Generalized Global Symmetries · Anton Kapustin (Simons Center for Geometry and Physics, Stony Brook)Generalized Global Symmetries April 9, 2015 20 / 33. Electric and Magnetic symmetries

Generalized Global Symmetries

Anton Kapustin

Simons Center for Geometry and Physics, Stony Brook

April 9, 2015

Anton Kapustin (Simons Center for Geometry and Physics, Stony Brook)Generalized Global Symmetries April 9, 2015 1 / 33

Questions I would like to have answers for

Is there extraterrestrial intelligence in our Galaxy?




Why does time always flow forward?




Why does time always flow forward?

Is there objective reality, or is it all just information?


Outline

Symmetries, phases, and anomalies

p-form symmetries

Examples

’t Hooft anomalies for 1-form symmetries

Application: phases of gauge theories

Symmetry d-groups

Some open questions

Based on arXiv:1412.5148 (with D. Gaiotto, N. Seiberg and B. Willett)and arXiv:1309.4721, 1308.2926 (with R. Thorngren).


Motivation

Explore the notion of symmetry in QFT. Claim: symmetries in ad-dimensional QFT form a d-group (a connected homotopy d-type),not a group.

A finer classification of gapped phases of matter.

Are there critical points governed by d-group symmetries?

How does one compute ’t Hooft anomalies for d-group symmetries?

What are universality classes of surface phase transitions?


Symmetries in quantum theory

A d-dimensional quantum system with Hilbert space V and HamiltonianH is said to have a global symmetry G when for every g ∈ G we have aunitary transformation U(g) such that all U(g) commute with H and forma representation of G .

More precisely, this is true if G is ”internal” (does not act on space-time).If g reverses time, then U(g) must be anti-unitary.

Sometimes one allows U(g) to form a projective representation.


Symmetries in QFT

In a local QFT, one is usually interested in symmetries which are ”local”(do not mix local and nonlocal observables).

If G is a connected Lie group, this means there is a conserved currenttaking values in g∗ (the dual of the Lie algebra of G ). This is automaticon the semi-classical level if the action is invariant under G .

A conserved current jµ defines a codimension-1 operator depending ong = exp(ia) ∈ G and a (d − 1)-dimensional closed submanifold:

Ug (M(d−1)) = exp(i

∫M(d−1)

?j(a))

It is topological: M(d−1) can be deformed freely if no other operatorinsertions are crossed.


Discrete symmetries in QFT

Similarly, a ”local” discrete symmetry defines a codimension-1 topologicaloperator Ug (M(d−1)), g ∈ G .

These operators must obey

Ug (M(d−1))Uh(M(d−1)) ' Ugh(M(d−1)).

The meaning of this equation is somewhat tricky if M(d−1) has boundaries,

because then it is not an operator but an object of a (d − 1)-category.


Spontaneous symmetry breaking

If a symmetry is unbroken if the vacuum is invariant under all U(g). Ingeneral, only a subgroup G0 ⊂ G remains unbroken. The rest of G is saidto be spontaneously broken.

Mermin-Wagner-Hohenberg-Coleman theorem: a continuous symmetrycannot be spontaneously broken if d ≤ 2.

Finite symmetry can be spontaneously broken in d = 2.

Goldstone theorem: when a continuous symmetry is spontaneously broken,there are masses scalar fields (Goldstone bosons) whose number is equal todimG − dimG0.


Phases and continuous phase transitions

Landau-Ginzburg theory:

Phases are characterized by dimension, symmetries of H and patternof symmetry breaking.

Continuous phase transitions separate phases with the samesymmetry but different pattern of symmetry breaking.

The physics near a continuous phase transition is universal (unlessone does additional fine-tuning).

In high enough dimensions fluctuations of the order parameter aresmall, and one can neglect them (mean-field theory).


Gauging symmetries

Gauging a global symmetry means modifying the theory so that it isinvariant under spatially-varying symmetry transformations. Usually oneneeds to add a gauge field for G . A gauge field is (locally) a 1-form withvalues in g.

Gauging a discrete symmetry means coupling the theory to aG -connection. Such connections are necessarily flat in a continuum QFT.

If the space-time is discretized (”lattice QFT”), e.g. the manifold istriangulated, this means the gauge field is a 1-cocycle A with values in G .

