Anomalies in the Space of Coupling Constants Nathan Seiberg IAS

C. Córdova, D. Freed, H.T. Lam, NS

1 Introduction

• Long history of exploring a classical or quantum theory as a function of its parameters. Two complementary applications: – Family of theories (e.g. Berry phase) – Make the coupling constants spacetime dependent • More generally, we should view every theory as a function of – Spacetime dependent coupling constants – Spacetime dependent background gauge fields for global symmetries – Spacetime geometry (and associated choices like spin- structure, etc.) – Explore defects (e.g. interfaces) as these vary in spacetime

2 Introduction

• Traditionally, we study anomalies in – global symmetries by coupling the system to spacetime dependent background gauge fields (these can be continuous or discrete and the symmetries can be ordinary symmetries or higher form symmetries) – the geometry by placing the system in arbitrary spacetime metric (and associated choices like spin-structure, etc.) • Then, the can be described by a classical field theory in one higher dimension (and sometimes it is convenient to study it in two higher dimensions). • Many applications starting with ‘t Hooft anomaly matching. • Since we will study spacetime dependent coupling constants,

the anomaly action can depend on them. 3 Introduction

This “generalized anomaly” has two classes of applications • An anomaly in a (multi-parameter) family of theories can lead, à la ‘t Hooft, to constraints on its long-distance behavior, e.g. predict phase transitions. – This is also useful as a test of dualities • 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 Introduction

• This view will streamline, unify, and strengthen many known results, and will lead to new ones. • We will start by demonstrating them in very simple examples (QM of a particle on a ring, 2d (1) ). – Since these examples are elementary, the more powerful formalism is not essential. But𝑈𝑈 the examples provide a good pedagogical way to demonstrate the formalism. • Then we will briefly turn to more advanced and more recently studied examples. • We’ll end by revisiting the free 4d .

5 of a particle on a ring

= + with + + 1 2 𝑖𝑖 ℒ 2 𝑞𝑞̇ 2𝜋𝜋 𝜃𝜃𝑞𝑞̇ 𝑞𝑞 ∼ 𝑞𝑞 2𝜋𝜋 Spectrum𝜃𝜃 ∼ 𝜃𝜃 2𝜋𝜋 = 1 𝜃𝜃 2 Symmetries:𝐸𝐸 𝑛𝑛 2 𝑛𝑛 − 2𝜋𝜋 • 1 : + • : 𝑈𝑈 𝑞𝑞 → 𝑞𝑞 𝛼𝛼 • = 0, also : (and therefore also ) combining to 𝒞𝒞𝒞𝒞2𝑞𝑞 𝜏𝜏 → −𝑞𝑞 −𝜏𝜏 𝜃𝜃 𝜋𝜋 𝒞𝒞 𝑞𝑞 → −𝑞𝑞 𝒯𝒯 • = the Hilbert space is in a projective representation. 𝑂𝑂 – A mixed - (1) anomaly [Gaiotto, Kapustin, Komargodski, 𝜃𝜃 NS𝜋𝜋]. This guarantees level-crossing there. 𝒞𝒞 𝑈𝑈 6 Quantum mechanics of a particle on a ring [Gaiotto, Kapustin, Komargodski, NS]

