M-theory and Matrix Theory Stan Seibert

Steve Young

May 5, 2004 Abstract In this paper, we review string theory dualities and discuss compactification of 11D supergravity as a motivation for the construction of M-theory. M-theory is defined, and the matrix theory Hamiltonian is derived both from a matrix regularization of the supermembrane action in 11D and from the low energy effective action of a system of D0-branes. The D0-brane picture is explored, leading to the BFSS conjecture.

1

Introduction

The study of branes and non-perturbative effects in string theory over the past decade is leading to a fascinating understanding of the relationships between the 5 superstring theories. Dualities between 10D string theories have been known for some time, but it has become clear that the introduction of an 11D theory, known as M-theory, is necessary for a complete picture.

2

Dualities in String Theory

M-theory arises from the study of dualities in string theory and their low energy effective supergravity actions. Because of the importance of dualities in this derivation, it will be worthwhile to begin by review the dualities between various string theories.

2.1

T-duality

T-duality relates two strings theories compactified on circles with different radii. The picture usually given is that of a closed string propagating on a cylinder. The string will have both excitation modes, enumerated by the integer n, and winding modes, enumerated by integer m. The momentum of the left and right-movers will be n + mR 2R n = − mR. 2R

pL = pR

1

(1)

T-duality arises because we can exchange 1 2R in conjunction with relabeling m ↔ n, and preserve the spectrum of states. If we are working with the NSR string with the 9th dimension compactified on a circle, this transformation is equivalent to R↔

ψL9 → ψL9 ψR9 → −ψR9 .

(2)

This flips the sign of the right-moving chirality operator Γ11 = ψR0 ψR1 · · · ψR9 → −Γ11

(3)

which flips the GSO projection of the right movers: IIA IIB FL FL (−1) = +1 → (−1) = +1 (−1)FR = +1 → (−1)FR = −1 So,we have a related non-chiral type IIA string theory with type IIB chiral string theory through the T-duality: T : IIA ↔ IIB By considering the moduli space of vacua, one can also show (See Kaku[1], pg. 463-465 for details) that the two heterotic superstring theories are T-dual: T : SO(32) ↔ E8 × E8

2.2

S-duality

Unlike T-duality, S-duality is non-perturbative. It relates theories at weak coupling to theories at strong coupling, i.e. e → e− . This makes it a very powerful tool for gaining insight into the non-perturbative behavior of string theory. One example of S-duality can be found in IIB superstring theory. If we consider the BPS states, which are protected at all couplings by non-renormalization theorems, we can write down their brane tensions: 1 F1 : TF ∼ 0 α 1 D1 : T1 ∼ 0 αg 1 D3 : T3 ∼ 0 2 (α ) g 1 NS5 : TNS5 ∼ 0 3 2 (α ) g 1 D5 : T5 ∼ 0 3 (α ) g 2

Under the transformation 1 g → α0 g

g → α0

(4)

we exchange F1 ↔ D1 D3 ↔ D3 NS5 ↔ D5

This is an indication that type IIB string theory at strong coupling is dual to type IIB string theory at weak coupling: S : IIB ↔ IIB One can also find a S-duality in the type I theories (again, see [1], pg. 474 for details): S : I ↔SO(32)

3

Compactifying Supergravity

Now that we have cataloged the dualities between the string theories, we turn our attention to supergravity actions. Following Bilal[2] section 3, we begin by considering the 11D supergravity action Z Z √ 1 (11) 11 2 S = d x G(R + |dA3 | ) + A3 ∧ dA3 ∧ dA3 + fermionic terms with Ψ (5) 2 (Note that we have set α0 = 1 and dropped relative constant factors for this discussion.) The massless fields in this action are: • GM N , the 11D metric, • A3 ≡ AM N P , the 3-form potential, and • ΨM , the 32 component Majorana gravitino. The indices M , N , and P all run over the range 1–11. If we count the number of massless states, we find that G has 44 dofs, A3 has 84, and Ψ has 128, for a total of 256 massless states in the 11D supergravity multiplet. If we now compactify this action on a circle (take 0 ≤ x11 < 2π) and let µ, ν, and ρ be indices that run from 1–10, then we can break the massless states up into the following components: 3

ΨM ≡

AM N P

GM N

1
ψM 2 ψM

⇒

ψµ1 , ψµ2 2 1 , ψ 2 ≡ ψ11 ψ 1 ≡ ψ11

pair of Majorana-Weyl gravitinos pair of Majorana-Weyl spinors

⇒

Aµνρ Bµν ≡ Aµν,11

10D 3-form 10D 2-form

e2γ A ⇒ eµ G

≡ ≡ ≡ =

G11,11 scalar −2γ −e Gµ,11 vector −2γ Gµν − e Gµ,11 G11,ν 10D metric Gµν − e2γ Aµ Aν

