Three-Algebra Theories for

Supersymmetric M2 Branes

Jonathan Bagger

Neil Lambert

Introduction

In this talk I will describe a new three-dimensional maximally

supersymmmetric conformal field theory ...

... which is thought to capture the low-energy dynamics of a

stack of M2 branes pinned to an orbifold fixed point.

I will be motivated by three-dimensional maximally supersymm-

metric Yang-Mills theory ...

... which is known to describe the low-energy dynamics of a

stack of D2 branes.

1

D2 branes

The effective low-energy theory of N D2 branes contains

• 7 scalars Xi

• 1 vector Aµ

• 8 spinors Ψ.

This gives a total of 8 + 8 degrees of freedom.

All the fields are Lie algebra valued:

Xi = XiaT

a Aµ = AµaTa Ψ = ΨaT

a.

2

Lie algebras

The hermitian generators T a obey a two-algebra relation,

[T a, T b] = fabcTc.

The structure constants fabc obey the Jacobi identity:

fgedfabg = faegf

gbd + fbegf

agd.

There is also an invariant metric on the algebra,

Tr(T a, T b) = hab.

This implies

fabc = f [abc].

3

Super Yang-Mills

In maximally supersymmetric Yang-Mills theory, the super-

symmetry transformations are as follows,

δXi = iǫΓiΨ

δAµ = iǫΓµΓ10Ψ

δΨ = DµXiΓµΓiǫ+

1

2FµνΓ

µνΓ10ǫ+i

2[Xi, Xj]ΓijΓ10ǫ.

Note the explicit Γ10, which interferes with a straightforward lift

to M theory.

Also note that the theory is not conformal: [g] = 12.

4

D2 branes

The two-algebra structure appears because D-brane interactions

are mediated by open strings:

D2

5

M2 branes

How does this change for M2 branes?

• 8 scalars XI

• 8 spinors Ψ.

There are 8 + 8 degrees of freedom, so if there is a gauge field,

it cannot be dynamical.

Moreover, the fields cannot be Lie algebra valued.

To see this, let ZA = XA + iXA+4. Then

W = εABCDTr(ZAZBZCZD) = 0.

6

M2 branes

How can interactions be introduced? Let’s follow our nose:

δXI = iǫΓIΨ

δΨ = ∂µXIΓµΓIǫ+ iκ[XI , XJ , XK]ΓIJKǫ,

where [κ] = 0. Then just close the algebra:

[δ1, δ2]XI = 2iǫ1Γ

µǫ2∂µXI + 6κǫ1Γ

JKǫ2[XJ , XK, XI].

The first term is supersymmetry. The second is a weird gauge

transformation:

δXI = [α, β,XI].

Let’s take this seriously!

7

Three-algebra A

We define [·, ·, ·] to be a totally antisymmetric three-product on

a three-algebra A: A×A×A → A.

We choose a (hermitian) basis T a, and write

[T a, T b, T c] = fabcdTd.

We then impose the fundamental identity

fgefdfabc

g = faefgfgbc

d + fbefgfagc

d + fcefgfabg

d,

which ensures that the symmetry acts as a derivation.

We also assume the existence of an invariant metric,

Tr(T a, T b) = hab,

in which case

fabcd = f [abcd].

8

Gauge transformations

With this notation, a gauge transformation is given by

δXIa = Λcdf

cdbaX

Ib ≡ ΛbaX

Ib ,

where Λab = Λ[ab] ∈ so(N). In the closure,

Λab ∝ iǫ1ΓJKǫ2XJc X

Kd f

cdab.

We gauge this symmetry by introducing a (nondynamical) gauge

field Aµcd, which appears in the covariant derivatives

DµXIa = ∂µX

Ia − Aµ

baX

Ib

DµΨa = ∂µΨa − AµbaΨb,

where Aµba ≡ fcdbaAµcd. The curvature is defined to be

[Dµ, Dν]XIa = Fµν

baX

Ib .

9

Supersymmetry transformations

We take the supersymmetry transformations to be

δXIa = iǫΓIΨa

δΨa = DµXIaΓ

µΓIǫ−1

6XIbX

Jc X

Kd f

bcdaΓ

IJKǫ

δAµba = iǫΓµΓIX

IcΨdf

cdba.

