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Anomalies in the Space of Coupling Constants
Nathan SeibergIAS
C. Cรณrdova, D. Freed, H.T. Lam, NS, to appear
1
Introduction
โข Long history of exploring a classical or quantum theory as a function of its parameters. Two complementary applications:โ Family of theories (e.g. Berry phase)โ Make the coupling constants spacetime dependent
โข More generally, we should view every theory as a function of โ Spacetime dependent coupling constantsโ Spacetime dependent background gauge fields for global
symmetriesโ Spacetime geometry (and associated choices like spin-
structure, etc.)โ Explore defects (e.g. interfaces) as these vary in spacetime
2
Introduction
We will discuss โgeneralized anomaliesโโ in this large space of backgrounds.โข This view will streamline, unify, and strengthen many known
results, and will lead to new ones.โข We will start by demonstrating them in very simple examples
(QM of a particle on a ring, 2d ๐๐(1) gauge theory).โ Since these examples are elementary, the more powerful
formalism is not essential. But the examples provide a good pedagogical way to demonstrate the formalism.
โข Then we will briefly turn to more advanced and more recently studied examples.
โข Weโll end by revisiting the free 4d fermion.3
Introduction
The generalized anomaly has two classes of applicationsโข An anomaly in a (multi-parameter) family of theories can lead,
ร la โt Hooft, to constraints on its long-distance behavior, e.g. predict phase transitions. โ This is also useful as a test of dualities
โข The coupling constants can vary in spacetime leading to a defect. The anomaly constrains the dynamics on the defect.
Since this view unifies many different, recently-studied phenomena, some of the results may seem familiar to many of you.
4
Quantum mechanics of a particle on a ring
โ = 12๏ฟฝฬ๏ฟฝ๐2 + ๐๐
2๐๐๐๐๏ฟฝฬ๏ฟฝ๐ with ๐๐ โผ ๐๐ + 2๐๐
๐๐ โผ ๐๐ + 2๐๐
Spectrum ๐ธ๐ธ๐๐ = 12๐๐ โ ๐๐
2๐๐
2
Symmetries: โข ๐๐ 1 : ๐๐ โ ๐๐ + ๐ผ๐ผโข ๐๐๐๐: ๐๐ ๐๐ โ โ๐๐ โ๐๐โข ๐๐ = 0,๐๐ also ๐๐: ๐๐ โ โ๐๐ (and therefore also ๐๐) combining to ๐๐ 2
โข ๐๐ = ๐๐ the Hilbert space is in a projective representation. โ A mixed ๐๐-๐๐(1) anomaly [Gaiotto, Kapustin, Komargodski,
NS]. This guarantees level-crossing there.5
Quantum mechanics of a particle on a ring[Gaiotto, Kapustin, Komargodski, NS]
Couple to a background ๐๐(1) gauge field ๐ด๐ด
โ =12
(๏ฟฝฬ๏ฟฝ๐ + ๐ด๐ด)2+๐๐2๐๐
๐๐ ๏ฟฝฬ๏ฟฝ๐ + ๐ด๐ด + ๐๐ ๐๐๐ด๐ด
๐๐ is a coupling for the background field only โ a counterterm.With nonzero ๐ด๐ด the ๐๐ periodicity is modified
๐๐,๐๐ โผ (๐๐ + 2๐๐,๐๐ โ 1)
โข Related to that, the spectrum ๐ธ๐ธ๐๐ = 12๐๐ โ ๐๐
2๐๐
2is invariant
under ๐๐ โ ๐๐ + 2๐๐ , but the states are rearranged ๐๐ โ ๐๐ + 1 .
โข We can restore the 2๐๐ periodicity by adding a โbulk termโ ๐๐2๐๐๐๐๐๐๐ด๐ด.
