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Closed Topological CP1 Sigma Model
Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67 www.elsevierphysics.com
Open/Closed Topological CP1 Sigma Model Shmuel Elitzur a , Yaron Oz b , Eliezer Rabinovici
a
and Johannes Walcher
c
a
Racah Institute of Physics, The Hebrew University of Jerusalem, 91904, Israel Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University, Ramat-Aviv 69978, Israel c Department of Physics, CERN - Theory Division, CH-1211 Geneva 23, Switzerland b
We consider the topological sigma-model on Riemann surfaces with genus g and h holes, and target space CP1 ∼ = S 2 . We calculate the correlations function of bulk and boundary operators, and study their symmetries. We study the open/closed TFT correspondence by summing up the boundaries. We argue that this summation can be understood as a renormalization of the closed TFT.
1. Introduction In first quantized string theory one considers in many cases a string moving in a given geometrical background. One then obtains S-matrix elements by adding up contributions of worldsheet calculations on the different world sheet genera. In order to obtain a target space picture one suggests a certain effective Lagrangian defined on the world volume, which was the target space in the worldsheet formulation. This candidate is validated by comparing the S-matrix elements it produces to those obtained from the worldsheet procedure. This straightforward manner does not address various questions such as: the uniqueness of the effective lagrangian and its world volume and topology. One is actually familiar with symmetries such as T-duality and dynamics such as holography which reflect ambiguities of the effective Lagrangian. In the lecture we address this issue in a very simple case, which does allow one to obtain the exact world sheet results. This is the case of topological theories of matter, which allow both closed and open strings. Topological field theories (TFTs) [1] can be used as a simple setup to study open/closed duality properties of string theory. One class of TFTs are the topological σ-models. In order to construct these models we start with a (2, 2) supersymmetric non-linear σ-model in two-dimensions. This is a theory of maps Φ from a two-dimensional 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.07.045
worldsheet Σ to a target space X, which is a Kahler manifold. The two-dimensional theory has a U (1) × U (1) R-symmetry. One can twist the theory by by adding to the stress tensor of the theory a derivative a U (1) R-current. There are two ways to do that, i.e. twisting with the vector symmetry U (1)V or twisting with the axial symmetry U (1)A . The first leads to the topological A-model, while the second to the topological B-model. Due to the axial anomaly, the B-model is well defined only when the target space is a Calabi-Yau manifold. We will consider the topological A-model. After the twisting, the supersymmetry transformation becomes a transformation under a nilpotent operator Q. The action becomes 2 S∼ d z{Q, Λ} + t Φ∗ (K) (1) Σ
Σ
where Φ∗ (K) is the pullback to the worldsheet of the target space Kahler two-form. We will normalize Φ∗ (K) = n , (2) Σ
where n is an integer, the degree of the instanton. The path integral of the theory localizes on holomorphic maps, and the correlation functions of the model depend only on the cohomology class of the Kahler form K.
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
In this lecture we will consider the topological sigma-model on Riemann surfaces with genus g and h holes, and target space CP1 ∼ = S 2 . We calculate the correlations function of bulk and boundary operators, and study their symmetries. We study the open/closed TFT correspondence. More details on these results and a discussion of the coupling to topological gravity will appear elsewhere [2]. 2. The computational scheme We will denote by O1 ...On g,h the correlation function of the operators O1 , ...On on a Riemann surface with genus g and h boundaries. In this section we will outline the computational scheme that we will use in order to calculate these correlation functions. 2.1. The sphere Consider first the correlators on the sphere with no boundaries. There are two operators of the topological CP1 σ-model, the identity operator, 1, and the operator corresponding to the second cohomology class of the sphere, which we shall denote by H. It is represented by a δ-function two-form on CP1 ∼ = S 2 . On the worldsheet H is a zero-form. We have 10,0 = 0 H0,0 = 1 H 2 0,0 = 0
derives the non-trivial ring relation (OPE) H 2 = β1
(4)
2.2. The disk We now want to consider the CP1 model with branes included in the background. As shown in [3], there are two possible branes that preserve topological invariance. Geometrically, both of them correspond to the equator of CP1 , viewed as the two-sphere. The two branes are distinguished by the value = ± of a Wilson line. We consider correlators on the disc, with boundary condition corresponding to one of the two branes. Both branes support, in addition to the identity (which we continue to denote by 1), a boundary operator corresponding to the first cohomology class of the equator circle, and we will denote this operator by E. It is represented by a δ-function one-form on the equator, and is a zero-form on the worldsheet. It is shown in [3] that 10,1 = 0 E0,1 = 1 H0,1 = 0 E 2 0,1 = 0
(5)
E 3 0,1 = β 1/2 EH0,1 = β 1/2
(3)
H 3 0,0 = β where β = e−t comes from the classical action of the CP1 σ-model and is the contribution of one-instanton, i.e. a degree one holomorphic map from the worldsheet to the CP1 target space. The one-point correlator of H is constant since it gets contribution only from the constant map: we map the worldsheet two-sphere to the target space two-sphere with the point where H is inserted on the worldsheet being mapped to a given point in the target space. The three-point correlator of H gets contribution from a degree one holomorphic map: we map the three insertion points to three given points on the target space. From (3), one
Here, it is understood that E will be inserted on the boundary, while H is inserted in the bulk of the disc. The one-point correlator of E on the disk is constant since it receives contribution only from the constant map: we map the disk worldsheet to the target space S 2 , such that the boundary of the disk is mapped to the equator and the insertion point on the boundary of the disk is mapped to a given point on the equator. This holomorphic map is the constant map. The three-point correlator of E receives contribution from the disk one-instanton (or half-instanton in the closed string sense), which is a degree one map from the disk to S 2 , where the boundary of the disk is mapped to the equator. The three insertion points on the boundary of the disk are mapped to three points on the equator.
