- Email: [email protected]
Loop Variables and Gauge Invariant Exact Renormalization Group Equations for (Open) String Theory
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116 www.elsevier.com/locate/npbps
Loop Variables and Gauge Invariant Exact Renormalization Group Equations for (Open) String Theory B. Sathiapalan Institute of Mathematical Sciences Taramani, Chennai 600041 India
Abstract The sigma model renormalization group formalism is manifestly background independent and is a possible way of obtaining a background independent string field theory. An exact renormalization group equation is written down for the world sheet theory describing the bosonic open string in general backgrounds and loop variable techniques are used to make the equation gauge invariant. The equations are quadratic in fields as in open string field theory. Some explicit examples are given and results are also given for curved space time. In contrast to BRST string field theory, the gauge transformations are not modified by the interactions. As in the Dirac-Born-Infeld action for massless fields, the interactions for massive fields can also be written in terms of gauge invariant field strengths. Keywords: gauge invariance, string field theory, renormalization group, higher spin
1. Introduction The renormalization group (RG) β- functions for the world sheet action for a string propagating in nontrivial backgrounds gives equations of motion for those fields[[1]-[14]]. These are typically non polynomial. However the Exact Renormalization Group (ERG) gives quadratic equations and these are related to string field theory equations.The first systematic attempt to connect string field theory with the ERG was made in [9, 8]. Loop Variable techniques have been developed to make implement gauge invariance of these RG equations [15, 16, 17]. The free equations were written down long back. However the interacting equations that were obtained, while gauge invariant, were not in a form convenient for a space time field theory interpretation. The aim of this work is to write down a gauge invariant exact renormalization group (ERG) and the equations are quadratic. The free part is the same as in the earlier works. The interaction terms are manifestly gauge invariant because they are written in terms of gauge invariant field strengths. In this aspect it differs http://dx.doi.org/10.1016/j.nuclphysbps.2014.04.019 0920-5632/© 2014 Elsevier B.V. All rights reserved.
from BRST string field theory. BRST string field theory [23, 24, 25] also gives quadratic equations. However (unlike in BRST string field theory), the gauge transformations of the interacting theory is the same as that of the free theory. It is also shown that there is a relatively straightforward generalization to curved space. In Section 2 we give a description of the Exact Renormalization Group. In Section 3 we describe the basic aspects of loop variables. In Section 4 we apply these ideas and give examples of interacting equations of motion of some massive higher spin fields of string theory. Section 5 describes a generalization to curved space. Section 6 gives a summary and lists some open questions. This talk is based on [26, 27]. These papers are referred to below as I and II respectively. More details and references are given there. 2. RG in Position Space In this section we derive the exact RG in position space. This is essentially a repetition of Wilson’s orig-
112
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
By choosing a, b, B suitably ( b = 2a f, B =
inal derivation [19]. Note that usual discussions of the ERG use momentum space rather than position space. We start with point particle quantum mechanics. (This section is a review of ERG and also of some results from [18] which is reproduced here for convenience.)
∂ψ ∂ ∂ = B (a + bx)ψ(x, τ) ∂τ ∂x ∂x we get ∂L ∂L f˙ ∂2 L = 2 [ 2 + ( )2 ] ∂τ 2 f ∂x ∂x
2.1. Quantum Mechanics
∂ψ ∂2 ψ =− 2 ∂t ∂y
(2.1.1) i
∂ψ ∂ ∂ = ( + x)ψ ∂τ ∂x ∂x
(2.1.2)
with Green’s function G(x2 , τ2 ; x1 , 0) = e 2π(1 − e−2τ2 )
2
−
(x2 −x1 e−τ2 ) 2(1−e−2τ2 )
(2.1.3)
Thus as τ2 → ∞ it goes over to √12π e− 2 x2 . As τ2 → 0 it goes to δ(x1 − x2 ). dx1G(x2 , τ2 ; x1 , 0)ψ(x1 , 0) ψ(x2 , τ2 ) = 1 2
So ψ(x2 , τ2 ) goes from being unintegrated ψ(x1 ) to 1 2 completely integrated √12π e− 2 x2 dx1 ψ(x1 ). Thus consider ∂ ∂ ∂ ( + x2 )ψ(x2 , τ) ψ(x2 , τ) = ∂τ ∂x2 ∂x2
(2.1.5)
