Anomaly polynomial of general 6d SCFTs

Anomaly polynomial ofgeneral 6d SCFTs

Kazuya Yonekura, IAS

• arXiv:1408.5572 with K. Ohmori, H. Shimizu, Y. Tachikawa, (U.Tokyo)

• Related work by Intriligator [1408.6745]

1

( : non-compact)

Introduction6d N=(2,0) theories have been very important in recentdevelopments of SUSY gauge theories.

They may be as interesting as N=(2,0) theories!2

There are also infinitely many 6d N=(1,0) SCFTs.

R6 ⇥ CY3

Large classes (possibly all) of N=(1,0) theories are realizedby F-theory on singularities of

CY3

[Heckman,Morrison,Vafa,2013]

Introduction

However, properties of N=(1,0) theories are not well studiedbecause of the lack of UV Lagrangians.

What we have studied: ’t Hooft anomalyAnomaly polynomials of the theories under background gravity and global symmetry gauge fields.

I would like to explain a general method to compute the anomaly polynomial of any known N=(1,0) theory.

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Contents

1. Introduction

2. Basics of 6d theory in IR and anomaly matching

3. Computation of Green-Schwarz contribution

4.General case in F-theory

5.Summary

4

6d N=(1,0) free multiplets

5

There are three types of free multiplets with spin 1

• Vector multiplet

• Hyper multiplet

• Tensor multiplet

gauge fields

two complex scalars(doublet of SU(2) R-symmetry)

Bosonic components

one scalar (singlet of any symmetry)+ two form (field strength: self-dual)

6d N=(1,0) free multiplets

6

The vev of the scalar in a tensor multiplet does not breakany global symmetry.

(Remark: in the case of N=(2,0), it actually breaks SO(5)R-symmetry to SO(4). But SO(4) is large enough to determine the anomaly polynomial of SO(5) completely.)

Interacting N=(1,0) SCFTs have tensor (Coulomb) branch as part of the moduli space of vacua. In the IR limit, thetheory consists of vector, hyper and tensor multiplets.

Moduli space of SCFT

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UV SCFT (no Lagrangian)

IR effective theory (almost free)

on tensor branch No global symmetry broken:The anomaly must matchbetween UV and IR

The anomaly of UV SCFT is computable by IR effective theory.

(’t Hooft anomaly matching)

Anomaly matching

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UV SCFT anomaly = IR one-loop anomaly+Green-Schwarz contribution

• Tensor multiplet one scalar (singlet of any symmetry)+ 2-form (field strength: self-dual)

ZB2 ^ I4

: 4-form constructed from background gravity and flavor fields (as well as dynamical fields).

I4

B2 : 2-form in tensor multiplet

I4 ⇠ atrR2 + btrF 2R + · · ·

Anomaly matching

9

Green-Schwarz contribution to anomaly polynomial:

Rough explanation (not really a derivation)E.O.M:

Gauge invariance of requiresH3

�gaugeB2 = I(1)2 , dI(1)2 = �gaugeI(0)3

�gauge

ZB2 ^ I4 = I(1)2 ^ I4

Uplifting to 6+2 dim via the usual descendant eqs.:

H3 = dB2 � I(0)3 , dI

(0)3 = I4

(I4)2 (up to coefficient)

d ⇤H3 = �I4 dH3 = �I4

d(I(1)2 I4) = �gauge(I(0)3 I4), d(I(0)3 I4) = (I4)

2

Anomaly matching

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UV anomaly polynomial

IR one-loop anomaly ofvector, hyper and tensor multiplets(explicitly known)

Green-Schwarz

We need to determine to compute the UV anomaly. I4

IUV = IIRone�loop

+ I24

Contents

1. Introduction




5.Summary

11

Gauge anomaly cancellation

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The gauge anomaly must be cancelled by Green-Schwarz contribution.

On tensor branch, fermions (e.g. gauginos) are charged under gauge groups and have gauge anomalies.

• Vector multiplet• Hyper multiplet

• Tensor multiplet

left-handed

right-handed

fermionic components

right-handedgauge anomaly


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Remark: The fact that the first term is a perfect square is requiredby the Green-Schwarz mechanism to work.It is a nontrivial constraint on the gauge group G and matter content of the IR effective theory.