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
The equations (5) imply the non-trivial relation on the boundary E 2 = β 1/2
(6)
as well as the important relation H = E 2 = β 1/2 1
(7)
between the bulk field H and the boundary field E 2 . In other words, when computing a correlator on a Riemann surface with boundary, an insertion of H in the bulk is equivalent to inserting E 2 on the boundary (or equivalently, inserting β 1/2 1, in the bulk or on the boundary). It is also important to note that there are no boundary condition changing operators between = + and = −. 2.3. Axial R-charge We assign axial R-charges to the operators R[H] = 2,
R[E] = 1
(8)
In general, an amplitude H E g,h will be proportional to β k if k is an integer (without boundaries) or half-integer (with boundaries) satisfying Ri = 2n + m = 4k + 2 − 2g − h (9) n
m
i
If there is no such k the amplitude vanishes. Note that a (g, h) amplitude will vanish if both boundary conditions appear at the boundaries of the surface. This is because there are no boundary condition changing operators. Therefore, we will choose equal for all boundaries and fix it. Moreover, the amplitude will vanish unless E is inserted an odd number of times on each boundary. 2.4. Handle and boundary states There exists a universal operator W , that relates correlators on surfaces of different topology and it called the handle operator. It has the property Og,h = W Og−1,h
(10)
One can compute W at g = 1 by degenerating the torus into a sphere. Here, the relations W 0,0 = 11,0 = 2H0,0 = 2 W H0,0 = H1,0 = 2H 0,0 = 0 2
(11)
imply W = 2H
(12)
When considering surfaces with boundaries, we have to fix boundary conditions a1 , a2 , . . . , ah on each of them. Moreover, we have to allow for a dependence of the boundary state on the boundary insertions. We will then label the boundary states as Va,θ to indicate dependence on the boundary condition and the boundary insertion. It satisfies Oθ1 · · · θh g,h = Oθ1 · · · θh−1 Vah ,θh g,h−1 (13) it being understood that the operator θi is inserted on the i-th boundary on the left and on the right hand side of the equation. The boundary states can be computed on the disc. From (5), we learn V,1 = 0
(14)
V,E = β 1/2 1 + H
2.5. Frobenius algebra Field theories can be axiomatized by the algebra structure provided by their operators. For a TFT on closed Riemann surfaces, the relevant structure is that of a Frobenius algebra. For the CP1 model, the Frobenius algebra has a basis of idempotents, which are given by H± =
1 (1 ± β −1/2 H) 2
(15)
and satisfy the algebra 2 H+ = H+
2 H− = H−
H+ H− = 0 (16)
The trace on the algebra is given by 1 2β 1/2
(17)
H+ H− + = 2β 1/2 (H+ − H− ) η+ η−
(18)
H± 0,0 = η± = ± The handle operator is W =
Abstractly, branes should correspond to modules over the algebra of bulk operators, in other
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
words, to (irreducible) representations of this algebra. Indeed, using (7), we learn that, in presence of the boundary , H = 1
H− = 0 ,
(19)
which are indeed the possible representations of (16). Note also that H η
V,E = 2β 1/2 H =
(20)
3. Solving the CP model 1
In this section we will compute the exact correlation functions of the CP1 model. 3.1. Summing over genera We weight a closed Riemann surface of genus g by a factor λ2g−2 , where λc is the closed string c coupling. Without boundaries we have 1 ≡
∞
λ2g−2 1g,0 = c
g=0
=
=
n=0 ∞
∞
2n+1 n+s λ4n β = c 2
2β s (22) 1 − 4λ4c β
λ4n−2 22n β n+s =
s λ−2 c β (23) 4 1 − 4λc β
λ2g−2 H W g 0,0 = c
g=0
λ−2 c η = = 1 − λ2c η−1
