A SUPER VERSION OF
ZHU'S THEOREM
by
ALEX JORDAN
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2008
11
University of Oregon Graduate School
Confirmation of Approval and Acceptance of Dissertation prepared by:
Alexander Jordan
Title:
"A Super Version of Zhu's Theorem"
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics by:
Arkady Vaintrob, Chairperson, Mathematics
Alexander Kleshchev, Member, Mathematics
Alexander Polishchuk, Member, Mathematics
Hal Sadofsky, Member, Mathematics
Nilendra Deshpande, Outside Member, Physics
and Richard Linton, Vice President for Research and Graduate Studies/Dean of the
Graduate School for the University of Oregon.
June 14, 2008
Original approval signatures are on file with the Graduate School and the University of
Oregon Libraries.
III
An Abstract of the Dissertation of
Alex Jordan for the degree of
in the Department of Mathematics to be taken
Title: A SUPER VERSION OF ZHU'S THEOREM
Approved: _
Dr. Arkady Vaintrob
Doctor of Philosophy
June 2008
We generalize a theorem of Zhu relating the trace of certain vertex algebra
representations and modular invariants to the arena of vertex super algebras. The
theorem explains why the space of simple characters for the Neveu-Schwarz
minimal models NS(p, q) is modular invariant. It also expresses negative products
in terms of positive products, which are easier to compute. As a consequence of
the main theorem, the subleading coefficient of the singular vectors of NS(p, q) is
determined for p and q odd. An interesting family of q-series identities is
established. These consequences established here generalize results of Milas in this
field.
CURRICULUM VITAE
NAME OF AUTHOR: Alex Jordan
PLACE OF BIRTH: Eugene, OR
DATE OF BIRTH: 23 September 1979
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Reed College, Portland, OR
DEGREES AWARDED:
Doctor of Philosophy, University of Oregon, 2008
Master of Arts, University of Oregon, 2004
Bachelor of Arts, Reed College, 2001
AREAS OF SPECIAL INTEREST:
Lie Algebras, Mathematical Physics, Mathematical Art, Brewing
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2002-2008
GRANTS, AWARDS AND HONORS:
Fulbright Fellowship, Eotvos Lorand University, 2001
iv
vACKNOWLEDGMENTS
I thank my family for enabling me to accomplish so much, my advisor for
introducing me to this problem, the U of 0 math department and the GTFF for
giving me wonderful teaching experience, and all my friends in Eugene who helped
to make these past six years so memorable.
To my grandpa Irvine
VI
Vll
TABLE OF CONTENTS
Chapter Page
I. PRELIMINARIES 1
1.1
1.2
1.3
1.4
Vertex Algebras and the Virasoro Algebra
A Change of Coordinates for Vertex Algebras
Modularity . . . . . . . . . . . . . . . . . . .
Trace, Zhu's Theorem, and the work of Milas
1
5
6
8
II. VERTEX SUPER ALGEBRAS ... 10
11.1 The Neveu-Schwarz Algebra. 10
II.2 Super Conformal Vertex Algebras and Representations of N S 12
III. A SUPER VERSION OF ZHU'S THEOREM
III. 1 A Lemma . . . . . .
III. 2 The Main Theorem
IV. THE MAIN THEOREM APPLIED TO NS(5, 3)
IV.1 The Application to NS(5, 3)
V. THE CASE OF NS(p, q) ....
V.1 A Differential operator .
V.2 Singular Vectors and N S(p, q)
V.3 NS(6k ± 1,3) ..
VI. FUTURE INQUIRIES.
VI. 1 Improving Upon the Differential Operator
VI.2 Quasi-Bernoulli Numbers
REFERENCES .
14
14
17
22
22
28
28
33
36
38
38
39
40
1CHAPTER I
PRELIMINARIES
In this chapter we recall some preliminary information about vertex
algebras and their representations, including Zhu's Theorem. The main result of
this paper is an augmentation to Zhu's Theorem that makes it applicable to super
algebras.
For ease of reference the basics of vertex algebra theory are laid out here,
however it is assumed that the reader is familiar with this information, which is
the topic of introductory texts [6] and [5].
1.1 Vertex Algebras and the Virasoro Algebra
The definition of a vertex algebra is complicated, being motivated by
theoretical small-scale physics. For the lay-mathematician, there are some notable
features separating vertex algebras from other algebras. In place of a single
bilinear multiplication, there are infinitely many products for any two vectors a
and b. These products are indexed by Z and are written anb. Rather than
studying the nth multiplication by a individually, we study all products with a at
once in the form of the generating series I: anz-n- 1 . In physics, this series is
thought of as a meromorphic expansion of a quantum field in EndV around a
point on a Riemann surface with local coordinate z ([12]).
2In [2], some four or five equivalent definitions for vertex algebras are
proposed. We present here the defintion of [6].
Definition 1.1. A vertex algebra consists of the following data:
• a graded vector space V
• a distinguished vector 1 E va
• the state-field correspondence Y : V ----+ EndV[[z, Z-l]], with conventional
notation Y(a, z) = I::z anz-n- 1 . (So an is in EndV.) Even more restrictive,
Y(a) must be a field. That is, for any b E V there is an integer N with
anb = 0 for n > N. Finally, Y should respect degree; an must be a degree
deg a - n - 1 endomorphism.
subject to the following axioms:
• For all a E V, Y(a, z)l E azo + zV[[z]] and Y(l, z) = Idzo. (vacuum or
identity axiom)
• Define a degree 1 homogeneous operator T by Ta = a-21. For all a E V,
[T, Y(a,z)] = 8zY(a,z). (translation covariance)
• For all a, bE V, (z - w)N[Y(a, z), Y(b, w)] = 0 for large enough N. (locality)
The vacuum axiom provides for a vacuous state whose field interacts
trivially with other fields. The translation operator and the translation axiom
provide a structured way to boost states one level higher. Locality is a reasonable
generalization for the commutativity of two fields.