Under this parameterization, the 11D line element becomes ds2 = GM N dxM dxN eµν dxµ dxν + e2γ Aµ Aν dxµ dxν − 2e2γ Aµ dxµ dx11 + e2γ dx11 dx11 = G eµν dxµ dxν + e2γ (dx11 − Aµ dxµ ). = G The dx11 term shows that the radius of compactification is eγ . Now we want to rewrite the 11D supergravity action in terms of these fields. The pae has been selected1 to give rameterization of G eµν . det GM N = e2γ det G

(6)

The 11D curvature R contains terms like G11,11 ∂µ Gν,11 ∂ρ Gσ,11 = e−2γ ∂µ (−e2γ Aν ) ∂ρ (e2γ Aσ ). This gives the field strength of Aµ when we explicitly separate out the 10D curvature: R(11) → R(10) + e−2γ e+2γ e+2γ |dA|2 + |∇γ|2 + · · · → R(10) + e2γ |dA|2 + |∇γ|2 + · · ·

(7)

Similarly, the field strength of the 11D 3-form A3 contains terms like G11,11 ∂Aµν,11 ∂Aρσ,11 = e−2γ ∂Bµν ∂Bρσ which gives rise to (11)

(10)

|dA3 |2 → |dA3 |2 + e−2γ |dB|2 + · · · 1

(8)

To see why, write out G as a matrix and add multiples of row 11 to rows 1–10 (which preserves the e µν , and the determinant determinant) to zero out all of column 11 except G11,11 . You will have constructed G e of GM N can be trivially expanded by minors in terms of Gµν .

4

in the action. Putting these results into the 11D action and integrating out dx11 gives a 10D action (dropping the wedge product and fermionic parts): Z p   (10) 10 e R + |∇γ|2 + e2γ |dA|2 + |dA3 |2 + e−2γ |dB|2 + · · · S = d x e2γ G Z p   e eγ (R + |∇γ|2 + |dA3 |2 ) + e3γ |dA|2 + e−γ |dB|2 + · · · (9) = d10 x G However, the usual form of this 10D supergravity action when derived as the low energy effective action of IIA string theory is Z  √  (10) S = d10 x g e−2φ (R + |∇φ|2 + |dB|2 ) + |dA3 |2 + |dA|2 + · · · (10) In order to get Equation 9 into this form, we need to Weyl rescale the metric eµν = e−γ gµν G

(11)

eµν = e−10γ det gµν . det G

(12)

which changes the determinant,

e in the old metric becomes R[G] e = eγ R[g] + · · · in terms of the new The curvature R[G] metric. The field strengths of the p-forms contain p + 1 factors of the inverse metric, so |dA|2 → e2γ |dA|2 |dB|2 → e3γ |dB|2 |dA3 |2 → e4γ |dA3 |2 Finally, the |∇γ|2 term picks up a factor of eγ . Substituting these rescaled quantities into Equation 9 gives Z  √  (10) S = d10 x g e−3γ (R + |∇φ|2 + |dB|2 ) + |dA3 |2 + |dA|2 + · · ·

(13) (14) (15)

(16)

which has the desired form under the identification eγ = e2φ/3 .

4

(17)

Emergence of M-theory

So far we have connected 11D supergravity to 10D type IIA supergravity by compactifying one dimension on a circle. We can also relate the 10D type IIA supergravity action back to type IIA string theory by recalling that φ in the action is the dilaton and the string coupling 5

constant gs = eφ . But in the previous derivation we also showed how eγ was related both to eφ and the radius of compactification R11 . If we put α0 back in, we get the relation √ √ √ R11 = α0 eγ = α0 e2φ/3 = α0 gs2/3 . (18) However, this measures the radius with the 11D metric, and not the Weyl-rescaled string metric, so we need to use φ/3 2 ) ds2(G) (19) ds2(g) = eγ ds2(G) e = (e e to compute the correct radius of compactification (g)

√

e (G)

R11 = eφ/3 R11 =

α0 gs .

(20)

In the supergravity theory, the presence of a compactified dimension means that there will be Kaluza-Klein modes, and the massless states will all acquire a mass M=

n (g) R11

=√

n α0 gs

(21)

where n is a positive integer. So for each n, there will be a multiplet of 256 supergravity states with a particular mass. Jumping back to type IIA string theory, we might wonder what states couple to the 1-form fields in the 10D low energy effective action. The fundamental strings will not, but Dp-branes naturally couple to a p + 1-form, so D0-branes are the obvious candidate for the objects that couple to A. Since D0-branes are BPS states in type IIA string theory, they saturate the mass bound M = |Z|, (22) where |Z| is the central charge of the supersymmetry algebra. The central charge √ of a D0brane with brane tension T0 is Z0 = T0 /gs , and the brane tension is T0 = 1/ α0 . So, we have found states in type IIA string theory with masses M=√

