nLab QFT with defects

Idea

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.

Definition

General

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

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

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

$$Z : Bord_n^{Def} \to \mathcal{C}$$

a TQFT with defects. A general formalization is in (Lurie 09, section 4.3), see at Cobordism theorem – For cobordisms with singuarities (boundaries/branes and defects/domain walls). See also (Davydov, Runkel & Kong 2011 and Carqueville-Runkel-Schaumann).

Such a morphism carries data as follows:

• for each label in $Def$ of codimension 0 there is an ordinary bulk field theory;

• for each label in $Def$ of codimension 1 data on how to “connect” the two TQFTs on both sides

• etc.

So one may think of the codimension $k$ 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 $n=2$ QFT such boundary conditions are also called branes, defects are also called bi-branes.

The statement of the cobordism theorem with singularities (Lurie 09, theorem 4.3.11) is essentially the following:

Given a symmetric monoidal (∞,n)-category $\mathcal{C}^\otimes$, then for every choice of pasting diagram of k-morphisms for all $k$, there is a type of manifolds with singularity $Def$, such that $Bord_n^Def$ is the free symmetric monoidal $(\infty,n)$-category on this data, hence such that TFTs with defects $(Bord_n^Def)^\otimes \to \mathcal{C}^\otimes$ are equivalently given by realizing such a pasting diagram in $\mathcal{C}$, where each of the given k-morphism appears as the value of a codimension $(n-k)$-defect. (See also Lurie 09, 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 $G$ be a Lie group, to be thought of as the (local or global) gauge group of some gauge theory. Let $H \hookrightarrow G$ a subgroup, to be thought of as the subgroup of global symmetries preserved by some vacuum configuration (which “spontaneously breaks” the symmetry from $G$ to $H$, the archetypical example is the Higgs mechanism).

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

The functions $X \to G/H$ 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 $X$, then these are called defects in the sense of broken gauge symmetry.

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

• if $X \to G/H$ is not smooth but is smooth on the pre-image of each element of $\pi_0(G/H)$ and becomes a smooth function on $X-S_1$ where $S_1 \hookrightarrow X$ is a codimension-1 submanifold, then $S_1$ is said to be a domain wall for vacuum configurations.

• if $X \to G/H$ is not smooth but becomes smooth on $X-S_2$, where $S_2$ is a codimension-2 submanifold, then $S_2$ is calld a cosmic string-defect of the vacuum configurations;

• if $X \to G/H$ is not smooth but becomes smooth on $X-S_3$, where $S_3$ is a codimension-3 submanifold, then $S_3$ is calld a monopole-defect of the vacuum configurations.

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

So

• domain walls can appear when $\pi_0(G/H)$ is non-trivial;

• cosmic strings can appear when $\pi_1(G/H)$ is non-trivial;

• monopoles can appear when $\pi_2(G/H)$ is non-trivial.

Next consider a sequence of subgroups

$$H_2 \hookrightarrow H_1 \hookrightarrow H_0 \coloneqq G$$

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

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

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

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

we have a homotopy fiber sequence

$$\Pi(H_1/H_2) \to \Pi(H_0/H_2) \to \Pi(H_0/H_1) \,.$$

This induces a long exact sequence of homotopy groups

$$\cdots \to \pi_{k+1}(H_0/H_1) \to \pi_k(H_1/H_2) \to \pi_k(H_0/H_2) \to \pi_k(H_0/H_1) \to \pi_{k-1}(H_1/H_2) \to \cdots \,.$$

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

(…)

In order to formalize this we introduce, following the cobordism theorem with singularities, cells in $Span_n(\mathbf{H})$ 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

$$\array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\rightarrow& \ast } \,.$$

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

$$\array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast }$$

shows that the former models a $k$-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

$$\array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(S^{k}), \Pi(H_0/H_1)] &\rightarrow& [\ast, \Pi(H_0/H_2)] } \,.$$