6
โ =12๏ฟฝฬ๏ฟฝ๐2 +
๐๐2๐๐
๐๐๏ฟฝฬ๏ฟฝ๐
Break ๐๐ 1 โ โค๐๐, e.g. by adding a potential โ cos ๐๐๐๐The qualitative conclusions are unchanged.โข For ๐๐ even, we still have a similar ๐๐ โ โค๐๐ anomaly at ๐๐ = ๐๐
and hence the ground state is degenerate there.โข For ๐๐ odd, there is no anomaly โ not a projective
representation. But there is still degeneracy at ๐๐ = ๐๐.โ In terms of background fields, we can preserve ๐๐ at ๐๐ = ๐๐
by adding a counterterm ๐๐ ๐๐โ12๐ด๐ด for the โค๐๐ gauge field ๐ด๐ด.
But then there is no ๐๐ at ๐๐ = 0 (referred to as โglobal inconsistencyโ).
7
Quantum mechanics of a particle on a ring[Gaiotto, Kapustin, Komargodski, NS]
Quantum mechanics of a particle on a ringโ =
12๏ฟฝฬ๏ฟฝ๐2 โ cos(๐๐๐๐) +
๐๐2๐๐
๐๐๏ฟฝฬ๏ฟฝ๐
Can also break ๐๐ for all ๐๐, e.g. by adding another degree of freedom โ โ โ + 1
2๏ฟฝฬ๏ฟฝ๐ฅ2 + ๐๐ ๐ฅ๐ฅ + ๐๐ ๐ฅ๐ฅ ๏ฟฝฬ๏ฟฝ๐ with generic ๐๐(๐ฅ๐ฅ).
Now the symmetry is only โค๐๐ (no ๐๐(1) and no ๐๐).Add a โค๐๐ background field ๐ด๐ด and a counterterm ๐๐๐๐๐ด๐ด (with ๐๐ an integer modulo ๐๐). Again
๐๐,๐๐ โผ (๐๐ + 2๐๐,๐๐ โ 1)Continuously shifting ๐๐ by 2๐๐ the states are rearranged โ a state with โค๐๐ charge ๐๐ is mapped to a state with a โค๐๐ charge ๐๐ + 1 ๐๐๐๐๐๐ ๐๐.
The ground state must jump (level-crossing) at least once in ๐๐ โ0,2๐๐ . There is a โphase transition.โ
8
Quantum mechanics of a particle on a ring
โ =12๏ฟฝฬ๏ฟฝ๐2 โ cos ๐๐๐๐ +
๐๐2๐๐
๐๐๏ฟฝฬ๏ฟฝ๐ +12๏ฟฝฬ๏ฟฝ๐ฅ2 + ๐๐ ๐ฅ๐ฅ + ๐๐ ๐ฅ๐ฅ ๏ฟฝฬ๏ฟฝ๐
๐๐,๐๐ โผ (๐๐ + 2๐๐,๐๐ โ 1)Two views on the parameter spaceโข Either ๐๐ โผ ๐๐ + 2๐๐๐๐ and preserve the โค๐๐ symmetry โข Or ๐๐ โผ ๐๐ + 2๐๐, but violate the โค๐๐ symmetry
Generalized anomaly between the โค๐๐ symmetry and ๐๐ โผ ๐๐ +2๐๐ (related discussion in [Thorngren; NS, Tachikawa, Yonekura]).
The anomaly is characterized by the 2d bulk term ๐๐2๐๐โซ ๐๐๐๐๐ด๐ด.
9
Quantum mechanics of a particle on a ring
Anomaly between the global โค๐๐ symmetry and the ๐๐ periodicity.
It is characterized by the 2d bulk term ๐๐2๐๐โซ ๐๐๐๐๐ด๐ด
โข Viewing our system as a one-parameter family of theories labeled by ๐๐ the anomaly means that there must be level-crossing โ โa phase transition.โ (Alternatively, this follows from tracking the โค๐๐ charge of the ground state.)
โข โinterfacesโ where ๐๐ changes as a function of time carry โค๐๐charge.
โข In order to discuss the interfaces we need to define the action more carefully.