2.2. Field Theory We now apply this to a Euclidean field theory.
(y2 −y1 )2
for which the Green’s function is √2π(t1 −t ) e 2(t2 −t1 ) , and 2 1 change variables :y = xeτ , it = e2τ and ψ = eτ ψ to get the differential equation
1
) in
Note that if f = G−1 (G is like the propagator) then f˙ = −G˙ f2
We start with the Schrodinger equation i
f˙ bf
(2.1.4)
with initial condition ψ(x, 0) Thus if we define Z(τ) = dx2 ψ(x2 , τ), where ψ obeys the above equation, we d Z = 0. Also for τ = 0 ψ is the unintegrated see that dτ ψ(x, 0). At τ = ∞ it is proportional to the integrated object dxψ(x, 0). Z(τ) has the same value. Thus as τ increases the integrand in Z is more completely integrated. We need to repeat this for the case where the initial i wave function is replaced by e S [x] where x denotes the space-time coordinates. Then for τ = ∞ ψ ≈ DxeiS [x] the integrated partition function. At τ = 0 it is the unintegrated eiS [x] . Z(τ) is the fully integrated partition function for all τ. We shall also split the action into a kinetic term and interaction term as in [22]. Thus in the quantum mechanical case discussed above we write 1 2 ψ = e− 2 x f (τ)+L(x)
ψ = e− 2 1
dz dz X(z)G−1 (z,z )X(z )+ dzL[X(z),X (z)]
(2.2.6)
Here X (z) = ∂z X(z). In general there could be higher derivatives X (z), X (z).... The equations can easily be generalized to include those cases. We apply the operator δ δ dz dz B(z, z ) [ + b(z, z )X(z )](2.2.7) δX(z ) δX(z) to ψ and require, as before, that this should be equal to ∂ψ ∂τ . Using δ du L[X(u), X (u)] δX(z) ∂L ∂L = du [ δ(u − z) + ∂u δ(u − z)](2.2.8) ∂X(u) ∂X (u) dropping terms independent of X (an unimportant over˙ z ) all constant), choosing b = 2G−1 , and B = − 12 G(z, we get:
∂L =− dz ∂τ
−∂z [
dz
⎛ ⎜⎜ 2 ˙ z )⎜⎜⎜⎜⎜ ∂ L[X(z), X (z)] δ(z − z ) dz G(z, 2 ⎝ 2 ∂X(z) 1
∂2 L[X(z), X (z)] ∂2 L[X(z), X (z)] ]δ(z − z ) + ∂z ∂z [ δ(z − z )] 2 ∂X(z)∂X (z) ∂X (z) ∂L[X(z), X (z)] ∂L[X(z), X (z)] − ∂z ] +[ ∂X(z) ∂X (z) ⎞ ∂L[X(z ), X (z )] ∂L[X(z ), X (z )] ⎟⎟⎟⎟⎟ [ − ∂z ] ⎟⎟⎠ (2.2.9) ∂X(z ) ∂X (z )
If we now interpret τ as ln a this becomes easy to interpret as an RG equation diagrammatically as done in [22]: the first curved bracket in the RHS which is linear in L represents contractions of fields at the same point - self contractions within an operator, and the second curved bracket represents contractions between fields at two different points - between two different operators. In terms of space-time fields, first term gives the free equations of motion and the second gives the interactions.
113
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
• Step 3: Make the loop variable gauge covariant: Introduce ”einbein” α(s),
3. Loop Variables • There are three steps involved in this formalism:
L[k(s)] = eik0 .X(z)+
• Step 1: Define the Loop Variable: L[k(s)] = e ik0 X(z)+i
c
ds k(s)∂z X(z+as)
where α(s) is:
X(z + as) = X(z) + as∂z X(z) +
α(s) = 1 +
1 2 2 2 a s ∂z X(z) + .... 2! k(s) = k0 +
+
k1 k2 + 2 + .... s s
L[k(s)]
=
eik0 .X(z)+
=
ei[k0 .X(z)+k1 ∂z X(z)+k2 ∂
=
eik0 .X(z) {1 + ik1 ∂z X(z) + ik2 ∂2 X(z) −
X(z)+...+kn
∂nz X(z) (n−1)! +...]
k1μ k1ν ∂z X μ (z)∂z X ν (z) + ...} 2 This contains all the open string vertex operators. The ERG will be written in terms of these k’s. But at the end of the day we have to map k’s to space time fields. • Step 2: Define space-time fields in terms of a “string field” Φ[kn ]: 1 = φ(k0 )
scalar tachyon
k1μ = Aμ (k0 ) ”photon” μν (k0 ) k1μ k1ν = S 1,1
massive spin 2
k2μ = S 2μ (k0 ) where ... =
[dkn ]...Φ[kn ]
α3 + ... s3
Y
=
X + α1 ∂z X + α2 ∂2z X + α3
∂3z X + + ... 2!
Let us now introduce xn by the following:
−m α(s) = αn s−n = e m≥0 s xm n≥0
They satisfy the property, ∂αn = αn−m , n ≥ m ∂xm
ds k(s)∂z X(z+s)
2
α1 α2 + 2 s s
• Let us define
• The index μ in knμ runs over all the space time dimensions plus one internal dimension. This component of kn is denoted by qn . Thus q0 plays the role of a mass parameter. Formally the free equations are written as massless equations in one higher dimension and dimensional reduction gives mass. We will not give these details here. We refer the reader to I and II.
ds α(s)k(s)∂z X(z+s)
Using this we see that if we define: Yn =
∂Y ∂xn
Then L[k(s)] = ei
n k n Yn