Two consequences of the axioms will be used in the proof of the main
theorem, so we present them here. These are proved in [5].
3Proposition 1.1. Associativity.
For any a, b in a vertex algebra V,
Resw-Aw - z)iY(Y(b, w - z)a, z)
= Resw ~w,Aw - z)iY(b, w)Y(a, z) - Resw ~z,w(w - z)iY(a, z)Y(b, w).
Here, ~w,z is the expansion of a rational series in wand z in the domain with
Iwl> Izl·
Proposition 1.2. Borcherd's Identity.
For any a, b in a vertex algebra V,
Consider the ring of complex-valued Coo functions on the circle 51, All such
functions have Fourier expansions Lz cnzn in the complex unit circle coordinate z.
Some such expressions in C[[z, Z-l]] converge, and some do not. We consider finite
sums, where convergence is not an issue. So our attention is restricted to C[z, Z-l]
and its Lie algebra of derivations V = Der (C[z, Z-l]). The Lie algebra V has a
basis consisting of the Virasoro elements Ln = _zn+1 oz ' In V, the Virasoro
elements satisfy the commutation relations
(1.1 )
The Virasoro Lie algebra Vir is the unique nontrivial central extension of V by the
4one-dimensional space CG, with commutation relations
(1.2)
(The cocycle m31;m is a traditional choice - all other choices give isomorphic or
trivial extensions.) Vir is of essential importance in defining a conformal vertex
algebra.
Definition 1.2. A conformal vertex algebra is a vertex algebra V containing a
distinguished vector W E 112 (called a conformal vector) such that
(a) Under the identification W n = L n - 1 , the Wn satisfy the relations 1. 2,
with G acting as multiplication by a complex number c.
(b) WI (or Lo) is the grading operator for V.
(c) Wo (or L_1) is the translation operator for V.
The number c is called the central charge of V.
The first example of a conformal vertex algebra comes directly from Vir
itself. Vir has the Z-grading with deg(Lm ) = -m and deg(G) = O. We allow Vir::;o
to act on C in an almost trivial manner, and then induce the action to Vir.
Choose two numbers c and h, and define an action of Vir::;o on Cl by
Gl c1
Lol hI
and Lm l = 0 for m > O. Define ~h to be the minimal quotient of the Vir module
Vir ®Virso C1. For certain choices of c, ~ = ~o has the structure of a rational
conformal vertex algebra, that is, one with finitely many simple representations.
The finite list of simple representations can be given formulaically as a list of ~h
5for certain h. All of this is established in [4] and [11]. Specifically, this happens
when
c
h
1 _ 6(p - q)
pq
(pr - qs)2 _ (p _ q)2
4pq
where p and q are integers greater than 1, and r (resp. s) ranges between 1 and
q-1 (resp. 1 andp-1).
1.2 A Change of Coordinates for Vertex Algebras
Zhu recognized in [12] that there is a change of coordinates we may apply to
the fields in the state-field correspondence of a vertex algebra V. If the given state
field correspondence is denoted by Y(a, z) = L: anz-n-l, let Y[a, z] be defined by
La[n]zn-l
(1.3)
(1.4)
Geometrically, Y ( ,z) uses z along the complex unit circle to coordinatize a string.
Y[ ,z] uses z along a much less standard cirlce - one that passes through In(2),
In(l + i), -00, and In(l - i).
It is known (see [12]) that V with the state-field correspondence Y( ,z) is
isomorphic as a vertex algebra to V with the state-filed correspondence Y[ ,z].
The isomorphism does not change the vacuum vector 1. However, the two vertex
algebras have different conformal vectors. If w is the conformal vector for V with
Y( ,z), then w= w - {41 is the conformal vector for V with Y[ ,z].
6Define c(d,i,m) by (d-~+X) = L~=oc(d,i,m)xm. Then the arm] may be
expressed in terms of the ai:
00
arm] = m! L c(d, i, m)an
i=s
(1.5)
This formula will be used in the proof of the main theorem, and is proved in [12].
I.3 Modularity
A basic familiarity with the theory of the modular group SL2 ('lL) and its
action on C is useful to understand the importance of Zhu's theorem and the main
theorem. The texts [8] and [3] are good introductions to the subject. The action of
SL2 (Z) on the complex plane may be shortly described as follows. Identify
C U {oo} with the projective space Cpl according to
T +--t (:)
00 +--t C).
Under this identification, SL2 (Z) acts on C U {oo} via matrix multiplication. This
action clearly preserves lR U {oo}, and the positive determinant of any matrix 'Y
ensures that the upper half plane is mapped to itself as well.
A meromorphic function on C is called a modular function of weight k if
k
fhT) = ((0 1) 'Y (:)) f(T) for all 'Y E SL2 (Z). For subgroups r of SL2 (Z), we
can have modular functions of weight k for r defined in the exact same way. Some
7important modular functions are given by the (normalized) Eisenstein series:
for k ;::: 1where q = e27fiT and the Bk are the Bernoulli numbers. The weight of G2k
is 2k. These series appear in Zhu's theorem below, which demonstrates the
modularity of conformal vertex algebra traces.
The following series appear in the main theorem:
B 2k+1,d 2 00 (n + 1/2)2k+l qn+l/2
(2k + 1)1 - (2k + 1)1~ 1 + qn+l/2
for k ;::: 0 and where Bm,d are defined for m ;::: 1 and d E ~ + Z by the conditionally
convergent double series
The Cd,i in the above series are defined by (1 + z)d- 1 ln (1 + Z)-l = L Cd,iZi. For
convenience, we set Qtk(q) = ~~:)f. (Although we hypothesize that these numbers
are 0.) The series Qt appear in the super version of Zhu's theorem that is our
main result.
