Global Anomalies In
Six-Dimensional Supergravity
Gregory MooreRutgers University
Work with DANIEL PARK & SAMUEL MONNIER
Work in progress with SAMUEL MONNIERStrings 2018, Okinawa
June 29, 2018

1Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3

Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6

F-Theory Check

7

Concluding Remarks

2Motivation
Relation of apparently consistent theories
of quantum gravity to string theory.
From W. Taylor’s TASI lectures:

State of art summarized in
Brennan, Carta, and Vafa 1711.00864

Brief Summary Of Results
Focus on 6d sugra(More) systematic study of global anomalies
Result 1: NECESSARY CONDITION:unifies & extends all previous conditions
Result 2: NECESSARY & SUFFICIENT:
A certain 7D TQFT 𝑍#$% must be trivial.
But effective computation of 𝑍#$% inthe general case remains open.
Result 3: Check in F-theory:(Requires knowing the global form of theidentity component of the gauge group.)1

Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6F-Theory Check

7

Concluding Remarks

5

(Pre-) Data For 6d Supergravity(1,0) sugra multiplet + vector multiplets +hypermultiplets + tensor multiplets
VM: Choose a (possibly disconnected) compact 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6d Sugra - 2Can write multiplets, Lagrangian,equations of motion. [Riccioni, 2001]
Fermions are chiral(symplecticMajorana-Weyl)
2-form fieldstrengths
are (anti-)self dual

The Anomaly Polynomial
Chiral fermions & (anti-)self-dual tensor
fields ⇒ gauge & gravitational anomalies.
From (𝐺, ℛ, Λ) we compute,following textbook procedures,𝐼/ ∼dimℍ ℛ − dim 𝐺 + 29 𝑇 − 273 𝑇𝑟 𝑅 > + ⋯+ 9 − 𝑇 𝑇𝑟 𝑅 @ @ + (𝐹 > −𝑡𝑦𝑝𝑒) + ⋯6d Green-Schwarz mechanism requires
1 @𝐼/ = 𝑌𝑌 ∈ Ω> 𝒲; Λ ⊗ ℝ2Standard Anomaly Cancellation
Interpret 𝑌 as background magnetic
current for the tensor-multiplets ⇒𝑑𝐻 = 𝑌⇒ 𝐵 transforms underdiff & VM gauge transformations…
Add counterterm to sugra action
XQR
TU V@WT@ ∫ Z[e →𝑒 𝑒So, What’s The Big Deal?

Definition Of Anomaly Coefficients
Let’s try to factorize:
1 @𝐼/ = 𝑌2>
𝑌 ∈ Ω 𝒲; Λ ⊗ ℝ𝔤 = 𝔤ff ⊕ 𝔤hijk ≅⊕T 𝔤T ⊕c 𝔲 1 cGeneral form of 𝑌:

𝑎1dTc
𝑌 = 𝑝X − ` 𝑏T 𝑐@ + ` 𝑏cd 𝑐X 𝑐X
421@𝑝X ≔𝑇𝑟𝑅
8𝜋 @ wjx𝑐@T1@≔𝑇𝑟𝐹
16𝜋 @ ℎT∨ stu TT

cdAnomaly coefficients:𝑎, 𝑏T , 𝑏cd ∈ Λ ⊗ ℝ

The Data Of 6d Sugra
XThe very existence of a factorization 𝐼/ = @ 𝑌 @puts constraints on 𝐺, ℛ, Λ . These have beenwell-explored. For example….
dimℍ ℛ − dim 𝐺 + 29 𝑇 − 273 = 0

𝑎@ = 9 − 𝑇, …
Also: There are multiple choices of anomaly
coefficients 𝑎, 𝑏T , 𝑏cd factoring the same 𝐼/
Full data for 6d sugra:
𝑎, 𝑏T , 𝑏cd ∈ Λ ⊗ ℝAND𝐺, ℛ, Λ

Standard Anomaly Cancellation -2/2
For any ( 𝐺, ℛ, Λ, 𝑎, 𝑏) adding theGS term cancels all perturbativeanomalies.