• xn are an infinite number of proper time coordinates. (Reminiscent of the KP hierarchy). They generalize the t of the open string vertex operator location. They (or αn ) parametrize the reparametrizations of the boundary - open string. We need to integrate over these. So we assume the vertex operators are accompanied by Dα(s) ≈ [dxn ]. • Gauge transformations take a very simple form k(s) → λ(s)k(s) - scale transformations! This is equivalent to discarding total derivatives in xn . 4. Applications Let us apply ERG to loop variable. Note that in the ERG expression we have used a compact notation: X (z) ∂Y stands for all first derivatives ∂x n
114
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
4.1. Free Equations We get for the Linear term for massless spin one : [k02 ik1μ − k1 .k0 ik0μ ]
∂Y =0 ∂x1
and for spin 2: −k02 k1μ k1ν
+
k1(μ k0ν) k1 .k0
−
k0μ k0ν k1 .k1
μ
∂ Aμ − ∂ ∂.A = ∂ν F
μν
A
B
dzA
= 0
These equations are gauge invariant under k(s) → λ(s)k(s). Thus kn → kn + p λ p kn−p . e.g. k2 → k2 + λ1 k1 + λ2 k0 We can now map to space time fields using the map given in Step 2. This gives: 2
Explicit expressions for all n are given in II. Thus the quadratic terms in the ERG takes the following form ˙ A , zB ) (Lμ (zA ) + Lμ (zA ))(Lμ (zB ) + Lμ (zB )) = G(z 1 2 1 2 ˙ A , zB ) dzBG(z
[Fρν Y1ν (zA ) + V2ρν Y2ν (zA ) + V11ρμν Y1μ (zA )Y1ν (zA )]eik0 (A).Y(zA ) ρ β α ik0 (B).Y(zB ) [F ρα Y1α (zB ) + V2ρ α Y2α (zB ) + V11 αβ Y1 (z B )Y1 (z B )]e
where V2μν ≡ [−k0μ k2ν + k1μ k1ν − k2μ k0ν + q2 k0μ k0ν ]
= 0
1 1 1 μν μ)ρ [ (∂2 − 1)S 11 − ∂ρ ∂(ν S 11 + ∂(μ S 2μ) + 2 2 2
1 1 1 ρμν V11 ≡ [− k0ρ k1μ k1ν + k1ρ (k1μ k0ν + k1ν k0μ ) − (k2ρ − q2 k0ρ )k0μ k0ν ] 2 2 2
1 μ ν ρ 1 ∂ ∂ S 11ρ − ∂μ ∂ν S 2 ] = 0 2 2
are gauge invariant ”field strengths” for the massive fields. An operator product expansion can now be made. The details are in I and II. Final Result for massive spin 2 gauge invariant interaction equation is the following (there are an infinite number of other interaction terms involving higher modes): ˙ A , zA )( 1 (∂2 − 1)S μν − 1 ∂ρ ∂(ν S μ)ρ + 1 ∂(μ S μ) [ dzAG(z 11 11 2 2 2 2
and mapping the gauge transformations gives for e.g k1 → k1 + k0 λ1 , which becomes Aμ → Aμ + ∂μ Λ. For the massive field we have performed a dimensional reduction to obtain a mass term. 4.2. Interactions For the interactions, if we use the same loop variable action it turns out the quadratic term is not gauge invariμ ant! We need to rewrite ik2μ ∂Y ∂x2 as iK2μ
1 1 ρ + ∂μ ∂ν S 11ρ − ∂μ ∂ν S 2 ])+ 2 2 ˙ A , zB ) V λρμ (∂ρ F λν )G10 (zA , zB ) dzA dzB G(z 11
2 μ ∂Y μ μ ∂ Y + iK11 ∂x2 ∂x12
μ where δK2μ = λ2 k0μ , δK11 = λ1 k1μ q¯ 21 μ 2 )k0 ,
K2μ
μ K11
k2μ
λρσ −(zB − zA )V11 (∂ρ ∂σ ∂ν Fλμ )
K2μ
= (q¯ 2 − = − The requirement is that the gauge variation of the above expression give derivatives of lower dimensional vertex operators. This is easily seen to be the case, for the gauge transformation is: ∂ μ ∂Y μ ∂ λ1 ik + λ2 (ikμ Y μ ) ∂x1 1 ∂x1 ∂x2 0
λμν ρ Fλ +∂ρ V11
+
The analogous construction for level 3: Write k3μ = μ μ + K21 + K111 and
+
2 μ 3 μ ∂Y μ μ ∂ Y μ ∂ Y + iK21 + iK111 − K2μ K1ν Y2μ Y1ν ] ∂x3 ∂x2 ∂x1 ∂x13
μ −K11 K1ν
k1μ k1ν k1ρ μ ν ρ ik0 Y ∂2 Y μ ν Y − i Y1 Y1 Y1 ]e 3! ∂x12 1
G0,1 λρμ ν ∂ ∂ρ Fλσ (G1,0G0,1 ) − (zB − zA )∂σ V11 2
(zB − zA )2 λρσ )(∂ρ ∂σ ∂μ ∂ν Fλδ )G21,0G0,1 (∂δ V11 4
λρσ μν λρμ σν −G21,0 (∂σ V11 )(∂ρ ∂σ V11λ ) − G21,0 (∂σ V11 )(∂ρ V11λ )
K3μ
L = [iK3μ
G210 2
(zB − zA ) 2 λρσ )(∂ν ∂ρ ∂σ V11λαμ )+ G1,0G0,1 (∂α V11 2
(zB − zA ) λρσ αμ )(∂ρ ∂σ V11λ ) G1,0G20,1 (∂ν ∂α V11 2 −
G21,0G20,1 (zB − zA )2 λρσ αβ )(∂μ ∂ν ∂ρ ∂σ V11λ ) (∂α ∂β V11 + ... = 0 2 4
115
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
δ[S 2ρ −
5. Loop Variables in Curved Space Ordinarily flat space expressions can be taken over to curved space by covariantizing derivatives. Here there is a problem: k0μ k0ν = k0ν k0μ , whereas [Dμ Dν ] 0. So one cannot map k0μ → Dμ . At first sight a solution is to define Riemann Normal Coordinates (RNC) yμ at the point and define k0μ = ∂y∂μ . These commute, and when expressed in terms of covariant derivatives there are terms involving Rανρμ . So we seem to get a map to space time. However there are problems: Consider the following example: k0ρ k1μ k1ν → ∂ρ S 1,1μν ↓ G ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) → ∂ρ (∂μ Λ1,1ν + μ ↔ ν) We can conclude from the above diagram that δS 1,1μν = ∂(μ Λ1,1ν) In curved space then this should become:
But consider the curved space analog of the diagram: Dρ S 1,1μν ↓ G Dρ (Dμ Λ1,1ν + μ ↔ ν) + 13 (Rανρμ + Rαμρν )Λ1,1α
This is clearly not consistent with the above gauge transformation law! Something has to give! Let us modify the top horizontal arrow in the diagram: Modify the map to space time of terms that contain k0 but no λ (i.e. do not modify the lower horizontal arrow): k0ρ k1μ k1ν → ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) →