Couple to a background (1) gauge field 1 = ( + ) + + + 2 𝑈𝑈 𝐴𝐴 2 𝑖𝑖 is a couplingℒ for the𝑞𝑞 ̇ background𝐴𝐴 field𝜃𝜃 𝑞𝑞̇ only𝐴𝐴 – a𝑖𝑖 counterterm.𝑘𝑘𝑘𝑘 With nonzero the periodicity2𝜋𝜋 is modified 𝑘𝑘 , ( + , 1) 𝐴𝐴 𝜃𝜃 • Related to that, the𝜃𝜃 spectrum𝑘𝑘 ∼ 𝜃𝜃 2𝜋𝜋=𝑘𝑘 − is invariant 2 under + , but the states are1 rearranged𝜃𝜃 𝐸𝐸𝑛𝑛 2 𝑛𝑛 − 2𝜋𝜋 + 1 . 𝜃𝜃 → 𝜃𝜃 2𝜋𝜋 • We can restore the periodicity by adding a “bulk term” 𝑛𝑛 → 𝑛𝑛 . 𝑖𝑖 2𝜋𝜋 7 2𝜋𝜋 𝜃𝜃𝜃𝜃𝜃𝜃 Quantum mechanics of a particle on a ring [Gaiotto, Kapustin, Komargodski, NS] 1 = + 2 2 𝑖𝑖 Break 1 with ℒa potential,𝑞𝑞̇ e.g. 𝜃𝜃 𝑞𝑞̇ = cos . The qualitative conclusions are unchanged.2𝜋𝜋 𝑈𝑈 → ℤ𝑁𝑁 𝑉𝑉 𝑞𝑞 𝑁𝑁𝑁𝑁 • For even, we still have a similar anomaly at = and hence the ground state is degenerate there. 𝑁𝑁 𝒞𝒞 − ℤ𝑁𝑁 𝜃𝜃 𝜋𝜋 • For odd, there is no anomaly – not a projective representation. But there is still degeneracy at = . 𝑁𝑁 – In terms of background fields, we can preserve at = 𝜃𝜃 𝜋𝜋 by adding a counterterm for the gauge field . 𝒞𝒞 𝜃𝜃 𝜋𝜋 But then there is no at 𝑁𝑁=−10 (referred to as “global 2 𝑁𝑁 inconsistency”). 𝑖𝑖 𝐴𝐴 ℤ 𝐴𝐴 𝒞𝒞 𝜃𝜃 8 Quantum mechanics of a particle on a ring 1 = + ( ) + 2 2 𝑖𝑖 Break and for allℒ , e.g𝑞𝑞̇. by making𝑉𝑉 𝑞𝑞 𝜃𝜃𝑞𝑞generiċ (but invariant) and adding degrees of freedom.2𝜋𝜋 𝑁𝑁 Now, the𝒞𝒞 symmetry𝒯𝒯 is𝜃𝜃 only (no (1)𝑉𝑉and𝑞𝑞 no ). ℤ

Add a background field 𝑁𝑁and a counterterm (with an integer modulo ). Again ℤ 𝑈𝑈 𝒞𝒞 ℤ𝑁𝑁 , 𝐴𝐴( + , 1) 𝑖𝑖𝑖𝑖𝑖𝑖 𝑘𝑘 Continuously shifting𝑁𝑁 by , the states are rearranged – a state with charge is mapped𝜃𝜃 𝑘𝑘 ∼ to𝜃𝜃 a state2𝜋𝜋 with𝑘𝑘 − a charge + 1 mod . 𝜃𝜃 2𝜋𝜋 𝑁𝑁 𝑁𝑁 The groundℤ state𝑙𝑙 must jump (level-crossing) atℤ least once in 0𝑙𝑙 , . There𝑁𝑁 is a “phase transition.” 𝜃𝜃 ∈ 9 2𝜋𝜋 Quantum mechanics of a particle on a ring 1 = + ( ) + 2 , 2 ( + , 𝑖𝑖 1) ℒ 𝑞𝑞̇ 𝑉𝑉 𝑞𝑞 𝜃𝜃𝑞𝑞̇ Two views on the parameter space 2𝜋𝜋 𝜃𝜃 𝑘𝑘 ∼ 𝜃𝜃 2𝜋𝜋 𝑘𝑘 − • Either + and preserve the symmetry • Or + , but violate the symmetry 𝜃𝜃 ∼ 𝜃𝜃 2𝜋𝜋𝜋𝜋 ℤ𝑁𝑁 𝜃𝜃 ∼ 𝜃𝜃 2𝜋𝜋 ℤ𝑁𝑁 Generalized anomaly between the symmetry and + (related discussion in [Thorngren; NS, Tachikawa, Yonekura]). ℤ𝑁𝑁 𝜃𝜃 ∼ 𝜃𝜃 2𝜋𝜋 The anomaly is characterized by the 2d bulk term . (Below it will be defined carefully.) 𝑖𝑖 2𝜋𝜋 ∫ 𝑑𝑑𝑑𝑑𝑑𝑑 10 Quantum mechanics of a particle on a ring

Anomaly between the global symmetry and the periodicity.