The supersymmetries close into a translation and a gauge trans-

formation,

[δ1, δ2]XIa = vµ∂µX

Ia + (Λba − vνAν

ba)X

Ib

[δ1, δ2]Ψa = vµ∂µΨa + (Λba − vνAνba)Ψb

[δ1, δ2]Aµba = vν∂νAµ

ba +Dµ(Λ

ba − vνAν

ba).

10

Supersymmetry transformations

In these expressions,

vµ = −2iǫ2Γµǫ1, Λba = −iǫ2ΓJKǫ1X

Jc X

Kd f

cdba.

For closure we need the following equations of motion:

ΓµDµΨa +1

2ΓIJX

IcX

JdΨbf

cdba = 0

Fµνba + εµνλ(X

Jc D

λXJd +

i

2ΨcΓ

λΨd)fcdb

a = 0.

Note that the fermion EOM is appropriate for a dynamical field.

The gauge field EOM is a constraint. Therefore the gauge field

is not dynamical, as required by supersymmetry.

11

N = 8 Lagrangian

The equations of motion can, in fact, be derived from a

supersymmetric and gauge-invariant Lagrangian:

L = −1

2(DµX

aI)(DµXIa) +

i

2ΨaΓµDµΨa +

i

4ΨbΓIJX

IcX

JdΨaf

abcd

−V +1

2εµνλ(fabcdAµab∂νAλcd +

2

3fcdagf

efgbAµabAνcdAλef).

The potential is a perfect square:

V =1

12fabcdfefgdX

IaX

Jb X

Kc X

IeX

JfX

Kg

=1

2 · 3!Tr([XI , XJ , XK], [XI , XJ , XK]).

12

N = 8 theory

The N = 8 theory is gauge invariant, conformally invariant and

parity invariant.

It has 16 supersymmetries, SO(8) R-symmetry, and no free

parameters, modulo rescaling...

...as expected for the low-energy theory of multiple interacting

M2 branes.

It is plausibly the gYM → ∞ limit of three-dimensional super

Yang-Mills theory....

13

Example

A nontrivial example is the so-called SO(4) theory, where

fabcd = ǫabcd.

It describes two M2 branes in a Z2 orbifold background.

It turns out that this is the only nontrivial theory with 16

supersymmetries and a positive definite metric.

There are lots of theories with a Lorentzian metric, but their

physical interpretation is not clear....

14

N = 6 theory

To construct more theories, Aharony, Bergmann, Jafferis and

Maldecena proposed considering theories with 12 supersymme-

tries. As we shall see, such theories also are also described by

three-algebras.

A complex notation is better suited to this case, adapted to the

SO(6) × SO(2) ∼= SU(4) × U(1) R-symmetry.

Let us decompose SO(8) → SU(4) × U(1):

• 8v → 4(1) + 4(−1)

• 8s → 4(1) + 4(−1)

• 8s′ → 6(0) + 1(2) + 1(−2).

In this notation, the fundamental objects are the scalar field ZAa ,

the fermion ΨAa, and the supersymmetry parameter

ǫAB =1

2εABCDǫCD.

15

Complex three algebra

It is a matter of work to construct the most general theory with

these symmetries. We find that the three algebra is complex,

with a three-bracket of the form

[T a, T b;Tc] = fabcd T

d.

We also introduce a metric

hab = Tr(Ta, T b).

With these definitions, we find that the structure constants obey

the relations

fabcd = −fbacd and fabcd = f∗cdab

as well as the following fundamental identity

fefgbfcba

d + ffeabfcbg

d + f∗gaf bfceb

d + f∗agebfcf b

d = 0.

16

N = 6 supersymmetry

With these conventions, the supersymmetry transformations

δZAd = iǫABψBd

δψBd = γµDµZAd ǫAB + fabcdZ

Ca Z

Ab ZCcǫAB + fabcdZ

Ca Z

Db ZBcǫCD

δAµcd = −iǫABγµZ

Aa ψ

Bbfcabd + iǫABγµZAbψBaf

cabd,

with the covariant derivative

DµZAd = ∂µZ

Ad − Aµ

cdZ

Ac ,

close into a supersymmetry transformation plus a gauge trans-

formation

[δ1, δ2]ZAd = vµDµZ

Ad + ΛadZ

Aa .

