Noncommutative Frobenius algebras and open-closed duality

0.27 MB·

by  Yusuke Kimura

OIQP-16-04
Noncommutative Frobenius algebras and open-closed
duality
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n Yusuke Kimura
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Okayama Institute for Quantum Physics (OIQP),

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2 Furugyocho 1-7-36, Naka-ku, Okayama, 703-8278, Japan
londonmileend at gmail.com
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[ Abstract

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2 Some equivalence classes in symmetric group lead to an interesting class of noncom-
8 mutive and associative algebras. From these algebras we construct noncommutative
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8 Frobenius algebras. Based on the correspondence between Frobenius algebras and two-
0 dimensional topological field theories, the noncommutative Frobenius algebras can be
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1 interpreted as topological open string theories. It is observed that the centre of the
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algebras are related to closed string theories via open-closed duality. Area-dependent
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1 two-dimensional field theories are also studied.
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1 Introduction
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a
Topological field theories are the simplest field theories whose amplitudes are indepedendent
of the local Riemann structure. Two-dimensional topological field theories may attract a
particular interest because they can be regarded as toy models of string theory. If we ignore
the conformal structure of Riemann surfaces corresponding to worldsheets, we have two-
dimensional topological field theories [1, 2].
In this paper we will focus on the one-to-one correspondence between two-dimensional
topological field theories and Frobenius algebras [3, 4, 5, 6]. This may suggest an interesting
story that two-dimensional topological field theories can be re-constructed from a certain
kind of algebras, which reminds us of some recent approaches of string theory that it is
re-constructed from discrete data like matrices in AdS/CFT correspondence and the matrix
models.
A Frobenius algebra is defined as a finite-dimensional associative algebra equipped with
a non-degenerate bilinear form. Given an associative algebra, it can be a Frobenius algebra
1
by specifying the bilinear form. It is emphasised that being Frobenius is not a property of
the algebra, but it is a structure we provide. We can construct different Frobenius algebras
by giving different Frobenius structures, even though the underlying algebra is the same. For
a Frobenius algebra, we can associate the bilinear form and the structure constant with the
propagator and the three-point vertex of a two-dimensional topological field theory. Topolog-
ical properties of the theory can be explained from the fact that it is an associative algebra
with a non-degenerate bilinear form.
Our interest on the equivalence comes from the recent development that correlation func-
tions of the Gaussian matrix models, encoding the space-time independent part of correlation
functions of N = 4 Super Yang-Mills, are expressed in terms of elements in associative alge-
bras obtained from symmetric group.
In section 2 we introduce some associative algebras obtained from the group algebra of
the symmetric group. Sums of all elements in the conjugacy classes are central elements in
the group algebra, and the product is closed. They form an associative and commutative
algebra. We can also construct more general class of algebras by introducing equivalence
relations in terms of elements in subgroups of the symmetric group [9, 10, 11]. Such algebras
are associative but in general noncommutative. Because the noncommutativity is related to
the restriction from the symmetric group to the subalgebra, the noncommutative structure
is sometimes referred to as restricted structure. Some algebras with restricted structure were
recently studied in [12].
In section 3, we will explain how the algebras introduced in section 2 appear in the
description of gauge invariant operators in N = 4 Super Yang-Mills. The elements in the
algebras have an one-to-one correspondence with the gauge invariant operators. Two-point
functions of the gauge invariant operators give a map from two elements in the algebra to
a c-number. Regarding the map as a bilinear form, we can introduce Frobenius algebras
naturally from the matrix models. The equivalence between Frobenius algebras and two-
dimensional topological field theories leads to an interpretation of correlation functions of
the matrix models in terms of two-dimensional topological field theories [13].
In section 4, we will review concretely how Frobenius algebras describe topological field
theories. Commutative Frobenius algebras correspond to topological closed string theories,
while noncommutative Frobenius algebras can be associated with open string theories [3, 4,
7, 8]. Because the closed propagator is related to a non-planar one-loop diagram of an open
string theory, the closed string theory and the open string theory should be described in a
unified algebraic framework. Our review will be given from this point of view.