There are two more series with modular properties important to us:
00
n=l
'TJo(q)
00 1 + qn-l/2
II 1- qn
n=l
8The series q rJ(q)24 is famously a modular form of wieght 12, also known as
the discriminant modular form 6(q).
1.4 Trace, Zhu's Theorem, and the work of Milas
Starting with any space M and a diagonalizable operator L whose
eigenspaces are finite dimensional, then given any operator B we can define its
q-trace. The idea is to collect the trace data for the restriction-projection of B to
each of L's eigenspaces. If the eigenvalues of L run through Z for example, then B
has restriction-projections B n for nEZ, and we write:
tr IMBqL = 2:: tr Bnqn
nEZ
The first example of a q-trace would be the case when B is the identity
(1.6)
map. Then tr B n is just the dimension of the n-eigenspace, and the q-trace of B is
just the character of M. Standard trace has the commutativity property
tr(AB) = tr(BA), and q-trace has an analogous commutativity property. If A is of
degree k and B of degree -k, then tr /MABqL = qktr /MBAqL.
In a vertex algebra V, consider a homogeneous vector v. Then only one of
the V n has degree 0 as an endomorphism of V, and therefore this is the only V n
that could have non-zero trace. In light of this Zhu introduced the notation o(v) to
mean Vdeg(v)-l, which is the one V n that might have interesting trace. We
emphasize that o(v) = Vdeg(v)-l, and not v[deg(v) - 1]. Zhu proved the following
theorem in [12].
9Theorem 1.1. Zhu's Theorem
Let a and b be homogeneous vectors in a conformal vertex algebra V with
representation M. Then
tr 1M o(b[O]a)qLo
tr 1M o(b[-l]a)qLo
o
tr 1M o(b) o(a)qLO +L GZk(q) tr 1M 0(b[2k - l]a)qLo.
k:2:1
This theorem is the basis for proving the modular invariance of the span of
V-module characters.
In [11] the characters of the rational vertex algebra ~o and of its simple
representations ~h were determined. We do not present them here, since we later
present these characters for the corresponding super algebras which are of more
importance to this text. In [?, ? , Milas] Milas discovered that he could apply Zhu's
Theorem to the singular vector of ~o, with M ranging through the modules ~h.
Given the knowledge of characters provided by [11] and some basic differential
equation theory, he proved anew a family of Ramanjan-style q-series identities and
proved modular invariance of the product of Vch-characters.
Noting that [11] had also established characters for some vertex super
algebras, we set out to extend Milas's work. The first obstacle was that Zhu's
theorem was inapplicable when odd vectors were in play. That led to the main
result of this paper, an augmentation that deals with odd vectors. Once this was
established, we could continue in the style of Milas searching for Ramanujan-style
identities. We also learned information about the singular vector in Neveu-Schwarz
algebras.
10
CHAPTER II
VERTEX SUPER ALGEBRAS
In this chapter we discuss the super analogs of the fundamentals discussed
in the introduction. Although I make an attempt to motivate the topics, a full
explanation of what follows would obscure the main result.
11.1 The Neveu-Schwarz Algebra
The Neveu-Shwarz Lie algebra NS is of fundamental importance to us. It is
the super version of Vir. The idea is to adjoin a square root of each L m to Vir, i.e.
we would like to have GT!!:. in the algebra with GT!!:.G!!!:. = Lm . But in a Lie algebra222
these kinds of products make no sense. If we allow these G T!!:. to be odd, then two
2
problems are solved at once. First, the relation we are looking for becomes
[G ~, G~ ] = 2Lm in the universal enveloping algebra. Second, this suggests a
geometric motivation as to from where such an algebra might naturally arise.
The super circle SIll is an object of study in theoretical physics - see [7].
By definition, it is the topological space Sl together with a super algebra denoted
000(Sl I1). This super algebra is an extension of the familiar ooo(Sl), acheived by
tensoring with the exterior algebra on one odd variable (. That is
000(S1I1) = OOO(Sl) 0 A((). This defines a ring of super functions on S1I1.
11
Consider derivations on the super circle, Der(COO(Slll )). The set of such
things is topologically spanned by derivations of the form znoz , zno(, zn(oz, and
zn(o(. As before, we first restrict our attention to finite sums from the canonical
topological basis. As a C[z, z-l]-module, this object is 4-dimensional. There is a
2-dimensional subalgebra we are interested in. For n E Z and r E Z + ~, we define:
L _ n+ I!=) n + 1 nI" !=)n - -z U z - --z ,>U(2
Gr = izr+~((oz + or;)
(ILl)
(II.2)
(Note we will only have square roots Gr for Ln with n odd. The algebra with
square roots of the other L n is called the Ramond algebra.) Any V-module is also
a representation of Der(Coo(Slll)), whereby ( and o( act as O. As such, the L n
we've just defined project to the L n in V.
Let's denote the Lie algebra spanned by the L n and the Gr by V . In V we
have the commutation relations:
[Lm , Ln ] = (m - n)Lm +n
m[Gr , L m ] = (r - 2" )Gr +m
[Gr , Gs ] = 2Lr+s
We define the Neveu-Scwarz algebra NS to be the one-dimensional central
extension of V with relations:
m 3 -m[Lm , Ln] = (m - n)Lm +n + 6m +n,O 12 C
[Gn Lm ] = (r - ;) Gr +m
4r2 -1[Gr, Gs ] = 2Lr+s + 6r+s ,o 12 C.