1 α0 gs

(23)

that come in a short multiplet of 256 states. This is in direct correspondence with the first Kaluza-Klein mode in compactified 11D supergravity. Various people (including Witten[3], Sethi and Stern[4], and Porrati and Rosenberg[5]), have done work to show that it is possible to form bound states of n D0-branes at threshold, giving a total mass of n M=√ . (24) α0 gs Thus, there is a direct correspondence between all the Kaluza-Klein modes of compactified 11D supergravity and bound states of D0-branes in IIA string theory. Moreover, in the strong coupling limit, gs → ∞, all of the n D0-brane √ states become (g) very light, and therefore low energy states. At the same time, R11 = α0 gs → ∞, and 6

we decompactify back to full 11D supergravity. This leads to the important observation that the low energy effective action of 10D type IIA string theory at strong coupling is 11D supergravity. But 11D supergravity is not expected to be a fully-consistent quantum theory at all energy scales, but rather the low energy effective action of some theory, given the name “M-theory.” Specifically, M-theory is defined to be the 11D theory which (g) √ • compactified on a circle gives type IIA string theory with gs = R11 / α0 • and has 11D supergravity as its low energy effective action. There is no known definition of M-theory in the traditional sense of a Lorentz-invariant action with dynamical degrees of freedom. However, there is a Hamiltonian which is conjectured to describe M-theory in the infinite momentum frame (IMF) that will be described in later sections. With this definition in hand, we can create a diagram that shows how the various supergravity and string theories all relate to each other. The figure below shows all of the dualities discussed so far, as well as some connections between string theory and other 10D supergravity theories. It also includes the fact that M-theory compactified in a different way will yield heterotic E8 × E8 string theory (see [1], page 473 for a derivation).

11D

M-Theory

tify

S

Heterotic SO(32)

Heterotic E8 X E8

N=1 SUGRA + E8 X E8 Super Yang-Mills

7

IIA SUGRA

T

IIB

IIB SUGRA

11D

11D Supergravity

IIA

10D Supergravity

N=1 SUGRA + SO(32) Super Yang-Mills

T

Low Energy Effective Action

Type I

10D String Theory

ac

mp

Co

S

5

Matrix Theory from the Quantized Membrane

With the discovery of M-theory there was great interest in finding the degrees of freedom and the action which would produce it. Given that M-theory is an 11D and not a 10D theory, string-like degrees of freedom are ruled out. However, looking at the 11D supergravity action (Equation 5), one might conjecture that membranes are the fundamental degrees of freedom since they would couple naturally to the A3 3-form field. This is the motivation for the next section in which we follow the review of membrane quantization given in section II of Taylor[6].

5.1

The Bosonic Membrane

We begin the quantization of the bosonic membrane in a manner very similar to that used for the bosonic string. We parameterize a 3 dimensional world-volume now with coordinates {τ, σ 1 , σ 2 } = {σ 0 , σ 1 , σ 2 }. Then we can write an analog of the Nambu-Goto action for a membrane Z Z p √ 3 µ (25) S = −T d σ − det ∂α X ∂β Xµ ≡ −T d3 σ −h where the indices α, β ∈ {0, 1, 2} and we have used the notational conventions hαβ ≡ ∂α X µ ∂β Xµ and h ≡ det hαβ . Just as in the string case, the square root is cumbersome, so an auxiliary metric γαβ is introduced to produce a Polyakov-like action Z √ T d3 σ −γ(γ αβ hαβ − 1). (26) S=− 2 Varying this action to get equations of motion for γαβ will show that γαβ = hαβ .

(27)

However, since these metrics are 3-dimensional, the extra −1 is needed in the action to recover the Nambu-Goto action. This term eliminates the scale invariance of the theory. This will limit some of our freedom when gauge fixing γαβ . Since γαβ is a 3x3 symmetric matrix, there are 6 independent components. We have 3 diffeomorphism symmetries and no Weyl-rescaling symmetries, so the metric can only be partially gauge fixed. Assuming the world volume is of the form Σ×R, where Σ is a Riemann surface, the action can be written fairly simply using the following choice   4 ¯ 0 0 − ν2 h , γ= 0 (28) ¯ ab h 0 ¯ ab is the unfixed 2x2 metric and h ¯ is its determinant. The constant where a, b ∈ {1, 2}, so h ν is a normalization constant which will be fixed later. Substituting this definition of γ into

8

the action, and using Equation 27 simplifies the action to r Z
4 ¯2 T 00 ab S = − d3 σ h γ h + γ h − 1 00 ab 2 ν2  2  Z T ν ˙µ ˙ 2¯ 3 ab = − d σ h − ¯ X Xµ + h hab − 1 2 ν 4h   Z Tν 4¯ 3 µ ˙ ˙ d σ X Xµ − 2 h . (29) = 4 ν ¯ ab can be used interchangeably.) Using the stan(Note that by Equations 27 and 28, hab and h dard definition of Poisson brackets {f, g} ≡ ab ∂a f ∂b g, we can rewrite the 2x2 determinant to produce the usual form of this action   Z Tν 2 3 µ ν µ S= d σ X˙ X˙ µ − 2 {X , X }{Xµ , Xν } . (30) 4 ν Varying the action in this form yields the equations of motion 4 X¨µ = 2 {{X µ , X ν }, Xν }. ν In addition, the system has constraints 4¯ 2 γ00 = − 2 h ⇒ X˙ µ X˙ µ = − 2 {X µ , X ν }{Xµ , Xν } ν ν µ ˙ γ0a = 0 ⇒ {X , Xµ } = 0

