Generalised Permutation Branes on a product of cosets $G_{k_1}/H\times G_{k_2}/H$

0.19 MB·English

by  Gor Sarkissian

Generalised Permutation Branes on a product of
cosets G /H × G /H
k k
1 2
6
0
0
2

n
a Gor Sarkissian ∗
J

0
1 Dipartimento di Fisica

1 Universita di Roma ”Tor Vergata”,
v
1 I.N.F.N sezione di Roma ”Tor Vergata”
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0 via della Ricerca Scientifica 1, 00133 Roma, Italy
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0
6
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h Abstract
t
-
p
We study the modifications of the generalized permutation branes de-
e
h
finedinhep-th/0509153,whicharerequiredtogiverisetothenon-factorizable
:
v
branes on a product of cosets G /H ×G /H. We find that for k 6= k
i k1 k2 1 2
X
there exists big variety of branes, which reduce to the usual permutation
r
a branes, when k = k and the permutation symmetry is restored.
1 2
∗
E–mail: [email protected]
1
1 Introduction and summary
In the recent paper [1] were suggested the generalized permutation branes on a
product of the WZW models G ×G with the not necessarily equal levels k
k1 k2 1
and k . Geometrically the branes wrap the following submanifolds:
2
(g ,g ) = {(h fh−1)k2′, (h fh−1)k1′|h ,h ∈ G} (1)
1 2 1 2 2 1 1 2
′
where k = k /k and k = gcd(k ,k ). Obviously for k = k (1) reduces to the
i i 1 2 1 2
usual permutation branes [2–8]
(g ,g ) = {h fh−1, h fh−1} (2)
1 2 1 2 2 1
It is well known that for a submanifold to serve as D-brane in the WZW model
the boundary two-form ω should exist trivializing the Wess-Zumino three-form
C
on the brane [9–11]:
ωWZW| = dω . (3)
brane C
Itwasfoundin[1]thattherestrictionoftheWZWformk ωWZW(g )+k ωWZW(g )
1 1 2 2
(f)
to the submanifold (1) indeed satisfies to this condition with ω given by the
C
equation:
k k
ω(f)(h ,h ) = 1 2{tr(h−1dh fh−1dh f−1)+tr(h−1dh fh−1dh f−1)}
C 1 2 k 1 1 2 2 2 2 1 1
k′−1
2
+ k1 (k2′ −j)tr(gj(g−1dg)g−jg−1dg)g=h1fh−21
j=1
X
k′−1
1
+ k2 (k1′ −j)tr(gj(g−1dg)g−jg−1dg)g=h2fh−11 (4)
j=1
X
From the consideration of the global issues [9–11] it is deduced in [1] that f =
exp πiλ, where λ is an integral weight of G Lie algebra, and κ = lcm(k ,k ).
κ 1 2
Comparing the formulae (1) and (2), describing generalized and usual permu-
tation branes respectively one can deduce that the generalized branes preserve
less symmetry. The usual permutation branes preserve two different twisted ad-
joint actions:
(g ,g ) → (mg ,g m−1), (g ,g ) → (g m−1,mg ), (5)
1 2 1 2 1 2 1 2
while the generalized branes preserve only the diagonal subgroup:
(g ,g ) → (mg m−1,mg m−1) (6)
1 2 1 2
1
Using the form (4) boundary equations of motion have been analyzed in [1] and
indeed found to be
J +J = J¯ +J¯ . (7)
1 2 1 2
Motivated by the recently established connection between the permutation D-
branes in the minimal N = 2 supersymmetric models and the Landau-Ginzburg
superpotential factorisation [12–14] it was suggested in [1] that the generalized
permutation branes should exist also for cosets of the form G /H ×G /H for
k1 k2
the different levels k and k . To produce such D-brane from the generalized D-
1 2
branes (1) one should modify them in a way to preserve adjoint action of product
H ×H:
(g ,g ) → (m g m−1,m g m−1) (8)
1 2 1 1 1 2 2 2
where m ∈ H. This problem is remained unsolved in [1].
i
Aim of this paper is to study the modifications of the ansatz (1) giving rise
to the required symmetry. In the next section we show that one can propose two
kinds of boundary conditions possessing with the necessary symmetries.
