Progress in String Field Theory

Progress in

String Field Theory

Syoji Zeze

Yokote Seiryo Gakuin High School

2 April, 2010 National Taiwan U.

• A very simple solution for tachyon condensation in string field theory is found

[Erler-Schnabl, arXiv: 0906.0979]

• We apply their solution to various problems of tachyon condensation

[S.Z., JPS in Okayama, 2010, to appear in arXiv]

• Classical potential

• Effective potential

• Interpretation from world sheet geometry

They are exact result

Claim

Tachyon

Condensation

in SFT

Sen s conjecture

unstable D-brane no open string

1998

t V

tension of unstable

D-brane t₀

V (t0) SFT can derive

this potential

10 yeas history

year brane tension

(ratio) method ^authors

1999 0.9864 level 4 Sen-Zwiebach, (Kostelecky-

Samuel)

2000 0.9912 level 10 Moeller-Taylor

2002 1.00049 level 18 Gaiotto-Rastelli

2005 1 exact Schnabl

2009 1 exact Erler-Schnabl

Exact era

Level truncation era

Erler-Shnabl solution

• Analytic solution of the classical SFT

• Lorentz invariant, i.e. scalar field

with no space time dependence

Schnabl Erler Schnabl form of the

solution

Infinite

sum Integral

`Phantom

term exists none

D-brane

tension OK OK

Homotopy

operator exists exists

Gauge equivalent

K B c subalgebra

B, c, K: string fields (identity based)

{B, c} = 1 ^{[K, B] = 0}

Q _B c = cKc

Q

K = 0

Q

B = K

π c(1) |I�

B = π

2 B_L |I�

K = π

2 K_L |I�

Okawa

= �

0

dt e

c(1 + K)BcΩ

= �

0

dt c(1 + K)Bce

Ψ = c(1 + K)Bc 1

1 + K

Ω

midpoint

π 2

Ω ^t

π 2 t

Level expansion in

dressed gauge

Level expansion

Ψ = cf(K)Bc 1

1 + K

f (K) = 1 + K

Erler-Schnabl

f (K) = t₀ + t₁K + t₂K² + · · ·

S[Ψ] = T r[ 1

2 ΨQ

Ψ + 1

3 Ψ

]

Dressed gauge condition

[B₀⁻{Ψ(1 + K)}] 1

1 + K = 0

3 The Tachyon Potential

In this section, we evaluate the tachyon potential V = 1

2Tr[ΨQ_BΨ] + 1

3Tr[Ψ³] (10)

for the string field (8). All of the calculation can be preformed by employing a procedure developed in Ref. [1]. We evaluate the tachyon potential order by order with respect to the expansion (9). We assign ‘level’ n to the nth order term in f (K), since the corresponding string field is an eigenstate of ‘dressed’ L₀ operator with eigenvalue n − 1.¹ Let us write the potential up to level N as

V = 1

2t_mK_mnt_n + 1

3V_mnpt_mt_nt_p, (11) where the sum over m,n, p is taken from 0 to N. The coefficients K_mn and V_mnp can be obtained by plugging the level n field

ψ_n ^{= cKnBc 1}

1 + K (12)

= lim

s→0(∂_s^{)n ! ∞}

0 e^−tdt cΩ^sBcΩ^t. (13) into the classical action. It is easily found that both K_mn and V_mnp can be evaluated in terms of the the well-known trace[2, 3]

g(r₁,r₂, r₃,r₄) ≡ Tr[BcΩ^r1cΩ^r2cΩ^r3cΩ^r4]. (14) More explicitly,

K_mn = lim

s₁→0,s2→0(−∂_s1^)m₍₋∂_s2^{)n ! ∞}

0 dt₁ ^{! ∞}

0 dt₂e^−t1−t2h₂(t₁,t₂, s₁,s₂), (15) where

h₂(t₁,t₂, s₁,s₂) = − lim

u→0∂_u^"g(t₂^,s₁ ^+t₁^,u,s_{2) − g(t2}^{, s}₁^,t₁ ^{+ u,s}₂⁾ + g(t₂,s₁,t₁, u + s₂) − g(t1,s₂ + u,t₂, s₁)

+ g(u,t₂,s₁ +t₁, s₂) − g(u,t²,s₁,t₁ + s₂)#

(16)

1The ‘dressed’ L0 is given by L Ψ = ¹₂L₀⁻ {Ψ(1 + K)}, where L₀⁻ = L₀ − L₀^†.