(31)

(32) (33)

which come from our prior gauge choices. Next, we go to light-front coordinates, using the convention
1 X ± = √ X 0 ± X D−1 (34) 2 where D is the dimensionality of the space the membrane is propagating in. The indices i, j will be used to denote the transverse directions. In these coordinates, the constraints can be solved to set X+ = τ (35) and the constraints on X − become 1 1 ˙i ˙i X X + 2 {X i , X j }{X i , X j } X˙ − = 2 ν − i i ˙ ∂a X = X ∂a X . These relations can be used to calculate the Hamiltonian for this system   Z Tν 2 2 i ˙i i j i j ˙ H= d σ X X + 2 {X , X }{X , X } . 4 ν

(36) (37)

(38)

The non-linearity of the equations of motion would normally make this system very difficult to quantize, however Goldstone and Hoppe[7] found a clever way to regularize the membrane action (or Hamiltonian, in this case) using a matrix representation. 9

5.2

Matrix Regularization

Hoppe’s technique was to change basis in such a way that preserved the algebra of Poisson brackets. He specialized on a membrane that was topologically equivalent to a sphere, but Fairlie, Fletcher and Zachos[8] and Floratos[9] have extended it to a torus, and Bordemann, Meinrenken,and Schlichenmaier[10] have implicitly show it is possible for higher genus membranes. In the case of of a membrane with the topology of a sphere, one can express functions on the sphere in terms of Cartesian coordinates ξ1 , ξ2 , ξ3 , which are constrained to a unit sphere and have a symplectic form such that {ξA , ξB } = εABC ξC . Hoppe realized that this algebra could be reproduced using the generators of SU (2) in N -dimensional matrix representation. Given the correspondence 2 (39) ξA ⇒ JA , N one can transform the spherical harmonics X (lm) (40) Ylm (ξ1 , ξ2 , ξ3 ) = tA1 ...Al ξA1 · · · ξAl (lm)

(where the tA1 ...Al are constants) into a matrix representation  Ylm =

2 N

l X

(lm)

tA1 ...Al JA1 · · · JAl .

(41)

Note that only spherical harmonics with l < N can be represented this way, so to have a complete basis, we will need to take the large N limit. Since functions defined on the membrane can be expanded in spherical harmonics, there is a mapping from coordinate functions to matrices defined by X X f (ξ1 , ξ2 , ξ3 ) = clm Ylm (ξ1 , ξ2 , ξ3 ) ⇒ F = clm Ylm . (42) l,m

l,m

Hoppe showed that the structure constants in Poisson brackets of the spherical harmonics 00

00

l m {Ylm , Yl0 m0 } = glm,l 0 m0 Yl00 m00

(43)

and the structure constants of the commutators of their matrix equivalents 00

00

m [Ylm , Yl0 m0 ] = Gllm,l 0 m0 Yl00 m00

(44)

are related in the large N limit by −iN l00 m00 l00 m00 Glm,l0 m0 = glm,l 0 m0 . N →∞ 2 lim

(45)

Thus, an equation with the Poisson brackets of two functions has a matrix form {f, g} ⇒

−iN [F, G]. 2 10

(46)

Moreover, it can be shown that the integral of the function f is related to the trace of the matrix F by Z 1 1 lim Tr F = d2 σ f. (47) N →∞ N 4π Equations 39, 46, 47 provide a prescription for transforming continuum equations into matrix equations. If we now fix our normalization constant ν to be N , then the bosonic membrane Hamiltonian becomes   1 ˙i˙i 1 i i i j (48) X X − [X , X ][X , X ] H = 2πT Tr 2 4

5.3

Supersymmetric Membranes

Finally, we need to go back and make this theory supersymmetric. Unlike string theory, where there are formalisms to add supersymmetry either to the worldsheet or to the target spacetime (NSR and Green-Schwartz, respectively), for the membrane, only a spacetime approach is known. In light-front coordinates, with κ-symmetry gauge-fixed away, we introduce θ, a 16-component Majorana spinor of SO(9), to make the supermembrane Hamiltonian   Z Tν 2 2 2 i i i j i j T i d σ X˙ X˙ + 2 {X , X }{X , X } − θ γi {X , θ} . (49) H= 4 ν ν Applying the matrix regularization procedure yields the matrix Hamiltonian   1 ˙i˙i 1 i i i j 1 T i H = 2πT Tr X X − [X , X ][X , X ] − θ γi [X , θ] 2 4 2

(50)

where θ is now a matrix-valued SO(9) spinor. Equation 50 is known as the Matrix Theory Hamiltonian, and describes the dynamics of supermembranes in terms of the quantum mechanics of N -dimension matrix degrees of freedom in the large N limit. This Hamiltonian is conjectured to describe M-theory as well, however the conjecture was arrived by considering the low energy effective action of D0-branes, which will be described in the next section.