I.
(g ,g ) = {(h fh−1)k2′ptp−1, L−1(h fh−1)k1′nrn−1L} (9)
1 2 1 2 2 1
where L,p,n ∈ H and run all the H subgroup, r and t are fixed quantized
elements of H. In other words we multiply both elements of the ansatz (1) by
the (quantized) conjugacy classes of the subgroup H, and then smear the derived
object along the adjoint action of H. The chain of the transformations:
h → m h , h → m h , L → m Lm−1, p → m p, n → m n (10)
1 1 1 2 1 2 1 2 1 1
reproduces (8).
II.
(g ,g ) = {(h fh−1)k2′(s ls−1)k2′, L−1(h fh−1)k1′(s ls−1)k1′L} (11)
1 2 1 2 1 2 2 1 2 1
where L,s ,s ∈ H run all the subgroup H and l is fixed quantized element.
1 2
By words, we multiply the generalized permutation brane of G × G by the
k1 k2
generalized permutation brane of H × H ( k = x k , k = x k , x is the
k3 k4 3 e 1 4 e 2 e
embedding index of H in G [15]) and then again smear derived object along the
adjont action of H. We show in the next section that for k = k (9) and (11)
1 2
are equivalent.
2
Using the Polyakov-Wiegmann identity
ωWZW(gh) = ωWZW(g)+ωWZW(h)−d(tr(g−1dgdhh−1)) (12)
one obtains that the branes (9) satisfy to the condition (3) with the following
boundary two-form:
Ω(h ,h ,L,p,n) = ω(f)(h ,h )+k ω(2)(C C ,L)+k ωt(p)−k (tr(C−1dC dC C−1))
1 2 C 1 2 2 2 4 1 1 1 1 3 3
− k (tr(C−1dC dC C−1))+k ωr(n) (13)
2 2 2 4 4 2
where we denoted
C = (h fh−1)k2′ ,
1 1 2
C = (h fh−1)k1′ ,
2 2 1
C = ptp−1,
3
C = nrn−1, (14)
4
ω(f) isthetwo-formgivenbytheformula(4),ωt(p)isthetwo-formfoundin[10,11]
C
ωt(p) = tr(p−1dptp−1dpt−1) (15)
and ω(2) is the following useful two-argument two-form
ω(2)(g,U) = tr(dUU−1(gdUU−1g−1+g−1dg +dgg−1)) (16)
which we frequently encounter in this paper.
For an abelian subgroup H = U(1), and equal levels k = k the brane
1 2
(9) was studied in [16]. (In that paper this brane was written in the form
(h fh−1L−1,h fh−1L), but after the redefinition L−1h′ = h we derive (9).)
1 2 2 1 2 2
It was checked for this case in [16], that the full Lagrangian with boundary term
(13) enjoys with the symmetry (8). It was also shown in [16] that the geometry of
this brane coincides with the shape corresponding to the permutation boundary
state of the parafermions product. This serves for us as the hint that we found
the correct solution.
One can also check that the branes (11) as well satisfy (3) with the two-form
Ω(h ,h ,s ,s ,L) = ω(f)(h ,h )+ω(l)(s ,s )+k ω(2)(C C ,L)−k (tr(C−1dC dC C−1))
1 2 1 2 C 1 2 C 1 2 2 2 6 1 1 1 5 5
− k (tr(C−1dC dC C−1)) (17)
2 2 2 6 6
3
where
C = (s ls−1)k2′ ,
5 1 2
C = (s ls−1)k1′ . (18)
6 2 1
The rest of the paper is organized as follows.
In the section 2 we obtain boundary conditions (9) and (11) by analyzing the
action of the gauged WZW model on a world-sheet with boundary.
Inthesection3weconsiderthesebranesforproductSU(2) /U(1)×SU(2) /U(1).
k1 k2
In the appendix A we deliver some algebraical calculations proving gauge
invariance of action with boundary term given by (13).
In the appendix B we review different coordinates systems on S3 sphere used
in the section 3.