(11.3)
(II.4)
(II.5)
(II.6)
(II.7)
(II.8)
12
II.2 Super Conformal Vertex Algebras and Representations of NS
Just as the definition of a conformal vertex algebra centered around the
algebra Vir, the definition of a conformal vertex super algebra centers around NS.
Definition ILL A conformal vertex super algebra is a graded vertex super
algebra V containing a distinguished vector T E \13/2 (called a superconformal
vector) and a conformal vector W equal to T(O)T such that
(a) Under the identifications Tn = Gn - I / 2 and Wn = Ln - I ,
the Tn and Wn satisfy the relations II.6 - II.8
with C acting as multiplication by a complex number c.
(b) WI (or Lo) is the grading operator for V.
(c) W-I (or L_2 ) is the translation operator for V.
The number c is called the central charge of V.
Most of what follows in this section is established in [4] and [11]. NS has
the obvious ~Z-grading with deg(Lm ) = -m, deg(Gr ) = -r, and deg(C) = O. If
we allow NSso to act in a simple way on C and induce to all of NS, we get an
important NS representation. Specifically, choose acE C. Let C1 be a
NS::;o-module where Lm and Gr act trivially for m, r ~ 0 and C· 1 = cl. Then
define the Verma module:
(11.9)
This NS-module has a unique minimal quotient which we call NS(c). It will
be important to us that this module NS(c) has the structure of a vertex super
algebra (see [11]). The superconformal vector T is taken to be G-3/ 21. From now
on when referring to NS(c), we will be referring to it as a vertex super algebra, not
just an NS-module.
13
For most values of c, NS(c) has infinitely many irreducible representations.
However, there is a two parameter family of c's for which NS(c) has finitely many
irreducible representations, all of which can be described. Let
C =~(1_2(P-q)2)p,q 2 pq (11.10)
where q - p E 22. and gcd (q;p ,p) = 1. These are precisely the c's for which NS (c)
has finitely many irreducible representations. From now on, we write
NS(p, q) = NS(cp,q).
Once p and q are fixed, we define the numbers
(pr - qS)2 _ (p _ q)2hr,s = --'---------'--------'---------'----
8pq
(lI.ll)
where 1 :::; r < q, 1 :::; s < p, and rand s have matching parity. These numbers are
called the energies of the irreducible modules of NS(p, q), and they determine its
irreducible modules. Specifically, we once again take a representation of N S-:;,o on
Cl. This time, 01 = cp,q1, L m and Gr act trivially for m, r > 0, but L 0 1 = hr,sl.
The Verma module obtained this way,
has a unique minimal quotient as an N S-module, which we call N S(p, q)hT ,8.
These NS(p, q)hT ,8 are the irreducible representations of NS(p, q).
14
CHAPTER III
A SUPER VERSION OF ZHU'S THEOREM
In this chapter we follow the consequences of commuting two odd vectors in
a vertex superalgebra, and obtain a super version of Zhu's trace identity.
IlL 1 A Lemma
We need the following lemma.
Lemma IlL 1. Let a and b be two odd vectors in the vertex super algebra V, and
let M be any V -module. Then
00L zdb-m-lQm(":", q) tr IMY(b[m]a, z)qLO
m=O W
(b) Wdb tr IMY(b, w)Y(a, z)qLO =
00L (_l)mzdb-m-lQm( w, q) tr 1MY(b[m]a, z)qLO
Z
m=O
where
Proof. For the first part of the lemma, we begin by expanding the state-field
(IIL1 )
(IIL2)
15
correspondence and observing that most terms cannot contribute to the trace.
Wdb tr 1MY(a, z)Y(b, w)qLO - Wdb tr 1M L a(k)b(l)z-k-lw-l-lqLo (III.3)
k,l
W-da+lZ- 1 L Z-kWk tr IMa(k)b(l)qLO (IlIA)
k
where l = da + db - k - 2 in the last term (so that the degree of a(k)b(l) is 0.)
Keeping in mind that a and b are odd, and using the commutativity of tr,
(III.5)
qda-k-l tr 1M ([b(l), a(k)] - a(k)b(l)) qLo (IIL6)
So we may solve for tr IMa(k)b(l)qLO and then apply the Borcherds identity
(proposition 1.2):
(III.7)
(III.8)
Combining Eq. (IlIA) with Eq. (IIL8), we have:
Wdb tr 1MY(a, z)Y(b, w)qLO
= (W)l-daz-da """"" (W)k qda-k-l tr 1M~ (l) (b(i)a) qLo (IIL9)
z L...: z 1 + qda-k-l ~ i l+k-i
~ triM (ct'" z-d.~~ C)k 1 ~~::~:-l G) (b(i)a),+k_i) qLO
(IILI0)
16
Recalling Eq. (1.5), we have:
Wdb tr IMY(a, z)Y(b, w)qLO
00 1 ( (Z )da-k-l qda-k-l )
=~ m! ~ w 1 + qda-k-l (da - k _l)m z-da tr 1M 0 (b[m]a) qLo
= ~~ (~ (!..-) r -L (r )m) Zdb-m- 1tr 1MY (b[m]a, Z)) qLo~ m! ~ w l+qr
m=O rEZ+~
00
= L Qm( ~, q)Zdb-m-l tr 1MY (b[m]a, z)) qLo
m=O
This proves part (a) of the lemma. For the second part, we swap the roles of (a, z)
and k with (b, w) and l. In Eq. (II1.7), since a and b are odd,
[a(k), b(l)] = [b(l), a(k)]. Many of the terms in Eq. (II1.9) remain unchanged,
despite the swapping of roles. We have:
17
00
= L (-l)rnQrn (~, q)Zdb-rn-l tr 1MY (b[m]a, Z)) qLO.
m=O Z
o
III. 2 The Main Theorem
Now we are ready to prove the main theorem.
