1. Comments On Global Anomalies In Six-Dimensional Supergravity
Gregory Moore
Rutgers University
DANIEL PARK & SAMUEL MONNIER
Work in progress with SAMUEL MONNIER
MIT, May 16, 2018

2. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

3. Motivation Relation of consistent theories of
quantum gravity to string theory.
Recall the Taylor-Venn diagram:
State of art summarized in
Brennan, Carta, and Vafa

4. Brief Summary Of Results
Focus on 6d sugra
(More) systematic study of global anomalies
Result 1: Quantization of anomaly coefficients
Result 2: 𝑎-coefficient is a characteristic vector in
lattice of string charges.
Result 3: Mathematically precise formulation of
the 6d Green-Schwarz anomaly cancellation
Result 4: Check in F-theory: Requires knowing
the global form of the (identity component of)
the gauge group.

5. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

6. (Pre-) Data For 6d Supergravity
(1,0) sugra multiplet + vector multiplets +
hypermultiplets + tensor multiplets
VM: Choose a compact (reductive) Lie group G.
HM: Choose a quaternionic representation ℛ of G
TM: Choose an integral lattice Λ of signature (1,T)
Pre-data:
𝐺,ℛ,Λ

7. 6d Sugra - 2 Can write multiplets, Lagrangian, equations
of motion.[Riccioni, 2001]
Fermions are chiral (symplectic Majorana-Weyl)
2-form fieldstrengths
are (anti-)self dual

8. The Anomaly Polynomial
Chiral fermions & (anti-)self-dual tensor fields ⇒ gauge & gravitational anomalies.
From 𝐺 and ℛ we compute,
following textbook procedures,
𝐼 8 ∼ dim ℍ ℛ − dim 𝐺 +29 𝑇 −273 𝑇𝑟 𝑅 4 +⋯
+ 9−𝑇 𝑇𝑟 𝑅 (𝐹 4 −𝑡𝑦𝑝𝑒) +⋯
6d Green-Schwarz mechanism requires (Sagnotti)
𝐼 8 = 1 2 𝑌 2
𝑌∈ 𝐻 4 𝒲;Λ⊗ℝ

9. Standard Anomaly Cancellation
Interpret 𝑌 as background magnetic
current for the tensor-multiplets ⇒
𝑑𝐻=𝑌
⇒𝐵 transforms under diff & VM gauge
transformations…
Add counterterm to sugra action
e iS → 𝑒 𝑖𝑆 𝑒 −2𝜋𝑖 1 2 ∫𝐵𝑌

10. So, What’s The Big Deal?

11. Definition Of Anomaly Coefficients
Let’s try to factorize:
𝐼 8 = 1 2 𝑌 2
𝑌∈ 𝐻 4 𝒲;Λ⊗ℝ
𝔤= 𝔤 𝑠𝑠 ⊕ 𝔤 𝐴𝑏𝑒𝑙 ≅ ⊕ 𝑖 𝔤 𝑖 ⊕ 𝐼 𝔲 1 𝐼
𝑌= 𝑎 4 𝑝 1 − 𝑖 𝑏 𝑖 𝑐 2 𝑖 𝐼𝐽 𝑏 𝐼𝐽 𝑐 1 𝐼 𝑐 1 𝐽
General form of 𝑌:
𝑝 1 ≔ 1 8 𝜋 2 𝑇 𝑟 𝑣𝑒𝑐 𝑅 2
Anomaly coefficients:
𝑎, 𝑏 𝑖 , 𝑏 𝐼𝐽 ∈Λ⊗ℝ
𝑐 2 𝑖 ≔ 𝜋 2 ℎ 𝑖 ∨ 𝑇 𝑟 𝑎𝑑𝑗 𝐹 𝑖 2

12. The Data Of 6d Sugra 𝐺,ℛ,Λ Factorization ⇒ constraints on 𝑎, 𝑏 𝑖 𝑏 𝐼𝐽
The very existence a factorization 𝐼 8 = 1 2 𝑌 2 puts strong constraints on 𝐺,ℛ,Λ . These have been well-explored. See Taylor’s TASI lectures.
Factorization ⇒ constraints on 𝑎, 𝑏 𝑖 𝑏 𝐼𝐽
Example: 𝑎 2 =9 −𝑇
There can be multiple choices of anomaly coefficients 𝑎, 𝑏 𝑖 , 𝑏 𝐼𝐽 factoring the same 𝐼 8
Full data for 6d sugra:
𝐺,ℛ,Λ
𝑎, 𝑏 𝑖 , 𝑏 𝐼𝐽 ∈Λ⊗ℝ
AND