Section 5 will be devoted to a detailed study of algebras with the restricted structure
from the point of view of the open-closed duality. In our previous paper [14] correlation
functions of topological field theories related to noncommutative Frobenius algebras were
studied. In this paper, we reinterpret the noncommutative Frobenius algebras as open string
theories. Elements in the noncommutative algebras are related to open string states, while
the centre of the algebra describes closed string states. The restricted structure may cause
an area-dependence in correlation functions of the closed string theories.
In section 6 we summarise this paper. In appendices, we give formulas, detailed com-
putations and additional explanations. Appendices of our previous paper [14] may also be
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helpful to check equations of this paper.
This paper is based on the talk “ Noncommutative Frobenius algebras and open strigns,”
given inworkshop “Permutations andGaugeString duality,” heldatQueen Mary, University
of London at July 2014. The relationship to our recent work [15] is also briefly mentioned.
2 Equivalence classes in symmetric group
We consider some types of equivalence classes in symmetric group S . One is the conjugacy
n
classes. The others are introduced by considering subgroups, which leads to an interesting
class of algebras.
Consider the group algebra of S over C, which is called C[S ]. For elements in S ,
n n n
introduce the equivalence relation σ ∼ gσg−1 by any element g in S . Because of the
n
equivalence relation, the set of elements in S are classified into conjugacy classes. It is then
n
convenient to consider
1
[σ] = τστ−1. (1)
n!
τX∈Sn
Elements in a conjugacy class give the same value of [σ], i.e. we have
[σ] = [gσg−1] (2)
for any g in S . In fact [σ] is the sum of all elements in the conjugacy class the σ belongs to.
n
The sums are central elements in C[S ],
n
[σ]τ = τ[σ], [σ][τ] = [τ][σ] (3)
for any σ,τ ∈ S . Because σ and σ−1 are in the same conjugacy class, we also have
n
[σ] = [σ−1]. (4)
The product of two central elements gives a central element of the form (1),
1
[σ ][σ ] = τσ [σ ]τ−1. (5)
1 2 n! 1 2
τX∈Sn
As above the [σ] span a basis of the centre of the group algebra. Central elements form a
closed commutative and associative algebra, which is denoted by Z(C[S ]).
n
We next introduce the equivalence relation determined by a subgroup H of S as σ ∼
n
hσh−1, whereh ∈ H andσ ∈ S . Generalising(1), theequivalenceclassesunderthesubgroup
n
H are classified by
1
[σ] = hσh−1, (6)
H
|H|
h∈H
X
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where |H| denotes the dimension of the subgroup. They have the following properties
[σ] = [hσh−1] , h[σ] = [σ] h (h ∈ H), (7)
H H H H
and also
[σ] 6= [σ−1] . (8)
H H
The product of two elements of the form (6) can be expressed by the form (6),
1
[σ ] [σ ] = hσ [σ ] h−1. (9)
1 H 2 H |H| 1 2 H
h∈H
X
The closed algebra formed by the elements (6) is called C[S ] in this paper. Our notation
n H
is that C[S ] = Z(C[S ]) for H = S . A big difference from the algebra relevant for the
n H n n
conjugacy classes is that the elements given in (6) span a noncommutative algebra,
[σ] [τ] 6= [τ] [σ] . (10)
H H H H
There are several interesting subgroups we can choose for H. For instance we choose
H = S ×S ×···, where n +n = ··· = n [9, 11, 19]. It is also interesting to consider
n1 n2 1 2
the wreath product group H = S [S ], which has played a role recently in [15].
n/2 2
Brauer algebras can also be similarly considered. The walled Brauer algebra B(n ,n )
1 2
with the subgroup H = S ×S gives equivalence classes determined by
n1 n2
1
[b] = hbh−1, (11)
H n !n !
1 2
h∈SXn1×Sn2
where b is an element in the walled Brauer algebra. These quantities can classify a certain
class of gauge invariant operators [10, 16].
Before closing this section, we shall give a remark about another basis labelled by a set
of representation labels. We may use the projection operators as a basis of Z(C[S ]). The
n
change of basis is given by
1
[σ] = χ (σ)PR, (12)
d R
R
R⊢n
X
where PR andχ aretheprojectionoperatorandthecharacter associatedwith anirreducible
R
representation R, and d is the dimension of R of S . The sum is over all Young diagrams
R n
with n boxes. The projectors by definition satisfy
PRPS = δ PR. (13)
RS
In this paper we call this basis a representation basis.