All is sweetness and light…

There are solutions of thefactorizations conditions thatcannot be realized in F-theory!

Global anomalies ?

Does the GS counterterm evenmake mathematical sense ?

1Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3

Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6F-Theory Check

7

Concluding Remarks

15The New Constraints
Global anomalies have been considered before.
We have just been a little more systematic.
To state the best result we note that 𝑏 = 𝑏T , 𝑏cddetermines a Λ ⊗ ℝ −valued quadratic form on 𝔤:
Vector space 𝒬 of such quadratic forms
arises in topology: 𝒬 ≅ 𝐻> 𝐵𝐺X ; Λ ⊗ ℝ>

𝐻 𝐵𝐺X ; Λ ⊂ 𝒬
1>𝑏 ∈ 𝐻 𝐵𝐺X ; Λ2

A Derivation
A consistent sugra can be put on anarbitrary spin 6-fold with arbitrary gauge bundle.
Cancellation of background string charge incompact Euclidean spacetime ⇒ ∀ Σ ∈ 𝐻> (ℳ• ; ℤ)
Because the background stringƒ𝑌 ∈ Λchargemustbecancelledbystrings.
„This is a NECESSARY but not (in general)
SUFFICIENT condition for cancellation of all
global anomalies…

6d Green-Schwarz Mechanism
Revisited
Goal: Understand Green-Schwarz anomaly
cancellation in precise mathematical terms.
Benefit: We recover the constraints:

1𝑎∈Λ𝑏 ∈ 𝐻> 𝐵𝐺X ; ΛΛ∨ ≅ Λ2and derive a new constraint:
𝑎 is a characteristic vector:∀𝑣 ∈ Λ𝑣 ⋅ 𝑣 = 𝑣 ⋅ 𝑎 𝑚𝑜𝑑 2

What’s Wrong With Textbook GreenSchwarz Anomaly Cancellation?
What does 𝐵 even mean when ℳ• has
nontrivial topology? (𝐻 is not closed!)
How are the periods of 𝑑𝐵 quantized?
Does the GS term even make sense?
11
ƒ 𝐵 𝑌 = ƒ 𝑑𝐵 𝑌
2 ℳ‹
2 𝒰Š
must be independent of extension to 𝒰Œ !

But it isn’t ….
Even for the difference of two B-fields,𝑑 𝐻X − 𝐻@ = 0
we can quantize‘
𝐻X − 𝐻@ ∈ 𝐻 (𝒰Œ ; Λ)1exp( 2𝜋𝑖 ƒ 𝐻X − 𝐻@ 𝑌 )2 𝒰Š

is not well-defined because
Xof the factor of @ .

1Introduction & Summary Of Results

2Six-dimensional Sugra & Green-Schwarz Mech.

3Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6F-Theory Check

7

Concluding Remarks

21

Geometrical Formulation Of Anomalies
Space of all fields in 6d sugra is fiberedover nonanomalous fields:ℬ = 𝑀𝑒𝑡 ℳ• ×𝐶𝑜𝑛𝑛 𝒫 ×{𝑆𝑐𝑎𝑙𝑎𝑟 𝑓𝑖𝑒𝑙𝑑𝑠}

Partitionfunction:
Ψh¢£

ƒƒ

ℬ
žjŸ T–Z𝒢sk¤ 𝐴, 𝑔§¨ , 𝜙 ≔ ƒ𝑒 U•–U—˜™š›œ•𝑒 U—˜™š›œ•žjŸ T–Z

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.

Approach Via Invertible Field Theory
Definition [Freed & Moore]:An invertible field theory 𝑍 has∗Partition function ∈ ℂ

One-dimensional Hilbert spaces of states ...
satisfying natural gluing rules.
Freed: Geometrical interpretation of anomalies in ddimensions = Invertible field theory in (d+1) dimensions

Invertible Anomaly Field Theory
Interpret anomaly as a 7D invertible fieldtheory 𝑍h¢£ sk¤ constructed from 𝐺, ℛ, Λ, ℬ
Data for the field theory: 𝐺-bundles 𝒫 with gauge
connection, Riemannian metric, spin structure 𝔰.
(it is NOT a TQFT!)