It is characterized by the 2d bulkℤ𝑁𝑁 term 𝜃𝜃 𝑖𝑖 • Viewing our system as a one-parameter family of theories 2𝜋𝜋 ∫ 𝑑𝑑𝑑𝑑𝑑𝑑 labeled by the anomaly means that there must be level- crossing – “a phase transition.” (Alternatively, this follows from tracking𝜃𝜃 the charge of the ground state.)

• “Interfaces” where 𝑁𝑁 changes as a function of time carry charge. ℤ 𝑁𝑁 • In order to discuss the𝜃𝜃 interfaces we need to define the actionℤ more carefully.

11 Quantum mechanics of a particle on a ring Differential Cohomology for pedestrians

When is Euclidean time dependent should be defined more1 carefully. Divide 𝜃𝜃the∈ Euclidean𝕊𝕊 -time circle into patches∮ 𝜃𝜃 𝜏𝜏=𝑞𝑞(̇ 𝜏𝜏, 𝑑𝑑𝑑𝑑 ), where is a continuous function to (no jump) and it jumps 𝐼𝐼 𝐼𝐼 𝐼𝐼+1 by with on overlaps of patches.𝒰𝒰 𝜏𝜏 𝜏𝜏 1 𝜃𝜃 1 ℝ 2𝜋𝜋 2𝜋𝜋𝑚𝑚𝐼𝐼 𝑚𝑚𝐼𝐼 ∈ ℤ + mod

𝐼𝐼 𝐼𝐼 �𝜃𝜃 𝜏𝜏 𝑞𝑞̇ 𝜏𝜏 𝑑𝑑𝑑𝑑 ≡ � � 𝐼𝐼𝜃𝜃𝜃𝜃𝜃𝜃 𝑚𝑚 𝑞𝑞 𝜏𝜏 2𝜋𝜋 2𝜋𝜋 𝐼𝐼 2𝜋𝜋 𝒰𝒰

12 Quantum mechanics of a particle on a ring Differential Cohomology for pedestrians

1 1 + mod

𝐼𝐼 𝐼𝐼 �𝜃𝜃 𝜏𝜏 𝑞𝑞̇ 𝜏𝜏 𝑑𝑑𝑑𝑑 ≡ � � 𝐼𝐼𝜃𝜃𝜃𝜃𝜃𝜃 𝑚𝑚 𝑞𝑞 𝜏𝜏 2𝜋𝜋 • Independent2𝜋𝜋 of the trivialization𝐼𝐼 2𝜋𝜋 𝒰𝒰 • Invariant under + and under + • If ( ) has nonzero winding = , this term is not invariant under 𝑞𝑞 →1 𝑞𝑞: 2𝜋𝜋 + . 𝜃𝜃 → 𝜃𝜃 2𝜋𝜋 ∑𝐼𝐼 𝐼𝐼 This is𝜃𝜃 our𝜏𝜏 anomaly. 𝑀𝑀 𝑚𝑚 𝑈𝑈 𝑞𝑞 → 𝑞𝑞 𝛼𝛼

13 3 kinds of “interfaces”

Smooth, universal = 2

= 0 𝜃𝜃 𝜋𝜋𝜋𝜋 𝜃𝜃 Discontinuous, not universal = 2 (can add an operator) = 0 𝜃𝜃 𝜋𝜋𝜋𝜋

𝜃𝜃 Discontinuous with . It is transparent 𝑖𝑖𝑖𝑖𝑖𝑖 = 2 𝑒𝑒 = 0 𝜃𝜃 𝜋𝜋𝜋𝜋 14 𝜃𝜃 A steep but smooth “interface”

= 2 = 0 𝜃𝜃 𝜋𝜋𝜋𝜋 𝜃𝜃 This is a rapid change in the Hamiltonian at some Euclidean time. In the “sudden approximation” it relabels the states + , which can be achieved by multiplying by . This means that 𝑛𝑛 → 𝑛𝑛 𝑚𝑚 the interface has (1) charge . 𝑖𝑖𝑖𝑖𝑖𝑖 When winds times around the Euclidean𝑒𝑒 circle, the (1) symmetry is violated𝑈𝑈 by unites.𝑚𝑚 𝜃𝜃 𝑚𝑚 𝑈𝑈 𝑚𝑚 15 This anomaly is related to a “symmetry”

