && \searrow \\ {low\;energy \atop codim-k\;defects} && (pb) && {high\;energy \atop decay\;processes} \\ & {}_{\mathllap{tunnel}}\searrow & & \swarrow_{\mathrlap{apply}} \\ && {codim-k\;defects \atop raised\;to\;higher\;energy} }


singularityfield theory with singularities
boundary condition/braneboundary field theory
domain wall/bi-braneQFT with defects



A general formulation via an (∞,n)-category of cobordisms with defects is in section 4.3 of

Defect TQFTs as 1-functors on stratified decorated bordisms are discussed in

Details in dimension 2 and 3 are discussed in

Discussion of defects in prequantum field theory, hence for coefficients in an (∞,n)-category of spans is in

DIscussion with emphasis on Koszul duality such as in holography as Koszul duality:



Examples in physics of interaction of defects of various dimension is discussed in

In 2d field theory

Lecture notes include

In Chern-Simons theory

Defects in Chern-Simons theory and related systems are discussed in

In the context of the 3d-3d correspondence:

Defects in higher dimensional Chern-Simons theory on manifolds with corners are discussed in

Topological defects in gauge theories with broken symmetry

The following references discuss the traditional notion of topological defects in the vacuum structure of gauge theory with spontaneous symmetry breaking such as domain walls, cosmic strings and monopoles.

Discussion of “topological defects in gauge theory” in higher codimension is in

In solid state physics

Defects field theory motivated from solid state physics is discussed in