∞
λ2g−2 η1−g c
g=0 λ2c η 1
1 −
λ2c η
(24) This result is seemingly invariant under [4] λ2 λ2c ↔1− c η± η±
−
λ2c η ˜
(27)
μ ) η
(28)
When μ = 0, η˜+ and η˜− are independent parameters and we have an exact symmetry of the theory generated by
In the idempotent basis, we have ∞
1
where
2 = 1 − 4λ4c β
n=0
H =
λ2c η ˜ 1
(21)
n=0
H 2s+1 =
H =
η˜ = η exp(
λ2g−2 (2H)g 0,0 c
2n+1 n λ4n β c 2
H =
λ2g−2 W g 0,0 c
where R is the worldsheet scalar curvature. Such a deformation is a weighted change of the size of the target CP1 , depending the worldsheet Euler number. Thus, we generalize the topological σ-model to a one-parameter family of models depending on μ. Repeating the above computation we have
g=0
g odd ∞
2s
∞
Note, however, that η+ = −η− and the transformations (25) are not compatibe. Thus, only correlators of one type, say H+ are invariant and, in particular, 1 is not invariant as can be easily checked. One can, however, deform the theory as to make it completely invariant. Consider a deformation of the action by the μ √ gRH (26) 8π Σ
(25)
λ2c λ2 → c, η˜ε η˜ε
λ2c λ2 ↔1− c η˜−ε η˜−ε
(29)
This symmetry is valid only after summing over all genera and is a prototype of the the nonperturbative S-duality symmetry. The theory exhibits also a perturbative T-duality like symmetry η+ → η− , which is valid genus by genus [4]. Note that although there are three independent parameters (λc , β, μ), the correlation functions on closed Riemann surfaces depend only on two parameλ2 ters, which are the combinations ( η˜εc , ε = ±). 3.2. The annulus Let us check the factorization properties of the annulus correlator (see Fig. 1). We put equal boundary conditions on the two boundaries. There are then three amplitudes to consider: 1
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
θ2
θ1
requires ± = +, in which case, using (5), we obtain agreement with (30). We conclude that the relative sign between the two terms in the last line of Fig. 1 depends on the boundary insertions. 3.3. General amplitudes
θ2
H1
θ1
+
θ2
1H
θ1 E m H n
1 E θ2
θ1
+
E 1 θ2
= θ1
∞
=
g=0,h=1 m ∞
λ2g−2 λho E m H n g,h c
(34)
λ2g−2 λho E m−h E h H n g,h . c
h=1 g=0
The outer sum is restricted to those h with the same parity as m. We have
E m H n =
Figure 1. Factorizing the annulus
m ∞
(m−h)/2 λ2g−2 λho β 1/2 c
h=1 g=0
×Cm,h VEh W g H n 0,0 m ∞ (m−h)/2 = λ2g−2 λho β 1/2 c
inserted on both boundaries, 1 on one, E on the other boundary, or E on both boundaries. Factorizing via boundary states (middle of Fig. 1), we find
h=1 g=0
×Cm,h (β 1/2 1 + H)h (2H)g H n 0,0 m ∞ (m−h)/2 = λ2g−2 λho β 1/2 c
110,2 = 0 1E0,2 = 0
h=1 g=0
(30)
EE0,2 = 2β 1/2
×Cm,h 2g
h h (β 1/2 )l H n+h−l+g 0,0 l l=0
We can also factorize as in the bottom of Fig.
(35)
1. 110,2 = 1E0,1 ± E10,1
(31)
where ± is a sign that appears to be not so well understood in the general axiomatics of openclosed TFT (see,e.g., [5]). To be consistent, we here need this sign to be −. The second correlator 1E0,2 = 1E 2 0,1 ± E1E0,1 = 0
(32)
does not suffer from this ambiguity and is consistent with (30) in any case. On the other hand, EE0,2 = E 3 0,1 ± E 3 0,1
(33)
Cm,h is combinatorial factor counting the ways to distribute k = m−h elements in h groups. It is 2 given by
Cm,h
1 m+h 1 k+h−1 −1 2 (36) = = h−1 h! h−1 h!
1 where the h! term is put since we do not distinguish between the boundaries, and one needs to take into account the symmetry factor of the diagram. The innermost sum gets restricted to n+h−l+g
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
odd, and the outer sum is still over (m − h) even.