On the lattice, a gauge transformation depends on a 0-cochainλ ∈ C 0(X ,G ). The gauge field transforms by a coboundary of λ. If G isabelian, we can write

A 7→ A + δλ.


Obstructions for gauging

Gauging a global symmetry may not be possible. Obstructions to gaugingare called ’t Hooft anomalies.

Example: in even dimensions, a symmetry which acts differently onleft-handed and right-handed fermions is typically ungaugeable.

’t Hooft anomalies are robust under RG flow. Important constraint on RGflows.

For free theories or theories which are free in the UV , all ’t Hooftanomalies arise from chiral fermions and their general form is well-known.

In general, ’t Hooft anomalies can be rather complicated. More on thislater.


Ambiguities of gauging

Even if a gauging exists, it is typically not unique. For continuous groups,this is referred to as ”non-minimal coupling”.

The ambiguity arises from the freedom to choose weights in thesummation over various gauge field configurations. The weights must be”local” and gauge-invariant.

For example, one can always modify a weight by a gauge-invariant ”local”function of the gauge field. For finite G , such weights are topologicalterms analogous to Chern-Simons terms. In the 2d case, gauging a finiteG is called orbifolding. Ambiguity is referred to as discrete torsion.


Gauging and defects

For finite G , one can reformulate gauging in terms of defects operatorsUg (M(d−1)).

A G gauge field can be represented by a network of defects.

Gauge-invariance requires the correlators to be invariant underrearrangements of the defect network (merging and splitting of defects).

Ambiguities arise from the freedom to assign weights to intersection ofdefects.

Sometimes one cannot choose weights to satisfy the requirement oftopological invariance. This means there is an ’t Hooft anomaly.


p-form global symmetries

A global p-form symmetry, p > 0, has a parameter which is a closedp-form λ (continuous case).

More generally, it is a closed p-cochain with values in an abelian group G .

For example, for p = 1 (most common case) the parameter of a 1-formsymmetry G is a flat gauge field for gauge group G .

A gauge p-form symmetry has a parameter which is an arbitrary p-cochainwith values in G (discrete case).

In general, it is a Cheeger-Simons differential character, or equivalently aDeligne-Belinson p-cocycle. For p = 1 the DB cocycle is the same as anabelian gauge field.


Charged objects and generators

An observable charged under p-form symmetry is supported on ap-dimensionsl sub-manifold (or more general, a p-cycle). The charge takesvalues in G (the Pontryagin dual of G ).

“Generators” of a p-form symmetry are supported on sub-manifolds ofco-dimension p + 1.

One way to think about the generator is assume that the p-form symmetryis gauged, so that the parameter λ is closed everywhere except at asub-manifold M of co-dimension p + 1. After excising M, we get awell-defined transformation of fields outside M. This defines a defect.

For example, for p = 1 and d = 4 charged objects are loop operators,while generators are surface operators.

For continuous G , a p-form symmetry has a conserved p + 1-form current.Generators are integrals of the Hodge dual d − (p + 1)-form over M.


Gauge p-form symmetries in string theory

Continuous p-form gauge symmetries are ubiquitous in string theory.

A B-field of oriented string theory is a gauge field for a 1-form U(1) gaugesymmetry:

B 7→ B + dλ.

An operator creating a string would be charged under this symmetry. Butsince this is a gauge symmetry, this operator does not act in the physicalHilbert space.

Similarly, a p + 1-form RR field is a gauge fields for p-form gaugesymmetry.

The heterotic B-field is a bit different: it transforms under the 0-formgauge symmetry H = E8 × E8 or H = SO(32). This happens because thegauge symmetry is not merely a product of 0-form H symmetry and1-form U(1) symmetry. More on this later.


Example: Maxwell theory

Consider U(1) gauge theory without charged matter and an action

S =1

2g2

∫XF ∧ ?F , F = dA.

The action is invariant under A 7→ A + λ, where λ is a flat U(1) gaugefield. This is a global 1-form U(1) symmetry.

Charged object: Wilson loop

Wn(C ) = exp(in

∮CA)

Generator: codimension-2 vortex (Gukov-Witten defect operator) labeledby α ∈ U(1).


Generalized Mermin-Wagner-Hohenberg-Coleman theorem

For d > 3 the global 1-form U(1) symmetry is spontaneously broken,because large Wilson loops have nonzero expectation values:

〈WC 〉 ∼ exp(−g2/Ld−4C ).