2 Lagrangian and symmetries
In this section we obtain the boundary conditions (9) and (11) by considering
gaugedWZWmodelactiononaworld-sheet withboundaryalongthewayworked
out in [17] and [18]. First of all we remind the bulk action of the gauged WZW
model [19–23]:
k
SG/H(g,A) = SG+ G d2ztr{A ∂ gg−1−A g−1∂ g+A gA g−1−A A } (19)
z¯ z z z¯ z¯ z z z¯
2π
ZΣ
where
k
SG = G d2zLkin + ωWZW(g) (20)
4π
(cid:18)ZΣ ZB (cid:19)
is bulk WZW action, Lkin = tr(∂ g∂ g−1), ωWZW = 1tr(dgg−1)3, B is a three-
z z¯ 3
dimensional manifold bounded by Σ, and A is a gauge field taking values in the
H Lie algebra.
Defining
A = ∂ U˜U˜−1, A = ∂ UU−1 (21)
z z z¯ z
we can write (19) in the form:
SG/H(g˜,h˜) = SG(g˜)−SH(h˜) (22)
where g˜ = U−1gU˜ and h˜ = U−1U˜. The level k of the SH is related to k
H G
through the embedding index x of H in G: k = x k [15]. The expression (22)
e H e G
4
is manifestly gauge invariant under the transformation:
g → mgm−1, U → mU, U˜ → mU˜ (23)
For the case under consideration the bulk action is:
SG/H(g˜ ,g˜ ,h˜ ,˜h ) = SG(g˜ )+SG(g˜ )−SH(h˜ )−SH(h˜ ) (24)
bulk 1 2 1 2 1 2 1 2
where
g˜ = U−1g U˜
1 1 1 1
g˜ = U−1g U˜
2 2 2 2
h˜ = U−1U˜
1 1 1
h˜ = U−1U˜ (25)
2 2 2
The levels k and k of the third and forth terms in (24), as explained above, are
3 4
related to the levels k and k of the first and second terms respectively through
1 2
the embedding index x of H in G
e
k = x k
3 e 1
k = x k . (26)
4 e 2
Considertheaction(24)inthepresenceofboundary. Weshouldspecifyboundary
conditions. Let us choose for the sum of the first two actions the generalized
permutation boundary conditions (1) and impose the arguments of the third and
forth terms to take their values in the quantized conjugacy classes:
g˜ = ((U−1h )f(U−1h )−1)k2′
1 1 1 1 2
h˜ = (U−1p)t−1(U−1p)−1
1 1 1
g˜ = ((U−1h )f(U−1h )−1)k1′
2 1 2 1 1
h˜ = (U−1n)r−1(U−1n)−1 (27)
2 1 1
It is easy to see that (27) brings to us conditions (9) for g and g with
1 2
L = U U−1 (28)
1 2
Using the forms (4) and (15) one can write the full action with the boundary
term:
1
S = SG/H(g˜ ,g˜ ,h˜ ,˜h )− (ω(f)(U−1h ,U−1h )−k ω(t−1)(U−1p)−k ω(r−1)(U−1n)
bulk 1 2 1 2 4π C 1 1 1 2 1 1 2 1
ZD
(29)
5
where ∂B = Σ+D.
The action (29) is manifestly gauge invariant under the gauge transformation:
h → m h , h → m h , U → m U , U → m U , p → m p, n → m n
1 1 1 2 1 2 1 1 1 2 2 2 1 1
(30)
implying, recalling the definition (28),
L → m Lm−1. (31)
1 2
We derived the transformation rules (10). Using the Polyakov-Wiegmann identi-
ties the action (29) can be written as :
1
S = SG/H(g ,A )+SG/H(g ,A )− Ω (32)
1 1 2 2
4π
ZD
where
Ω = ω(f)(U−1h ,U−1h )+k ω(t)(U−1p)+k ω(r)(U−1n)+k ω(g ,U ,U˜ )+k ω(g ,U ,U˜ )
C 1 1 1 2 1 1 2 1 1 1 1 1 2 2 2 2
(33)
where
ω(g ,U ,U˜ ) = tr(g−1dg dU˜ U˜−1−dU U−1dg g−1−dU U−1g dU˜ U˜−1g−1+dU U−1dU˜ U˜−1)
i i i i i i i i i i i i i i i i i i i i i
(34)
It is cumbersome but straightforward to check that the form (33) coincides with
(13) with L given by (28). The details of the calculations are delivered in the
appendix A.