plus closed system depends on four parameters (λc , λo , β, μ), the correlation functions in the ∞ m 1/2 (m−h)/2 m n 2g−2 h presence of boundaries depend only on two paλc λo β E H = λ2 rameters ( η˜c , λ2o η˜ε ). Note that in the presence h=1 g=0 of boundaries both signs of ε are not allowed sih h (β 1/2 )l β (n+h−l+g−1)/2 multaneously. It is easy to see now, that the ×Cm,h 2g n+h+g−1 l symmetry (29) of the closed TFT is preserved by l=0 m ∞ the open plus closed system, provided we require (m−h)/2 = λ2g−2 λho β 1/2 that we keep λ2o η˜ε invariant. This works, whether c g=0 h=1 or not we relate the open string coupling to the closed coupling, such as implied by unitarity in ×Cm,h 2g n+h+g−1 β (n+h+g−1)/2 2h−1 m string theory. (m+h)/2−1 = 2h−1 λho β 1/2 Cm,h 4. Open/Closed duality h=1 ∞ g A way to view open/closed string theory dual1/2 n (37) × 2λ2c β 1/2 λ−2 ) c (β ity is that summing up the open string degrees g=0 of freedom results in modifying the closed string We have background. Schematically, m n m 1/2 n (38) E H = E (β ) . Fopen+closed (to , tc ) = Fclosed (tc ) (42) open Note, however, that there is no factorization m n m n E H = E H . Here to and tc denote all the open and closed We get for m = 2s even string moduli, and tc are the modified closed string moduli due to the open strings back res−1 1/2 s+l 2s n 2l+1 2l+2 action. A natural question is whether we can see E H = 2 λo C2s,2l+2 β such a duality in our open/closed topological field l=0 −2 1/2 n theory. We will see that in the absence of gravity, λc (β ) × , (39) summing up the open string degrees of freedom 1 − 2λ2c β 1/2 results in a renormalization of the closed TFT and for m = 2s + 1 odd operator. s The generating functional Fg,h (tH , tE ) for the 1/2 s+l β 22l λ2l+1 C E 2s+1 H n = correlators of the type E m Hn g,h is 2s+1,2l+1 o l=0
×
1/2 n ) λ−2 c (β 2 1/2 1 − 2λc β
∂ n+m Fg,h (tH , tE ) |tH =0,tE =0 = E m Hn g,h (43) (40) ∂ n tH ∂ m tE
.
As noted before, H is defined with the same as the D-brane boundary condition. There are also the observables H− . However, correlators H− with the other observables have only the disconnected parts. Define
In order to check whether the symmetry (29) remains in the presence of boundaries, it is useful to compute E m Hε with the deformation (26). We get E m Hε =
λ2c η˜
1
1−
λ2c η˜
m
h
m
Cm,h (2λo )h (2˜ ηε ) 2 (2ηε )− 2
h=1
(41) We can rescale the boundary operator by E → 1 We see that although the open (2ηε )− 2 E.
Fopen+closed (λc , λo , tH , tE , μ, β) ∞ ∞ λ2g−2 λho Fg,h (tH , tE , μ, β) = c =
g=0 h=0 ∞ ∞ m=0 n=0
E m Hn
tn tm E H . m! n!
(44)
S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
The generating functional for the correlators of the closed topological σ-model reads Fclosed (λc , tHε ) =
η λ2c
1−
λ2c η
exp[tH ]
(45)
In particular, using η− = −η we see that 1Exact = H + H− Exact =
1−
2
2 2 (46) λc η
We can obtain a simple expression for the closed generating functional after summing up over the boundaries. Recall that we can replace a boundary with an insertion of the boundary operator E by using the operator Vε,E as described by equations (13) and (14). We need to sum over the boundaries with any number of (odd) E s inserted on each boundary. Consider first one boundary. We have ∞
t2m+1 E (47) (2m + 1)! m=1
1 1 1 1 = H 0,0 2λo (β 2 ) 2 Sinh (β 2 ) 2 tE λo
H E 2m+1 0,1
Now we need to sum over the number of boundaries, which exponentiates (48). This can be achieved in the generating functional by modifying
1 1 1 1 tH → t˜H = tH + 2λo (β 2 ) 2 Sinh (β 2 ) 2 tE (48) Thus, the generating functional for the open plus closed topological σ-model (44) is obtained by using the change (48) in the generating functional of the closed topological σ-model (45). We see that summing up the open string degrees of freedom results in a renormalization of the closed string operator H by adding to it an infinite series of the boundary operator E, weighted by β and the open string coupling λo . Acknowledgements The work of Y.O. is supported in part by the Israeli Science Foundation center of excellence, by the Deutsch-Israelische Projektkooperation (DIP), by the US-Israel Binational Science
67
Foundation (BSF), and by the German-Israeli Foundation (GIF). The work of S.E. was partially supported by the Israel Science Foundation, the Einstein Center in the Hebrew University, and by a grant of DIP (H.52). The work of E.R. was partially sup- ported by the European Union Marie Curie RTN network under contract MRTN-CT-2004-512194, the American- Israel BiNational Science Foundation, the Israel Science Foundation, The Einstein Center in the Hebrew Univer- sity, and by a grant of DIP (H.52). REFERENCES 1. E. Witten, “On The Structure Of The Topological Phase Of Two-Dimensional Gravity,” Nucl. Phys. B 340, 281 (1990). 2. S. Elitzur, Y. Oz, E. Rabinovici and J. Walcher, to appear. 3. K. Hori, “Linear models of supersymmetric D-branes,” arXiv:hep-th/0012179. 4. S. Elitzur, A. Forge and E. Rabinovici, “On effective theories of topological strings,” Nucl. Phys. B 388, 131 (1992). 5. C. I. Lazaroiu, “On the structure of openclosed topological field theory in two dimensions,” Nucl. Phys. B 603 (2001) 497 [arXiv:hep-th/0010269]. 6. P. Horava, Nucl. Phys. B 418, 571 (1994) [arXiv:hep-th/9309124].