10
Quantum mechanics of a particle on a ringWhen ๐๐ โ ๐๐1 is time dependent โฎ ๐๐ ๐๐ ๏ฟฝฬ๏ฟฝ๐ ๐๐ ๐๐๐๐ should be defined more carefully.Divide the Euclidean-time circle into patches ๐ฐ๐ฐ๐ผ๐ผ = (๐๐๐ผ๐ผ , ๐๐๐ผ๐ผ+1),where ๐๐ is a continuous function in โ (no 2๐๐ jump) and it jumps by 2๐๐๐๐๐ผ๐ผ with ๐๐๐ผ๐ผ โ โค on overlaps of patches.
12๐๐
โฎ ๐๐ ๐๐ ๏ฟฝฬ๏ฟฝ๐ ๐๐ ๐๐๐๐ โก ๏ฟฝ๐ผ๐ผ
12๐๐
๏ฟฝ๐ฐ๐ฐ๐ผ๐ผ๐๐๐๐๐๐ + ๐๐๐ผ๐ผ๐๐ ๐๐๐ผ๐ผ mod 2๐๐
โข Independent of the trivializationโข Invariant under ๐๐ โ ๐๐ + 2๐๐ and under ๐๐ โ ๐๐ + 2๐๐โข If ๐๐(๐๐) has nonzero winding ๐๐ = โ๐ผ๐ผ๐๐๐ผ๐ผ, this term is not
invariant under ๐๐ 1 : ๐๐ โ ๐๐ + ๐ผ๐ผ. This is our anomaly.
11
Smooth, universal
Discontinuous, not universal (can add an operator)
Discontinuous with ๐๐๐๐๐๐๐๐.It is transparent
3 kinds of โinterfacesโ
12
๐๐ = 2๐๐๐๐
๐๐ = 0
๐๐ = 0
๐๐ = 2๐๐๐๐
๐๐ = 0
๐๐ = 2๐๐๐๐
This is a rapid change in the Hamiltonian at some Euclidean time. In the โsudden approximationโ it relabels the states
๐๐ โ ๐๐ + ๐๐ ,which can be achieved by multiplying by ๐๐๐๐๐๐๐๐. This means that the interface has ๐๐(1) charge ๐๐. When ๐๐ winds ๐๐ times around the Euclidean circle, the ๐๐(1)symmetry is violated by ๐๐ unites.
๐๐ = 0
A steep but smooth โinterfaceโ
13
๐๐ = 2๐๐๐๐
This anomaly is related to a โsymmetryโ
Normally an anomaly is associated with a global symmetry: 0-form, 1-form, etc.We can think of this anomaly as associated with a โ-1-form symmetry.โJust as -1-branes (instantons) are not branes, a -1-form global symmetry is not a symmetry.If we view it as a global symmetry,โข ๐๐ is its gauge field (its transition functions involve jumps by 2๐๐โค)
โข ๐๐๐๐ is the field strength (well defined even across overlaps)โข Then, this anomaly follows the standard anomaly picture.
14
2d ๐๐(1) gauge theory
Consider a 2d ๐๐(1) gauge theory with ๐๐ charge-one scalars ๐๐๐๐
and impose that the potential ๐๐ ๐๐ 2 is ๐๐๐๐(๐๐) invariant.
Include ๐๐2๐๐๐๐๐๐๐๐.
We are not going to assume charge conjugation symmetry.โข The system has a ๐๐๐๐๐๐ ๐๐ global symmetry and we couple it
to a background ๐๐๐๐๐๐ ๐๐ gauge field ๐ต๐ต with the counterterm
2๐๐๐๐๐๐๐๐๐ค๐ค2 ๐ต๐ต
๐ค๐ค2 ๐ต๐ต is the second Stiefel-Whitney class of the ๐๐๐๐๐๐(๐๐)bundle. ๐๐ is an integer modulo ๐๐.
โข Now ๐๐,๐๐ โผ (๐๐ + 2๐๐,๐๐ โ 1) [Gaiotto, Kapustin, Komargodski, NS].