Normally an anomaly is associated with a global symmetry: 0-form, 1-form, etc. We can think of this anomaly as associated with a “-1-form symmetry.” Just as -1-branes (instantons) are not branes, a -1-form global symmetry is not a symmetry. If we view it as a global symmetry, • is its gauge field (its transition functions involve jumps by ) 𝜃𝜃 • is the field strength (well defined even across overlaps) 2𝜋𝜋� • Then, this anomaly follows the standard anomaly picture. 𝑑𝑑𝜃𝜃

16 2d (1) gauge theory

Consider a 2d (1) gauge theory with charge-one scalars and impose that the 𝑈𝑈potential is ( ) invariant. 𝑖𝑖 𝑈𝑈 2 𝑁𝑁 𝜙𝜙 Include . 𝑉𝑉 𝜙𝜙 𝑆𝑆𝑆𝑆 𝑁𝑁 𝑖𝑖 For spacetime-dependent this leads to a background current 2𝜋𝜋 𝜃𝜃𝜃𝜃𝜃𝜃 . 1 𝜃𝜃 Specifically, when depedends on space, there is background 2𝜋𝜋 𝑑𝑑𝑑𝑑 charge density. And when winds𝜃𝜃 around space 1 𝜃𝜃 = there is total background charge𝑚𝑚 �. 𝑑𝑑𝑑𝑑 2𝜋𝜋 17 𝑚𝑚 2d (1) gauge theory

We are not going to assume charge conjugation symmetry. • The system has a 𝑈𝑈 global symmetry and we couple it to a background gauge field with the counterterm 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁 𝐵𝐵 𝑘𝑘 is the second Stiefel2𝜋𝜋𝑖𝑖 -Whitney𝑤𝑤2 𝐵𝐵 class of the ( ) bundle. is an integer modulo𝑁𝑁 . 𝑤𝑤2 𝐵𝐵 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁 • Now , ( + , 1) [Gaiotto, Kapustin, Komargodski,𝑘𝑘 NS]. 𝑁𝑁 𝜃𝜃 𝑘𝑘 ∼ 𝜃𝜃 2𝜋𝜋 𝑘𝑘 −

18 2d (1) gauge theory • Now , ( + , 1) [Gaiotto, Kapustin, Komargodski, NS].𝑈𝑈 • This can𝜃𝜃 𝑘𝑘be viewed∼ 𝜃𝜃 as2𝜋𝜋 a 𝑘𝑘mixed− anomaly between the periodicity of and the global symmetry. • It is described by the 3d anomaly term (see also [Thorngren]) 𝜃𝜃 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁

𝑖𝑖 �𝑑𝑑𝑑𝑑𝑤𝑤2 𝐵𝐵 𝑁𝑁

19 2d (1) gauge theory

𝑈𝑈 𝑖𝑖 Use the anomaly as in the QM� example.𝑑𝑑𝑑𝑑𝑤𝑤2 𝐵𝐵 𝑁𝑁 • Using the anomaly in a one-parameter family of theories: since the system is gapped, it must have a phase transition at some . – Note that we do not assume charge conjugation symmetry. 𝜃𝜃 • A smooth interface, where + – carries background electric charge 𝜃𝜃 → 𝜃𝜃 2𝜋𝜋𝜋𝜋 – has an anomalous QM system on it – a projective representation – an representation𝑚𝑚 with -ality . 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁 – Can interpret as charged particles due to Coleman’s mechanism. 𝑆𝑆𝑆𝑆 𝑁𝑁 𝑁𝑁 𝑚𝑚

𝑚𝑚 20 2d (1) gauge theory

𝑈𝑈 𝑖𝑖 When is spacetime dependent� and𝜃𝜃𝜃𝜃𝜃𝜃 + this integral 2𝜋𝜋 needs to be defined more carefully. 𝜃𝜃 𝜃𝜃 ∼ 𝜃𝜃 2𝜋𝜋 As above, we divide spacetime to patches, integrate in each patch. But add a “correction term” of the form 𝑖𝑖 when 2𝜋𝜋 𝜃𝜃𝑑𝑑𝑑𝑑 is shifted by between two patches. ∫ This guarantees that the answer is independent of the𝑖𝑖𝑖𝑖 ∫ 𝑎𝑎 trivialization𝜃𝜃 and2𝜋𝜋 hence𝜋𝜋 makes it well defined when winds.