Boundaryconditions(11)canbereceived inthesameway, butnowoneshould
take the generalized boundary conditions as for the sum of the first two actions,
as well for the sum of actions for the gauge groups:
g˜ = ((U−1h )f(U−1h )−1)k2′
1 1 1 1 2
h˜ = ((U−1s )l−1(U−1s )−1)k2′
1 1 2 1 1
g˜ = ((U−1h )f(U−1h )−1)k1′
2 1 2 1 1
h˜ = ((U−1s )l−1(U−1s )−1)k1′ (35)
2 1 1 1 2
˜
where we have taken into account that k = gcd(k ,k ) = x gcd(k ,k ), and
3 4 e 1 2
′ ˜ ′ ′ ˜ ′
k = k /k = k , k = k /k = k .
3 3 1 4 4 2
One can easily check that these conditions are equivalent to (11) with again
L = U U−1.
1 2
6
The full action with boundary term is:
S = SG/H(g˜ ,g˜ ,h˜ ,h˜ )− 1 (ω(f)(U−1h ,U−1h )−ω(l−1)(U−1s ,U−1s ) (36)
bulk 1 2 1 2 4π C 1 1 1 2 C 1 2 1 1
ZD
where ∂B = Σ + D. The action (36) is manifestly gauge invariant under the
gauge transformation:
h → m h , h → m h , U → m U , U → m U , s → m s , s → m s
1 1 1 2 1 2 1 1 1 2 2 2 1 1 1 2 1 2
(37)
implying, again,
L → m Lm−1. (38)
1 2
Using Polyakov-Wiegmann identities the action (36) can be written as :
1
S = SG/H(g ,A )+SG/H(g ,A )− Ω (39)
1 1 2 2
4π
ZD
where
Ω = ω(f)(U−1h ,U−1h )+ω(l)(U−1s ,U−1s )+k ω(g ,U ,U˜ )+k ω(g ,U ,U˜ )
C 1 1 1 2 C 1 1 1 2 1 1 1 1 2 2 2 2
(40)
Repeating the same steps as outlined in the appendix A one can show that (40)
coincides with (17).
Some comments:
Let us consider the branes (9) and (11) for equal levels k = k , when permu-
1 2
tation symmetry is restored:
(g ,g ) = (C C ,L−1C C L) = (h fh−1ptp−1,L−1h fh−1nrn−1L) (41)
1 2 1 3 2 4 1 2 2 1
(g ,g ) = (C C ,L−1C C L) = (h fh−1s ls−1,L−1h fh−1s ls−1L) (42)
1 2 1 5 2 6 1 2 1 2 2 1 2 1
By the redefinition
h−1C = h′, L′ = C−1L (43)
2 5 2 5
one can write the brane (42) in the form (C ,L−1C C C L). Taking into account
1 2 6 5
that C C in this case is the usual conjugacy class, we see that at the point of the
6 5
restored symmetry the family of the branes (42) coincides with the family (41).
Performing the same kind of redefinition in (41)
h−1C = h′, L′ = C−1L (44)
2 3 2 3
7
one can write all the branes (41) in the form
(g ,g ) = (C ,L−1C C C L) (45)
1 2 1 2 4 3
Presumably we can multiply both elements in (9) by the chain of conjugacy
classes, as in [8], but for the case of k = k , when permutation symmetry is
1 2
restored, we see, that one can cover all the family already multiplying just one
of them. The conclusion is, that generically when k 6= k we have two families
1 2
of branes (9) and (11) , which reduce to the branes of the form (45), when the
permutation symmetry is restored.