15
2d ๐๐(1) gauge theoryโข Now ๐๐,๐๐ โผ (๐๐ + 2๐๐,๐๐ โ 1) [Gaiotto, Kapustin,
Komargodski, NS]. โข This can be viewed as a mixed anomaly between the
periodicity of ๐๐ and the global ๐๐๐๐๐๐ ๐๐ symmetry.โข It is described by the 3d anomaly term (see also [Thorngren])
๐๐๐๐โซ ๐๐๐๐๐ค๐ค2 ๐ต๐ต
16
2d ๐๐(1) gauge theory๐๐๐๐โซ ๐๐๐๐๐ค๐ค2 ๐ต๐ต
Use the anomaly as in the QM example.โข Using the anomaly in a one-parameter family of theories:
since the system is gapped, it must have a phase transition at some ๐๐.โ Note that we do not assume charge conjugation symmetry.
โข A smooth interface, where ๐๐ โ ๐๐ + 2๐๐๐๐ must have an anomalous QM system on it โ a projective ๐๐๐๐๐๐ ๐๐representation with ๐๐-ality ๐๐. โ Can interpret as ๐๐ charged particle due to Colemanโs
mechanism.
17
4d S๐๐(๐๐) gauge theoryThis system has a one-form โค๐๐ global symmetry.
Couple it to a classical two-form โค๐๐ gauge field ๐ต๐ต. It twists the ๐๐๐๐ ๐๐ bundle to a ๐๐๐๐๐๐(๐๐) bundle with [Kapustin, NS]
๐ค๐ค2 ๐๐ = ๐ต๐ต ,
where ๐ค๐ค2 ๐๐ is the second Stiefel-Whitney class of the ๐๐๐๐๐๐(๐๐) bundle โ โt Hooft twisted configurations.
Mixed anomaly between the โค๐๐ one-form symmetry and ๐๐ โผ๐๐ + 2๐๐ characterized by the 5d term
๐๐๐๐ โ 12๐๐
โซ ๐๐๐๐ ๐ซ๐ซ(๐ต๐ต)
with ๐ซ๐ซ ๐ต๐ต the Pontryagin square of ๐ต๐ต.18
4d S๐๐(๐๐) gauge theory
๐๐๐๐ โ 12๐๐
โซ ๐๐๐๐ ๐ซ๐ซ(๐ต๐ต)
This leads to earlier derived results:
โข In ๐ฉ๐ฉ = 1 SUSY, a mixed anomaly between the โค2๐๐ R-symmetry and the โค๐๐ one-form symmetry. Hence, a TQFT on domain walls [Gaiotto, Kapustin, NS, Willett] in agreement with the string construction of [Acharya, Vafa].
โข โฆ
19
4d S๐๐(๐๐) gauge theory
๐๐๐๐ โ 12๐๐
โซ ๐๐๐๐ ๐ซ๐ซ(๐ต๐ต)
โข Without SUSY, a mixed anomaly between ๐๐ (equivalently, ๐๐๐ซ๐ซ)and the โค๐๐ one-form symmetry at ๐๐ = ๐๐ (for even ๐๐) implies a transition at ๐๐ = ๐๐ (or elsewhere) [Gaiotto, Kapustin, Komargodski, NS]
โ Can use the new anomaly in similar systems without ๐๐, (e.g. add a massive scalar with coupling ๐๐๐๐ ๐๐๐๐(๐น๐น โง ๐น๐น)). The gapped system must have a transition for some ๐๐.
โข Nontrivial TQFT on interfaces where ๐๐ changes by 2๐๐๐๐[Gaiotto, Komargodski, NS; Hsin, Lam, NS]
20
Smooth, universal
Discontinuous, not universal (can add d.o.f on the interface)
Discontinuous, with special QFT(e.g. a CS term) it is transparent
3 kinds of interfaces
21
๐๐ = 2๐๐๐๐
๐๐ = 0
๐๐ = 0
๐๐ = 2๐๐๐๐
๐๐ = 0
๐๐ = 2๐๐๐๐
A free massive Weyl fermion in 4dโข The parameters: a complex mass ๐๐.โข Upon compactification of the ๐๐ plane, a nontrivial 2-cycle in
the parameter space.โข The phase of ๐๐ can be removed by a chiral rotation ๐๐ โ ๐๐๐๐๐๐๐๐. But because of a ๐๐ 1 -gravitational anomaly, the
Lagrangian is shifted by ๐๐๐๐192๐๐2
๐๐๐๐(๐ ๐ โง ๐ ๐ ).