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

$$\array{ [\Pi(S^{k-1}), \Pi(H_1/H_2)] &\leftarrow& [\Pi(D^{k}), \Pi(H_1/H_2)] &\rightarrow& \ast }$$

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

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

$$\array{ && [\Pi(S^{k}), \Pi(H_0/H_1)] \\ && \downarrow \\ && [\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)] \\ & \swarrow && \searrow \\ [\Pi(S^{k-1}), \Pi(H_1/H_2)] && (pb) && [\Pi(D^k), \Pi(H_0/H_2)] & \simeq & [\ast, \Pi(H_0/H_2)] \\ & \searrow & & \swarrow \\ && [\Pi(S^{k-1}), \Pi(H_0/H_2)] }$$

This may be read as follows:

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

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

3. the bottom right map exhibt the possible “decays”: a lift through this map takes a field configuration that winds around a $k$-ball and contracts it through that $k$-ball, hence going forth and back through the bottom two maps corresponds to carrying a defect over the energy barrier from $E_2$ to $E_1$ 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 $[\Pi(S^{k-1}), \Omega\Pi(H_0/H_1)]$, as indicated. But this space is really given by field configurations at energy scale $E_1$ that wind around a $(k+1)$-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-$k$ defect of the low energy theory decays at higher energy, and the decay is witnessed by the appearance of a codimension $(k+1)$-defect of the high energy theory.

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

(…)

Related concepts

• defect brane

• anyon, braid group statistics, loop braid group

• generalized global symmetry

singularity	field theory with singularities
boundary condition/brane	boundary field theory
domain wall/bi-brane	QFT with defects

References

General

A general formulation via an (∞,n)-category of cobordisms with defects:

• Jacob Lurie, section 4.3 of: On the Classification of Topological Field Theories, Current Developments in Mathematics Volume 2008 (2009) 129-280 [arXiv:0905.0465, doi:10.4310/CDM.2008.v2008.n1.a3]

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

• Alexei Davydov, Ingo Runkel, Liang Kong, Field theories with defects and the centre functor [arXiv:1107.0495], in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory AMS (2011) [ams:pspum-83]

• Nils Carqueville, Ingo Runkel, Gregor Schaumann, Orbifolds of $n$-dimensional defect TQFTs, Geom. Topol. 23 (2019) 781-864 lbrack;arXiv:1705.06085, doi:10.2140/gt.2019.23.781]

Details in dimension 2 and 3 are discussed in

• Chris Schommer-Pries, Topological Defects and Classification of Local TQFTs in Low Dimension, Oberwolfach Workshop, June 2009 – Strings, Fields, Topology (pdf)

• Jürgen Fuchs, Christoph Schweigert, Alessandro Valentino, Bicategories for boundary conditions and for surface defects in 3-d TFT, Commun. Math. Phys. 321 (2013) 543–575 [arXiv:1203.4568, doi:10.1007/s00220-013-1723-0]

• Nils Carqueville, Catherine Meusburger, Gregor Schaumann, 3-dimensional defect TQFTs and their tricategories, Advances in Mathematics 364 (2020), [arXiv:1603.01171, doi:10.1016/j.aim.2020.107024]

Review:

• Andrew Neitzke, Some uses of defects in quantum field theory (2016?) [html]

• Nils Carqueville, Michele Del Zotto, Ingo Runkel, Topological defects, in: Encyclopedia of Mathematical Physics 2nd ed [arXiv:2311.02449]

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

• Domenico Fiorenza, Alessandro Valentino, Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, Commun. Math. Phys. 338 (2015) 1043–1074 [arXiv:1409.5723, doi:10.1007/s00220-015-2371-3]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber et. al, Higher Chern-Simons local prequantum field theory

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

• Natalie Paquette, Brian R. Williams, Koszul duality in quantum field theory (arXiv:2110.10257)