6 6.1

The D-Brane Low Energy Effective Action D-brane actions

In the last section, we saw the Hamiltonian for the 11D quantized membrane. This Hamiltonian can also be derived from a Lagrangian which is the low energy effective action of a system of D0-branes. The reason we cover this alternate derivation of the Hamiltonian is to motivate the BFSS conjecture, which will be covered in the next section. We will first review the basics of deriving the low energy effective action for a Dp-brane.

11

This is done in much the same way as one derives the low energy effective action for a string. First one introduces the possibility of strings propagating in curved background fields. The simplest example replaces the flat space-time metric with one dependent on the embedding fields X µ : η µν → Gµν (X). The worldsheet action becomes Z 1 d2 σ g 1/2 g ab Gµν (X)∂a X µ ∂b X ν . (51) Sσ = 4πα0 This now has the form of a field theory with a field-dependent kinetic term – we are now dealing with a non-linear sigma model, which is the reason for the subscript on the action. We will look at the bosonic case for simplicity. Recall that the closed bosonic string had more than just the graviton Gµν in its massless spectrum. There is also the dilaton φ and the Kalb-Ramond field Bµν . We should naturally include the possibility of a background field for these as well: Z 
 1 d2 σ g 1/2 g ab Gµν (X) + iab Bµν (X) ∂a X µ ∂b X ν + α0 φR . (52) Sσ = 0 4πα

For the conformal field theory on the world sheet to be consistent when quantized, we will need the action to be Weyl invariant, which is equivalent to tracelessness of the twodimensional stress energy tensor, Taa = 0. This is in turn equivalent to the vanishing of a set of β functions, which resemble spacetime field equations for Gµν , Bµν , and φ. These field equations can be derived from a spacetime action which is the low energy effective action we are looking for. When we consider a theory with open strings (which will include the closed string sector as well), we can consider the effect of putting in a background field Aµ corresponding to the vector arising from the massless open string excitations. Aµ will couple to the boundary of the open string worldsheet via Z Sboundary = dτ Aµ (X)∂τ X µ . (53) ∂M

When we look at the low energy effective action on the Dp-brane world volume, we will split this up into the p-dimensional vector living on the world volume and the d − (p + 1) remaining scalars δi corresponding to the dualized components of Aµ . Thus Z Sboundary =

ds

p X

0

p

m

Am (x , . . . , x )∂τ X +

m=0

Z ds

25 X

ˆi . δi (x0 , . . . , xp )∂n X

(54)

i=p+1

Adding this to the closed string terms in Equation 52, and demanding Weyl invariance of the 2D worldsheet theory as before gives us the low energy effective action for the Dp-brane:

12

ef f SDp−brane

Z = −Tp

dp+1 ξ e−φ det1/2 (gmn + bmn + 2πα0 Fmn ) .

(55)

Here we have changed to a new notation, writing the coordinates on the Dp-brane worldvolume as ξ m . Tp is the Dp-brane tension √ (2π α0 )1−p . (56) Tp = 2πα0 Also gmn =

∂X µ ∂X ν Gµν ∂ξ m ∂ξ n

(57)

and likewise for bmn and Fmn are the pullbacks of Gµν , Bµν , and Fµν to the Dp-brane world volume. For Gµν = ηµν , Bµν = 0, and eφ = gs , this is the Born-Infield action for a single Dp-brane. Expanding this to lowest non-trivial order in F, the gauge field portion is Z −Tp (2πα0 )2 ef f dp+1 ξ(Fmn )2 . (58) SDp−brane = 4gs

6.2

Multiple D-branes

We will want to describe a system consisting of an arbitrary number N of Dp-branes. Consider N space-filling D25-branes, necessarily coincident, with open strings stretching in between them. This is no different from open strings propagating in the full 26 dimensional spacetime, with extra Chan-Paton degrees of freedom λij introduced to label which of the N space-filling branes the open strings begin and end on. For N Dp-branes, we have N 2 sectors, each containing one massless vector Aµ , and so we represent them as N ×N matrices (Aµ )ij transforming in the adjoint of a U (N ) Yang-Mills theory. Now let d be the dimensionality of the full spacetime, and consider Dp-branes with p < (d − 1). The massless vector Aµ is broken into a (p + 1) dimensional vector on the Dp-brane plus d − (p + 1) scalars, which we previously called δi , but will now call X i , in anticipation of their interpretation as the oscillations of the branes in the normal directions. We will label these as Am (ξ) and X i (ξ), where m = 0, . . . , p and i = p + 1, . . . , d − 1. Am and X i depend only on ξ m , the coordinates on the Dp-brane world volume, and not on the any of the normal directions. This is because the Dirichlet boundary conditions have removed the zero-modes in the directions normal to the branes. For p < (d − 1), and non-coincident branes, we will only have a U (1)N subgroup of U (N ), with the massless vectors associated to strings beginning and ending on the same brane. The strings that stretch between separated branes must have a non-vanishing minimal length, and as a result will have a mass M=