13. Standard Anomaly Cancellation -2/2
For any ( 𝐺,ℛ,Λ,𝑎,𝑏) adding the
GS term cancels all perturbative anomalies.
All is sweetness and light…

14. There are solutions of the factorizations conditions that cannot be realized in F-theory!
Global anomalies ?
Does the GS counterterm even make mathematical sense ?

15. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

16. The New Constraints Example of new constraints:
Global anomalies have been considered before.
Λ is self-dual [Seiberg & Taylor ]
Examples:
𝑎⋅ 𝑏 𝑖 , 𝑏 𝑖 ⋅ 𝑏 𝑗 ∈ℤ [Kumar, Morrison, Taylor ]
We have just been a little more systematic.
Example of new constraints:
𝑎, 𝑏 𝑖 , 1 2 𝑏 𝐼𝐼 , 𝑏 𝐼𝐽 ∈Λ
But these are not the strongest constraints…

17. Quadratic Forms On 𝔤 And 𝐻 4 (𝐵𝐺)
To state the best result we note that 𝑏= 𝑏 𝑖 , 𝑏 𝐼𝐽
determines a Λ⊗ℝ−valued quadratic form on 𝔤:
𝔤= 𝔤 𝑠𝑠 ⊕ 𝔤 𝐴𝑏𝑒𝑙 ≅ ⊕ 𝑖 𝔤 𝑖 ⊕ 𝐼 𝔲 1 𝐼
𝑏 𝑓,𝑓 = 𝑖 𝑏 𝑖 1 2 ℎ 𝑖 ∨ 𝑇 𝑟 𝔤 𝑖 𝑓 𝑖 𝐼𝐽 𝑏 𝐼𝐽 𝑓 𝐼 𝑓 𝐽 ∈Λ⊗ℝ
1→ 𝐺 1 →𝐺→ 𝜋 0 𝐺 →1
𝐺 1 ≅ 𝐺 1 /Γ
Γ finite group
𝐺 1 ≔ 𝐺 1,𝑠𝑠 ×𝑈 1 𝑟
Vector space 𝒬 of Λ⊗ℝ -valued quadratic forms on 𝔤: 𝒬≅ 𝐻 4 𝐵 𝐺 1 ;Λ⊗ℝ ≅ 𝐻 4 𝐵 𝐺 1 ;Λ⊗ℝ

18. Strongest Constraints - 2
𝐻 4 𝐵 𝐺 1 ;Λ ⊂ 𝐻 4 𝐵 𝐺 1 ;Λ ⊂𝒬
Lattices!
𝑏 𝑖 , 1 2 𝑏 𝐼𝐼 , 𝑏 𝐼𝐽 ∈Λ
1 2 𝑏∈ 𝐻 4 𝐵 𝐺 1 ;Λ ⇔
1 2 𝑏∈ 𝐻 4 𝐵 𝐺 1 ;Λ

19. First Derivation Σ 𝑌∈Λ Assume Completeness Hypothesis:
A consistent sugra can be put on an
arbitrary spin 6-fold with arbitrary gauge bundle.
Cancellation of background string charge in
compact Euclidean spacetime ⇒ ∀ Σ∈ 𝐻 4 ( ℳ 6 ;ℤ)
Σ 𝑌∈Λ
Because the background string
charge must be cancelled by strings.
Suffices to consider gauge bundles on
ℳ 6 =ℂ ℙ 3 . Note that 𝐺 1 bundles that do not
lift to 𝐺 1 -bundles give the ``most fractional’’ 𝑐 2 𝑖 .

20. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

21. And What About F-Theory ?
F-Theory compactifications ?

22. F-Theory: Some Definitions
Smooth resolution of a
Weierstrass model over 2-fold 𝐵
𝜋: 𝑋 →𝐵
𝔤 is determined from the
discriminant locus [Morrison & Vafa 96]
Wrapped 3-branes give self-dual strings .
Λ= 𝐻 2 𝑓𝑟𝑒𝑒 𝐵;ℤ
𝑏 : 𝐻 4 𝑓𝑟𝑒𝑒 𝑋 ;ℤ → 𝐻 2 𝐵;ℤ
𝑏 𝑥 1 , 𝑥 2 =− 𝜋 ∗ ( 𝑥 1 ∩ 𝑥 2 )
Can show: 𝑏 is even.