โข As above, we should add the counterterm ๐๐ ๐๐๐บ๐บ384 ๐๐2
๐๐๐๐(๐ ๐ โง ๐ ๐ ),
and then we should identify ๐๐,๐๐๐บ๐บ โผ ๐๐๐๐๐๐๐๐,๐๐๐บ๐บ + ๐ผ๐ผ
22
A free massive Weyl fermion in 4d๐๐,๐๐๐บ๐บ โผ ๐๐๐๐๐๐๐๐,๐๐๐บ๐บ + ๐ผ๐ผ
โข For nonzero ๐๐ we can remove the anomaly, i.e. absorb the phase of ๐๐, in a redefinition: ๐๐๐บ๐บ โ ๐๐๐บ๐บ โ arg(๐๐). But we cannot do it for all complex ๐๐.
โข This can be described as an anomaly using the 5d term๐๐โซ ๐ฟ๐ฟ 2 ๐๐ ๐๐2๐๐ ๐ถ๐ถ๐๐๐๐
with ๐ถ๐ถ๐๐๐๐ is the gravitational Chern-Simons term.
Equivalently, this can be written as the 6d term๐๐
192๐๐โซ ๐ฟ๐ฟ 2 ๐๐ ๐๐2๐๐ ๐๐๐๐(๐ ๐ โง ๐ ๐ )
โข Related discussion in [NS, Tachikawa, Yonekura].23
A free massive Weyl fermion in 4d1
192๐๐๐ฟ๐ฟ 2 ๐๐ ๐๐2๐๐ ๐๐๐๐(๐ ๐ โง ๐ ๐ )
Applications:โข A 2-parameter family of theories. At some point in the
complex ๐๐ plane the theory is not trivially gapped.โ For the free fermion this is trivial. But the same conclusion
in more complicated models, with additional fields and interactions. Note, we do not need any global symmetry.
โข Defects. ๐๐ โผ ๐ฅ๐ฅ + ๐๐๐๐ is a string along the ๐ง๐ง axis. As in [Jackiw, Rossi], the anomaly means that there are Majorana-Weyl fermions (๐๐๐ฟ๐ฟ โ ๐๐๐ ๐ = 1/2) on the defect. โ The same conclusion in a more complicated theory with
more fields and interactions.24
โข To study ordinary anomalies we couple every global symmetry to a classical background field and place the theory on an arbitrary spacetime. Denote all these background fields by ๐ด๐ด. An anomaly is the statement that the partition function might not be gauge invariant or coordinate invariant.
โข This is described by a higher dimensional classical (invertible) field theory for ๐ด๐ด.
โข Similarly, we make all the coupling constants ๐๐ background fields and then the partition function might not be invariant under identifications in the space of coupling constants.
โข This is described by a generalized anomaly theory โ a higher dimensional classical (invertible) field theory for ๐ด๐ด and ๐๐.
Conclusions
25
โข This anomaly theory is invariant under renormalization group flow. Therefore, it can be used, ร la โt Hooft, to constrain the long-distance dynamics and to test dualities. For example, it can force the low-energy theory to have some phase transitions.
โข Defects are constructed by making ๐ด๐ด and ๐๐ spacetime dependent. Using these values in the anomaly theory, we find the anomaly of the theory along the defect.
Conclusions
26
โข We presented examples of these anomalies (and their consequences) in various number of dimensions. The parameter space in the examples has one-cycles or two-cycles.
โข We have studied many other cases with related phenomena.โ 4d ๐๐๐๐ ๐๐ , ๐๐๐๐๐๐๐๐ ๐๐ , ๐๐๐๐(๐๐) with quarks
โข There are many other cases, we have not yet studied.
Conclusions
27
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