𝜃𝜃 (We needed such a definition for the bulk term in the particle on the ring example.) 21 4d S ( ) gauge theory This system has a one-form global symmetry. 𝑈𝑈 𝑁𝑁 Couple it to a classical two-formℤ𝑁𝑁 gauge field . It twists the bundle to a ( ) bundle with [Kapustin, NS] ℤ𝑁𝑁 𝐵𝐵 = , 𝑆𝑆𝑆𝑆 𝑁𝑁 𝑃𝑃𝑃𝑃𝑃𝑃 𝑁𝑁 where is the second𝑤𝑤 Stiefel2 𝑎𝑎 -Whitney𝐵𝐵 class of the ( ) bundle – ‘t Hooft twisted configurations. 𝑤𝑤2 𝑎𝑎 𝑃𝑃𝑆𝑆𝑈𝑈 𝑁𝑁 Mixed anomaly between the one-form symmetry and + 2 characterized by the 5d term ℤ𝑁𝑁 𝜃𝜃 ∼ 1 𝜃𝜃 𝜋𝜋 ( ) 𝑁𝑁 − with the Pontryagin𝑖𝑖 square�𝑑𝑑𝑑𝑑 of 𝒫𝒫. 𝐵𝐵 2𝑁𝑁 22 𝒫𝒫 𝐵𝐵 𝐵𝐵 4d S ( ) gauge theory 1 𝑈𝑈 𝑁𝑁 ( ) 𝑁𝑁 − This leads to earlier derived𝑖𝑖 results:�𝑑𝑑𝑑𝑑 𝒫𝒫 𝐵𝐵 2𝑁𝑁 • In = 1 SUSY, a mixed anomaly between the R- symmetry and the one-form symmetry. Hence, a TQFT on 𝒩𝒩 ℤ2𝑁𝑁 domain walls [Gaiotto, Kapustin, NS, Willett] in agreement ℤ𝑁𝑁 with the string construction of [Acharya, Vafa]. • …

23 4d S ( ) gauge theory 1 𝑈𝑈 𝑁𝑁 ( ) 𝑁𝑁 − • Without SUSY, a mixed𝑖𝑖 anomaly�𝑑𝑑𝑑𝑑 between𝒫𝒫 𝐵𝐵 (equivalently, ) 2𝑁𝑁 and the one-form symmetry at = (for even ) implies 𝒯𝒯 𝒞𝒞𝒞𝒞 a transition at = (or elsewhere) [Gaiotto, Kapustin, ℤ𝑁𝑁 𝜃𝜃 𝜋𝜋 𝑁𝑁 Komargodski, NS] 𝜃𝜃 𝜋𝜋 – Can use the new anomaly in similar systems without , (e.g. add a massive scalar with coupling ( )). 𝒯𝒯 The gapped system must have a transition for some . 𝑖𝑖𝑖𝑖 𝑇𝑇𝑇𝑇 𝐹𝐹 ∧ 𝐹𝐹 • Nontrivial TQFT on interfaces where changes by 𝜃𝜃 [Gaiotto, Komargodski, NS; Hsin, Lam, NS] 𝜃𝜃 2𝜋𝜋𝜋𝜋 24 3 kinds of interfaces

Smooth, universal = 2

= 0 𝜃𝜃 𝜋𝜋𝜋𝜋 𝜃𝜃 Discontinuous, not universal = 2 (can add d.o.f on the interface) = 0 𝜃𝜃 𝜋𝜋𝜋𝜋