23. F-Theory: More Definitions
Define a lattice: ℋ⊂ 𝐻 4 𝑓𝑟𝑒𝑒 𝑋 ;ℤ
ℋ ≔ { 𝑥 𝑥∩𝑓=0 & 𝑥 𝑍 =0 }
= 𝑏 ⊥ 𝑡𝑜 𝑆𝑝𝑎𝑛{ 𝑍, 𝜋 −1 Σ 𝑖 }
ℋ⊗ℝ ≅𝔱⊂𝔤
D. Park:
In order to check 1 2 𝑏∈ 𝐻 4 (𝐵 𝐺 1 ;ℤ)
we clearly need to know 𝐺 1 .
𝑏 ≔ 𝑏 ​ ℋ⊗ℝ

24. Bonus: Global Form Of F-Theory Gauge Group
All we know from the Lie algebra 𝔤 is that
𝐺 1 =( 𝐺 1,𝑠𝑠 ×𝑇)/Γ
where Γ⊂𝑍 𝐺 1,𝑠𝑠 is a finite subgroup of the center of the universal cover of the ss part.
Knowing Γ is same as knowing Λ 𝑐𝑜𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝐺 1
Claim: ℋ= Λ 𝑐𝑜𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝐺 1
Proof is omitted here, but we believe a very similar
argument also determines the (identity component of)
the gauge group in 4d F-theory compactifications.

25. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

26. 6d Green-Schwarz Mechanism Revisited
Goal: Understand Green-Schwarz anomaly cancellation in precise mathematical terms.
Benefit: We recover the constraints:
1 2 𝑏∈ 𝐻 4 𝐵 𝐺 1 ;Λ
𝑎∈Λ
and derive a new constraint:
𝑎 is a characteristic vector:
∀𝑣∈Λ 𝑣⋅𝑣=𝑣⋅𝑎 𝑚𝑜𝑑 2

27. What’s Wrong With Textbook Green-Schwarz Anomaly Cancellation?
What does 𝐵 even mean when ℳ 6 has
nontrivial topology? (𝐻 is not closed!)
How are the periods of 𝑑𝐵 quantized?
Does the GS term even make sense?
1 2 ℳ 6 𝐵 𝑌= 𝒰 7 𝑑𝐵 𝑌
must be independent of extension to 𝒰 7 !

28. But it isn’t …. 𝑑 𝐻 1 − 𝐻 2 =0 is not well-defined because
Even for the difference of two B-fields,
𝑑 𝐻 1 − 𝐻 2 =0
we can quantize 𝐻 1 − 𝐻 2 ∈ 𝐻 3 ( 𝒰 7 ;Λ)
exp⁡( 2𝜋𝑖 𝒰 𝐻 1 − 𝐻 2 𝑌 )
is not well-defined because
of the factor of

29. Geometrical Anomaly Cancellation - 1/2
If 𝐷:𝑉→𝑉 is a linear transformation
then det 𝐷 is a well-defined number.
If 𝐷:𝑉→𝑊 is a linear transformation
then det 𝐷 is not a well-defined number!
Rather, it is an element of a line
(= one-dimensional vector space):
det 𝐷 ∈𝐷𝐸𝑇 𝐷 ≔ Λ 𝑡𝑜𝑝 𝑊⊗ Λ 𝑡𝑜𝑝 𝑉 ∨
Determinants of chiral Dirac operators 𝐷:Γ 𝑆 − ⊗𝐸 𝑆 − ⊗𝐸 →Γ 𝑆 + ⊗𝐸 are sections of line bundles over spaces of metrics and connections.