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The algebras with the restricted structure also admit a similar expansion,
1
[σ] = χR ([σ] )PR , (14)
H d A,µν H A,µν
A
R,A,µ,ν
X
where R is an irreducible representation of S and A is an irreducible representation of H.
n
The PR are obtained by decomposing the projector PR
A,µν
PR = PR = PR . (15)
A A,µµ
A A,µ
X X
The indices µ,ν behave like matrx indices,
PR PR′ = δ δ δ PR , (16)
A,µν A′,ρσ RR′ AA′ νρ A,µσ
and the trace part with respect of these indices is denoted by PR. Let MR be the number
A A
of times A appears in R under the embedding of H in S . We call µ,ν multiplicity indices
n
because they run over 1,··· ,MR. See [17, 18, 11, 19] for the construction of the operators
A
PR and the coefficients χR for H = S × ···× S . The noncommutativity in (10) is
A,µν A,µν n1 n2
reflected to the noncommutativity of the matrix structure in (16). The PR commute with
A,µν
all elements in the subgroup,
PR h = hPR , h ∈ H. (17)
A,µν A,µν
In general there may exist some representation bases in the algebras. Elements in C[S ]
n H
allow a different expansion from (14), which will be given in Appendix D.
3 Frobenius algebras and matrix models
Intheprevioussectionwehaveintroducedsomeequivalence relationsinthesymmetricgroup.
The equivalence classes also play a role in the description of gauge invariant operators in
N = 4 SYM, where it can be found that the number of the equivalence classes is equal to the
number of the gauge invariant operators. Two-point functions of the gauge theory naturally
give a map from W⊗W → C, where W is a basis of the algebra, and the map can be adopted
to make the algebra Frobenius.
Consider the matrix model
hf(X,X†)i = [dX dX†]e−2tr(XaXa†)f(X,X†), (18)
a a
Z a
Y
where the measure is normalised to give
h(X ) (X†) i = δ δ δ . (19)
a ij b kl il kl ab
Here f is an invariant quantity under X → gX g−1, X† → gX g−1 where g ∈ U(N), and
a a a a
N is the size of the matrices. Such invariant quantities are in general linear combinations of
multi-traces. They are called gauge invariant operators.
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When we say holomorphic operators, we consider (linear combinations of) multi-traces
built from only X . By two-point functions of holomorphic operators, we mean
a
hf(X)g(X†)i = [dX dX†]e−2tr(XaXa†)f(X)g(X†). (20)
a a
Z a
Y
As a shorthand, we denote it by hfgi.
We first consider holomorphic gauge invariant operators built from one kind of complex
matrix. We use n to express the number of matrices involved in the operators. For example
at n = 3 we have three operators, tr(X3), tr(X2)trX, (trX)3. Counting the number of gauge
invariant operators is a combinatorics problem, and it has been known that the number of
matrix gauge invariant operators involving n X’s is equivalent to the number of conjugacy
classes of the symmetric group S .1
n
From the correspondence, it is expected that elements in Z(C[S ]) can classify the gauge
n
invariant operators. Consider
i i i
O(σ) := (X)σ(1)(X)σ(2) ···(X)σ(n). (21)
i1 i2 in
Regarding X as an endormorphism acting on a vector space V, we can express it as
O(σ) = tr (σX⊗n), (22)
n
where the σ permutes the n factors in V⊗n and tr is a trace over the tensor space V⊗n. We
n
can show that
O(σ) = O(τστ−1) (23)
for any τ ∈ S , then we have
n
1
O([σ]) = O(τστ−1) = O(σ). (24)
n!
τX∈Sn
Two-point functions of the gauge invariant operators are computed as
hO([σ])O([τ])i = tr(r)(Ω [σ][τ]). (25)
n
Here the trace is a trace over regular representation,
tr(r)(σ) = d χ (σ), (26)
R R
R⊢n
X
1 At n = 3 we have three multi-traces tr(X3), trXtr(X2) and (trX)3, while we have three conjugacy
classes in S , [1,1,1], [1,2] and [3]. Each conjugacy class corresponds to an integer partition of n. This
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correspondencedoesnotworkwhenthematrixsizeN islessthann. ForN =2,wehavethe identityamong
the operators, tr(X3)− 3trXtr(X2)+ 1(trX)3 = 0, which means that only two of the three operators are
2 2
independent. This is an example of finite N constraints. Operators with finite N constraints taken into
account for n>N can be correctly counted by considering a representation basis.