Open/Closed Topological CP1 Sigma Model Shmuel Elitzur a , Yaron Oz b , Eliezer Rabinovici
a
and Johannes Walcher
c
a
Racah Institute of Physics, The Hebrew University of Jerusalem, 91904, Israel Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University, Ramat-Aviv 69978, Israel c Department of Physics, CERN - Theory Division, CH-1211 Geneva 23, Switzerland b
We consider the topological sigma-model on Riemann surfaces with genus g and h holes, and target space CP1 ∼ = S 2 . We calculate the correlations function of bulk and boundary operators, and study their symmetries. We study the open/closed TFT correspondence by summing up the boundaries. We argue that this summation can be understood as a renormalization of the closed TFT.
1. Introduction In first quantized string theory one considers in many cases a string moving in a given geometrical background. One then obtains S-matrix elements by adding up contributions of worldsheet calculations on the different world sheet genera. In order to obtain a target space picture one suggests a certain effective Lagrangian defined on the world volume, which was the target space in the worldsheet formulation. This candidate is validated by comparing the S-matrix elements it produces to those obtained from the worldsheet procedure. This straightforward manner does not address various questions such as: the uniqueness of the effective lagrangian and its world volume and topology. One is actually familiar with symmetries such as T-duality and dynamics such as holography which reflect ambiguities of the effective Lagrangian. In the lecture we address this issue in a very simple case, which does allow one to obtain the exact world sheet results. This is the case of topological theories of matter, which allow both closed and open strings. Topological field theories (TFTs) [1] can be used as a simple setup to study open/closed duality properties of string theory. One class of TFTs are the topological σ-models. In order to construct these models we start with a (2, 2) supersymmetric non-linear σ-model in two-dimensions. This is a theory of maps Φ from a two-dimensional 0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.07.045
worldsheet Σ to a target space X, which is a Kahler manifold. The two-dimensional theory has a U (1) × U (1) R-symmetry. One can twist the theory by by adding to the stress tensor of the theory a derivative a U (1) R-current. There are two ways to do that, i.e. twisting with the vector symmetry U (1)V or twisting with the axial symmetry U (1)A . The first leads to the topological A-model, while the second to the topological B-model. Due to the axial anomaly, the B-model is well defined only when the target space is a Calabi-Yau manifold. We will consider the topological A-model. After the twisting, the supersymmetry transformation becomes a transformation under a nilpotent operator Q. The action becomes 2 S∼ d z{Q, Λ} + t Φ∗ (K) (1) Σ
Σ
where Φ∗ (K) is the pullback to the worldsheet of the target space Kahler two-form. We will normalize Φ∗ (K) = n , (2) Σ
where n is an integer, the degree of the instanton. The path integral of the theory localizes on holomorphic maps, and the correlation functions of the model depend only on the cohomology class of the Kahler form K.
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
In this lecture we will consider the topological sigma-model on Riemann surfaces with genus g and h holes, and target space CP1 ∼ = S 2 . We calculate the correlations function of bulk and boundary operators, and study their symmetries. We study the open/closed TFT correspondence. More details on these results and a discussion of the coupling to topological gravity will appear elsewhere [2]. 2. The computational scheme We will denote by O1 ...On g,h the correlation function of the operators O1 , ...On on a Riemann surface with genus g and h boundaries. In this section we will outline the computational scheme that we will use in order to calculate these correlation functions. 2.1. The sphere Consider first the correlators on the sphere with no boundaries. There are two operators of the topological CP1 σ-model, the identity operator, 1, and the operator corresponding to the second cohomology class of the sphere, which we shall denote by H. It is represented by a δ-function two-form on CP1 ∼ = S 2 . On the worldsheet H is a zero-form. We have 10,0 = 0 H0,0 = 1 H 2 0,0 = 0
derives the non-trivial ring relation (OPE) H 2 = β1
(4)
2.2. The disk We now want to consider the CP1 model with branes included in the background. As shown in [3], there are two possible branes that preserve topological invariance. Geometrically, both of them correspond to the equator of CP1 , viewed as the two-sphere. The two branes are distinguished by the value = ± of a Wilson line. We consider correlators on the disc, with boundary condition corresponding to one of the two branes. Both branes support, in addition to the identity (which we continue to denote by 1), a boundary operator corresponding to the first cohomology class of the equator circle, and we will denote this operator by E. It is represented by a δ-function one-form on the equator, and is a zero-form on the worldsheet. It is shown in [3] that 10,1 = 0 E0,1 = 1 H0,1 = 0 E 2 0,1 = 0
(5)
E 3 0,1 = β 1/2 EH0,1 = β 1/2
(3)
H 3 0,0 = β where β = e−t comes from the classical action of the CP1 σ-model and is the contribution of one-instanton, i.e. a degree one holomorphic map from the worldsheet to the CP1 target space. The one-point correlator of H is constant since it gets contribution only from the constant map: we map the worldsheet two-sphere to the target space two-sphere with the point where H is inserted on the worldsheet being mapped to a given point in the target space. The three-point correlator of H gets contribution from a degree one holomorphic map: we map the three insertion points to three given points on the target space. From (3), one
Here, it is understood that E will be inserted on the boundary, while H is inserted in the bulk of the disc. The one-point correlator of E on the disk is constant since it receives contribution only from the constant map: we map the disk worldsheet to the target space S 2 , such that the boundary of the disk is mapped to the equator and the insertion point on the boundary of the disk is mapped to a given point on the equator. This holomorphic map is the constant map. The three-point correlator of E receives contribution from the disk one-instanton (or half-instanton in the closed string sense), which is a degree one map from the disk to S 2 , where the boundary of the disk is mapped to the equator. The three insertion points on the boundary of the disk are mapped to three points on the equator.