Dρ S 1,1μν + ΔS ↓ G Dρ (Dμ Λ1,1ν + μ ↔ ν) + 13 (Rανρμ + Rαμρν )Λ1,1α
Choose ΔS such that: δΔS =
1 α (R + Rαμρν )Λ1,1α 3 νρμ
This is always possible because this is a massive spin two (after dimensional reduction) we have the following gauge transformations: δS 2ρ = k0ρ Λ2 + Λρ1,1 δS 25 = 2k05 Λ2 = 2mΛ2
So ΔS =
1 α 1 (R + Rαμρν )[S 2α − ∂α S 25 ] 3 νρμ 2m
Note that the m → 0 result is not well defined. This can be done term by term in the loop variable equations. The result is that the map becomes well defined and is consistent with gauge invariance and general covariance. The result for a massive spin 2 is: μν F ρρμν + (k05 )2 S 1,1 − Fρ(νμ)ρ + Fμνρ ρ +
D(μ S 2ν) (k05 )2 − D(μ Dν) S 2 k05 = 0 Dρ Dρ S 2μ − F ρμρ + Fμρρ − Dμ Dρ S 2ρ = 0
(5.0.10)
ρ − 2Dρ S 2ρ (k05 )2 − Dρ Dρ S 2 k05 = 0 −F ρσρσ + F ρρ σσ − (k05 )2 S 1,1ρ
Here: 1 Fρσμν = Dρ Dσ S μν + (Rαμρσ + Rασρμ )S αν + 3
δS 1,1μν = D(μ Λ1,1ν)
k0ρ k1μ k1ν → ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) →
1 ρ 5 ∂ S 2 ] = Λρ1,1 2k05
1 α + Rασρν )S μα + fρσμν (R 3 νρσ and fρσμν =
5
i fρσμν
i=1
Dα S 25 1 1 fρσμν = − (Rαμρσ + Rασρμ + Rαρσμ )Dν (S 2α − ) 6 2k05 Dα S 25 1 2 fρσμν = − (Rανρσ + Rασρν + Rαρσν )Dμ (S 2α − ) 6 2k05 3 fρσμν =
Dα S 25 2 α ) (R νρμ + Rαμρν )Dσ (S 2α − 3 2k05
4 fρσμν =
Dα S 25 2 α ) (R νσμ + Rαμσν )Dρ (S 2α − 3 2k05
5 fρσμν =[
7 (Dρ Rαμσν + Dρ Rανσμ )+ 12
1 1 1 (Dσ Rαμρν + Dσ Rανρμ ) + (− Dμ Rαρνσ + Dμ Rασνρ ) 4 4 12 Dα S 25 1 1 ) +(− Dν Rαρμσ + Dν Rασμρ )](S 2α − 4 12 2k05
116
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
Thus we finally get 1 Fρσμν = Dρ Dσ S μν + (Rαμρσ + Rασρμ )S αν + 3 1 α + Rασρν )S μα + fρσμν (R 3 νρσ By construction the equations (5.0.10) are gauge invariant under δS 1,1μν = D(μ Λ1,1ν) , δS 2μ = Λ1,1μ + ∂μ Λ2 , δS 2 = Λ2 k05 . 6. Summary and Conclusions • In this talk we have shown how the loop variable approach can be used to obtain Gauge and Generally Covariant interacting equations for the open string modes. • The approach is manifestly background independent at least for gravitational backgrounds. • The equations seem to have an origin in one higher dimension. • The gauge transformations are not modified by interactions - as befits an Abelian theory. The interactions are expressed in terms of gauge invariant ”field strengths” - similar to Born-Infeld. We conclude with some open questions: • The Higher Dimensional Structure is very intriguing. It needs to be understood better. • Can we do closed strings using the same techniques? • Is there an an Action? In curved space time the requirement that the free equations come from an action introduces extra terms into the equation gauge invariant terms that vanish in flat space. • What does this imply for non critical string theory? References [1] C. Lovelace, Phys. Lett. B135,75 (1984). [2] C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B262,593 (1985). [3] A. Sen, Phys. Rev. D32,2102 (1985). [4] E. Fradkin and A.A. Tseytlin, Phys. Lett. B151,316 (1985). [5] C. Callan and Z. Gan, Nucl. Phys. B272, 647 (1987) [6] S. Das and B. Sathiapalan, Phys. Rev. Lett. B183,65 (1985). [7] B. Sathiapalan, Nucl. Phys. B294, (1987) 747.
[8] J. Hughes,J. Liu and J. Polchinski, Nucl. Phys. B316 (1989). [9] T. Banks and E. Martinec, Nucl. Phys. B294, 733 (1987). [10] V.A. Kostelecky, M.J. Perry, and R. Potting, Phys. Rev. Lett. 84 (2000) 4541, hepth/9912243. [11] I.L. Buchbinder, O.M.Gitman,V.A. Krykhtin and Y.D. Pershin, hepth/9910188, Nucl. Phys. B584 (2000) 615. [12] A. Abouelsaood, C. G. Callan, C. R. Nappi and S. A. Yost, Nucl. Phys. B280,(1989) 599. [13] A. Polyakov, “Gauge Fields and Strings”, Harwood Academic Publishers, New York, 1987. [14] A.A. Tseytlin, Int. J. Mod. Phys. A4:4249 (1989) and references therein. [15] B. Sathiapalan, Nucl. Phys. B326 (1989)376. [16] B. Sathiapalan, hepth/0207098, Int.J.Mod. Phys. A18 (2003) 767-810. [17] B. Sathiapalan, hep-th/0405119 ,Mod.Phys.Lett. A19 (2004) 2857. [18] B. Sathiapalan, Mod. Phys. Lett A 19 (2007) 1701. [19] K.G. Wilson and J. Kogut, Phys. Reports 12 , (1974)75. [20] K.G.Wilson, Rev. Mod. Phys. 47, (1975)773. [21] K.G. Wilson, Phs. Rev. D7 (1973)2911. [22] J. Polchinski, Nucl. Phys. B231 (1984) 269. [23] W. Siegel and B. Zwiebach, Nucl. Phys. B263 (1986) 105. [24] W. Siegel, Phys. Lett B149, 157; 162 (1984);B151 391;396 (1985). [25] E. Witten, Nucl. Phys B268, (1986) 513. [26] B. Sathiapalan, arXiv:1202.4298, Nucl.Phys. B862 (2012) 4374. [27] B. Sathiapalan, arXiv:1209.1447, Nucl.Phys. B868 (2013) 1637.
Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116 www.elsevier.com/locate/npbps
Loop Variables and Gauge Invariant Exact Renormalization Group Equations for (Open) String Theory B. Sathiapalan Institute of Mathematical Sciences Taramani, Chennai 600041 India
Abstract The sigma model renormalization group formalism is manifestly background independent and is a possible way of obtaining a background independent string field theory. An exact renormalization group equation is written down for the world sheet theory describing the bosonic open string in general backgrounds and loop variable techniques are used to make the equation gauge invariant. The equations are quadratic in fields as in open string field theory. Some explicit examples are given and results are also given for curved space time. In contrast to BRST string field theory, the gauge transformations are not modified by the interactions. As in the Dirac-Born-Infeld action for massless fields, the interactions for massive fields can also be written in terms of gauge invariant field strengths. Keywords: gauge invariance, string field theory, renormalization group, higher spin
1. Introduction The renormalization group (RG) β- functions for the world sheet action for a string propagating in nontrivial backgrounds gives equations of motion for those fields[[1]-[14]]. These are typically non polynomial. However the Exact Renormalization Group (ERG) gives quadratic equations and these are related to string field theory equations.The first systematic attempt to connect string field theory with the ERG was made in [9, 8]. Loop Variable techniques have been developed to make implement gauge invariance of these RG equations [15, 16, 17]. The free equations were written down long back. However the interacting equations that were obtained, while gauge invariant, were not in a form convenient for a space time field theory interpretation. The aim of this work is to write down a gauge invariant exact renormalization group (ERG) and the equations are quadratic. The free part is the same as in the earlier works. The interaction terms are manifestly gauge invariant because they are written in terms of gauge invariant field strengths. In this aspect it differs http://dx.doi.org/10.1016/j.nuclphysbps.2014.04.019 0920-5632/© 2014 Elsevier B.V. All rights reserved.
from BRST string field theory. BRST string field theory [23, 24, 25] also gives quadratic equations. However (unlike in BRST string field theory), the gauge transformations of the interacting theory is the same as that of the free theory. It is also shown that there is a relatively straightforward generalization to curved space. In Section 2 we give a description of the Exact Renormalization Group. In Section 3 we describe the basic aspects of loop variables. In Section 4 we apply these ideas and give examples of interacting equations of motion of some massive higher spin fields of string theory. Section 5 describes a generalization to curved space. Section 6 gives a summary and lists some open questions. This talk is based on [26, 27]. These papers are referred to below as I and II respectively. More details and references are given there. 2. RG in Position Space In this section we derive the exact RG in position space. This is essentially a repetition of Wilson’s orig-
112
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
By choosing a, b, B suitably ( b = 2a f, B =
inal derivation [19]. Note that usual discussions of the ERG use momentum space rather than position space. We start with point particle quantum mechanics. (This section is a review of ERG and also of some results from [18] which is reproduced here for convenience.)
∂ψ ∂ ∂ = B (a + bx)ψ(x, τ) ∂τ ∂x ∂x we get ∂L ∂L f˙ ∂2 L = 2 [ 2 + ( )2 ] ∂τ 2 f ∂x ∂x
2.1. Quantum Mechanics
∂ψ ∂2 ψ =− 2 ∂t ∂y
(2.1.1) i
∂ψ ∂ ∂ = ( + x)ψ ∂τ ∂x ∂x
(2.1.2)
with Green’s function G(x2 , τ2 ; x1 , 0) = e 2π(1 − e−2τ2 )
2
−
(x2 −x1 e−τ2 ) 2(1−e−2τ2 )
(2.1.3)
Thus as τ2 → ∞ it goes over to √12π e− 2 x2 . As τ2 → 0 it goes to δ(x1 − x2 ). dx1G(x2 , τ2 ; x1 , 0)ψ(x1 , 0) ψ(x2 , τ2 ) = 1 2
So ψ(x2 , τ2 ) goes from being unintegrated ψ(x1 ) to 1 2 completely integrated √12π e− 2 x2 dx1 ψ(x1 ). Thus consider ∂ ∂ ∂ ( + x2 )ψ(x2 , τ) ψ(x2 , τ) = ∂τ ∂x2 ∂x2
(2.1.5)
2.2. Field Theory We now apply this to a Euclidean field theory.
(y2 −y1 )2
for which the Green’s function is √2π(t1 −t ) e 2(t2 −t1 ) , and 2 1 change variables :y = xeτ , it = e2τ and ψ = eτ ψ to get the differential equation
1
) in
Note that if f = G−1 (G is like the propagator) then f˙ = −G˙ f2
We start with the Schrodinger equation i
f˙ bf
(2.1.4)
with initial condition ψ(x, 0) Thus if we define Z(τ) = dx2 ψ(x2 , τ), where ψ obeys the above equation, we d Z = 0. Also for τ = 0 ψ is the unintegrated see that dτ ψ(x, 0). At τ = ∞ it is proportional to the integrated object dxψ(x, 0). Z(τ) has the same value. Thus as τ increases the integrand in Z is more completely integrated. We need to repeat this for the case where the initial i wave function is replaced by e S [x] where x denotes the space-time coordinates. Then for τ = ∞ ψ ≈ DxeiS [x] the integrated partition function. At τ = 0 it is the unintegrated eiS [x] . Z(τ) is the fully integrated partition function for all τ. We shall also split the action into a kinetic term and interaction term as in [22]. Thus in the quantum mechanical case discussed above we write 1 2 ψ = e− 2 x f (τ)+L(x)
ψ = e− 2 1
dz dz X(z)G−1 (z,z )X(z )+ dzL[X(z),X (z)]
(2.2.6)
Here X (z) = ∂z X(z). In general there could be higher derivatives X (z), X (z).... The equations can easily be generalized to include those cases. We apply the operator δ δ dz dz B(z, z ) [ + b(z, z )X(z )](2.2.7) δX(z ) δX(z) to ψ and require, as before, that this should be equal to ∂ψ ∂τ . Using δ du L[X(u), X (u)] δX(z) ∂L ∂L = du [ δ(u − z) + ∂u δ(u − z)](2.2.8) ∂X(u) ∂X (u) dropping terms independent of X (an unimportant over˙ z ) all constant), choosing b = 2G−1 , and B = − 12 G(z, we get:
∂L =− dz ∂τ
−∂z [
dz
⎛ ⎜⎜ 2 ˙ z )⎜⎜⎜⎜⎜ ∂ L[X(z), X (z)] δ(z − z ) dz G(z, 2 ⎝ 2 ∂X(z) 1
∂2 L[X(z), X (z)] ∂2 L[X(z), X (z)] ]δ(z − z ) + ∂z ∂z [ δ(z − z )] 2 ∂X(z)∂X (z) ∂X (z) ∂L[X(z), X (z)] ∂L[X(z), X (z)] − ∂z ] +[ ∂X(z) ∂X (z) ⎞ ∂L[X(z ), X (z )] ∂L[X(z ), X (z )] ⎟⎟⎟⎟⎟ [ − ∂z ] ⎟⎟⎠ (2.2.9) ∂X(z ) ∂X (z )
If we now interpret τ as ln a this becomes easy to interpret as an RG equation diagrammatically as done in [22]: the first curved bracket in the RHS which is linear in L represents contractions of fields at the same point - self contractions within an operator, and the second curved bracket represents contractions between fields at two different points - between two different operators. In terms of space-time fields, first term gives the free equations of motion and the second gives the interactions.