(brane separation) = (length of stretched string) · (string tension) 2πα0 13

(59)

where the string tension is T1 = 1/2πα0 . The massive strings will be represented in the off-diagonal (Aµ )ij components. If the branes come to coincide, all N 2 sectors will again become massless, and we will recover the full U (N ) gauge symmetry. The low energy effective action for N coinciding branes will be non-Abelian U (N ) YangMills dimensionally reduced to the (p + 1)-dimensional brane world volume. Now the world volume has the same form for all N branes, coinciding or not, so the non-Abelian U (N ) YangMills theory will remain the correct effective action even when the branes are separated. We will work with supersymmetric strings, so what we will actually be interested in is the U (N ) Super-Yang-Mills (SYM) action, but we will just look at the bosonic part for now, and add the fermionic terms later. Recall the standard definition Fµν =

∂Aµ ∂Aν − + i[Aµ , Aν ] ∂xµ ∂xν

(60)

of the gauge field strength. Now whereas the xµ run over all 10 spacetime dimensions, let xm , m = 0, . . . , p be the coordinates ξ m on the brane world volume, and xi , i = p + 1, . . . , d − 1 be the coordinates in the normal directions. Also, split the Aµ into the vector on the world volume Am and the remaining scalars X i . Neither the Am nor the X i depend on the normal coordinates xi , so Fµν can be split up into 3 terms Fmn =

∂An ∂Am − + i[Am , An ] ∂xm ∂xn

(2πα0 )Fmj =

∂Xj + i[Am , Xj ] ≡ Dm Xj ∂xm

(2πα0 )2 Fij = i[Xi , Xj ] The dimensionally reduced action becomes Z
Tp (2πα0 )2 (p+1) 2 2 SY M = − dp+1 ξ Tr Fmn + 2Fmj + Fij2 4gs

(61)

(62)

which after substitution becomes   Z Z Tp (2πα0 )2 Tp 1 1 ([X i , X j ])2 (p+1) p+1 2 p+1 i 2 d ξ TrFmn + d ξ Tr − (Dm X ) + . (63) SY M = − 4gs gs 2 4 (2πα0 )2 The first term here is the p + 1 dimensional Yang-Mills action on the brane, and is the non-Abelian generalization of the Born-Infield action, Equation 58. The second and third terms are the effective action describing the D-brane dynamics. We have left out the supersymmetric terms for now, but will put them in when we study the D0-brane action in the next section.

14

6.3

The D0-brane action

We now want to dimensionally reduce the 10D SYM action to 0+1 dimensions. This will give us a 0+1 dimensional supersymmetric QFT, i.e. a supersymmetric quantum mechanics. We then have a collection of N D0-branes, which, being 0-dimensional, can be thought of as point particles. All 9 spatial directions are now normal to the branes, so we have Dirichlet BCs in all spatial directions. This means that we have no gauge fields Am , and instead have 9 scalars X µ , describing the ’positions’ of the D0-branes. For N D0-branes, these X µ are N × N U (N ) matrices, and we will see later that the eigenvalues of these matrices, at large brane separations, can be interpreted as the branes’ coordinates in spacetime. In addition to the X µ , the SYM action also contains a 16-real-component spinor Ψ, which will also be an N × N matrix in the adjoint of U (N ). The SYM action dimensionally reduced to 0+1 dimensions is then   Z 1 µν µ D0 ¯ Dµ Ψ Fµν F + iΨΓ (64) S = T0 dt Tr − 4gs c2 with

1 c≡ 2πα0

0



Γ =

0 1 −1 0



j



Γ =

0 γj

γj 0

 (65)

where γ i , i = 1, . . . , 9 are  realsymmetric 16 × 16 gamma matrices of SO(9). Ψ is a θ Majorana-Weyl spinor Ψ = and θ is a 16 component real spinor. T0 is the D0-brane 0 √ tension T0 = 1/ α0 ≡ 1/ls .We have Fij = ic2 [X i , X j ]

F0j = cD0 X j ≡ c∂0 X j + ic[A0 , X j ]

Dj θ = ic[Xj , θ]

D0 θ = ∂0 θ + i[A0 , θ]

(66)

so that finally S

D0



Z = T0

dt Tr


2 1 c2 i 2 T i T j (D0 X ) − iθ D0 θ + [X , Xj ] + cθ γ [Xj , θ] . 2gs 4gs

(67)

This is our final form for the D0-brane action. It is a supersymmetric matrix quantum mechanics for X i and θ in the adjoint of U (N ) (N × N hermitian matrices), and each component of the θ matrix is a real 16 component spinor.