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
The equations (5) imply the non-trivial relation on the boundary E 2 = β 1/2
(6)
as well as the important relation H = E 2 = β 1/2 1
(7)
between the bulk field H and the boundary field E 2 . In other words, when computing a correlator on a Riemann surface with boundary, an insertion of H in the bulk is equivalent to inserting E 2 on the boundary (or equivalently, inserting β 1/2 1, in the bulk or on the boundary). It is also important to note that there are no boundary condition changing operators between = + and = −. 2.3. Axial R-charge We assign axial R-charges to the operators R[H] = 2,
R[E] = 1
(8)
In general, an amplitude H E g,h will be proportional to β k if k is an integer (without boundaries) or half-integer (with boundaries) satisfying Ri = 2n + m = 4k + 2 − 2g − h (9) n
m
i
If there is no such k the amplitude vanishes. Note that a (g, h) amplitude will vanish if both boundary conditions appear at the boundaries of the surface. This is because there are no boundary condition changing operators. Therefore, we will choose equal for all boundaries and fix it. Moreover, the amplitude will vanish unless E is inserted an odd number of times on each boundary. 2.4. Handle and boundary states There exists a universal operator W , that relates correlators on surfaces of different topology and it called the handle operator. It has the property Og,h = W Og−1,h
(10)
One can compute W at g = 1 by degenerating the torus into a sphere. Here, the relations W 0,0 = 11,0 = 2H0,0 = 2 W H0,0 = H1,0 = 2H 0,0 = 0 2
(11)
imply W = 2H
(12)
When considering surfaces with boundaries, we have to fix boundary conditions a1 , a2 , . . . , ah on each of them. Moreover, we have to allow for a dependence of the boundary state on the boundary insertions. We will then label the boundary states as Va,θ to indicate dependence on the boundary condition and the boundary insertion. It satisfies Oθ1 · · · θh g,h = Oθ1 · · · θh−1 Vah ,θh g,h−1 (13) it being understood that the operator θi is inserted on the i-th boundary on the left and on the right hand side of the equation. The boundary states can be computed on the disc. From (5), we learn V,1 = 0
(14)
V,E = β 1/2 1 + H
2.5. Frobenius algebra Field theories can be axiomatized by the algebra structure provided by their operators. For a TFT on closed Riemann surfaces, the relevant structure is that of a Frobenius algebra. For the CP1 model, the Frobenius algebra has a basis of idempotents, which are given by H± =
1 (1 ± β −1/2 H) 2
(15)
and satisfy the algebra 2 H+ = H+
2 H− = H−
H+ H− = 0 (16)
The trace on the algebra is given by 1 2β 1/2
(17)
H+ H− + = 2β 1/2 (H+ − H− ) η+ η−
(18)
H± 0,0 = η± = ± The handle operator is W =
Abstractly, branes should correspond to modules over the algebra of bulk operators, in other
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words, to (irreducible) representations of this algebra. Indeed, using (7), we learn that, in presence of the boundary , H = 1
H− = 0 ,
(19)
which are indeed the possible representations of (16). Note also that H η
V,E = 2β 1/2 H =
(20)
3. Solving the CP model 1
In this section we will compute the exact correlation functions of the CP1 model. 3.1. Summing over genera We weight a closed Riemann surface of genus g by a factor λ2g−2 , where λc is the closed string c coupling. Without boundaries we have 1 ≡
∞
λ2g−2 1g,0 = c
g=0
=
=
n=0 ∞
∞
2n+1 n+s λ4n β = c 2
2β s (22) 1 − 4λ4c β
λ4n−2 22n β n+s =
s λ−2 c β (23) 4 1 − 4λc β
λ2g−2 H W g 0,0 = c
g=0
λ−2 c η = = 1 − λ2c η−1
∞
λ2g−2 η1−g c
g=0 λ2c η 1