Theorem 111.1. Let a and b be two odd vectors in the vertex super algebra V, and
let M be any V -module. Then
00
tr 1M 0 (b[-l]a) qLo = L Q~(q) tr 1M 0 (b[m]a) qLo
m=l
Proof. Recall the numbers Ci, defined by the equation
00
(1 + z)db - 11n (1 + z)-l = L Cizi .
i=-l
(IlL 11)
18
Then we may write:
CXl
tr\Mo(b[-I]a) q Lo = L Ci tr IMo(b(i)a) qLo
i=-l
CXlL Ci tr IMzda+db-i-ly (b(i)a, z) qLo
i=-l
00L Cizda+db-i--l Resw-z(w - Z)i tr 1MY (Y (b, w - z) a, z) qLo
i=-l
According to associativity for vertex super algebras (the super version of
proposition 1.1), this equals
(III. 12)
f Cizda+db-i-l ( Resw ~w,z(w - z)i tr 1MY(b, w)Y (a, z)qLO + Resw ~z,w (w - z)i tr 1MY(a, z)Y(b, w)qLo)
i=-l
= f Cizda+db-i-l (Resw ~w,z(w - z)iw-db f (-l)mQm C~~, q) tr 1MY(b[m]a, z)qLO
i=-l m=O Z
+ Resw ~z,w(w - Z)iw-db f Qm (';,q) trIMY(b[m]a,z)qLo).
m=O
Here we have used lemma III. 1. By Lz,w(Z - W)i we mean the power series
expansion of (z - W)i in the domain where Izi > Iwl. For most i In the expression
above, Lz,w(Z - W)i is simply (z - W)i. The exception is when i = -1. So we will
compute the i = -1 term of the sum separately. Note C-l = 1. We find the i = -1
term to be:
19
ith term of the sum is:
= 1 -db+i+l . m r
+Ci L --, L ( t )zda(_l)i-db-r+l~trIMY(blm]a,z)qLO
m=O m. r=-db+l db + r - 1 1 + q
= -db+i+l . ( )m-r
= ci L ~ L ( t )zda(_l)i-db-r+1+m -r ~r trIMY(b[m]a,z)qLo
m=O m. r=-db+ 1 db + r - 1 1 + q
= 1 -db+i+l . m r
+C; L --, L ( t )zda(_l)i-db-r+l~trIMY(b[m]a,z)qLO
m=O m. r=-db+l db + r - 1 1 + q
= 1 -db+i+l . -r r
= Ci L - L ( t ) zda(_l)i-db-r+l rm(_q__ + -q-) tr 1M Y(b[m]a, z)qLO
m=O m! r=-db+ 1 db + r - 1 1 + q-r 1 + qr
= 1 -db+i+1 .
=Ci L - L ( t )zda(_l)i-db-r+lrmtrIMY(b[m]a,z)qLo
m=O m! r=-db+l db + r - 1
Summing over all i not equal to -1, we have:
(III.13)
For fixed m the inner double series is not absolutely convergent. However,
the subseries of it in which r sums only through negative values is absolutely
convergent, and so we may switch the order of the inner summations, reindexing
appropriately. Then III.13 is rewritten:
20
An elementary examination of the generating series for the Ci reveals that
for k :::; db - 1, L~k (-1 )i-kci G) = 1. So III.13 reduces to:
And so the sum in Eq. (III. 12) is:
21
This completes the proof of the theorem. Since (L[-l]b)[-n] = nb[-n - 1], we
have:
Corollary 111.2. Let a and b be two odd vectors in the vertex super algebra V,
and let M be any V -module. Then
00
tr 1M 0 (b[-2]a) qLo = - L mQt(q) tr 1M 0 (b[m - l]a) qLo.
m=l
o
In the chapters that follow, it is this corollary that will be more often referred to.
22
CHAPTER IV
THE MAIN THEOREM APPLIED TO NS(5,3)
An application of the main result of the previous chapter to the
Neveu-Schwarz algebras of a certain type yields interesting combinatorial
identities. In the course of studying these examples, we discover the values for
some of the numbers Bm,db' (So far, we know them only as series). In this chapter,
we specifically examine NS(5, 3). In the following, we examine NS(p, q).
IV.l The Application to NS(5, 3)
The Verma module for NS with Cp,q = CS,3 = 170 has a unique (up to scalar)
singular vector Vsing, and the corresponding quotient yields NS(5, 3). In the case of
NS(5, 3),
Since the vertex algebra with state-field correspondence Y( , ) is isomorphic to
that with Y[ , ], we know that in NS(5, 3),
23
Since 2L[_4]1 = L[-1]L[-3]1 = w[0]L[_3]1, Zhu's theorem tells us that L[-4]1 does
not contribute to trace. So we have that:
Now L[_2]L[_2]1 = w[-l]W, and G[-S/2]G[-3/2] 1 = 7[-2]7. So we can apply
both Zhu's theorem and the corollary to the main theorem to obtain:
00
0= triM o(w) o(w)qLO + L G2m (q) tr 1M 0 (w[2m - 1]W) qLo
m=l
3 00
+2" L (mQ~ (q) tr 1M 0 (7 [m - 1]7) qLo) .
m=l
Given the filtered degrees of wand 7 (2 and 3/2) and the fact that only
nonnegative products appear in this expression, only a few m yield nonzero terms.