FG

X4, Y8 : a 4-form and an 8-form depending only on background fields.

: gauge field strength

Ione�loop

⇠ �(trF 2

G)2 � 2(trF 2

G)X4

� Y8

For the moment, let’s consider the case of a single gauge group and single tensor multiplet.G


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Terms involving must be cancelled by Green-SchwarzFG

(I4)2

This uniquely determines I4

So we getIUV = I

one�loop

+ (I4

)2

= (X4

)2 � Y8

Ione�loop

= �(trF 2

G)2 � 2(trF 2

G)X4

� Y8

I4 = trF 2G +X4

(Here I am using ) 2 = ⇡ = · · · = 1

IUV Ione�loop

is uniquely determined from !

Example: SU(N) with 2N flavors

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CY3 ! B2 B2 : base of elliptic fibration

Suppose the base contains 2-cycles with 7-branes(singular fibers) wrapped on them.

AN�1

flavor branes

SU(N) • SU(N) gauge theory• N+N flavors from brane intersections• The size of the compact 2-cycle

= VEV of the tensor multipletzero size: interacting SCFTfinite size: tensor branch{

• Generalization to quiver SU(N)’s is straightforward

Example: SU(N) with 2N flavors

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Ihyper = 2NtrfundF4G

Only SU(2) R-symm. considered for simplicity

�tradjF4G + 2NtrfundF

4G = �(trF 2

G)2

Igaugino

= �tradj

F 4

G � tradj

F 2

GtrF2

R � (N2 � 1)(trF 2

R)2

tradjF2G = 2NtrF 2

G

(Here I am using ) 2 = ⇡ = · · · = 1

Ione�loop

= �(trF 2

G +NtrF 2

R)2 + (N2 +O(1))(trF 2

R)2

Ione�loop

+ (I4

)2 = (N2 +O(1))(trF 2

R)2

Group theoryrelations: {

perfect square!

• If , anomaly cancellation does not determine Green-Schwarz contribution uniquely.

More general case

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Ione�loop

= �⌘IJtrF 2

I trF2

J � 2⌘IJXItrF2

J � Y

= �⌘IJ(trF 2

I +XI)(trF2

J +XJ) + ⌘IJXIXJ � Y

NT

:number of simple gauge groups:number of tensor multiplets

• If , gauge anomaly cannot be cancelled. • If , anomaly is cancelled in a unique way.

Ione�loop

+ ⌘IJIIIJ = ⌘IJXIXJ � Y

NG

NG > NT

NG = NT

NG < NT

Contents

1. Introduction




5.Summary

18

General 6d SCFTs

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All known 6d SCFTs can be constructed by F-theory on

R6 ⇥ CY3

CY3 : elliptically fibered Calabi-Yau 3-fold oversome base 2-fold B

The base contains intersecting 2-cyclesB

General 6d SCFTs

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• The size of each cycle = a scalar in a tensor multiplet.

• Some cycles have a simple gauge group G (due to singular fibers) and others have no gauge group at all.

• Blowing-down all the cycles: SCFT at the singularity Blowing-up the cycles: generic points of tensor branch

More detailed rules are in [Heckman,Morrison,Vafa,2013]and very recently in [Heckman,Morrison,Rudelius,Vafa,2015]

G1 G2 G3 G4 G5

General 6d SCFTs

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NT

:number of gauge groups:number of tensor multiplets = number of cycles

(e.g. in the figure, and )

Always

We want so that the anomaly cancellation by Green-Schwarz proceeds in a unique way.

NG

NG NT

NG = NT

G1 G2 G3 G4 G5

NG = 5 NT = 7

General 6d SCFTs

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Blowing-down the cycles which do not contain gauge groups.

The cycles shrunk become E-string theories (6d SCFT).

is realized in this way.

We consider E-string theories as “bifundamental matter”.

NG = NT

G1 G2 G3 G4 G5

G1 G2 G3 G4 G5

Gi ⇥Gi+1 ⇢ E8

General method

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IUV = Ione�loop

+ ⌘IJIIIJ

“One-loop” contains contributions of E-string theories. Green-Schwarz is uniquely

determined by gauge anomalycancellation.

IUVof any SCFT can be computed in this way!

