Generalized Global 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, Generalized StonyGlobal Brook) Symmetries April 9, 2015 1 / 33

Questions I would like to have answers for Is there extraterrestrial intelligence in our Galaxy? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 2 / 33

Questions I would like to have answers for Is there extraterrestrial intelligence in our Galaxy? Why does time always flow forward? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 3 / 33

Questions I would like to have answers for Is there extraterrestrial intelligence in our Galaxy? Why does time always flow forward? Is there objective reality, or is it all just information? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 4 / 33

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). Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 5 / 33

Motivation Explore the notion of symmetry in QFT. Claim: symmetries in a d-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? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 6 / 33

Symmetries in quantum theory A d-dimensional quantum system with Hilbert space V and Hamiltonian H is said to have a global symmetry G when for every g G we have a unitary transformation U(g) such that all U(g) commute with H and form a 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. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 7 / 33

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 current taking values in g (the dual of the Lie algebra of G). This is automatic on the semi-classical level if the action is invariant under G. A conserved current j µ defines a codimension-1 operator depending on g = exp(ia) G and a (d 1)-dimensional closed submanifold: U g (M (d 1) ) = exp(i j(a)) M (d 1) It is topological: M (d 1) can be deformed freely if no other operator insertions are crossed. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 8 / 33

Discrete symmetries in QFT Similarly, a local discrete symmetry defines a codimension-1 topological operator U g (M (d 1) ), g G. These operators must obey U g (M (d 1) )U h (M (d 1) ) U gh (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. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 9 / 33

Spontaneous symmetry breaking If a symmetry is unbroken if the vacuum is invariant under all U(g). In general, only a subgroup G 0 G remains unbroken. The rest of G is said to be spontaneously broken. Mermin-Wagner-Hohenberg-Coleman theorem: a continuous symmetry cannot 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 to dim G dim G 0. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 10 / 33

Phases and continuous phase transitions Landau-Ginzburg theory: Phases are characterized by dimension, symmetries of H and pattern of symmetry breaking. Continuous phase transitions separate phases with the same symmetry but different pattern of symmetry breaking. The physics near a continuous phase transition is universal (unless one does additional fine-tuning). In high enough dimensions fluctuations of the order parameter are small, and one can neglect them (mean-field theory). Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 11 / 33

Gauging symmetries Gauging a global symmetry means modifying the theory so that it is invariant under spatially-varying symmetry transformations. Usually one needs to add a gauge field for G. A gauge field is (locally) a 1-form with values in g. Gauging a discrete symmetry means coupling the theory to a G-connection. Such connections are necessarily flat in a continuum QFT. If the space-time is discretized ( lattice QFT ), e.g. the manifold is triangulated, this means the gauge field is a 1-cocycle A with values in G. On the lattice, a gauge transformation depends on a 0-cochain λ C 0 (X, G). The gauge field transforms by a coboundary of λ. If G is abelian, we can write A A + δλ. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 12 / 33

Obstructions for gauging Gauging a global symmetry may not be possible. Obstructions to gauging are called t Hooft anomalies. Example: in even dimensions, a symmetry which acts differently on left-handed and right-handed fermions is typically ungaugeable. t Hooft anomalies are robust under RG flow. Important constraint on RG flows. For free theories or theories which are free in the UV, all t Hooft anomalies arise from chiral fermions and their general form is well-known. In general, t Hooft anomalies can be rather complicated. More on this later. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 13 / 33

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 the summation 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 topological terms analogous to Chern-Simons terms. In the 2d case, gauging a finite G is called orbifolding. Ambiguity is referred to as discrete torsion. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 14 / 33

Gauging and defects For finite G, one can reformulate gauging in terms of defects operators U g (M (d 1) ). A G gauge field can be represented by a network of defects. Gauge-invariance requires the correlators to be invariant under rearrangements of the defect network (merging and splitting of defects). Ambiguities arise from the freedom to assign weights to intersection of defects. Sometimes one cannot choose weights to satisfy the requirement of topological invariance. This means there is an t Hooft anomaly. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 15 / 33

p-form global symmetries A global p-form symmetry, p > 0, has a parameter which is a closed p-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-form symmetry G is a flat gauge field for gauge group G. A gauge p-form symmetry has a parameter which is an arbitrary p-cochain with values in G (discrete case). In general, it is a Cheeger-Simons differential character, or equivalently a Deligne-Belinson p-cocycle. For p = 1 the DB cocycle is the same as an abelian gauge field. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 16 / 33