3

and

V_mnp = lim

s₁→0,s2→0,s3→0(−∂_s1^)m₍₋∂_s2⁾ⁿ₍₋∂_s3^{)p ! ∞}

0 dt₁ ^{! ∞}

0 dt₂ ^{! ∞}

0 dt₃

e^{−t1−t2−t3}h₃(t₁,t₂,t₃, s₁,s₂,s₃), (17) where

h₃(t₁,t₂,t₃,s₁,s₂,s₃) = g(t₃,s₁ +t₁, s₂ +t₂,s₃) − g(t3, s₁ +t₁,s₂,t₂ + s₃)

− g(t3, s₁,t₁ +t₂ + s₂,s₃) + g(t₃, s₁,t₁ + s₂,t₂ + s₃). (18) In [1], the integrals in the kinetic term (which corresponds to K_mn in our paper) is performed with the help of the reparametrization

t₁ = uv, t₂ = u(1 − v) (19)

where u = t₁ +t₂ parametrize the width of the strip world sheet in the sliver frame.

Under this reparametrization, the masure of the integral is transformed as

! _∞

0 dt₁ ^{! ∞}

0 dt₂ →

! _∞

0 du ^{! 1}

0 dvu. (20)

While the qubic term is not calculated in [1], we find that similar reparametrization also works. Thus the integrals in V_mnp can be preformed with the replacement

t₁ = uv₁, t₂ = uv₂, t₃ = u(1 − v1 − v2). (21) where u = t₁ +t₂ +t₃ also represents the width of the world sheet. The integration mesure is also be rewritten into

! _∞

0 dt₁ ^{! ∞}

0 dt₂ ^{! ∞}

0 dt₃ →

! _∞

0 du^{! 1}

0 dv₁ ^{! 1−v1}

0 dv₂ u². (22) After integrating out v₁, v₂ and v₃, we obtain the potentail as an integral with respetct to the width u as

V = ^{! ∞}

0 due^−uA(u,t_n), (23)

where A(u,t_n) is finite order in u, possibly includes negative powers of u. This prescription in terms of world sheet width u helps us to understand the relation between world sheet fluctuation and the tachyon potential.

4

Tr[ψ_mQ_Bψ_n]

Tr[ψ_mψ_nψ_p]

3 The Tachyon Potential

In this section, we evaluate the tachyon potential V = 1

2Tr[ΨQ_BΨ] + 1

3Tr[Ψ³] (10)

for the string field (8). All of the calculation can be preformed by employing a procedure developed in Ref. [1]. We evaluate the tachyon potential order by order with respect to the expansion (9). We assign ‘level’ n to the nth order term in f (K), since the corresponding string field is an eigenstate of ‘dressed’ L₀ operator with eigenvalue n − 1.¹ Let us write the potential up to level N as

V = 1

2t_mK_mnt_n + 1

3V_mnpt_mt_nt_p, (11) where the sum over m,n, p is taken from 0 to N. The coefficients K_mn and V_mnp can be obtained by plugging the level n field

ψ_n = cKⁿBc 1

1 + K (12)

= lim

s→0(∂_s)^{n ! ∞}

0 e^−tdt cΩ^sBcΩ^t. (13) into the classical action. It is easily found that both K_mn and V_mnp can be evaluated in terms of the the well-known trace[2, 3]

g(r₁, r₂, r₃, r₄) ≡ Tr[BcΩ^r1cΩ^r2cΩ^r3cΩ^r4]. (14) More explicitly,

K_mn = lim

s₁→0,s2→0(−∂^s₁)^m(−∂^s₂)^{n ! ∞}

0 dt₁ ^{! ∞}

0 dt₂e^−t1−t2h₂(t₁,t₂, s₁, s₂), (15) where

h₂(t₁,t₂, s₁, s₂) = − lim

u→0 ∂_u"

g(t₂, s₁ +t₁, u,s₂) − g(t2, s₁,t₁ + u,s₂) + g(t₂,s₁,t₁, u + s₂) − g(t¹, s₂ + u,t₂, s₁)

+ g(u,t₂, s₁ +t₁,s₂) − g(u,t², s₁,t₁ + s₂)#

(16)

1The ‘dressed’ L₀ is given by L Ψ = ¹₂L₀⁻ {Ψ(1 + K)}, where L₀⁻ = L₀ − L₀^†.