Varying metric and gauge connection ⇒𝑍h¢£ sk¤ (ℳ• ) is a LINE BUNDLEΨh¢£sk¤is a SECTION of 𝑍h¢£sk¤ (ℳ• )Anomaly Cancellation In Terms of
Invertible Field Theory
1.

Construct a ``counterterm’’
7D invertible field theory 𝑍¸#𝑍¸# ℳ• ≅ 𝑍h¢£

∗ℳsk¤
•

2. Using just the data of the local fields
in six dimensions, we construct a section:Ψ¸# ∈ 𝑍¸# (ℳ• )Then:U—˜™š›œ• Ψ𝑒∫žjŸ T–Z¸#is canonically a function on ℬ/𝒢Dai-Freed Field Theory
𝐷: Dirac operator in𝜂 𝐷 + dim ker(𝐷)𝜉 𝐷 ≔ODD dimensions.
2𝑒 @WT¼ ½ defines an invertible field theory[Dai & Freed, 1994]

If 𝜕𝒰 = ∅𝑍½sTžŸjjt 𝒰 = 𝑒@WT¼ ½

If 𝜕𝒰 = ℳ ≠ ∅ then 𝑒 @WT ¼ ½ is asection of a line bundle over thespace of boundary data.
Suitable gluing properties hold.

Anomaly Field Theory For 6d Sugra
On 7-manifolds 𝒰Œ with 𝜕𝒰Œ = ∅𝑍h¢£ sk¤ 𝒰Œ =exp[2𝜋𝑖 𝜉 𝐷žjŸ T + 𝜉 𝐷ZVÃTjkt]

On 7-manifolds with 𝜕𝒰Œ = ℳ• :The sum of 𝜉 −invariants defines a unitÅh¢£ sk¤ in a line ZÇÈÉÊËÌÍ (ℳ• )vector ΨSimpler Expression When (𝒰Œ , 𝒫)
Extends To Eight Dimensions
In general it is impossible to compute𝜂-invariants in simpler terms.
X

But if the matter content is such that 𝐼/ = @ 𝑌 @AND if (𝒰Œ, 𝒫) is bordant to zero:
1𝑠𝑖𝑔𝑛 Λ 𝜎 𝒲/@𝑍h¢£ sk¤ (𝒰Œ ) = exp( 2𝜋 𝑖 ( ƒ 𝑌 −))2 𝒲Ï
8When can you extend 𝒰Œ and its
gauge bundle 𝒫 to a spin 8-fold 𝒲/ ??

Spin Bordism TheoryfÐT¢ΩŒ = 0 :Can always extend spin 𝒰Œ to spin 𝒲/fÐT¢ΩŒ
𝐵𝐺 : Can be nonzero: There can beobstructions to extending a 𝐺-bundle 𝒫 → 𝒰Œto a 𝐺-bundle 𝒫Ñ → 𝒲/
fÐT¢
ΩŒ
𝐵𝐺

= 0 for many groups, e.g.
products of 𝑈(𝑛), 𝑆𝑈(𝑛), 𝑆𝑝(𝑛). Also 𝐸/
But for some 𝐺 it is nonzero!