𝜃𝜃 Discontinuous, with special QFT (e.g. a CS term). It is transparent = 2

= 0 𝜃𝜃 𝜋𝜋𝜋𝜋 25 𝜃𝜃 A free massive Weyl fermion in 4d • The parameters: a complex mass . • Upon compactification of the plane, a nontrivial 2-cycle in 𝑚𝑚 the parameter space. 𝑚𝑚 • The phase of can be removed by a chiral rotation . But because of a 1 -, the 𝑖𝑖𝑖𝑖 𝑚𝑚 Lagrangian𝜓𝜓 → 𝑒𝑒 𝜓𝜓 is shifted by 𝑈𝑈 ( ). 𝑖𝑖𝑖𝑖 2 • As above, we should add192 the𝜋𝜋 counterterm𝑇𝑇𝑇𝑇 𝑅𝑅 ∧ 𝑅𝑅 ( ), 𝑖𝑖 𝜃𝜃𝐺𝐺 and then we should identify , , 2 + 384 𝜋𝜋 𝑇𝑇𝑇𝑇 𝑅𝑅 ∧ 𝑅𝑅 𝑖𝑖𝑖𝑖 𝑚𝑚 𝜃𝜃𝐺𝐺 ∼ 𝑒𝑒 𝑚𝑚 𝜃𝜃𝐺𝐺 𝛼𝛼

26 A free massive Weyl fermion in 4d

, , + • For nonzero we can remove𝑖𝑖𝑖𝑖 the anomaly, i.e. absorb the 𝑚𝑚 𝜃𝜃𝐺𝐺 ∼ 𝑒𝑒 𝑚𝑚 𝜃𝜃𝐺𝐺 𝛼𝛼 phase of , in a redefinition: arg( ). But we cannot do it 𝑚𝑚for all complex . 𝑚𝑚 𝜃𝜃𝐺𝐺 → 𝜃𝜃𝐺𝐺 − 𝑚𝑚 • This can be described as an anomaly using the 5d term 𝑚𝑚

2 2 with is the gravitational𝑖𝑖 �𝛿𝛿 𝑚𝑚 Chern𝑑𝑑 𝑚𝑚-Simons𝐶𝐶𝑆𝑆𝑔𝑔 term. Equivalently, this can be written as the 6d term 𝐶𝐶𝑆𝑆𝑔𝑔 ( ) ( ) 𝑖𝑖 2 2 • Related discussion in� [NS𝛿𝛿 , Tachikawa,𝑚𝑚 𝑑𝑑 𝑚𝑚 𝑇𝑇𝑇𝑇 Yonekura𝑅𝑅 ∧ 𝑅𝑅 ]. 27 192𝜋𝜋 A free massive Weyl fermion in 4d 1 ( ) 2 2 Applications: 𝛿𝛿 𝑚𝑚 𝑑𝑑 𝑚𝑚 𝑇𝑇𝑇𝑇 𝑅𝑅 ∧ 𝑅𝑅 • A 2-parameter 192𝜋𝜋family of theories. At some point in the complex plane the theory is not trivially gapped. – For the free fermion this is trivial. But the same conclusion in more𝑚𝑚 complicated models, with additional fields and interactions. Note, we do not need any global symmetry. • Defects. + is a string along the axis. As in [Jackiw, Rossi], the anomaly means that there are Majorana- Weyl 𝑚𝑚 ∼ (𝑥𝑥 𝑖𝑖𝑖𝑖 = ) on the defect. 𝑧𝑧 1 – The same conclusion𝐿𝐿 𝑅𝑅 in a more complicated theory with 𝑐𝑐 − 𝑐𝑐 2 more fields and interactions. 28 Conclusions

• To study ordinary anomalies we couple every global symmetry to a classical background field and place the theory in an arbitrary spacetime. Denote all these background fields by . An anomaly is the statement that the partition function might not be gauge invariant or coordinate invariant. 𝐴𝐴 • This is described by a higher dimensional classical (invertible) field theory for . • Similarly, we make all the coupling constants background fields and then the𝐴𝐴 partition function might not be invariant under identifications in the space of coupling𝜆𝜆 constants. • This is described by a generalized anomaly theory – a higher dimensional classical (invertible) field theory for and .

29 𝐴𝐴 𝜆𝜆 Conclusions

• This anomaly theory is invariant under renormalization group flow. (More precisely, it can be deformed continuously). Therefore, it can be used, à la ‘t 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. • Defects are constructed by making and spacetime dependent. Using these values in the anomaly theory, we find the anomaly of the theory along𝐴𝐴 the defect.𝜆𝜆

30 Conclusions

• 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. • We have studied many other cases with related phenomena. – 4d , , ( ) with quarks • There are many other cases, we have not yet studied. 𝑆𝑆𝑆𝑆 𝑁𝑁 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑁𝑁 𝑆𝑆𝑆𝑆 𝑁𝑁

31