113
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
• Step 3: Make the loop variable gauge covariant: Introduce ”einbein” α(s),
3. Loop Variables • There are three steps involved in this formalism:
L[k(s)] = eik0 .X(z)+
• Step 1: Define the Loop Variable: L[k(s)] = e ik0 X(z)+i
c
ds k(s)∂z X(z+as)
where α(s) is:
X(z + as) = X(z) + as∂z X(z) +
α(s) = 1 +
1 2 2 2 a s ∂z X(z) + .... 2! k(s) = k0 +
+
k1 k2 + 2 + .... s s
L[k(s)]
=
eik0 .X(z)+
=
ei[k0 .X(z)+k1 ∂z X(z)+k2 ∂
=
eik0 .X(z) {1 + ik1 ∂z X(z) + ik2 ∂2 X(z) −
X(z)+...+kn
∂nz X(z) (n−1)! +...]
k1μ k1ν ∂z X μ (z)∂z X ν (z) + ...} 2 This contains all the open string vertex operators. The ERG will be written in terms of these k’s. But at the end of the day we have to map k’s to space time fields. • Step 2: Define space-time fields in terms of a “string field” Φ[kn ]: 1 = φ(k0 )
scalar tachyon
k1μ = Aμ (k0 ) ”photon” μν (k0 ) k1μ k1ν = S 1,1
massive spin 2
k2μ = S 2μ (k0 ) where ... =
[dkn ]...Φ[kn ]
α3 + ... s3
Y
=
X + α1 ∂z X + α2 ∂2z X + α3
∂3z X + + ... 2!
Let us now introduce xn by the following:
−m α(s) = αn s−n = e m≥0 s xm n≥0
They satisfy the property, ∂αn = αn−m , n ≥ m ∂xm
ds k(s)∂z X(z+s)
2
α1 α2 + 2 s s
• Let us define
• The index μ in knμ runs over all the space time dimensions plus one internal dimension. This component of kn is denoted by qn . Thus q0 plays the role of a mass parameter. Formally the free equations are written as massless equations in one higher dimension and dimensional reduction gives mass. We will not give these details here. We refer the reader to I and II.
ds α(s)k(s)∂z X(z+s)
Using this we see that if we define: Yn =
∂Y ∂xn
Then L[k(s)] = ei
n k n Yn
• xn are an infinite number of proper time coordinates. (Reminiscent of the KP hierarchy). They generalize the t of the open string vertex operator location. They (or αn ) parametrize the reparametrizations of the boundary - open string. We need to integrate over these. So we assume the vertex operators are accompanied by Dα(s) ≈ [dxn ]. • Gauge transformations take a very simple form k(s) → λ(s)k(s) - scale transformations! This is equivalent to discarding total derivatives in xn . 4. Applications Let us apply ERG to loop variable. Note that in the ERG expression we have used a compact notation: X (z) ∂Y stands for all first derivatives ∂x n
114
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
4.1. Free Equations We get for the Linear term for massless spin one : [k02 ik1μ − k1 .k0 ik0μ ]
∂Y =0 ∂x1
and for spin 2: −k02 k1μ k1ν
+
k1(μ k0ν) k1 .k0
−
k0μ k0ν k1 .k1
μ
∂ Aμ − ∂ ∂.A = ∂ν F
μν
A
B
dzA
= 0
These equations are gauge invariant under k(s) → λ(s)k(s). Thus kn → kn + p λ p kn−p . e.g. k2 → k2 + λ1 k1 + λ2 k0 We can now map to space time fields using the map given in Step 2. This gives: 2
Explicit expressions for all n are given in II. Thus the quadratic terms in the ERG takes the following form ˙ A , zB ) (Lμ (zA ) + Lμ (zA ))(Lμ (zB ) + Lμ (zB )) = G(z 1 2 1 2 ˙ A , zB ) dzBG(z
[Fρν Y1ν (zA ) + V2ρν Y2ν (zA ) + V11ρμν Y1μ (zA )Y1ν (zA )]eik0 (A).Y(zA ) ρ β α ik0 (B).Y(zB ) [F ρα Y1α (zB ) + V2ρ α Y2α (zB ) + V11 αβ Y1 (z B )Y1 (z B )]e
where V2μν ≡ [−k0μ k2ν + k1μ k1ν − k2μ k0ν + q2 k0μ k0ν ]
= 0
1 1 1 μν μ)ρ [ (∂2 − 1)S 11 − ∂ρ ∂(ν S 11 + ∂(μ S 2μ) + 2 2 2
1 1 1 ρμν V11 ≡ [− k0ρ k1μ k1ν + k1ρ (k1μ k0ν + k1ν k0μ ) − (k2ρ − q2 k0ρ )k0μ k0ν ] 2 2 2
1 μ ν ρ 1 ∂ ∂ S 11ρ − ∂μ ∂ν S 2 ] = 0 2 2
are gauge invariant ”field strengths” for the massive fields. An operator product expansion can now be made. The details are in I and II. Final Result for massive spin 2 gauge invariant interaction equation is the following (there are an infinite number of other interaction terms involving higher modes): ˙ A , zA )( 1 (∂2 − 1)S μν − 1 ∂ρ ∂(ν S μ)ρ + 1 ∂(μ S μ) [ dzAG(z 11 11 2 2 2 2