In fact we only need know the mth product of wwith itself for m = 1 and m = 3,
and of 7 with itself for m = 0, m = 1, and m = 2. This information can be
computed by hand, and is summarized:
w[l]W 72w--1120
w[3]W ~120
7[0]7 72w--1120
7[1]7 0
7[2]7 7-115
24
And so the trace equation reduces to:
Note the presence of the operator Lo - 2~0 = lo - 2c4 = 0 (w ). It suggests we
change our degree operator to L o = L o - 2~' Upon multiplication by q-c/24, the
equation then reads:
o= tr \MLo
2 qLo + 2G2 (q) tr IMLoqLo + G4 (q) tr 1M ;0 qLo
- 9 7 -
+3Qi(q) tr IMLoqLo + "2 Qt (q) tr 1M 15 qLo .
And so the Lo-character of M must be annihilated by the differential operator
NS(5,3) has two simple modules, whose characters are known. Their
energies are 0 and 110' This implies that their characters have lowest terms qO and
ql/l0. So their shifted characters have lowest terms q-7/240 and qI7/240. In light of
the above, this means:
(~~) \ (-~ + 3BL3/2) (-2:0) + Go 7~0 + :a B3,3/2) - 0
(21470) 2 + ( _~ + 3B1.3/2) (21:0) + (;0 7~0 + 270B3,3/2) 0
25
This allows for the solving of B1,3/2 and B3,3/2:
1
24
7
960
(IV.l)
(IV.2)
Finding these numbers is the important part of the example. We continue
studying it to arrive at an interesting q-series identity. The shifted characters of
NS(5, 3)° and NS(5, 3)1/10 are determined in [11]:
cho q-7/240TlO (q) L (q!(lSk 2+2k) _ q!(15k2 +Sk+l))
kE7L.
q17/240TlO (q) L (q!(15k 2 -4k) _ q!(15k2 +l4k+3)
kE7L.
An application of the quintuple product identity (see [1]) allows us to rewrite these
as:
00
cho = q-7/240TlO (q) II (1 - qSn) (1 - q5n-9/2) (1 - q5n-1/2) (1 _ qlOn-4) (1 _ qlOn-6)
n=l
00
ch I / 10 = q17/240Tlo (q) II (1 - q5n) (1 - q5n-7/2) (1 - q5n-3/2) (1 _ qlOn-2) (1 _ q10n-S)
n=l
As infinite products, their logarithmic derivatives are easy to establish:
(Ian - 2)qlOn-3 (Ian - 8)qlOn-9 nqn- 1 )
-'---------:c'::-"----=-- - + -,-----=-------,--,-1 - qlOn-2 1 - qlOn-S (1 _ qn)3
(5n - 9/2)q5n-ll/2
1 - q5n-9/2
5nq5n-1
1 - q5n
ch~ = _~ -1 + f-- ((n - 1/2)qn-3/2
ch 240 q LJ 1 + qn-1/2
° n=l
(5n - 1/2)q5n-3/2 (lOn - 4)q10n-5 (Ian _ 6)qlOn-7 nqn- 1 )
----'---- - +---I - q5n-1/2 1 - qlOn-4 1 - qlOn-6 (1 _ qn)3
Ch~/l0 _ ~ -1 LOO ((n - 1/2)qn-3/2 5nq5n-1 (5n - 7/2)q5n-9/2
- - 240 q + 1 1/2 5 5 7/2ch I / 10 n=l + qn- 1 - q n 1 - q n-
(5n - 3/2)q5n-5/2
1 - q5n-3/2
26
As fundamental solutions to the differential equation D f (q) = 0, the Wronskian of
cho and chl / IO is related to the subleading term of D. We compute the Wronskian
(where now f means the derivative with respect to T, where q = eT ):
cho chl / l0
-f -f
cho ch l / l0
ch ch (~ ~ ((~) (n/2)qn/2 _ (!.!.) (2n)q2n))
o 1/10 10 + L...J 10 1 - qn/2 5 1 _ q2n
n=l
Abel's theorem tells us that for some C dependent upon our choice of
antiderivative,
ch ch (~ ~ ((~) (n/2)qn/2 _ (!.!.) (2n)q2n))
o 1/10 10 + L...J 10 1 _ qn/2 5 1 _ q2n
n=l
The quantity 2G2(q) + 3Qt(q) has an easy-to-describe antiderivative. Since:
An antiderivative with respect to q is:
This is the same as an antiderivative for 2G2(q) + 3Qt(q) with respect to T. SO we
have that
ch ch (~ ~ ((~) (n/2)qn/2 _ (!.!.) (2n)q2n))
o 1/10 10 + L...J 10 1 _ qn/2 5 1 _ q2n
n=l
00
= Cql/24 II(1 - qn)4(1 + qn-l/2)6
n=l
Recalling the infinite product expressions for cho and Chi/la' we have that
Comparing constant terms, C must be l~' So we normalize, and rewrite the
right-hand side in terms of 1]-functions:
27
1 + 10~ [(.!!.-) (n/2)qn/2 _ (~) (2n)q2n]
L...J 10 1 - qn/2 5 1 _ q2n
n=l
1]2(5T)1]g(T)
1]10 (T)1]o(5T) . (IVA)
28
CHAPTER V
THE CASE OF NS(p, q)
For p and q odd, we can repeat the example of the previous chapter to a
point. For (p, q) = (6k ± 1,3), we can go further. Not only does this yield a family
of identities like IVA, but it gives us information about the constants Bm,d and
information about the form of singular vectors in a vertex superalgebra minimal
model.
V.1 A Differential operator
We would normally write an even monomial vector v of degree d in NS (p, q)
according to the PBW theorem as Lr~n]G[~(n-l/2)]L{:_(n-l)J'" Lr':'2JG[~3/2J1, with
ji ::::: 0, jo > °or to = 1, and ti is either °or 1. Here we would have
2.:7=0 (ji(n - i) + ti(n - i - 1/2)) = d. The evenness of v would imply ti = 1 for
an even number of i.