When 7D data extends to 𝒲/ the formula1
𝑠𝑖𝑔𝑛 Λ 𝜎 𝒲/
@𝑍h¢£ sk¤ (𝒰Œ ) = exp( 2𝜋 𝑖 ( ƒ 𝑌 −))2 𝒲Ï
8⇒ clue to constructing 𝑍¸# :ZÇÈÉÊËÌÍ 𝒰Œ
1
𝑋 = 𝑌 − 𝜆′21= exp( 2𝜋𝑖 ƒ𝑋 𝑋 + 𝜆′ )𝒲Ï 2×𝜆 = 𝑎⊗𝜆1
𝜆 ≔ 𝑝X
2Thanks to our quantization condition on 𝑏,𝑋 ∈ Ω> 𝒲/ ; Λ has coho class in 𝐻> 𝒲/ ; Λ1exp( 2𝜋𝑖 ƒ𝑋 𝑋 + 𝜆′ )𝒲Ï 2is independent of extension ONLY if𝑎 ∈ Λ is a characteristic vector:∀𝑣 ∈Λ𝑣 @ = 𝑣 ⋅ 𝑎 𝑚𝑜𝑑 2

This is the partition function of a 7D
topological field theory known as``Wu-Chern-Simons theory.’’Wu-Chern-Simons Theory
Generalizes spin-Chern-Simons to p-form gauge fields.
Developed in detail in great generality by
Samuel Monnier arXiv:1607.0139

Our case: 7D TFT 𝑍Ø¸U of a (locally defined) 3form gauge potential C with fieldstrength 𝑋 = 𝑑𝐶>

𝑋 ∈ 𝐻 ⋯;ΛInstead of spin structure we need a``Wu-structure’’: A trivialization 𝜔 of:

𝜈> = 𝑤> + 𝑤‘ 𝑤X + 𝑤@@ + 𝑤X>

Wu-Chern-Simons
In our case 𝜈> = 𝑤> will have a trivialization in 6
and 7 dimensions, but we need to choose one
to make sense of 𝑍Ø¸U 𝒰Œ and 𝑍Ø¸U ℳ•

1ZÜÝR (𝒰Œ) = exp( − 2𝜋𝑖 ƒ𝑋 𝑋 + 𝜆′ )𝒲Ï 2×𝜆 = 𝑎⊗𝜆𝑎 must be a characteristic vector of Λ∨Λ ≅ΛDefining 𝑍¸# From 𝑍Ø¸UTo define the counterterm line bundle 𝑍¸#we want to evaluate 𝑍Ø¸U on (ℳ• , 𝑌).
1
Problem 1: 𝑌 is shifted: 𝑌 = 𝑎 ⊗ 𝜆 + [𝑋]2
1
Tc d
𝑋 = ∑𝑏T 𝑐@ + ∑𝑏cd 𝑐X 𝑐X ∈ 𝐻> ⋯ ; Λ2ßneeds a choice of Wu-structure 𝜔.
Problem 2:𝑍Ø¸U!! We do not want to add a choice of Wu
structure to the defining set of sugra data
𝐺, ℛ, Λ, 𝑎, 𝑏

Defining 𝑍¸# From 𝑍Ø¸USolution: Given a Wu-structure 𝜔 we can shift 𝑌Xto 𝑋 = 𝑌 − @ 𝑣 𝜔 , an unshifted field, such thatXß𝑍Ø¸U (… ; 𝑌 − @ 𝑣 𝜔 ) is independent of 𝜔𝑍¸# ⋯ ; 𝑌ß≔ 𝑍Ø¸U1⋯; 𝑌 − 𝑣 𝜔2Thus, 𝑍¸# is independent of Wu structure 𝜔: So no needto add this extra data to the definition of 6d sugra.
𝑍¸# transforms properly under B-field, diff, and VM
gauge transformations: 𝑍¸# ℳ• ≅ 𝑍h¢£ sk¤ ℳ• ∗Anomaly Cancellation
𝑍#$% ≔ 𝑍h¢£ sk¤ ×𝑍¸# is a 7Dtopological field theory that is definedbordism classes of 𝐺-bundles.
7D partition function is a homomorphism:fÐT¢𝑍h¢£ sk¤ ×𝑍¸# ∶ ΩŒ
𝐵𝐺→𝑈 1

If this homomorphism is trivial then𝑍h¢£ sk¤ ×𝑍¸# ℳ• ≅ 1
is canonically trivial.