Charged objects and generators An observable charged under p-form symmetry is supported on a p-dimensionsl sub-manifold (or more general, a p-cycle). The charge takes values in Ĝ (the Pontryagin dual of G). Generators of a p-form symmetry are supported on sub-manifolds of co-dimension p + 1. One way to think about the generator is assume that the p-form symmetry is gauged, so that the parameter λ is closed everywhere except at a sub-manifold M of co-dimension p + 1. After excising M, we get a well-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. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 17 / 33

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) gauge symmetry: B B + dλ. An operator creating a string would be charged under this symmetry. But since this is a gauge symmetry, this operator does not act in the physical Hilbert space. Similarly, a p + 1-form RR field is a gauge fields for p-form gauge symmetry. The heterotic B-field is a bit different: it transforms under the 0-form gauge symmetry H = E 8 E 8 or H = SO(32). This happens because the gauge symmetry is not merely a product of 0-form H symmetry and 1-form U(1) symmetry. More on this later. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 18 / 33

Example: Maxwell theory Consider U(1) gauge theory without charged matter and an action S = 1 2g 2 F F, F = da. X The action is invariant under A A + λ, where λ is a flat U(1) gauge field. This is a global 1-form U(1) symmetry. Charged object: Wilson loop W n (C) = exp(in Generator: codimension-2 vortex (Gukov-Witten defect operator) labeled by α U(1). C A) Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 19 / 33

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: W C exp( g 2 /L d 4 C ). The photon can be regarded as a Goldstone boson for the 1-form symmetry. For d 3 the 1-form symmetry is unbroken: { exp( g W C 2 L C log L C ), d = 3 exp( g 2 L 2 C ), d = 2 Generalized MWHC theorem: U(1) p-form symmetry cannot be broken if d p + 2. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 20 / 33

Electric and Magnetic symmetries In d = 4 electric-magnetic duality maps the Maxwell theory to itself and maps Wilson loop to t Hooft loop. Therefore d = 4 Maxwell theory must also have another U(1) 1-form symmetry. Let us call it magnetic symmetry, and the original one the electric symmetry. t Hooft loop is charged with respect to the magnetic 1-form symmetry. Generator of the magnetic 1-form symmetry: exp(iη F ), η R/Z. M (2) The magnetic 1-form symmetry is also broken in the Coulomb phase, because t Hooft loop has a perimeter law. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 21 / 33

Generalized Goldstone theorem Photon is simultaneously a Goldstone boson for two broken 1-form symmetries. Thus the generalized Goldstone theorem for p-form symmetries is a bit subtle. In general, the count of Goldstone bosons depends not only on the relative size of the UV symmetry and the unbroken symmetry, but also on the t Hooft 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 t Hooft anomaly. Gauging the electric symmetry A A + λ gives a massive vector theory which does not have any 1-form symmetries. For general d, the Maxwell theory has 1-form electric symmetry and (d 3)-form magnetic symmetry. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 22 / 33

Adding matter Adding charged matter to Maxwell theory breaks electric U(1) symmetry explicitly. 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 Z n. The generator is a Gukov-Witten surface operator with α 2π n Z. A Wilson loop is still charged under this symmetry. In the Coulomb phase the magnetic 1-form symmetry is spontaneously broken because t Hooft loops have perimeter law. In the Higgs phase it is unbroken because all t Hooft loops have area law. The electric 1-form Z n symmetry is spontaneously broken in both phases, because all Wilson loops have perimeter law. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 23 / 33

Example: SU(N) Yang-Mills theory SU(N) Yang-Mills theory (no matter fields) has electric 1-form Z N symmetry. Adding matter which transforms trivially under the center of SU(N) preserves this symmetry. Adding fundamental matter destroys electric Z N. Generators: special Gukov-Witten surface operators (holonomy in the center of SU(N)). Charged objects: Wilson loops in the fundamental representation. Confining phase: 1-form Z N unbroken. Higgs and Coulomb phases: 1-form Z N broken. If N is not prime, Z N can be broken down to a subgroup Z N/m. Then (N/m) th power of the fundamental Wilson loop has a perimeter law (nontrivial confinement index ). Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 24 / 33