30. Geometrical Anomaly Cancellation -2/2
Space of all fields in 6d sugra is fibered
over nonanomalous fields:
ℬ=𝑀𝑒𝑡 ℳ 6 ×𝐶𝑜𝑛𝑛 𝒫 ×{𝑆𝑐𝑎𝑙𝑎𝑟 𝑓𝑖𝑒𝑙𝑑𝑠}
ℬ 𝒢 𝐹𝑒𝑟𝑚𝑖+𝐵 𝑒 𝑆 0 + 𝑆 𝐹𝑒𝑟𝑚𝑖+𝐵
Partition function:
Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝐴, 𝑔 𝜇𝜈 ,𝜙 ≔ 𝐹𝑒𝑟𝑚𝑖+𝐵 𝑒 𝑆 𝐹𝑒𝑟𝑚𝑖+𝐵
is a section of a line bundle over ℬ/𝒢
You cannot integrate a section of a line bundle over ℬ/𝒢 unless it is trivialized.

31. Approach Via Invertible Field Theory
Definition: An invertible 𝑑 – dimensional
field theory 𝑍 is a ``map’’ (functor)
𝒞 ≤𝑑 ={≤𝑑 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 −𝑚𝑎𝑛𝑖𝑓𝑜𝑙𝑑𝑠 𝑤𝑖𝑡ℎ 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒 𝒮}
𝑍: 𝒞 𝑑 →𝑈 1 ⊂ ℂ ∗ ⊂ℂ
𝑍: 𝒞 𝑑−1 → 𝑑𝑖𝑚=1 𝐻𝑒𝑟𝑚𝑖𝑡𝑖𝑎𝑛 𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑝𝑎𝑐𝑒𝑠
𝑍: 𝒞 𝑑−2 → ⋯
satisfying natural gluing rules.

32. The Anomaly Field Theory
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 is a 7D invertible field theory constructed from 𝐺,ℛ,Λ + ℬ
Structure 𝒮: 𝐺-bundles 𝒫 with gauge connection, Riemannian metric, spin structure 𝔰
Varying metric and gauge connection ⇒
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 ( ℳ 6 , 𝒫, 𝔰) is a LINE BUNDLE
Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 is a SECTION of 𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 ( ℳ 6 ,𝒫, 𝔰)

33. Anomaly Cancellation In Terms of Invertible Field Theory
This field theory must be trivialized by a
``counterterm’’ 7D invertible field theory 𝑍 𝐶𝑇
𝑍 𝐶𝑇 ℳ 6 ,𝒮 ≅ 𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 ℳ 6 ,𝒮 ∗
and, using just the data of the local fields
in six dimensions we construct a section:
Ψ 𝐶𝑇 ∈ 𝑍 𝐶𝑇 ( ℳ 6 ,𝒮)
𝐹𝑒𝑟𝑚𝑖+𝐵 𝑒 𝑆 𝐹𝑒𝑟𝑚𝑖+𝐵 Ψ 𝐶𝑇
so that
is canonically a function on ℬ/𝒢

34. APS 𝜂-Invariant
Recall APS 𝜂− invariant of Dirac operator.
𝜂 𝐷 = 𝐸≠0 𝑠𝑖𝑔𝑛(𝐸)
𝜉 𝐷 ≔ 𝜂 𝐷 + dim ker⁡(𝐷) 2
𝑒 2𝜋𝑖 𝜉 𝐷 is relevant to anomalies: It is the holonomy of an anomalous path integral
around loops in ℬ/𝒢 [Witten, 1982]
We’ll say later just HOW it fits in with the
anomaly field theory.

35. Important Facts About 𝜉
If 𝐷 is a Dirac operator on a odd-diml
manifold 𝒰, with 𝜕𝒰=∅ then 𝜉(𝐷) is a number
- in general impossible to compute explicitly ...
But, if 𝒰, and D extend to 𝒲 with 𝜕𝒲=𝒰 then
𝑒 2𝜋𝑖 𝜉 𝐷 is given by APS index theorem:
𝑒 2𝜋𝑖 𝜉 𝐷 = 𝑒 2𝜋 𝑖 𝒲 𝐼𝑛𝑑𝑒𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑦

36. Dai-Freed Field Theory
𝑒 2𝜋𝑖𝜉 𝐷 defines an invertible field theory
[Dai & Freed, 1994]
If 𝜕𝒰=∅
𝑍 𝐷𝑎𝑖𝐹𝑟𝑒𝑒𝑑 𝒰 = 𝑒 2𝜋𝑖𝜉 𝐷
If 𝜕𝒰=ℳ≠∅ then 𝑒 2𝜋𝑖 𝜉 𝐷 is a section of a line bundle over the
space of boundary data.
Suitable gluing properties hold.