The anomaly of E-string theory has been already computed.[Ohmori,Shimizu,Tachikawa][Ohmori,Shimizu,Tachikawa,KY]

Appendix 1: E-string Anomaly

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The anomaly polynomial of the E-string theory can becomputed by either

• Anomaly inflow

• Anomaly matching of this talk

[Ohmori,Shimizu,Tachikawa]

[Ohmori,Shimizu,Tachikawa, KY]

I would like to discuss the second method.


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E-string theory (of rank 1 for concreteness) is an interacting6d SCFT realized by

1. Blowing-down a 2-cycle of self-intesection -1 in F-theory

2. Placing an M5 brane on Horava-Witten M9 brane

S^1 compactification: a D4 brane on an O8 plane.5d theory is SU(2) gauge theory with 8 flavors.


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ZB2 ^ I4

Compactification on a circleZA1 ^ I4

We can determine by computing the Chern-Simons term of 5d gauge field with background fields.

: Chern-Simons in 5d

I4A1

A1 :

(Remark: tensor mult. in 6d → vector mult. in 5d)

U(1) ⇢ SU(2) gauge field of 5d vector multipleton the Coulomb branch


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U(1) field A1

background field

background fieldZ

A1 ^ I4

I4 is computed just by one-loop computation of the triangle diagram in 5d.

IUV = IIRone�loop

+ I24

Total anomalyeasily computed.

in 6d is

Remark

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• N=(2,0) theory• E-string theory

5d maximal Super-Yang-MillsSp(N) with an anti-symmetricand 8 fundamental hypers

By the same method, we can completely determine anomaly polynomials of N=(2,0) and E-string theories.

(rank N)

S^1 reduction

The results reproduce the known/conjectured ones inthe literature.

Appendix 2: Gauge anomaly

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More precise form of the Green-Schwarz contribution is as follows:

IGS =1

2⌘IJIIIJ

c2(FG) / trF 2G such that one instanton gives c2(FG) ! 1

II = nGI c2(FG)

nGI : Charge of string constructed by gauge instanton.

: Integer due to Dirac quantization of strings.

One-loop anomaly must be consistent with this condition.

⌘IJnGI n

G0

J

Appendix 2: Gauge anomaly

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Example: single G=SO(8) without hypers

Ione�loop = �4

2(c

2

(FG))2

if the string charge is normalized as

This reproduce F-theory construction -4

IGS =⌘n2

2(c2(FG))

2

Theories with non-integer are excluded by the field theory consistency condition.

⌘n2

( are intersection numbers in F-theory)�⌘IJ

⌘ = 4 n = 1

Contents

1. Introduction




5.Summary

31

Summary

32

• General method is described to compute anomaly polynomial of any known 6d SCFT realized in F-theory.

• Anomaly matching of UV and IR theories on tensor branch and Green-Schwarz contribution are the crucial ingredient.

IUV = IIRone�loop

+ I24

Future directions

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• Compactification of 6d SCFT to 4 or other dimensions

• 6d central charges, a-theorem, etc.

Anomalies of lower dimensional theories are related to anomalies in 6d.

In 4d, central charges are related to coefficients of anomaly polynomial in N=1 theories. Similarly in6d N=(1,0) theories?

a, c

Thank you very much!

Example: U(N) N=(2,0)

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U(N) N=(2,0) (N M5-branes) → 5d U(N) SYM (N D4-branes)

Coulomb branch U(N) → U(1)^N� = diag(�1, · · · ,�N )

�1 > · · · > �N

adjoint scalar

About the k-th U(1) field , there are

• k-1 gauginos of positive charge +1• N-k gauginos of negative charge -1

with positive masses.

Ak

(There are also gauginos with mass and charge flipped.)

(�)kj(�)jk

(1 j < k)(k < j N)

Example: U(N) N=(2,0)

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Ak: k-th U(1) field SU(2) R-symmetry

SU(2) R-symmetry

Green-Schwarz contribution to 6d anomaly polynomial:

/NX

k=1

[(k � 1)� (N � k)]2(trF 2R)

2

/ (N3 �N)(trF 2R)

2

N^3 behavior of N=(2,0) theory is reproduced!

Ak

/ [(k � 1)� (N � k)]trF 2R ^Ak

= I(k)4 ^Ak

Documents

Anomaly polynomial of general 6d SCFTs