Examples

General

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

• Muneto Nitta, Defect formation from defect–anti-defect annihilations, Phys. Rev. D85:101702,2012 (arXiv:1205.2442)

In 2d field theory

Defects in 2-dimensional conformal field theory have a long history in real-world application, for instance in Kramers-Wannier duality. Formalization by higher categorical algebra via the FRS theorem on rational 2d CFT is due to:

• Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [doi:10.1103/PhysRevLett.93.070601, arXiv:cond-mat/0404051]

with a comprehensive review in

• Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354-430 [arXiv:hep-th/0607247, doi:10.1016/j.nuclphysb.2006.11.017]

Realization in lattice field theory:

• David Aasen, Roger S. K. Mong, Paul Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A: Math. Theor. 49 (2016) 354001 [arXiv:1601.07185, doi:10.1088/1751-8113/49/35/354001]

• David Aasen, Paul Fendley, Roger S. K. Mong, Topological Defects on the Lattice: Dualities and Degeneracies [arXiv:2008.08598, inspire:1812559]

• Mao Tian Tan, Yifan Wang, Aditi Mitra, Topological Defects in Floquet Circuits [arXiv:2206.06272]

and simulation on a quantum computer:

• Sutapa Samanta, Derek S. Wang, Armin Rahmani, Aditi Mitra, Isolated Majorana mode in a quantum computer from a duality twist [arXiv:2308.02387]

Defects in 2-dimensional topological field theory have been studied a lot in the context of genus-0 TFT, where they are described using the language of planar algebras. See discussion at Planar Algebras, TFTs with Defects.

Lecture notes:

• Nils Carqueville, Lecture notes on 2-dimensional defect TQFT [arXiv:1607.05747]

On the derivation of the relevant topological term in 2d theories with defects through the use of bi-branes and inter-bi-branes:

• Ingo Runkel, Rafal R. Suszek. Gerbe-holonomy for surfaces with defect networks (2008). (arXiv:0808.1419).

• Jürgen Fuchs, Thomas Nikolaus, Christoph Schweigert, Konrad Waldorf. Bundle Gerbes and Surface Holonomy (2009). (arXiv:0901.2085).

• Rafal R. Suszek. Defects, dualities and the geometry of strings via gerbes. I. Dualities and state fusion through defects (2011). (arXiv:1101.1126).

• Rafal R. Suszek. Defects, dualities and the geometry of strings via gerbes II. Generalised geometries with a twist, the gauge anomaly and the gauge-symmetry defect (2012). (arXiv:1209.2334).

Defects in Landau-Ginburg theory:

• Ilka Brunner, Daniel Roggenkamp. B-type defects in Landau-Ginzburg models (arXiv:0707.0922)

• Nils Carqueville, Daniel Murfet, Adjunctions and defects in Landau-Ginzburg models, (arXiv:1208.1481)

• Nils Carqueville, Ingo Runkel, Orbifold completion of defect bicategories, (arXiv:1210.6363)

In Chern-Simons theory

• An old example is the class of Turaev-Reshetikhin TQFT, which is a functor on 3-dimensional cobordisms with codimension 3 and 2 defects. This is supposed to be the would-be result of Chern-Simons theory, where the defect lines are the original Wilson lines in this context.

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

• Davide Gaiotto, Edward Witten, S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory (arXiv:0807.3720)

• Anton Kapustin, Mikhail Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 0911:006,2009 (arXiv:0904.0840)

• Anton Kapustin, Natalia Saulina, Surface operators in 3d TFT and 2d Rational CFT in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory AMS, 2011

• Nils Carqueville, Ingo Runkel, Gregor Schaumann, Line and surface defects in Reshetikhin-Turaev TQFT, Quantum Topology 10 (2019), 399-439 (arXiv:1710.10214)

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

• Dongmin Gang, Nakwoo Kim, Mauricio Romo, Masahito Yamazaki, Aspects of Defects in 3d-3d Correspondence, J. High Energ. Phys. (2016) (arXiv:1510.05011)