37. Chern-Simons On Manifold With Boundary
Simpler example: Consider gauge theory on a
two-dimensional manifold with G-bundle 𝒫→ ℳ 2
Ψ 𝐴 𝜕 ≔ exp 2𝜋𝑖 𝒰 3 𝐶𝑆( 𝐴 𝜕 )
where ( 𝒰 3 , 𝐴 𝜕 ) extends ℳ 2 , 𝐴 𝜕
Consider the set of all functions Ψ
{extensions of ℳ 2 , 𝐴 𝜕 to( 𝒰 3 , 𝐴 𝜕 ) } →ℂ,
s.t. if we glue two extensions to get 𝒰 , 𝐴 then
Ψ 𝐸𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛1 Ψ 𝐸𝑥𝑒𝑛𝑠𝑖𝑜𝑛2 = exp 2𝜋𝑖 𝒰 𝐶𝑆 𝐴

38. Chern-Simons On Manifold With Boundary - 2
This set of functions is
ONE DIMENSIONAL for each 𝐴 𝜕
Therefore we have defined a line bundle over the space of connections 𝐶𝑜𝑛𝑛(𝒫→ ℳ 2 )
A choice of extensions gives a
section of that line bundle.
Similarly, 𝑒 2𝜋𝑖 𝜉 𝐷 is a section of a line bundle on odd-dimensional 𝒰 with bdry

39. Anomaly Field Theory For 6d Sugra
On 7-manifolds 𝒰 7 with 𝜕 𝒰 7 =∅
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝒰 7 , 𝒮 7 =exp⁡[2𝜋𝑖 𝜉 𝐷 𝐹𝑒𝑟𝑚𝑖 +𝜉 𝐷 𝐵−𝑓𝑖𝑒𝑙𝑑 ]
On 7-manifolds with 𝜕𝒰 7 = ℳ 6 :
The sum of 𝜉−invariants defines a unit vector Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 in a line Z Anomaly ( ℳ 6 , 𝒮 6 )

40. Simpler Expression When ( 𝒰 7 , 𝒮 7 ) Extends To Eight Dimensions
In general it is essentially impossible to compute 𝜂-invariants in simpler terms.
But if the matter content is such that 𝐼 8 = 1 2 𝑌 2
AND if ( 𝒰 7 , 𝒮 7 ) is bordant to zero:
Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝒰,𝒫 =exp⁡( 2𝜋 𝑖 ( 𝒲 8 𝑌 2 − 𝑠𝑖𝑔𝑛 Λ 𝜎 𝒲 ))
When can you extend 𝒰 7 and its gauge bundle
to a spin 8-fold 𝒲 ??

41. Spin Bordism Theory Ω 7 𝑠𝑝𝑖𝑛 =0 :
Can always extend spin 𝒰 7 to spin 𝒲 8
Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 : Can be nonzero: There can be
obstructions to extending a 𝐺-bundle 𝒫→ 𝒰 7
to a G-bundle 𝒫 → 𝒲 8
But Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 =0 for many groups, e.g.
products of U(n), SU(n), Sp(n). Also E8
We will return to this key point later.

42. When 7D data extends to 𝒲 8 the formula
Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝒰,𝒫 =exp⁡( 2𝜋 𝑖 ( 𝒲 8 𝑌 2 − 𝑠𝑖𝑔𝑛 Λ 𝜎 𝒲 ))
gives a clue to construct the counterterm
invertible field theory, 𝑍 𝐶𝑇 . We can write it as:
exp ( 2𝜋𝑖 𝒲 𝑋 𝑋+𝜆′ )
𝑋= 𝑌− 1 2 𝜆′
𝜆 ′ =𝑎⊗𝜆
𝑎 2 =𝑠𝑖𝑔𝑛 Λ 𝑚𝑜𝑑 8
𝜆= 1 2 𝑝 1
𝜆 2 =𝜎 𝒲 𝑚𝑜𝑑 8
𝑋∈ Ω 4 𝒲;Λ has quantized coho class in 𝐻 4 𝒲;Λ

43. is independent of extension ONLY if 𝑎∈Λ is a characteristic vector:
exp ( 2𝜋𝑖 𝒲 𝑋 𝑋+𝜆′ )
is independent of extension ONLY if
𝑎∈Λ is a characteristic vector:
∀ 𝑣∈Λ 𝑣 2 =𝑣⋅𝑎 𝑚𝑜𝑑 2
𝜆 is a characteristic vector of 𝐻 4 ( 𝒲 8 ;ℤ)
Happily, a characteristic vector always satisfies 𝑎 2 =𝑠𝑖𝑔𝑛 Λ 𝑚𝑜𝑑 8
This action is the partition function of a 7-dimensional topological field theory known
as ``Wu-Chern-Simons theory.’’ WCS generalizes
spin Chern-Simons theory to higher form fields.