The photon can be regarded as a Goldstone boson for the 1-form

symmetry.

For d ≤ 3 the 1-form symmetry is unbroken:

〈WC 〉 ∼{

exp(−g2LC log LC ), d = 3exp(−g2L2C ), d = 2

Generalized MWHC theorem: U(1) p-form symmetry cannot be broken ifd ≤ p + 2.


Electric and Magnetic symmetries

In d = 4 electric-magnetic duality maps the Maxwell theory to itself andmaps Wilson loop to ’t Hooft loop.Therefore d = 4 Maxwell theory must also have another U(1) 1-formsymmetry. Let us call it magnetic symmetry, and the original one theelectric symmetry.

’t Hooft loop is charged with respect to the magnetic 1-form symmetry.

Generator of the magnetic 1-form symmetry:

exp(iη

∫M(2)

F ), η ∈ R/Z.

The magnetic 1-form symmetry is also broken in the Coulomb phase,because ’t Hooft loop has a perimeter law.


Generalized Goldstone theorem

Photon is simultaneously a Goldstone boson for two broken 1-formsymmetries. Thus the generalized Goldstone theorem for p-formsymmetries is a bit subtle.

In general, the count of Goldstone bosons depends not only on the relativesize of the UV symmetry and the unbroken symmetry, but also on the ’tHooft anomaly.

Separately, both electric and magnetic 1-form symmetries can be gauged.But they cannot be gauged at the same time, i.e. there is a mutual ’tHooft anomaly.

Gauging the electric symmetry A 7→ A + λ gives a massive vector theorywhich does not have any 1-form symmetries.

For general d , the Maxwell theory has 1-form electric symmetry and(d − 3)-form magnetic symmetry.


Adding matter

Adding charged matter to Maxwell theory breaks electric U(1) symmetryexplicitly. Does not affect magnetic symmetry.

More precisely, if the charges of all matter fields are integer multiples of n,the electric symmetry is reduced to Zn. The generator is a Gukov-Wittensurface operator with α ∈ 2π

n Z. A Wilson loop is still charged under thissymmetry.

In the Coulomb phase the magnetic 1-form symmetry is spontaneouslybroken because ’t Hooft loops have perimeter law. In the Higgs phase it isunbroken because all ’t Hooft loops have area law.

The electric 1-form Zn symmetry is spontaneously broken in both phases,because all Wilson loops have perimeter law.


Example: SU(N) Yang-Mills theory

SU(N) Yang-Mills theory (no matter fields) has “electric” 1-form ZN

symmetry.

Adding matter which transforms trivially under the center of SU(N)preserves this symmetry. Adding fundamental matter destroys electric ZN .

Generators: special Gukov-Witten surface operators (holonomy in thecenter of SU(N)). Charged objects: Wilson loops in the fundamentalrepresentation.

Confining phase: 1-form ZN unbroken. Higgs and Coulomb phases: 1-formZN broken.

If N is not prime, ZN can be broken down to a subgroup ZN/m. Then

(N/m)th power of the fundamental Wilson loop has a perimeter law(nontrivial ”confinement index”).


Example: SU(N)/ZN Yang-Mills theory

Can be obtained by gauging the 1-form ZN symmetry of the previousexample.

There is ’t Hooft anomaly in this case, but gauging is ambiguous.

Ambiguity arises from the fact that implicitly there is a 2-form gauge fieldwith values in ZN and one can write a nontrivial topological action for it.

The 2-form gauge field B ∈ H2(X ,ZN) is the ’t Hooft flux in SU(N)/ZN

theory. Different topological actions correspond to different discretetheta-angles.

The theory has magnetic 1-form ZN symmetry; ’t Hooft loops are chargedunder it.


Example: 6d (2, 0) theories

These theories are associated with self-dual Lie groups and have 2-formglobal symmetries. On the Coulomb branch, where there are self-dual2-form gauge fields Bi , these symmetries shift Bi by flat 2-form gaugefields.

For example, U(N) is self-dual; the associated 2-form symmetry is U(1).

Upon reduction to 4d on a torus, this 2-form symmetry gives rise to thefollowing global symmetries: a 2-form symmetry, a pair of 1-formsymmetries, and a 0-form symmetry.