3

Level expansion terminates at level 3 because

tr[K · · · ] → lim

s→0

∂

sin t

u + s ∼ t

u

cos t

u

higher order of K higher rank derivative integrand contains sin(s/u) factor

(Erler-Shnabl: solution of EOM terminates at level 1)

(m,n, p) v_m,n,p(u) (m,n, p) v_m,n,p(u) (0,0,0) _4π^3u5₄ (0,0,1) (−15+π²)^u4

6π⁴

(0,1,1) −(−15+π²)^u3

3π⁴ (0,0,2) −(−15+4π²)^u3

3π⁴

(1,1,2) ²(−15+π²)^u

π⁴ (0,2,2) ²(−15+4π²)^u

π⁴

(1,2,2) −⁴(−15+π²)

π⁴ (0,0,3) −²(−6+π²)^u2

3π²

(0,1,3) −²(45−9π²+π⁴)^u

3π⁴ (1,1,3) −⁴(−45+6π²+π⁴)

3π⁴

(0,2,3) ⁴(45−15π²+2π⁴)

3π⁴

Table 1: A complete list of v_m,n,p(u) up to level 3. Other coefficients are zero.

swer for the tachyon potential is obtained by performing integration with respect to u, together with the e^−u factor. We present the final expression below.

V = 1 π²

! 15

64π^{2t03 −}

15t₁

16π^{2t02 +} 1

16t₁t₀² + 15

16π^{2t2t02 −} 1

4t₂t₀² − 1

12π^2t₃^t₀² + 1

2t₃t₀² − 3

32t₀² + 15

16π^{2t12t0 −} 1

16t₁²t₀ − 15

4π^{2t22t0 +t22t0 +} 3

4t₂t₀ − 1

6 π²t₁t₃t₀

− 15

2π^{2t1t3t0 +} 3

2t₁t₃t₀ + 4

3π²t₂t₃t₀ + 30

π^{2t2t3t0 − 10t2t3t0 +} π²t₃t₀ − 3t³t₀ + 15

π^{2t1t22 −t1t22 −}

15

4π^{2t12t2 +} 1

4t₁²t₂ − 1

3π²t₁²t₃ + 15

π^{2t12t3 − 2t12t3}

"

(28)

3.2 The instability of the classical potential

In last section, we have seen that the level expansion in the dressed B₀ gauge termnates at level 3 within the KBc subalgegra. This implies that any result ob- tained from the potential (28) is exact within our setup, although the KBc subalge- bra covers very limited part of the whole spcae of the string fields. Therefore we can expect some exact result for the tachyon potential from our analysis. Since the potential is completely known, we only need to follow the proceder as in Ref. [5].

The closed string vacuum is given by the configuration (t₀,t₁,t₂,t₃) = (1,1,0,0).

We have cheked that it is a stationary point of (28), and furthermore gives correct D-brane tension −1/2π². To see whether the vacuum is stable or not, we evaluate

6

Level 3 potential

exact !

fields:

Is closed string vacuum stable ?

(t₀, t₁, t₂, t₃) = (1, 1, 0, 0)

Eigenvalues of Hessian

Hessian matrix at the closed string vacuum

H_{i j} = ∂ ²V

∂^t_i∂^t _j ^{. (29)}

The closed string vacuum is stable(unstable) if all eigenvalues of H_{i j} is posi- tive(negative). Otherwise, it is a saddle point.

The Hessian matrix for this configuration is







180 + 6π² _{−180 + 12}π² _{180 − 30}π² _{−180 + 48}π²

−180 + 12π² _{180 − 12}π² _{−180 + 12}π² _{180 − 12}π² _{− 12}π⁴ 180 − 30π² _{−180 + 12}π² 180 + 24π² _{360 − 120}π² + 16π⁴

−180 + 48π² _{180 − 12}π² _{− 12}π⁴ _{360 − 120}π² + 16π⁴ 0





, and eigenvalues is found to be (30)

1487.14, −1261.59, 412.919, 79.1831. (31) Hessian matrix has a negative eigenvalue while others are positive. This con- cludes that the closed string vacuum of Erler and Schnabl is a suddle point. It should be stressed that the four eigenvalues obtained here do not change even if we include other field outside KBc subalgebra or outside the dressed B₀ gauge, since such fields are set to zero on the closed string vacuum. Therefore our result is completely exact and does not depend on choice of basis for string fields.