17

N = 6 closure

We find

vµ =i

2ǫCD2 γµǫ1CD, Λad = i(ǫDE2 ǫ1CE − ǫDE1 ǫ2CE)ZDcZ

Cb f

abcd.

with Λcb ∈ u(N). The following equations of motion must also

be satisfied

0 = γµDµψCd + fabcdψCaZDb ZDc − 2fabcdψDaZ

Db ZCc

−εCDEFfabc

dψDc Z

Ea Z

Fb

Fµνcd = −εµνλ

(

DλZAa ZAb − ZAa DλZAb − iψA

bγλψAa

)

fcabd.

As before, the gauge field EOM is a constraint.

18

N = 6 Lagrangian

It is not hard to check that the EOM can be derived from the

following supersymmetric action

L = −DµZaADµZAa − iψAaγµDµψAa − V + LCS

−ifabcdψAdψAaZ

Bb ZBc + 2ifabcdψA

dψBaZ

Bb ZAc

+i

2εABCDf

abcdψAdψBc Z

Ca Z

Db −

i

2εABCDfcdabψAcψBdZCaZDb.

The potential is

V =2

3ΥCDBd ΥBd

CD,

where

ΥCDBd = fabcdZ

Ca Z

Db ZBc−

1

2δCBf

abcdZ

Ea Z

Db ZEc+

1

2δDBf

abcdZ

Ea Z

Cb ZEc.

The twisted Chern-Simons term is as before.

19

ABJM

What is the connection to ABJM?

Let X be a N1 ×N2 matrix, and define

[X,Y ;Z] = λ(XZ†Y − Y Z†X),

Tr(X,Y ) = tr(X†Y ).

Here † denotes the transpose conjugate, tr the matrix trace, and

λ is an arbitrary constant.

This three-bracket satisfies the fundamental identity.

20

ABJM gauge transformations

Consider

δX = XM1 −M†2X,

where M1, M2 are elements of u(N1) and u(N2) respectively.

In other words, X transforms as a bi-fundamental.

It is not hard to show that

δ[X, Y ; Z] = [X,Y ; Z]M1 −M†2[X, Y ; Z],

as a consequence of the fundamental identity.

We see that the three-bracket transforms covariantly.

21

ABJM gauge transformations

With this choice of three algebra, the N = 6 action takes the

following form:

L = −tr(DµZ†ADµZ

A) − itr(ψA†γµDµψA) − V + LCS

−iλtr(ψA†ψAZ†BZ

B − ψA†ZBZ†BψA)

+2iλtr(ψA†ψBZ†AZ

B − ψA†ZBZ†AψB)

+iλεABCDtr(ψA†ZCψB†ZD) − iλεABCDtr(Z†DψAZ

†CψB) .

For N1 = N2, this is the action of ABJM, written in SU(4)

covariant form.

22

Open questions

There are many open questions:

• What is the role of the Lorentzian N = 8 models?

Mukhi and Papageorgakis showed how to relate these models to

supersymmetric Yang-Mills theories by giving one scalar field a

vev. This “nonabelian duality” removes one scalar and gives rise

to one propagating vector, relating M2 to D2 branes. Is there a

deeper meaning behind their trick?

• Why N = 6?

Why is N = 8 supersymmetry manifest for two M2’s, but only

N = 6 for three or more?

23

Open questions

• Why three-algebras?

Are three-algebras merely fancy ways of writing theories with

bifundamental matter, a la ABJM? Or are they telling us more?

Alishahiha and Mukhi examined the higher-derivative corrections

to super Yang-Mills theory induced by D2 branes, and found

that they dualize to terms that can be written solely in terms of

three-brackets....

24

Open questions

• What does all this tell us about the microscopic dynamics of

M theory?

Previously, I argued that the strings connecting D2 branes give

rise to two-algebras.

Do the three-algebras that appear in the theory of M2 branes

hint that the fundamental objects are themselves M2’s?

M2

25