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

• Hisham Sati, Corners in M-theory, J.Phys.A44:255402,2011 (arXiv:1101.2793)

In Rozansky-Witten theory

• Anton Kapustin, Lev Rozansky, Natalia Saulina, Three-dimensional topological field theory and symplectic algebraic geometry I (arXiv:0810.5415)

• Ilka Brunner, Nils Carqueville, Daniel Roggenkamp, Truncated affine Rozansky-Witten models as extended TQFTs, Comm. Math. Phys. (2023), (arXiv:2201.03284)

• Ilka Brunner, Nils Carqueville, Pantelis Fragkos, Daniel Roggenkamp, Truncated affine Rozansky-Witten models as extended defect TQFTs, (arXiv:2307.06284)

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

• John Preskill, Alexander Vilenkin, Decay of Metastable Topological Defects, Phys. Rev. D47 : 2324-2342 (1993) (arXiv:hep-ph/9209210)

• Alexander Vilenkin, E.P.S. Shellard, Cosmic strings and other topological defects, Cambridge University Press (1994)

In solid state physics

Defects field theory motivated from solid state physics is discussed in

• Alexei Kitaev, Liang Kong, Models for gapped boundaries and domain walls Commun. Math. Phys. 313 (2012) 351-373 (arXiv:1104.5047)

Defect anyons

Often the concept of anyons is introduced as if a generalization of perturbative quanta like fundamental bosons and fermions. But many (concepts of) types of anyons are really solitonic$\;$defects such as vortices.

The general concept of braiding of defects in solid state physics:

• N. David Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591 $[$doi:10.1103/RevModPhys.51.591$]$

  (including a review of basic homotopy theory)

and more specifically for vortices:

• Hoi-Kwong Lo, John Preskill, Non-Abelian vortices and non-Abelian statistics, Phys. Rev. D 48 (1993) 4821 $[$doi:10.1103/PhysRevD.48.4821$]$

Explicit discussion in terms of anyons:

• Alexei Kitaev, Anyons in an exactly solved model and beyond, Annals of Physics 321 1 (2006) 2-111 $[$doi:10.1016/j.aop.2005.10.005$]$

  Anyonic particles are best viewed as a kind of topological defects that reveal nontrivial properties of the ground state. $[$p. 4$]$

Concrete vortex$\;$anyons in Bose-Einstein condensates:

• B. Paredes, P. Fedichev, J. Ignacio Cirac, Peter Zoller: 1/2-Anyons in small atomic Bose-Einstein condensates, Phys. Rev. Lett. 87 (2001) 010402 [doi:10.1103/PhysRevLett.87.010402, arXiv:cond-mat/0103251]

• Julien Garaud, Jin Dai, Antti J. Niemi, Vortex precession and exchange in a Bose-Einstein condensate, J. High Energ. Phys. 2021 157 (2021) $[$arXiv:2010.04549$]$

• Thomas Mawson, Timothy Petersen, Joost Slingerland, Tapio Simula, Braiding and fusion of non-Abelian vortex anyons, Phys. Rev. Lett. 123 (2019) 140404 $[$doi:10.1103/PhysRevLett.123.140404$]$

and in (other) superfluids:

• Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Half-Quantum Vortices in 3P2 Topological Superfluids $[$arXiv:2107.02448$]$

and in condensates of non-defect anyons:

• Michele Correggi, Romain Duboscq, Douglas Lundholm, Nicolas Rougerie, Vortex patterns in the almost-bosonic anyon gas, Europhys. Lett. 126 (2019), 20005 $[$arXiv:1901.10739, doi:10.1209/0295-5075/126/20005$]$

See also Ahn, Park & Yang 19 who refer to the band nodes in the Brillouin torus of a semi-metal as “vortices in momentum space”.

And see at defect brane.