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where d is the dimension of an irreducible representation R of symmetric group S , and χ
R n R
is the character associated with R. The sum is over all Young diagrams with n boxes. This
derivation is reviewed in section 2 of our previous paper [14]. The N-dependence is entirely
contained in the Omega factor
1
Ω = σNCσ−n = 1+ σNCσ, (27)
n Nn
σX∈Sn Xσ6=1
where C counts the number of cycles in the σ.
σ
The right-hand side of (25) can also be written as
n! DimR
tr(r)(Ω [σ][τ]) = χ (σ)χ (τ). (28)
n Nn d R R
R
R
X
Here DimR is the dimension of an irreducible representation R of U(N), which can be
written in terms of the Omega factor as (A.2). Note that characters are class functions. The
quantity (26), equivalently (28), gives a bilinear map W ⊗ W → C, where W is a basis of
Z(C[S ]), and we also find that it is non-degenerate i.e. it is invertible for N > n. We
n
denote the nondegenerate bilinear form by η, with which we obtain a commutative Frobenius
algebra. We denote the Frobenius algebra by (Z(C[S ]),η). The two-point function can be
n
diagonalised using the represetation basis [20]
n!
tr(r)(Ω PRPS) = d DimRδ . (29)
n Nn R RS
We next consider the two-matrix model. The number of holomorphic gauge invariant
operators built from m X’s and n Y’s is the same as the number of equivalence classes under
σ ∼ hσh−1 in S , where h ∈ S ×S . In fact the holomorphic gauge invariant operators
m+n m n
can be associated with the equivalence classes as
i i i i
O(σ) = (X)σ(1)···(X)σ(m)(Y)σ(m+1) ···(Y)σ(m+n)
i1 im im+1 im+n
= tr (σX⊗m ⊗Y⊗n), (30)
m+n
which have the invariance
O(σ) = O(hσh−1), h ∈ S ×S . (31)
m n
Two-point functions are
hO(σ)O(τ)i = tr(r)(Ω [σ] [τ] ), (32)
m+n H H
where H = S ×S . We now define
m n
θ := tr(r)(Ω [σ] [τ] ). (33)
σ,τ m+n H H
Because this gives a non-degenerate pairing W ⊗ W → C, where W is the space spanned
by elements in C[S ] , we can construct a Frobenius algebra with the bilinear form.
m+n H
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As we emphasised in (10), this algebra is noncommutative. The bilinear form (33) can be
diagonalised as [11]
m!n!
tr(r)(Ω PR PS ) = d DimR δ δ δ δ . (34)
m+n A,µν B,ρσ A Nm+n RS AB µσ νρ
Recently singlet gaugeinvariant operatorsunder the globalO(N ) symmetry were studied
f
in [15]. We can label the singlet operators in terms of some permutation as
S(σ) = (Φ )iσ(1)(Φ )iσ(2)(Φ )iσ(3)(Φ )iσ(4) ···(Φ )iσ(n−1)(Φ )iσ(n)
a1 i1 a1 i2 a2 i3 a2 i4 an/2 in−1 an/2 in
= tr (σΦ ⊗Φ ···Φ ⊗Φ ), (35)
n a1 a1 an/2 an/2
which are invariant under the conjugation by the wreath product group S [S ]
n/2 2
S(σ) = S(hσh−1), h ∈ S [S ]. (36)
n/2 2
The two-point functions are [15]
NnNn/2
hS(σ)S(τ)i = f ϕ
n! σ,τ
1
ϕ = W(ρ)tr(r)(Ω [σ] ρ[τ] ρ−1), (37)
σ,τ Nn/2 m+n H H
f ρX∈Sn
where H = S [S ]. The W(ρ) is a function of the form Nz(ρ), with z(ρ) = z(h ρh ),
n/2 2 f 1 2
h ,h ∈ H. They can not be diagonalised by the representation basis in (14) but can be
1 2
diagonalised by the basis in (D.24). The two point functions can also be regarded as a map
from W ⊗W → C, where W is a basis of C[S ] .
n Sn/2[S2]
It is very interesting that two-point functions are entirely expressed in terms of symmetric
group quantities. A consequence of this fact might be an interpretation of the correlation
functionsintermsoftwo-dimensionaltopologicalfieldtheories[13]. Associatingthetwo-point
functions with the non-degenerate bilinear form of the Frobenius algebra, we can interpret
the two-point functions of the matrix model as two-point functions of the two-dimensional
topological field theory.