44. Spin Chern-Simons Theory 1/2
If 𝐴 is a connection on a 𝑈(1) bundle
𝒫→ 𝒰 3 then we can define a (level one)
Chern-Simons invariant:
𝑋= 𝐹 2𝜋
𝐶𝑆 𝐴 = 𝒲 4 𝑋 2 mod ℤ ∈ℝ/ℤ
We want to divide by 2 but this is not well-defined.
If we give 𝒰 3 a spin structure we can make sense of
level ½ Chern-Simons: Choose integral lift of 𝑤 2 ( 𝒲 4 )
𝐶 𝑆 𝑠𝑝𝑖𝑛 𝐴 = 𝒲 𝑋 𝑋+ 𝑤 2 𝑚𝑜𝑑 ℤ ∈ℝ/ℤ

45. Spin Chern-Simons Theory – 2/2
Makes sense because 𝑤 2 is a characteristic
vector on 𝐻 4 𝒲 4 ;ℤ
A choice of spin structure, 𝜔 on 𝒰 3 can be viewed as a trivialization 𝑤 2 =𝑑 𝜂 at the boundary of 𝒲 4
Change of spin structure: 𝜔→𝜔+ 𝜖 , 𝜖 ∈ 𝐻 1 𝒰 3 ; 1 2 ℤ ℤ
𝜖∈ 𝑍 1 𝒰 3 ; 1 2 ℤ
𝜂→𝜂+𝜖
The value of the spin Chern-Simons action
changes:
𝐶 𝑆 𝜔+ 𝜖 𝐴 =𝐶 𝑆 𝜔 𝐴 + 𝒰 3 𝑋 𝜖

46. Wu-Chern-Simons Theory
Generalizes spin-Chern-Simons to p-form gauge fields.
Developed in detail in great generality by
Samuel Monnier arXiv:
Our case: 7D TFT 𝑍 𝑊𝐶𝑆 of a (locally defined) 3-form gauge potential C with fieldstrength 𝑋 = 𝑑𝐶
𝑋 ∈ 𝐻 4 ⋯;Λ
Instead of spin structure we need a
``Wu-structure’’: A trivialization of:
𝜈 4 = 𝑤 4 + 𝑤 3 𝑤 1 + 𝑤 𝑤 1 4

47. Wu-Chern-Simons e iS WCS = exp ( − 2𝜋𝑖 𝒲 1 2 𝑋 𝑋+𝜆′ ) 𝜆 ′ =𝑎⊗𝜆
On a spin manifold 𝑤 1 =0 and 𝑤 2 =0 and 𝑝 1 𝑇𝑀 has a canonical quotient by 2 : 𝜆= 1 2 𝑝 1 .
Moreover, 𝜆 is an integral lift of 𝑤 4 .
e iS WCS = exp ( − 2𝜋𝑖 𝒲 𝑋 𝑋+𝜆′ )
is the action when 𝒰 7 is spin-bordant to zero.
𝜆 ′ =𝑎⊗𝜆
𝑎 must be a characteristic vector of Λ

48. Dependence On Wu Structure
Action and partition function depend on choice of Wu structure.
Set of Wu-structures are a torsor for 𝐻 3 𝒰 7 ; 1 2 ℤ ℤ
𝑆 𝑊𝐶𝑆 𝜔+𝜖 𝑋 = 𝑆 𝑊𝐶𝑆 𝜔 𝑋 + 𝒰 7 𝜖𝑋
𝑍 𝑊𝐶𝑆 is an invertible 7D field theory: Evaluation on a six-manifolds ℳ 6 gives a line bundle over the space of boundary data for X