Anomaly Cancellation
Suppose the 7D TFT is indeed trivializable
Now need a section, Ψ¸# ℳ• which is local
in the six-dimensional fields.
This will be our Green-Schwarz counterterm:ƒžjŸ T–Z𝑒

U—˜™š›œ•Ψ¸# (ℳ• ; 𝐴, 𝑔§¨ , 𝐵)

The integral will be a function on ℬ/𝒢1

Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3

Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6

F-Theory Check

7

Concluding Remarks

38

Checks & Hats:
Differential Cohomology

á𝐻 𝑋

á
â
→𝐻 𝑋

Checks & Hats:
Differential Cohomology
Precise formalism for working with p-form fields ingeneral spacetimes (and p-form global symmetries)
Three independent pieces of gauge invariant information:
Wilson lines

FieldstrengthTopological class

Differential cohomology is an infinite-dimensional
Abelian group that precisely accounts for thesedata and nicely summarizes how they fit together.
Exposition for physicists: Freed, Moore & Segal, 2006

Construction Of The GreenSchwarz Counterterm:ã,ßΨ¸# = exp 2𝜋𝑖 ƒ𝑔𝑠𝑡

ℳ‹

1
𝑔𝑠𝑡 =
2

11
â − 𝜂̌ ∪ 𝑌æ + 𝜈̌𝐻221, ℎ@ − 𝜂 )2ç£kSection of the right line bundle & independentof Wu structure 𝜔.
Locally constructed in six dimensions, but
makes sense in topologically nontrivial cases.
Locally reduces to the expected answer

Conclusion: All Anomalies Cancel:
for 𝐺, ℛ, Λ, 𝑎, 𝑏 such that:
1 @𝐼/ = 𝑌2𝑎 ∈ Λ ≅ Λ∨ is characteristic & 𝑎 @ = 9 − 𝑇1>𝑏 ∈ 𝐻 𝐵𝐺X ; Λ2fÐT¢
ΩŒ
𝐵𝐺

=0

Except,…

What If The Bordism Group IsNonzero?
We would like to relax the last condition,
but it could happen thatfÐT¢𝑍h¢£ sk¤ ×𝑍¸# ∶ ΩŒ
𝐵𝐺

→𝑈 1

defines a nontrivial bordism invariant.
For example, if 𝐺 = 𝑂 𝑁 , for suitable representations,the 7D TFT might have partition function exp 2𝜋𝑖 ∫𝒰 𝑤XŒ
Š

Then the theory would be anomalous.

Future Directions
Understand how to compute
𝑍#$% ≔ 𝑍h¢£ sk¤ ×𝑍¸#fÐT¢When ΩŒ𝐵𝐺

is nonvanishing
there will be new conditions.
(Examples exist!!)Finding these new conditions in complete
generality looks like a very challenging problem…

1

Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3

Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6

F-Theory Check7Concluding Remarks

46And What About F-Theory ?
F-Theory
compactifications?

F-Theory: It’s O.K.
𝔤 is determined from thediscriminant locus [Morrison & Vafa 96]
X>
In order to check 𝑏 ∈ 𝐻 (𝐵𝐺X ; Λ)@

we clearly need to know 𝐺X.

We found a way
to determine 𝐺X.

F-theory passes
this test.

We believe a very similar argument also givesthe (identity component of) 4D F-theory.

1Introduction & Summary Of Results2

Six-dimensional Sugra & Green-Schwarz Mech.

3

Quantization Of Anomaly Coefficients.

4

Geometrical Anomaly Cancellation, 𝜂-Invariants& Wu-Chern-Simons

5

Technical Tools & Future Directions

6

F-Theory Check

7

Concluding Remarks

49

HUGE THANKS TO THE ORGANIZERS!
Hirosi Ooguri
おおぐり ひろし
Youhei Morita
もりた ようへい
Hitoshi Murayama
むらやま ひとし

Koji Hashimoto
はしもと こうじ
Yoshihisa Kitazawa
きたざわ よしひさ
Hirotaka Sugawara
すがわら ひろたか

and the OIST!!