Anomaly polynomial of general 6d SCFTs Kazuya Yonekura, IAS • arXiv:1408.5572 with K. Ohmori, H. Shimizu, Y. Tachikawa, (U.Tokyo) • Related work by Intriligator [1408.6745] 1
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
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6d N=(1,0) free multiplets
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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
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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
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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
2. Basics of 6d theory in IR and anomaly matching
3. Computation of Green-Schwarz contribution
4.General case in F-theory
5.Summary
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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
Gauge anomaly cancellation
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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
Gauge anomaly cancellation
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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
2. Basics of 6d theory in IR and anomaly matching
3. Computation of Green-Schwarz contribution
4.General case in F-theory
5.Summary
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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.
Appendix 1: E-string Anomaly
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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.
Appendix 1: E-string Anomaly
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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
Appendix 1: E-string Anomaly
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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
2. Basics of 6d theory in IR and anomaly matching
3. Computation of Green-Schwarz contribution
4.General case in F-theory
5.Summary
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Summary
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• 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