1 −
λ2c η
(24) This result is seemingly invariant under [4] λ2 λ2c ↔1− c η± η±
−
λ2c η ˜
(27)
μ ) η
(28)
When μ = 0, η˜+ and η˜− are independent parameters and we have an exact symmetry of the theory generated by
In the idempotent basis, we have ∞
1
where
2 = 1 − 4λ4c β
n=0
H =
λ2c η ˜ 1
(21)
n=0
H 2s+1 =
H =
η˜ = η exp(
λ2g−2 (2H)g 0,0 c
2n+1 n λ4n β c 2
H =
λ2g−2 W g 0,0 c
where R is the worldsheet scalar curvature. Such a deformation is a weighted change of the size of the target CP1 , depending the worldsheet Euler number. Thus, we generalize the topological σ-model to a one-parameter family of models depending on μ. Repeating the above computation we have
g=0
g odd ∞
2s
∞
Note, however, that η+ = −η− and the transformations (25) are not compatibe. Thus, only correlators of one type, say H+ are invariant and, in particular, 1 is not invariant as can be easily checked. One can, however, deform the theory as to make it completely invariant. Consider a deformation of the action by the μ √ gRH (26) 8π Σ
(25)
λ2c λ2 → c, η˜ε η˜ε
λ2c λ2 ↔1− c η˜−ε η˜−ε
(29)
This symmetry is valid only after summing over all genera and is a prototype of the the nonperturbative S-duality symmetry. The theory exhibits also a perturbative T-duality like symmetry η+ → η− , which is valid genus by genus [4]. Note that although there are three independent parameters (λc , β, μ), the correlation functions on closed Riemann surfaces depend only on two parameλ2 ters, which are the combinations ( η˜εc , ε = ±). 3.2. The annulus Let us check the factorization properties of the annulus correlator (see Fig. 1). We put equal boundary conditions on the two boundaries. There are then three amplitudes to consider: 1
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
θ2
θ1
requires ± = +, in which case, using (5), we obtain agreement with (30). We conclude that the relative sign between the two terms in the last line of Fig. 1 depends on the boundary insertions. 3.3. General amplitudes
θ2
H1
θ1
+
θ2
1H
θ1 E m H n
1 E θ2
θ1
+
E 1 θ2
= θ1
∞
=
g=0,h=1 m ∞
λ2g−2 λho E m H n g,h c
(34)
λ2g−2 λho E m−h E h H n g,h . c
h=1 g=0
The outer sum is restricted to those h with the same parity as m. We have
E m H n =
Figure 1. Factorizing the annulus
m ∞
(m−h)/2 λ2g−2 λho β 1/2 c
h=1 g=0
×Cm,h VEh W g H n 0,0 m ∞ (m−h)/2 = λ2g−2 λho β 1/2 c
inserted on both boundaries, 1 on one, E on the other boundary, or E on both boundaries. Factorizing via boundary states (middle of Fig. 1), we find
h=1 g=0
×Cm,h (β 1/2 1 + H)h (2H)g H n 0,0 m ∞ (m−h)/2 = λ2g−2 λho β 1/2 c
110,2 = 0 1E0,2 = 0
h=1 g=0
(30)
EE0,2 = 2β 1/2
×Cm,h 2g
h h (β 1/2 )l H n+h−l+g 0,0 l l=0
We can also factorize as in the bottom of Fig.
(35)
1. 110,2 = 1E0,1 ± E10,1
(31)
where ± is a sign that appears to be not so well understood in the general axiomatics of openclosed TFT (see,e.g., [5]). To be consistent, we here need this sign to be −. The second correlator 1E0,2 = 1E 2 0,1 ± E1E0,1 = 0
(32)
does not suffer from this ambiguity and is consistent with (30) in any case. On the other hand, EE0,2 = E 3 0,1 ± E 3 0,1
(33)
Cm,h is combinatorial factor counting the ways to distribute k = m−h elements in h groups. It is 2 given by
Cm,h
1 m+h 1 k+h−1 −1 2 (36) = = h−1 h! h−1 h!
1 where the h! term is put since we do not distinguish between the boundaries, and one needs to take into account the symmetry factor of the diagram. The innermost sum gets restricted to n+h−l+g
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S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
odd, and the outer sum is still over (m − h) even.