7

The BFSS conjecture

In this section, we will motivate a conjecture by Banks, Fischler, Shenker, and Susskind, which relates the physics of our D0-brane matrix model to that of 11 dimensional M-theory in a particular reference frame. We saw previously that in the strong coupling limit of type IIA string theory, an 11th dimension appears, and that the Kaluza-Klein states of 11 dimensional Supergravity correspond to bound states at threshold of N D0-branes. We have 15

also seen how the low energy effective action of a system of n D0-branes is described by a 10 dimensional U (N ) Super Yang-Mills theory reduced to 0+1 dimensions, in other words, an N×N matrix quantum mechanics. The principal idea behind the conjecture of BFSS is to interpret the 9 spatial coordinates i X of the matrix model as the transverse dimensions of the full 11D M-theory in what is called the light-cone or Infinite Momentum Frame (IMF), with the longitudinal direction compactified on a circle of radius R. Recall that in the light-cone quantization of the bosonic string, we had a Hamiltonian of the form H = H⊥ /p+ with H⊥ the Hamiltonian for the d − 2 transverse degrees of freedom. We found however that the string had full Lorentz invariance in all d directions. Presently, we would like to interpret the matrix model Hamiltonian as H⊥ for M-theory, and hope that the theory retains 11D Lorentz invariance. We will now define the infinite momentum frame, and then formulate M-theory in it. The IMF is a reference frame in which we have boosted to a very large momentum in a direction we will call P~ . This is our longitudinal direction, and the remaining transverse directions will be called p~a⊥ . After boosting in the P~ direction, the momentum of the ath particle in the system is p~a = ηa P~ + p~a⊥

(68)

with X

p~a⊥ · P~ = 0

ηa = 1,

(69)

The boost in the P~ direction, sufficiently large, will make all the ηa positive, so that all particles will have large momentum in the P direction.The energy of the ath particle is Ea =

p

p2a + m2a = ηa P +

(pa⊥ )2 + m2a (pa )2 + O(P −2 ) ' ⊥ + (ηa P + m2a ) 2ηa P 2ηa P

(70)

which has the form of a d − 2 dimensional system living in the transverse space, with a non-relativistic mass equal to ηa P . This is the infinite momentum frame. Let’s now look at M-theory in the IMF. We will have p0 , the transverse momenta pi (i = 1, . . . , 9) which we identify with p⊥ , and p11 which will be the total longitudinal momentum P a ηa P = p11 . After boosting p11  p⊥ , we compactify x11 on a circle of radius R, so that p11 = N/R is the non-relativistic 10D mass.The total energy becomes  X  (pa )2 N X (pa⊥ )2 ⊥ tot 2 + m = + (71) E = p11 + a a a 2p R 2p 11 11 a a where the 11D mass term m2a has been dropped since the 11D graviton multiplet is massless. The energy takes a non-relativistic form with an added term from the longitudinal momentum modes. We have full Galilean invariance in the 9D transverse space, and in fact will have super-Galilean invariance due to the 32 real supersymmetry generators. Previously, we saw how each D0-brane carried one unit of RR charge for Aµ , which corresponded to p11 = 1/R and was identified as the D0-brane mass. We had an infinite 16

tower of Kaluza-Klein modes, which when compactified on R11 had momenta p11 = N/R, and were identified with bound states of N D0-branes. We identify the D0-brane bound states with the p11 = N/R term in the energy, Equation 71. The type IIA string states are massless and hence have N = 0. Anti-D0-brane states have N < 0. When we go to the IMF by taking p11 → ∞ the degrees of freedom will not include these states, but rather only the D0-branes. Also, as p11 → ∞, we will take the uncompactified limit R → ∞ as well as take the number N of D0-branes to infinity. This will give us uncompactified M-theory in the IMF. All the groundwork is now laid for the BFSS conjecture: Conjecture (BFSS). M-theory in the IMF is a system where the only degrees of freedom are D0-branes, each with p11 = 1/R. The system is described by an effective action for N D0-branes, which is an N × N matrix quantum mechanics, taken in the N → ∞ limit.

8

Matrix Model Hamiltonian

We will now take the action of our D0-brane quantum mechanics, and go to a Hamiltonian formulation. We will see that we arrive at the same Hamiltonian as we obtained from quantizing the supermembrane (Equation 50). We first rescale the fields X i to agree with the Weyl rescaling of the metric done in Equation 11: X i = gs1/3 Y i (72) as well as the time t t = gs2/3 τ =

T0 R 1/3

τ

(73)

gs

so that we have a Hamiltonian with energy dimensions. We also define the conjugate mo∂ ) menta to Y i and θ (· is ∂τ Y˙ i Πi = π = −iT0 θT (74) R Plugging these into the action, we arrive at the Hamiltonian  
1 2 c2 T02 i j 2 2 T j H = R Tr Π − [Y , Y ] − cT0 θ γ [Yj , θ] 2 i 4