The PBW expression for v is inconvenient in what follows. We would prefer
to write v as a linear combination of monomial vectors that begin with L[-1J, L[-2J,
or G[-5/2J. Each such monomial would be a zeroth product with W, a minus first
product with W, or a minus second product with T. The following lemma allows us
to do this.
29
Lemma V.l. The vector v = L[~nJGf~(n-1/2)]LI~(n-1)] ... L{:' 2J Gf':'3/2] 1 can be
rewritten as a C-linear combination of vectors of the form L[_l]U) L[_2]W and
G[-5/2]X with u, w) and x PEW monomials. Furthermore, if v =1= L[_2] 1 and
v =1= G[-5/2]L[--=-2]G[-3/1] 1) then the leading terms of all v and x will be L[-n] or
G[-n-1/2] with n > 2. In other words)
NS(p, q) = L[_l]NS(p, q) + L[_2]NS(p, q) + G[-5/2] NS(p, q). (V.I)
Proof. We prove a slightly stronger claim: If a is a PBW monomial, the vectors
L[-nJa and G[-(n-1/2)]a can be rewritten as a C-linear combination of vectors of the
form L[_l]U, L[-2JV and G[-5/2]W with U, v, and w PBW monomials.
The claim is true for L[-2]a and G[-(3-1/2)]a. From here we can induct on n
using the identities:
L[-n]
G[-(n-1/2)]
I
--[L[-l)' L[-(n-1)]]
n-2
I
--[L[-l], G[-((n-1)-1/2)Jl.
n-2
After each application of the above identities, we may need to commute terms to
the right to write the tail of the vector as the sum of PBW monomials. If
v =1= L[_2] 1 and v =1= G[-5/2]L[--=-2]G[-3/2] 1, then the commutations above had to have
been applied at least once. This proves that leading terms of all v and x will be
L[-n] or G[_n_1/2]with n > 2.
D
The next lemma guarantees the existence of the differential operator
generalizing that which appeared in the previous chapter.
30
Lemma V.2. For any vector v,
where Fv is a polynomial with coefficients in C[02' 0 4"", QT, Qt,·· .].
Proof. The claim is immediately verifiable for v of low degree. For v of higher
degree, the claim first reduces to PBW monomials via the additivity of trace. The
previous lemma reduces it further to the case v = L[-llW, L[_2]w or G[-5!2]W.
Zhu's theorem tells us that
Using Zhu's theorem again, and inducting on the degree of v,
tr 1M o(L[_2]w)qLo tr 1M o(w[-l]w)qLo
tr 1M o(w) o(w)qLO +2: 02k(q) tr 1M o(w[2k - l]w)qLO
k=l
qaq tr 1M o(w)qLO + 2: 02k(q) tr 1M o(L[2k_2]w)qLo
k=l
qaq tr IMFw (La) qLo +2: 02k(q) tr IMFL[2k_2jW (La) qLo
k=l
= tr IMLoFw (La) qLo + 2: 02k(q) tr IMFL[2k_2jW (La) qLo,
k=l
recrsively defining FLr-2JW' Similarly, using the corollary to the main theorem and
inducting on the degree of v,
31
tr 1M 0(T[-2]w) qL o
00
- L mQ~(q) tr 1M 0 (T[m - l]w) qLo
m=l
00
- LmQ~(q)trIMo (G[m-3/2]]W) qLo
m=l
recrsively defining FCI_5/2jW' o
The next lemma tells us about the leading and subleading coefficients of Fv .
Lemma V.3. Let v = L[~n]G[~(n-l/2)]L{=_(n_l)] ... L{:.2]G[~3/2]1 be an even PEW
monomial of degree 2d.
• Otherwise! deg Fv :s; d - 2
Proof. The first part of this lemma was proved in [10], but we prove the entire
lemma here. We induct on the degree of v. First, let v = Lf-2]1. The claim is true
when d = 1, with Fv(X) = X. Now let d > 1, and apply Zhu's theorem in what
follows. (Appearances of ... indicate terms of low enough degree so that the
inductive hypothesis guarantees they do not contribute to the degree d or degree
d -1 coefficient of Fv .)
---------------- ---_.
32
tr 1M o(w[ -11Lf-=-2] 1 )qLO
tr 1M o(W) o( Lf-=-2] 1)qLO
+ L G2k (q) tr 1M 0(L[2k_l]Lf-=-2]1)qLO
k=l
tr 1M o(W) o(Lf-=-~ l) qLO+ G2(q) tr 1M 0(L[1]Lf-=-2]1)qLO+ ...
tr IMLo( Lod- 1+ (d -l)(d - 2)G2(q)Lod- 2)qLo
+(2d - 2)G2(q) tr 1M (Lod-l + (d - l)(d - 2)G2(q)Lod-2) qLo+ ...
trlM(Lod +d(d-l)G2(q)Lod-l)qLo + ... ,
inductively proving the claim for v = Lf-2] 1.
Next, if v = G[-5/2]Lf-=-~G[-3/2]1, then the corollary to main theorem shows
tr 1M 0(T[-21Lf-=-2JG[-3/2]1) qLo
- L mQ~(q) tr 1M 0 (T[m - llLf-=-~G[-3/2Jl) qLo
m=l
-Qt(q) tr 1M 0 (G[-1/2]Lf-=-~G[-3/2Jl) qLo + ....
We can commute the G[-1/2] all the way to the right and see inductively that
almost all terms arising have degree less than our concern. The exception is when
G[-1/2] commutes past G[-3/2]' producing 2L[_2]' So we have
inductively proving the claim for v = G[-5/2]Lf-=-~G[-3/2] 1.