49. Defining 𝑍 𝐶𝑇 From 𝑍 𝑊𝐶𝑆
To define the counterterm line bundle 𝑍 𝐶𝑇 we want to evaluate 𝑍 𝑊𝐶𝑆 on ( ℳ 6 , 𝑌).
𝑌 = 1 2 𝑎⊗𝜆+[𝑋]
Problem 1:
𝑌 is shifted:
𝑋 =∑ 𝑏 𝑖 𝑐 2 𝑖 ∑ 𝑏 𝐼𝐽 𝑐 1 𝐼 𝑐 1 𝐽 ∈ 𝐻 4 ⋯;Λ
Problem 2:
𝑍 𝑊𝐶𝑆 𝜔 needs a choice of Wu-structure 𝜔.
!! We do not want to add a choice of Wu structure to the defining set of sugra data
𝐺,ℛ,Λ,𝑎,𝑏

50. Defining 𝑍 𝐶𝑇 From 𝑍 𝑊𝐶𝑆
Solution: Given a Wu-structure we can shift 𝑌 to 𝑋=𝑌 − 1 2 𝑣 𝜔 , an unshifted field and then we show that 𝑍 𝑊𝐶𝑆 𝜔 (𝑌− 1 2 𝑣 𝜔 ) is independent of 𝜔
𝑍 𝐶𝑇 ⋯, 𝑌 ≔ 𝑍 𝑊𝐶𝑆 𝜔 ⋯, 𝑌− 1 2 𝑣 𝜔
𝑍 𝐶𝑇 is actually independent of Wu structure: So no need to add this extra data to the definition of 6d sugra.
𝑍 𝐶𝑇 transforms properly under B-field, diff, and VM gauge transformations: 𝑍 𝐶𝑇 ℳ 6 , 𝒮 6 ≅ 𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 ℳ 6 , 𝒮 6 ∗

51. Anomaly Cancellation If this homomorphism is trivial then
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 × 𝑍 𝐶𝑇 is a 7D topological field theory that is defined on spin bordism classes of
𝐺-bundles. It’s 7D partition function is a homomorphism:
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 × 𝑍 𝐶𝑇 : Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 →𝑈 1
If this homomorphism is trivial then
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 × 𝑍 𝐶𝑇 ℳ 6 ≅1
is canonically trivial.

52. Anomaly Cancellation Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝐴, 𝑔 𝜇𝜈 Ψ 𝐶𝑇 𝐴, 𝑔 𝜇𝜈 ,𝐵
Suppose the 7D TFT is indeed trivializable
Now need a section, Ψ 𝐶𝑇 ℳ 6 , 𝒮 6 which is local in the six-dimensional fields. This will be our Green-Schwarz counterterm:
Ψ 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝐴, 𝑔 𝜇𝜈 Ψ 𝐶𝑇 𝐴, 𝑔 𝜇𝜈 ,𝐵
The product will be a function on ℬ/𝒢

53. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

54. Important Technicalities 1: Differential Cohomology
Precise formalism for working with p-form fields in
general spacetimes.
Three independent pieces of gauge invariant information:
Wilson lines
Fieldstrength
Topological class
Differential cohomology is an infinite-dimensional
Abelian group that precisely accounts for these data and nicely summarizes how they fit together.
Noncanonically: ℋ×𝑈 1 𝑠 × 𝐻 𝑝 ℳ;ℤ
ℋ is an infinite-dimensional Hilbert space
of gauge-inequivalent local modes.

55. Differential Cochains For Torus-Valued p-Form Gauge Fields
Follow M. Hopkins & I. Singer:
Torus = V/Λ where 𝑉=Λ⊗ℝ
𝐶 𝑝 𝑀,𝑉 ≔ 𝐶 𝑝 𝑀;𝑉 × 𝐶 𝑝−1 𝑀;𝑉 × Ω 𝑝 (𝑀;𝑉)
𝑐 ≔ 𝑐 𝑐ℎ , 𝑐 ℎ𝑜𝑙 , 𝑐 𝑓𝑖𝑒𝑙𝑑𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ
Differential: 𝑑 𝑎,ℎ,𝐹 ≔(𝑑𝑎, 𝐹−𝑎−𝑑ℎ, 𝑑𝐹 )
Shifted cocycles: Let 𝜈∈ 𝐻 𝑝 𝑀;Λ
𝑍 𝑝 𝜈 𝑀;Λ ⊂ 𝑍 𝑝 𝑀 where 𝑎 =𝜈 𝑚𝑜𝑑 Λ