(75)

The second term in this equation is a non-negative potential, and its form contains a lot of interesting physics which we will now investigate. If [Y i , Y j ] = 0 for some i, j, we can diagonalize Y i and Y j . The eigenvalues will be the coordinates of the D0-branes. The interaction potential will then be minimized along these directions. If the branes are far apart, the Y i will be large and noncommutativity will cost much energy. In this case, for minimized potential, the positions of the D0-branes will then be more or less well defined. On the other hand, if the branes are “close”, non-commuting 17

Y i ’s will not cost significantly more energy than non-commuting ones, and the idea of the branes having a definite position is less meaningful. The concept of ordinary commutative space only has validity when the branes are sufficiently far apart. The matrix model Hamiltonian has the capability to describe any number of supergravitons, and is therefore a second quantized theory. Let’s first see how it describes one supergraviton. For N = 1, we have a single D0-brane, with p11 = R1 and H(N =1) =

R 2 R 2 p2 Πi = p ⊥ = ⊥ 2 2 2p11

(76)

which, comparing to Equation 71, we see is the relativistic energy-momentum relation of a massless particle in the IMF. The sixteen θ’s generate a 28 = 256 dimensional supermultiplet, so we see that for N = 1, the spectrum of the Hamiltonian is an eleven dimensional massless supermultiplet, which is exactly the supergravity multiplet. So we have identified a single supergraviton in the Hamiltonian’s spectrum. For N > 1, we separate the Y i and Πi into center-of-mass and relative coordinates: 1 i i i (77) Y i = Yrel + Ycm 1 Ycm = Tr Y i N 1 cm Πi = Πrel Pi 1 Pc mi = Tr Πi (78) i + N i where Tr Yrel = Tr Πrel i = 0. Plugging these in and separating the Hamiltonian gives H = Hcm + Hrel

(79)

with

2 (pcm i ) (80) 2p11 Hrel looks like Equation 75 but now the matrices are traceless, i.e. in the SU (N ) representation. It has been shown that Hrel has zero-energy bound states, in which case the total energy is just the center-of-mass energy

Hcm =

E = Ecm =

1 2 (pcm ⊥ ) 2p11

(81)

which is again the expression for a massless multiplet of 256 states, i.e. the supergraviton. We conclude that for any N , the matrix model Hamiltonian contains single supergraviton states in its spectrum. To describe more than one supergraviton, we consider the case in which the Y i and Πi are exactly block-diagonal. Now the total Hamiltonian may be split into a sum of n decoupled Hamiltonians Ha , one for each block of size Na . In each block matrix Yai , we can again separate into center-of-mass and relative coordinates and momenta, and identify a supergraviton residing in the spectrum of each block’s Hamiltonian Ha . As we go to the N → ∞ limit, we can get an arbitrarily large number of supergravitons. For this reason, matrix theory is second quantized theory. We can get interactions by looking at the off-blockdiagonal elements, which represent amplitudes for strings stretching between otherwise free supergravitons. 18

9

Conclusion

M-theory has provided great insight into the connections between various string theories and the non-perturbative aspects of these theories. It suggests that perhaps the true nature of string theory is found in the 11th dimension. Matrix theory is a remarkably simple-looking theory compared to typical quantum field theories, and amazingly enough describes the dynamics of both membranes and D0-branes. Through the BFSS conjecture, we see that matrix theory is a very promising candidate to describe M-theory in the IMF as well. Multiple gravitons can be constructed in the theory, and simple supergravity-style calculations have been performed in the matrix formalism which agree with the traditional supergravity results to the two-loop level. While perhaps not as satisfying as a fully Lorentz-invariant formulation of M-theory, matrix theory thus far has been very successful and has a promising future.

References [1] M. Kaku, Introduction to Superstrings and M-Theory, 2nd ed.(1999). [2] A. Bilal, “M(atrix) Theory: A Pedagogical Introduction,” hep-th/9710136. [3] E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B443 (1996) 85, hep-th/9510135. [4] S. Sethi and M. Stern, “D-brane bound states redux,” hep-th/9705046. [5] M. Porrati and A. Rozenberg, “Bound states at threshold in supersymmetric quantum mechanics,” hep-th/9708119. [6] W. Taylor, “M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory,” hep-th/0101126. [7] J. Hoppe, Quantum Theory of a Massless Relativistic Surface, MIT Ph.D Thesis, 1982. [8] D. Fairlie, P. Fletcher, and C. Zachos, “Trigonometric Structure Constants For New Infinite Algebras,” Phys. Lett. B218 (1989) 203. [9] E. Floratos, “The Heisenberg-Weyl Group on the Z(N ) × Z(N ) Discretized Torus Membrane,” Phys. Lett. B228 (1989) 335. [10] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, “Toeplitz Quantization of K¨ahler Manifolds and gl(N ) N → ∞,” Commun. Math. Phys. 165 (1994) 281, hep-th/9309134.

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