3.3 Effective Potential

The effective potential for tachyon can be obtained by eliminating fields other than tahyon field from the classical potential by solving equations of motion. In this paper, we identified t₀ as the tachyon mode, also such choice is ad-hoc, but seems to be natural. Therefore we have to solve

∂^V

∂ t_i = 0 (i = 0,1,2,3) (32)

for fields t₁,t₂,t₃. In general, there appear many branches since the equations of motion is cubic in each t_i. However, one can see that (28) is linear in t₃. Therefore, t₃ is an auxiliary field and can be eliminated by imposing

∂^V

∂ t₃ = 0. (33)

7

Hessian matrix at the closed string vacuum

H

= ∂

^V

∂ ^t

∂ ^t

^{. (29)}

The closed string vacuum is stable(unstable) if all eigenvalues of H

is posi- tive(negative). Otherwise, it is a saddle point.

The Hessian matrix for this configuration is



 



180 + 6 π

_{−180 + 12} π

_{180 − 30} π

_{−180 + 48} π

−180 + 12 π

_{180 − 12} π

_{−180 + 12} π

_{180 − 12} π

_{− 12} π

180 − 30 π

_{−180 + 12} π

180 + 24 π

_{360 − 120} π

+ 16 π

−180 + 48 π

_{180 − 12} π

_{− 12} π

_{360 − 120} π

+ 16 π

0



 

, (30)

and eigenvalues is found to be

1487.14, −1261.59, 412.919, 79.1831. (31) Hessian matrix has a negative eigenvalue while others are positive. This con- cludes that the closed string vacuum of Erler and Schnabl is a suddle point. It should be stressed that the four eigenvalues obtained here do not change even if we include other field outside KBc subalgebra or outside the dressed B

gauge, since such fields are set to zero on the closed string vacuum. Therefore our result is completely exact and does not depend on choice of basis for string fields.

3.3 Effective Potential

The effective potential for tachyon can be obtained by eliminating fields other than tahyon field from the classical potential by solving equations of motion. In this paper, we identified t

as the tachyon mode, also such choice is ad-hoc, but seems to be natural. Therefore we have to solve

∂ ^V

∂ ^t

^{= 0} (i = 0,1,2,3) (32)

for fields t

, t

, t

. In general, there appear many branches since the equations of motion is cubic in each t

. However, one can see that (28) is linear in t

. Therefore, t

is an auxiliary field and can be eliminated by imposing

∂ ^V

∂ ^t

^{= 0. (33)}

7

Unstable (classically)

Arroyo (2009)

�1.0 �0.5 0.0 0.5 1.0

�1.0

�0.5 0.0 0.5 1.0

t₀ t1

Effective Potential

• t_3 and t_2 can be integrated out without specifying branch

closed string vacuum

V = −T^p

Fake

vacuum ?

V = −0.71T^p

Effective potential

•

•

0.5 1.0 1.5 2.0 t₀

�0.05 0.05 0.10

V₁

• Closed string vacuum only

• Stable, no fall-off

• defined for positive

t_0

How identity

based solution

fails?

4 Analysys in terms of width of the strip 5 Identity based solution

Ψ = −cK (34)

= lim

s→0

∂

cΩ

. (35)

asd f (36)

A Formlas

B Summary and Discussion References

[1] T. Erler and M. Schnabl, JHEP 0910, 066 (2009) [arXiv:0906.0979 [hep- th]].

[2] Y. Okawa, JHEP 0604, 055 (2006) [arXiv:hep-th/0603159].

[3] T. Erler, JHEP 0705, 083 (2007) [arXiv:hep-th/0611200].

[4] T. Erler, JHEP 0705, 084 (2007) [arXiv:hep-th/0612050].

[5] E. Aldo Arroyo, JHEP 0910, 056 (2009) [arXiv:0907.4939 [hep-th]].

10

Equation of motion contracted with itself

4 Analysys in terms of width of the strip 5 Identity based solution

Ψ = −cK (34)

= lim

s→0 ∂_scΩ^s. (35)

TrΨQ_BΨ = − lim

s→0 lim

u→0Tr[cΩ^scΩ^ucΩ^s] (36) TrΨ³ = − lim

s→0 Tr[cΩ^sΩ^scΩ^s] (37)

A Formlas

B Summary and Discussion References

[1] T. Erler and M. Schnabl, JHEP 0910, 066 (2009) [arXiv:0906.0979 [hep- th]].