plus closed system depends on four parameters (λc , λo , β, μ), the correlation functions in the ∞ m 1/2 (m−h)/2 m n 2g−2 h presence of boundaries depend only on two paλc λo β E H = λ2 rameters ( η˜c , λ2o η˜ε ). Note that in the presence h=1 g=0 of boundaries both signs of ε are not allowed sih h (β 1/2 )l β (n+h−l+g−1)/2 multaneously. It is easy to see now, that the ×Cm,h 2g n+h+g−1 l symmetry (29) of the closed TFT is preserved by l=0 m ∞ the open plus closed system, provided we require (m−h)/2 = λ2g−2 λho β 1/2 that we keep λ2o η˜ε invariant. This works, whether c g=0 h=1 or not we relate the open string coupling to the closed coupling, such as implied by unitarity in ×Cm,h 2g n+h+g−1 β (n+h+g−1)/2 2h−1 m string theory. (m+h)/2−1 = 2h−1 λho β 1/2 Cm,h 4. Open/Closed duality h=1 ∞ g A way to view open/closed string theory dual1/2 n (37) × 2λ2c β 1/2 λ−2 ) c (β ity is that summing up the open string degrees g=0 of freedom results in modifying the closed string We have background. Schematically, m n m 1/2 n (38) E H = E (β ) . Fopen+closed (to , tc ) = Fclosed (tc ) (42) open Note, however, that there is no factorization m n m n E H = E H . Here to and tc denote all the open and closed We get for m = 2s even string moduli, and tc are the modified closed string moduli due to the open strings back res−1 1/2 s+l 2s n 2l+1 2l+2 action. A natural question is whether we can see E H = 2 λo C2s,2l+2 β such a duality in our open/closed topological field l=0 −2 1/2 n theory. We will see that in the absence of gravity, λc (β ) × , (39) summing up the open string degrees of freedom 1 − 2λ2c β 1/2 results in a renormalization of the closed TFT and for m = 2s + 1 odd operator. s The generating functional Fg,h (tH , tE ) for the 1/2 s+l β 22l λ2l+1 C E 2s+1 H n = correlators of the type E m Hn g,h is 2s+1,2l+1 o l=0
×
1/2 n ) λ−2 c (β 2 1/2 1 − 2λc β
∂ n+m Fg,h (tH , tE ) |tH =0,tE =0 = E m Hn g,h (43) (40) ∂ n tH ∂ m tE
.
As noted before, H is defined with the same as the D-brane boundary condition. There are also the observables H− . However, correlators H− with the other observables have only the disconnected parts. Define
In order to check whether the symmetry (29) remains in the presence of boundaries, it is useful to compute E m Hε with the deformation (26). We get E m Hε =
λ2c η˜
1
1−
λ2c η˜
m
h
m
Cm,h (2λo )h (2˜ ηε ) 2 (2ηε )− 2
h=1
(41) We can rescale the boundary operator by E → 1 We see that although the open (2ηε )− 2 E.
Fopen+closed (λc , λo , tH , tE , μ, β) ∞ ∞ λ2g−2 λho Fg,h (tH , tE , μ, β) = c =
g=0 h=0 ∞ ∞ m=0 n=0
E m Hn
tn tm E H . m! n!
(44)
S. Elitzur et al. / Nuclear Physics B (Proc. Suppl.) 192–193 (2009) 61–67
The generating functional for the correlators of the closed topological σ-model reads Fclosed (λc , tHε ) =
η λ2c
1−
λ2c η
exp[tH ]
(45)
In particular, using η− = −η we see that 1Exact = H + H− Exact =
1−
2
2 2 (46) λc η
We can obtain a simple expression for the closed generating functional after summing up over the boundaries. Recall that we can replace a boundary with an insertion of the boundary operator E by using the operator Vε,E as described by equations (13) and (14). We need to sum over the boundaries with any number of (odd) E s inserted on each boundary. Consider first one boundary. We have ∞
t2m+1 E (47) (2m + 1)! m=1
1 1 1 1 = H 0,0 2λo (β 2 ) 2 Sinh (β 2 ) 2 tE λo
H E 2m+1 0,1
Now we need to sum over the number of boundaries, which exponentiates (48). This can be achieved in the generating functional by modifying
1 1 1 1 tH → t˜H = tH + 2λo (β 2 ) 2 Sinh (β 2 ) 2 tE (48) Thus, the generating functional for the open plus closed topological σ-model (44) is obtained by using the change (48) in the generating functional of the closed topological σ-model (45). We see that summing up the open string degrees of freedom results in a renormalization of the closed string operator H by adding to it an infinite series of the boundary operator E, weighted by β and the open string coupling λo . Acknowledgements The work of Y.O. is supported in part by the Israeli Science Foundation center of excellence, by the Deutsch-Israelische Projektkooperation (DIP), by the US-Israel Binational Science
67
Foundation (BSF), and by the German-Israeli Foundation (GIF). The work of S.E. was partially supported by the Israel Science Foundation, the Einstein Center in the Hebrew University, and by a grant of DIP (H.52). The work of E.R. was partially sup- ported by the European Union Marie Curie RTN network under contract MRTN-CT-2004-512194, the American- Israel BiNational Science Foundation, the Israel Science Foundation, The Einstein Center in the Hebrew Univer- sity, and by a grant of DIP (H.52). REFERENCES 1. E. Witten, “On The Structure Of The Topological Phase Of Two-Dimensional Gravity,” Nucl. Phys. B 340, 281 (1990). 2. S. Elitzur, Y. Oz, E. Rabinovici and J. Walcher, to appear. 3. K. Hori, “Linear models of supersymmetric D-branes,” arXiv:hep-th/0012179. 4. S. Elitzur, A. Forge and E. Rabinovici, “On effective theories of topological strings,” Nucl. Phys. B 388, 131 (1992). 5. C. I. Lazaroiu, “On the structure of openclosed topological field theory in two dimensions,” Nucl. Phys. B 603 (2001) 497 [arXiv:hep-th/0010269]. 6. P. Horava, Nucl. Phys. B 418, 571 (1994) [arXiv:hep-th/9309124].