The 2-form symmetry current is the Hodge dual of ∂µσ, where σ is one ofthe scalars. The 0-form symmetry shifts this scalar by a constant.


’t Hooft anomalies for 1-form symmetries

How does one classify and compute ’t Hooft anomalies?

Rough idea: ’t Hooft anomalies can always be canceled by anomaly in flowfrom a theory in one dimension higher. This d + 1-dimensional theory canbe chosen to be a TQFT for a 2-form gauge field B ∈ H2(Y ,G ).

A 2-form gauge field is the same as a map Y 7→ K (G , 2). K (G , 2) is anyspace Z with π2(Z ) = G and πi (Z ) = 0 for i 6= 2.

TQFT actions are classified by elements of Hd+1(K (G , 2),U(1)). Theseare 2-form analogues of Dijkgraaf-Witten theories.

Taking into account gravitational effects, anomalies are classified by thecobordism groups of K (G , 2) (oriented cobordism groups for bosonictheories, spin cobordism groups for theories with fermions).

How does one actually compute these groups? What are examples oftheories with all these ’t Hooft anomalies?


The case d = 3

Eilenberg-MacLane: Let G be a finite group. Then H4(K (G , 2),U(1)) isisomorphic to the space of quadratic functions on G with values in U(1).

These anomalies are realized by Chern-Simons theories in 3d. Thesymmetry acts by

A 7→ A + λ,

where λ is a flat connection with suitably quantized holonomies.

What about the cobordism group and the corresponding anomalies? Notsure.

Eilenberg and MacLane also computed H5(K (G , 2),U(1)) for an arbitraryabelian G . The result is complicated. It describes candidate ’t Hooftanomalies for d = 4 QFT (neglecting gravitational effects).


Phases of gauge theories in 4d

Gauge theories in 4d typically have electric and magnetic 1-formsymmetries.

Phases can be classified by

UV symmetries and their ’t Hooft anomalies

Pattern of 1-form symmetry breaking

If the phase is gapped, then a TQFT describing the IR limit.

This refines the Wilson-’t Hooft classification of phases.


Examples of distinct massive phases in 4d

SU(2) Yang-Mills with fundamental matter: no symmetry (confining∼ Higgs).

SU(2) with adjoints: Z2 1-form symmetry (confining � Higgs).

SU(2) with an isospin j = 2 field: same as previous.

SU(2)× SU(2) with bi-fundamental matter: same as previous.

SU(2)× SU(2) with adjoint matter: Z2 × Z2 1-form symmetry.

SU(4)/Z2 with adjoint matter: Z2 × Z2 1-form symmetry, with ’tHooft anomaly.


Landau, Wilson, and ’t Hooft meet Eilenberg and MacLane

More generally, one needs to keep track of ordinary symmetries (0-formsymmetries) and 1-form symmetries.

They can ”interact” in a nontrivial way.

Key observation: in general, instead of a symmetry one has a (connected)homotopy type.

Ordinary symmetry G corresponds to K (G , 1) (space Z with π1(Z ) = Gand πi (Z ) = 0 for i 6= 1).

p-form symmetry G corresponds to K (G , p + 1) (space Z withπp+1(Z ) = G and πi (Z ) = 0 for i 6= p + 1).

In a d-dimensional QFT, symmetry is described by a homotopy type Zwith non-vanishing homotopy groups up to dimension d . Such a homotopytype is called a d-group.


2-group symmetry

Suppose we have only 0-form and 1-form symmetries. They can bedescribed by a 2-group (space Z with non-vanishing π1 = G0 andπ2 = G1).

Simplest possibility: Z = K (G0, 1)× K (G1, 2).

There are two ways to “deform” this.

G0 can act nontrivially on G1.

Further deformation by β ∈ H3(K (G0, 1),G1).

Would be interesting to find examples of phases where one or both ofthese is nontrivial.


Phase transitions and universality

Are there continuous phase transitions on the boundary betweenphases with broken and unbroken p-form symmetries?

Does universality holds for such phase transitions?

Do ’t Hooft anomalies affect the universality class of the phasetransition (I think the answer is ”yes”).

More generally, do different d-groups also correspond to differentuniversality classes of phase transitions?


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Generalized Global Symmetries · Anton Kapustin (Simons Center for Geometry and Physics, Stony Brook)Generalized Global Symmetries April 9, 2015 20 / 33. Electric and Magnetic symmetries