and mapping the gauge transformations gives for e.g k1 → k1 + k0 λ1 , which becomes Aμ → Aμ + ∂μ Λ. For the massive field we have performed a dimensional reduction to obtain a mass term. 4.2. Interactions For the interactions, if we use the same loop variable action it turns out the quadratic term is not gauge invariμ ant! We need to rewrite ik2μ ∂Y ∂x2 as iK2μ
1 1 ρ + ∂μ ∂ν S 11ρ − ∂μ ∂ν S 2 ])+ 2 2 ˙ A , zB ) V λρμ (∂ρ F λν )G10 (zA , zB ) dzA dzB G(z 11
2 μ ∂Y μ μ ∂ Y + iK11 ∂x2 ∂x12
μ where δK2μ = λ2 k0μ , δK11 = λ1 k1μ q¯ 21 μ 2 )k0 ,
K2μ
μ K11
k2μ
λρσ −(zB − zA )V11 (∂ρ ∂σ ∂ν Fλμ )
K2μ
= (q¯ 2 − = − The requirement is that the gauge variation of the above expression give derivatives of lower dimensional vertex operators. This is easily seen to be the case, for the gauge transformation is: ∂ μ ∂Y μ ∂ λ1 ik + λ2 (ikμ Y μ ) ∂x1 1 ∂x1 ∂x2 0
λμν ρ Fλ +∂ρ V11
+
The analogous construction for level 3: Write k3μ = μ μ + K21 + K111 and
+
2 μ 3 μ ∂Y μ μ ∂ Y μ ∂ Y + iK21 + iK111 − K2μ K1ν Y2μ Y1ν ] ∂x3 ∂x2 ∂x1 ∂x13
μ −K11 K1ν
k1μ k1ν k1ρ μ ν ρ ik0 Y ∂2 Y μ ν Y − i Y1 Y1 Y1 ]e 3! ∂x12 1
G0,1 λρμ ν ∂ ∂ρ Fλσ (G1,0G0,1 ) − (zB − zA )∂σ V11 2
(zB − zA )2 λρσ )(∂ρ ∂σ ∂μ ∂ν Fλδ )G21,0G0,1 (∂δ V11 4
λρσ μν λρμ σν −G21,0 (∂σ V11 )(∂ρ ∂σ V11λ ) − G21,0 (∂σ V11 )(∂ρ V11λ )
K3μ
L = [iK3μ
G210 2
(zB − zA ) 2 λρσ )(∂ν ∂ρ ∂σ V11λαμ )+ G1,0G0,1 (∂α V11 2
(zB − zA ) λρσ αμ )(∂ρ ∂σ V11λ ) G1,0G20,1 (∂ν ∂α V11 2 −
G21,0G20,1 (zB − zA )2 λρσ αβ )(∂μ ∂ν ∂ρ ∂σ V11λ ) (∂α ∂β V11 + ... = 0 2 4
115
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
δ[S 2ρ −
5. Loop Variables in Curved Space Ordinarily flat space expressions can be taken over to curved space by covariantizing derivatives. Here there is a problem: k0μ k0ν = k0ν k0μ , whereas [Dμ Dν ] 0. So one cannot map k0μ → Dμ . At first sight a solution is to define Riemann Normal Coordinates (RNC) yμ at the point and define k0μ = ∂y∂μ . These commute, and when expressed in terms of covariant derivatives there are terms involving Rανρμ . So we seem to get a map to space time. However there are problems: Consider the following example: k0ρ k1μ k1ν → ∂ρ S 1,1μν ↓ G ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) → ∂ρ (∂μ Λ1,1ν + μ ↔ ν) We can conclude from the above diagram that δS 1,1μν = ∂(μ Λ1,1ν) In curved space then this should become:
But consider the curved space analog of the diagram: Dρ S 1,1μν ↓ G Dρ (Dμ Λ1,1ν + μ ↔ ν) + 13 (Rανρμ + Rαμρν )Λ1,1α
This is clearly not consistent with the above gauge transformation law! Something has to give! Let us modify the top horizontal arrow in the diagram: Modify the map to space time of terms that contain k0 but no λ (i.e. do not modify the lower horizontal arrow): k0ρ k1μ k1ν → ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) →
Dρ S 1,1μν + ΔS ↓ G Dρ (Dμ Λ1,1ν + μ ↔ ν) + 13 (Rανρμ + Rαμρν )Λ1,1α
Choose ΔS such that: δΔS =
1 α (R + Rαμρν )Λ1,1α 3 νρμ
This is always possible because this is a massive spin two (after dimensional reduction) we have the following gauge transformations: δS 2ρ = k0ρ Λ2 + Λρ1,1 δS 25 = 2k05 Λ2 = 2mΛ2
So ΔS =
1 α 1 (R + Rαμρν )[S 2α − ∂α S 25 ] 3 νρμ 2m
Note that the m → 0 result is not well defined. This can be done term by term in the loop variable equations. The result is that the map becomes well defined and is consistent with gauge invariance and general covariance. The result for a massive spin 2 is: μν F ρρμν + (k05 )2 S 1,1 − Fρ(νμ)ρ + Fμνρ ρ +
D(μ S 2ν) (k05 )2 − D(μ Dν) S 2 k05 = 0 Dρ Dρ S 2μ − F ρμρ + Fμρρ − Dμ Dρ S 2ρ = 0
(5.0.10)
ρ − 2Dρ S 2ρ (k05 )2 − Dρ Dρ S 2 k05 = 0 −F ρσρσ + F ρρ σσ − (k05 )2 S 1,1ρ