If the algebras we are considering are commutative, they will describe two-dimensional
field theories that may be associated with closed string theories. On the other hand, noncom-
mutative algerbas may be relevant for open string theories. Because an one-loop non-planar
open string diagram is topologically equivalent to a closed propagator, both open string al-
gebras and closed string algebras should be described in an algebraic framework. We will
clarify how open string degrees of freedom and closed string degrees of freedom are described
in the algebras we have discussed in this section.
4 Simplest open-closed TFT
In this section we shall review how open-closed topological systems are described by semi-
simple algebras [3, 4]. As a prototype of the relation between two-dimensional topological
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field theories and Frobenius algebras, we consider the group algebra of the symmetric group
S . Let σ be a basis of C[S ], the group algebra of S . In order to make this algebra
n i n n
Frobenius, we now adopt the following bilinear form,
g = C lC k, (38)
ij ik jl
where C k is the structure constant of the algebra,
ij
σ σ = C kσ . (39)
i j ij k
This algebra is noncommutative C k 6= C k. We call this noncommutative Frobenius algebra
ij ji
(C[S ],g). The bilinear form can also be written as
n
g = tr(r)(σ σ ). (40)
ij i j
The nondegenerate bilinear form is also called a Frobenius pairing.
We now define the dual basis by
1
σi = σ−1, (41)
n! i
and the inverse of g is given by
ij
gij = tr(r)(σiσj). (42)
Then we have
g gjk = g k, g j = tr(r)(σ σj) (43)
ij i i i
and
g jg k = g k. (44)
i j i
By these properties, the bilinear form is called non-degenerate in the basis. The g j is
i
associated with the propagator of the two-dimensional theory. The property (44) expresses
the fact that the two-dimensional theory has the vanishing Hamiltonian, which is a general
feature of topological quantum field theories [2]. The dual basis also satisfies
σi = gijσ . (45)
j
A comment about the relation between (38) and (45) will be given in appendix B. The
difference between the basis and the dual basis corresponds to the different orientations of
the boundaries.
The structure constant can be interpreted as the three-point vertex of the theory, which
has the form
C k = tr(r)(σ σ σk). (46)
ij i j
Reflectingthenoncommutativity, thetheorymaybeassociatedwithtwo-dimensional surfaces
corresponding toopenstringworldsheets. WethenregardtheFrobeniusalgebraasdescribing
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(cid:1)
(cid:2)
Figure 1: The RHS of (47), C lC k, is shown in this figure.
ik lt
a topological open string theory. We also call the bilinear form an open string metric.
Indices are raised or lowered by the open string metric.
Let us next explain how the dual closed theory arises in this description. The dual closed
string propagator is given by computing the open string one-loop non-planar diagram (see
figure 1),
η = C lC k. (47)
it ik lt
We call it a closed string metric. From (46) with the orthogonality relation of represen-
tations (A.1), we also have
η = tr(r)([σ ][σ ]), (48)
ij i j
where [σ] is defined in (1). We can easily check that the closed string propagator is a
projection operator
η jη k = η k, (49)
i j i
where indices are raised by the open string metric,
η j = gjkη . (50)
i ik
The open string metric (38) secures the topological property of the closed string theory. In
this way we obtain a topological closed string theory from the topological open string theory.
The topological closed string theory is described by the centre of C[S ] with the bilinear form
n
η . This Frobenius algebra is indeed what we have introduced in the previous section, which
ij
we denoted by (Z(C[S ]), η), if we ignore the Ω by taking a large N limit.2
n n
For convenience, let us introduce a map from C[S ] to Z(C[S ]) by
n n
π(σ) = σ σσi = [σ]. (51)
i
i
X
2 Because the Ω has an 1/N expansion as in (27), by taking a large N limit, we ignore the contribution
n
of Ω in (25).
n
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