56. The background shifted differential cocycle
Given a Riemannian metric, gauge connection, and anomaly coefficients (a,b) we can
canonically construct a differential cocycle 𝑌
𝑌 𝑓𝑖𝑒𝑙𝑑𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ =𝑌
= 𝑎 4 𝑝 1 − 𝑖 𝑏 𝑖 𝑐 2 𝑖 𝐼𝐽 𝑏 𝐼𝐽 𝑐 1 𝐼 𝑐 1 𝐽
It is shifted by 𝜈= 1 2 𝑎⊗𝜆
𝑌 is NOT gauge invariant under diffeomorphisms
and gauge transformations:
𝑌 → 𝑌 +𝑑 𝑉

57. Model For The B-field 𝑌 =𝑑 𝐻 𝐻 ∈ 𝐶 3 𝑀;Λ 𝐻 → 𝐻 +𝑑 𝑅 𝑌 → 𝑌 +𝑑 𝑉
𝑌 =𝑑 𝐻
𝐻 ∈ 𝐶 3 𝑀;Λ
𝐻 → 𝐻 +𝑑 𝑅
B-field gauge transformations:
Diffeomorphisms + VM Gauge transformations:
𝑌 → 𝑌 +𝑑 𝑉
𝐻 → 𝐻 + 𝑉

58. Important Technicalities 2: E-Theory
A second subtlety is that to define the WCS action precisely in ≤7 dimensions we actually have to work with a generalization of cohomology known as E-theory.
Example: WZW term in 4d pion Lagrangian for
effective action of 4d 𝑆𝑈( 𝑁 𝑐 ) YM + 𝑁 𝑓 flavors.
𝑁 𝑓 odd: The action should
depend on spin structure!!
Freed (2006): Actually, the well-defined
integral is an integral in E-theory.

59. Construction Of The Green-Schwarz Counterterm:
Ψ 𝐶𝑇 = exp 2𝜋𝑖 ℳ 6 𝐸,𝜔 𝑔𝑠𝑡
𝑔𝑠𝑡= 𝐻 − 1 2 𝜂 ∪ 𝑌 𝜈 ℎ𝑜𝑙 , ℎ 2 − 1 2 𝜂 )
Section of the right line bundle & independent
of Wu structure 𝜔.
Locally constructed in six dimensions, but
makes sense in topologically nontrivial cases.
Locally reduces to the expected answer

60. 1
Introduction & Summary Of Results 2
Six-dimensional Sugra & Green-Schwarz Mech.
Quantization Of Anomaly Coefficients. 3 4
F-Theory Check 5
Geometrical Anomaly Cancellation, 𝜂-Invariants
& Wu-Chern-Simons 6
Technical Tools
Conclusion & Discussion 7

61. Conclusion: All Anomalies Cancel:
for 𝐺,ℛ,Λ,𝑎,𝑏 such that:
𝐼 8 = 𝑌 2
𝑎∈Λ is a characteristic vector & 𝑎 2 =9−𝑇
1 2 𝑏∈ 𝐻 4 𝐵 𝐺 1 ;Λ
Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 =0

62. Except,…

63. What If The Bordism Group Is Nonzero?
We would like to relax the last condition,
but it could happen that
𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 × 𝑍 𝐶𝑇 : Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 →𝑈 1
defines a nontrivial bordism invariant.
For example, if 𝐺=𝑂(𝑁) then we could have exp 2𝜋𝑖 𝒰 7 𝑤 In that case the theory is anomalous.
It is (just barely) conceivable that some more clever
anomaly cancellation mechanism could be invented,
but this seems extremely unlikely.

64. Future Directions Global form of VM gauge group of
6- and 4- dimensional F-theory compactifications.
Understand the spin bordism theories we can
get from 𝑍 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 × 𝑍 𝐶𝑇 for arbitrary 6d sugra
data: (𝐺,ℛ,Λ, 𝑎,𝑏)
We have only shown that our quantization conditions on (𝑎,𝑏) are complete for 𝐺 such
that Ω 7 𝑠𝑝𝑖𝑛 𝐵𝐺 =0. When it is nonvanishing
there will probably be new conditions.
Finding them looks like a very challenging problem…