Here: 1 Fρσμν = Dρ Dσ S μν + (Rαμρσ + Rασρμ )S αν + 3
δS 1,1μν = D(μ Λ1,1ν)
k0ρ k1μ k1ν → ↓ G λ1 k0ρ (k0μ k1ν + μ ↔ ν) →
1 ρ 5 ∂ S 2 ] = Λρ1,1 2k05
1 α + Rασρν )S μα + fρσμν (R 3 νρσ and fρσμν =
5
i fρσμν
i=1
Dα S 25 1 1 fρσμν = − (Rαμρσ + Rασρμ + Rαρσμ )Dν (S 2α − ) 6 2k05 Dα S 25 1 2 fρσμν = − (Rανρσ + Rασρν + Rαρσν )Dμ (S 2α − ) 6 2k05 3 fρσμν =
Dα S 25 2 α ) (R νρμ + Rαμρν )Dσ (S 2α − 3 2k05
4 fρσμν =
Dα S 25 2 α ) (R νσμ + Rαμσν )Dρ (S 2α − 3 2k05
5 fρσμν =[
7 (Dρ Rαμσν + Dρ Rανσμ )+ 12
1 1 1 (Dσ Rαμρν + Dσ Rανρμ ) + (− Dμ Rαρνσ + Dμ Rασνρ ) 4 4 12 Dα S 25 1 1 ) +(− Dν Rαρμσ + Dν Rασμρ )](S 2α − 4 12 2k05
116
B. Sathiapalan / Nuclear Physics B (Proc. Suppl.) 251–252 (2014) 111–116
Thus we finally get 1 Fρσμν = Dρ Dσ S μν + (Rαμρσ + Rασρμ )S αν + 3 1 α + Rασρν )S μα + fρσμν (R 3 νρσ By construction the equations (5.0.10) are gauge invariant under δS 1,1μν = D(μ Λ1,1ν) , δS 2μ = Λ1,1μ + ∂μ Λ2 , δS 2 = Λ2 k05 . 6. Summary and Conclusions • In this talk we have shown how the loop variable approach can be used to obtain Gauge and Generally Covariant interacting equations for the open string modes. • The approach is manifestly background independent at least for gravitational backgrounds. • The equations seem to have an origin in one higher dimension. • The gauge transformations are not modified by interactions - as befits an Abelian theory. The interactions are expressed in terms of gauge invariant ”field strengths” - similar to Born-Infeld. We conclude with some open questions: • The Higher Dimensional Structure is very intriguing. It needs to be understood better. • Can we do closed strings using the same techniques? • Is there an an Action? In curved space time the requirement that the free equations come from an action introduces extra terms into the equation gauge invariant terms that vanish in flat space. • What does this imply for non critical string theory? References [1] C. Lovelace, Phys. Lett. B135,75 (1984). [2] C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B262,593 (1985). [3] A. Sen, Phys. Rev. D32,2102 (1985). [4] E. Fradkin and A.A. Tseytlin, Phys. Lett. B151,316 (1985). [5] C. Callan and Z. Gan, Nucl. Phys. B272, 647 (1987) [6] S. Das and B. Sathiapalan, Phys. Rev. Lett. B183,65 (1985). [7] B. Sathiapalan, Nucl. Phys. B294, (1987) 747.
[8] J. Hughes,J. Liu and J. Polchinski, Nucl. Phys. B316 (1989). [9] T. Banks and E. Martinec, Nucl. Phys. B294, 733 (1987). [10] V.A. Kostelecky, M.J. Perry, and R. Potting, Phys. Rev. Lett. 84 (2000) 4541, hepth/9912243. [11] I.L. Buchbinder, O.M.Gitman,V.A. Krykhtin and Y.D. Pershin, hepth/9910188, Nucl. Phys. B584 (2000) 615. [12] A. Abouelsaood, C. G. Callan, C. R. Nappi and S. A. Yost, Nucl. Phys. B280,(1989) 599. [13] A. Polyakov, “Gauge Fields and Strings”, Harwood Academic Publishers, New York, 1987. [14] A.A. Tseytlin, Int. J. Mod. Phys. A4:4249 (1989) and references therein. [15] B. Sathiapalan, Nucl. Phys. B326 (1989)376. [16] B. Sathiapalan, hepth/0207098, Int.J.Mod. Phys. A18 (2003) 767-810. [17] B. Sathiapalan, hep-th/0405119 ,Mod.Phys.Lett. A19 (2004) 2857. [18] B. Sathiapalan, Mod. Phys. Lett A 19 (2007) 1701. [19] K.G. Wilson and J. Kogut, Phys. Reports 12 , (1974)75. [20] K.G.Wilson, Rev. Mod. Phys. 47, (1975)773. [21] K.G. Wilson, Phs. Rev. D7 (1973)2911. [22] J. Polchinski, Nucl. Phys. B231 (1984) 269. [23] W. Siegel and B. Zwiebach, Nucl. Phys. B263 (1986) 105. [24] W. Siegel, Phys. Lett B149, 157; 162 (1984);B151 391;396 (1985). [25] E. Witten, Nucl. Phys B268, (1986) 513. [26] B. Sathiapalan, arXiv:1202.4298, Nucl.Phys. B862 (2012) 4374. [27] B. Sathiapalan, arXiv:1209.1447, Nucl.Phys. B868 (2013) 1637.