[2] Y. Okawa, JHEP 0604, 055 (2006) [arXiv:hep-th/0603159].

[3] T. Erler, JHEP 0705, 083 (2007) [arXiv:hep-th/0611200].

[4] T. Erler, JHEP 0705, 084 (2007) [arXiv:hep-th/0612050].

[5] E. Aldo Arroyo, JHEP 0910, 056 (2009) [arXiv:0907.4939 [hep-th]].

10

take this first

They differ at finite s because width are different

4 Analysys in terms of width of the strip 5 Identity based solution

Ψ = −cK (34)

= lim

s→0

∂

cΩ

. (35)

TrΨQ

Ψ = − lim

s→0

lim

u→0

Tr[cΩ

cΩ

cΩ

] (36) TrΨ

= − lim

s→0

Tr[cΩ

cΩ

cΩ

] (37)

A Formlas

B Summary and Discussion References

[1] T. Erler and M. Schnabl, JHEP 0910, 066 (2009) [arXiv:0906.0979 [hep- th]].

[2] Y. Okawa, JHEP 0604, 055 (2006) [arXiv:hep-th/0603159].

[3] T. Erler, JHEP 0705, 083 (2007) [arXiv:hep-th/0611200].

[4] T. Erler, JHEP 0705, 084 (2007) [arXiv:hep-th/0612050].

[5] E. Aldo Arroyo, JHEP 0910, 056 (2009) [arXiv:0907.4939 [hep-th]].

10

Tr[cΩ^s1 cΩ^s2 cΩ^s3]

different limits of

4 Analysys in terms of width of the strip 5 Identity based solution

Ψ = −cK (34)

= lim

s→0

∂

cΩ

. (35)

TrΨQ

Ψ = − lim

s→0

lim

u→0

Tr[cΩ

cΩ

cΩ

] (36) Tr Ψ

= − lim

s→0

Tr[c Ω

c Ω

c Ω

] (37) TrΨQ

Ψ = 1

2 + 2

π

⁽³⁸⁾

TrΨ

= 2

9 + 9 √ 3

4 π

⁺

√ 3

2 π ⁽³⁹⁾

A Formlas

B Summary and Discussion References

[1] T. Erler and M. Schnabl, JHEP 0910, 066 (2009) [arXiv:0906.0979 [hep- th]].

[2] Y. Okawa, JHEP 0604, 055 (2006) [arXiv:hep-th/0603159].

[3] T. Erler, JHEP 0705, 083 (2007) [arXiv:hep-th/0611200].

[4] T. Erler, JHEP 0705, 084 (2007) [arXiv:hep-th/0612050].

[5] E. Aldo Arroyo, JHEP 0910, 056 (2009) [arXiv:0907.4939 [hep-th]].

10

4 Analysys in terms of width of the strip 5 Identity based solution

Ψ = −cK (34)

= lim

s→0

∂

cΩ

. (35)

TrΨQ

Ψ = − lim

s→0

lim

u→0

Tr[cΩ

cΩ

cΩ

] (36) TrΨ

= − lim

s→0

Tr[cΩ

cΩ

cΩ

] (37) TrΨQ

Ψ = 1

2 + 2

π

⁽³⁸⁾

TrΨ

= 2

9 + 9 √ 3

4 π

⁺

√ 3

2 π ⁽³⁹⁾

A Formlas

B Summary and Discussion References

[1] T. Erler and M. Schnabl, JHEP 0910, 066 (2009) [arXiv:0906.0979 [hep- th]].

[2] Y. Okawa, JHEP 0604, 055 (2006) [arXiv:hep-th/0603159].

[3] T. Erler, JHEP 0705, 083 (2007) [arXiv:hep-th/0611200].

[4] T. Erler, JHEP 0705, 084 (2007) [arXiv:hep-th/0612050].

[5] E. Aldo Arroyo, JHEP 0910, 056 (2009) [arXiv:0907.4939 [hep-th]].

10

Not a solution !

Finite value of action (D-brane tension) But they doesn t depend of s !

Tr[Ψ(Q_BΨ + Ψ²)] �= 0

Summary

•

•

•

stable in the effective potential

•

Speculation

V =

�

0

du e

A(u)

* u is length of the

boundaries of an open string

* small u divergence corresponds to

zero momentum closed string ?