3 Generalized permutation branes on SU(2) /U(1)×
k
1
SU(2) /U(1)
k
2
In this section we consider permutation branes on product of SU(2) /U(1) ×
k1
SU(2) /U(1) cosets. At the beginning we consider usual permutation brane for
k2
k = k and show that the geometrical description given above coincide with the
1 2
permutation boundary state [3] overlap with the graviton wave packet. Actually
this calculation was performed in [16], but for the completeness and the reader’s
convenience we repeat it (slightly generalized and with corrected typos) here.
Then we elaborate the geometry of the simplest generalized permutation brane.
With the U(1) subgroup generated by σ the brane (9) for k = k = k takes
3 1 2
the form:
(g1,g2) = (h1fh−21, eiασ23h2h−11e−iασ23eiπkM σ23). (46)
brane
where f = eiψˆσ23, ψˆ =(cid:12)(cid:12)2jπ, j = 0,..., k, and M is an integer. The factor eiπkM σ23
(cid:12) k 2
reflects Z symmetry of an abelian coset [24]. One can multiply with this factor
k
also the first element in (46) , but as explained above, performing the redefinition
(44), one gets again (46). We see that all the branes are labelled by two indices
ˆ
ψ and M, exactly as the permutation states of the parafermions product. The
elements g and g belong to the brane surface if the following equation admits
1 2
a solution for the parameter α,
tr g1e−iασ23g2eiασ23e−iπkM σ23 = 2cosψˆ. (47)
(cid:16) (cid:17)
This equation can be further elaborated in the Euler coordinates, reviewed in
the appendix B. The formulae for the Euler angles of a product of two elements
gˆ = g g are given in [25]
1 2
8
ˆ˜ ˜ ˜ ˜ ˜
cosθ = cosθ cosθ −sinθ sinθ cos(χ +ϕ ), (48)
1 2 1 2 2 1
eiφˆ˜ = eiχ1+2ϕ2 cos θ˜1 cos θ˜2eiχ2+2ϕ1 −sin θ˜1 sin θ˜2e−iχ2+2ϕ1 . (49)
cos ˆθ˜ 2 2 2 2 !
2
where the hatted variables refer to the product gˆ.
Denoting by Θ˜, Φ˜ Euler angles θ˜ and φ˜ of the product g1e−iασ23g2 and using
(48) and (49) we can rewrite (47) as
˜
Θ πM
˜ ˜ ˆ
cos cos(γ/2−ξ/2−φ −φ + ) = cosψ, (50)
1 2
2 2k
where
cosΘ˜ = cosθ˜ cosθ˜ −sinθ˜ sinθ˜ cosγ, (51)
1 2 1 2
and we have introduced new labels γ = χ +ϕ −α and ξ/2 = Φ˜ − χ1+ϕ2. The
2 1 2
variables ξ and γ are related to each other by the equation
˜ ˜ ˜ ˜
1 θ θ θ θ
eiξ2 = cos 1 cos 2eiγ2 −sin 1 sin 2e−iγ2 . (52)
cos Θ˜ 2 2 2 2 !
2
Let us recall that the vectorial gauging of U(1) symmetry is corresponding
to the translation of φ and the resulting target space of the SU(2) /U(1) model,
k
derived after the gauge fixing φ = 0 and integrating out of the gauge field, is the
˜
two-dimensional disc, parameterized by θ and φ. Inthe case of product thetarget
˜ ˜
space is parameterized by θ ,θ ,φ ,φ . Hence the brane consists of those points
1 2 1 2
for which equation (50) admits a solution for γ. Θ and ξ are considered here
˜ ˜
as the complicated functions of θ ,θ and γ given by (51) and (52) respectively.
1 2
ˆ
For ψ = 0 there are additional constraints, which imply that in this case the
brane is two dimensional and given by the equations
πM
˜ ˜ ˜ ˜
θ = −θ , φ = −φ + . (53)
1 2 1 2
2k
Nowwecalculatetheeffectivegeometrycorrespondingtothepermutationbound-
ary state [3]:
S
|L,Mi = LjeiπMm/k |j,m,N i ⊗|j,m,N i ⊗|j,m,N i ⊗|j,m,N i
1 1 1 2 2 2 2 1
S
0j
Xj,m NX1,N2
(54)
9