Example: SU(N)/Z N Yang-Mills theory Can be obtained by gauging the 1-form Z N symmetry of the previous example. 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 field with values in Z N and one can write a nontrivial topological action for it. The 2-form gauge field B H 2 (X, Z N ) is the t Hooft flux in SU(N)/Z N theory. Different topological actions correspond to different discrete theta-angles. The theory has magnetic 1-form Z N symmetry; t Hooft loops are charged under it. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 25 / 33

Example: 6d (2, 0) theories These theories are associated with self-dual Lie groups and have 2-form global symmetries. On the Coulomb branch, where there are self-dual 2-form gauge fields B i, these symmetries shift B i by flat 2-form gauge fields. 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 the following global symmetries: a 2-form symmetry, a pair of 1-form symmetries, and a 0-form symmetry. The 2-form symmetry current is the Hodge dual of µ σ, where σ is one of the scalars. The 0-form symmetry shifts this scalar by a constant. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 26 / 33

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 flow from a theory in one dimension higher. This d + 1-dimensional theory can be chosen to be a TQFT for a 2-form gauge field B H 2 (Y, G). A 2-form gauge field is the same as a map Y K(G, 2). K(G, 2) is any space Z with π 2 (Z) = G and π i (Z) = 0 for i 2. TQFT actions are classified by elements of H d+1 (K(G, 2), U(1)). These are 2-form analogues of Dijkgraaf-Witten theories. Taking into account gravitational effects, anomalies are classified by the cobordism groups of K(G, 2) (oriented cobordism groups for bosonic theories, spin cobordism groups for theories with fermions). How does one actually compute these groups? What are examples of theories with all these t Hooft anomalies? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 27 / 33

The case d = 3 Eilenberg-MacLane: Let G be a finite group. Then H 4 (K(G, 2), U(1)) is isomorphic to the space of quadratic functions on G with values in U(1). These anomalies are realized by Chern-Simons theories in 3d. The symmetry acts by A A + λ, where λ is a flat connection with suitably quantized holonomies. What about the cobordism group and the corresponding anomalies? Not sure. Eilenberg and MacLane also computed H 5 (K(G, 2), U(1)) for an arbitrary abelian G. The result is complicated. It describes candidate t Hooft anomalies for d = 4 QFT (neglecting gravitational effects). Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 28 / 33

Phases of gauge theories in 4d Gauge theories in 4d typically have electric and magnetic 1-form symmetries. 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. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 29 / 33

Examples of distinct massive phases in 4d SU(2) Yang-Mills with fundamental matter: no symmetry (confining Higgs). SU(2) with adjoints: Z 2 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: Z 2 Z 2 1-form symmetry. SU(4)/Z 2 with adjoint matter: Z 2 Z 2 1-form symmetry, with t Hooft anomaly. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 30 / 33

Landau, Wilson, and t Hooft meet Eilenberg and MacLane More generally, one needs to keep track of ordinary symmetries (0-form symmetries) 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 π 1 (Z) = G and π i (Z) = 0 for i 1). p-form symmetry G corresponds to K(G, p + 1) (space Z with π p+1 (Z) = G and π i (Z) = 0 for i p + 1). In a d-dimensional QFT, symmetry is described by a homotopy type Z with non-vanishing homotopy groups up to dimension d. Such a homotopy type is called a d-group. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 31 / 33

2-group symmetry Suppose we have only 0-form and 1-form symmetries. They can be described by a 2-group (space Z with non-vanishing π 1 = G 0 and π 2 = G 1 ). Simplest possibility: Z = K(G 0, 1) K(G 1, 2). There are two ways to deform this. G 0 can act nontrivially on G 1. Further deformation by β H 3 (K(G 0, 1), G 1 ). Would be interesting to find examples of phases where one or both of these is nontrivial. Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 32 / 33

Phase transitions and universality Are there continuous phase transitions on the boundary between phases with broken and unbroken p-form symmetries? Does universality holds for such phase transitions? Do t Hooft anomalies affect the universality class of the phase transition (I think the answer is yes ). More generally, do different d-groups also correspond to different universality classes of phase transitions? Anton Kapustin (Simons Center for Geometry and Physics, Generalized StonyGlobal Brook) Symmetries April 9, 2015 33 / 33