Finally, let v be otherwise. That is, let the leading term of v be L[-n] or
33
G[-n-l/2] with n > 2. Then lemma V.I allows us to rewrite v as a sum of terms of
the form L[_l]U, L[-l]w, and G[-S/2]X in such a way so that the leading terms of all
wand x are L[-n] or G[-n-l/2] with n > 2. Also, all wand x have smaller degrees
than v. Zhu's theorem and the main theorem inductively imply the third claim of
the lemma.
D
V.2 Singular Vectors and N S(p, q)
The Verma module for NS obtained in with charge Cp,q has a unique
singular vector Vsing of homogeneous degree 26 = (P-1)2(q-1) (see [11]). In the case
where p and q are both odd, we may normalize to write
where Ap,q is an as yet undetermined constant. Just as with NS(5, 3), we consider
the trace on Vsing' We can apply lemmas V.2 and V.3 to obtain
a trIMo(vSing)qLO
tr 1M FVSing ( La) qLo
tr 1M (LaO + [6(6 - I)G2k - 2Ap,qQi(q)] L OO- 1 + ...) qLo
And so the La characters of the 6 simple representations of NS(p, q) are
annihilated by a differential operator:
34
The charge and energies of NS(p, q) are given by
c
~ (1 _2(p - q)2))
2 pq
(pr-qs)2- (p_q)2
8pq
(V.2)
(V.3)
where 1 ::; s < p, 1 ::; r < q, and rand s have matching parity. In [11] the
characters of the NS(p, q)h were established to be
Chh = 'TJa(q) L (qb k - qak )
kEZ
where
(pr + qs + 2pqk)2 _ (p _ q)2
8pq
(pr - qs + 2pqk)2 _ (p - q)2
8pq
(V.4)
We will denote the La-character of NS(p, q)h by Chh' The collection of Chh
form a fundamental set of solutions to the differential equation DF(q) = 0, where
D is the differential operator established in the previous section. And so Abel's
theorem tells us:
(V.5)
Q(Q-1)+Ap,q 00 (1 _ qn)28(8-1)
C q 12 II (1 -1/2)4>-' (V.6)+ qn p,q
n=l
35
The leading exponent in ch~) is hi - 2c4' So the leading exponent in the
Wronskian on the left is L:i(hi - 2C4)' Using the formulas V.3 and Y.2, and
summing over values of rand s one can show'" (h· - ...f...) = (pq-p-q+1) (pq-p-q-6)
, Di t 24 92'
Equating the leading exponent on both sides of V.6 yields
(pq - p - q + l)(pq - p - q - 6)
It is not hard to see that 0 = (P-1~(q-1). So
160(0 - 1) + 16'\p,q.
(pq - p - q + 1) (pq - p - q - 6)
=} -3(pq - p - q + 1)
We have proved
(p - l)(q - l)(pq - p - q - 3) + 16'\p,q
16'\p,q
3
--(p - l)(q - 1).
16
Theorem V.l. For odd p, q, the singular vector of NS(p, q) has first few PBW
terms
or
We can also compare the leading coefficients of V.6 to determine the value
of C. Since chh; = qh;-"14 + ..., the leading coeffecient of ch~) is (hi - ;4)j. And so
------ ~------------
36
the leading coefficient in equation V.6 is the Vandermonde determinant:
1 1 1
We have proved
Theorem V.2. Let V = NS(p, q) with p, q odd. If hi denotes the energies of the
6 = (P-l~(q-l) simple represntations and chhi denotes the shifted (by c/24)
characters, then the Wronskian of the chhi
00
/pq-p-q+~~(fq-P-q-6)II (1- qn)28(8-1)(1 + qn-l/2)1(p-l)(q-l) II(hi - hj ).
n=l i>j
V.3 NS(6k ± 1,3)
The consequences of theorem V.2 can be followed through a bit further in
the special case q = 3. In this case all the energies hr,s can be chosen with r = 1,
s = 2i - 1. This in turn allows for an application of the Watson quintuple product
([1]) to chhi' We have
37
_ /24 () """' (P_ qS+2Pq lc)2_(p_q)2 (P+qS+2Pqlc)2_(p_q)2)
q C Tlo q L...J q 8pq - q 8pq
kE'L
-c/24+ (p_qs)2_(p_q)2 () """' (P-qS)Ic+Pq Ic 2 S+(P+Q(2i-l»Ic+PQ Ic 2 )
q 8pQ Tlo q L...J q 2 - q 2
kE'L
-c/24+ (p_Qs)2_(p_q)2 () """' (P-QS)Ic+PQ Ic 2 S+(P+QS)Ic+PqIc2)
q 8pQ Tlo q L...J q 2 - q 2
kE'L
-c/24+ (p_qs)2_(p_q)2 () """' (( =-")3k (-S )-1-3k) ( P) 1c(31c+l)
q 8pq Tlo q L...J q 2 - q 2 q 2
kE'L
qCSTlO(q) I1(1- qPn)(1- qPn-~)(l_ qp(n-l)+~)(1_qP(2n-l)-S)(1_ qP(2n-l)+S).
n~l
The point is that we can express Chhl,s as an infinite product, TIn As,no In that
-(1) ((1)) -
case, Chh1,s = TIn As,n L":m As,m/As,m = Chh1,s ·5s · Inductively, we can show
-(i)
chh l,s
So theorem V.2 and the above imply
Corollary V. 3.
00I1(h
l
,i - hl,j)q(2P-~)9~P-9) I1(1- qn)(P-l)iP- 3 ) (1 + qn-l/2)~(P-l).
i>j n=l
38
CHAPTER VI
FUTURE INQUIRIES
VI.1 Improving Upon the Differential Operator
Lemma V.3 used induction to determine the leading terms of Fv for two
types of PBW monomials. For the purposes of chapter V this was enough; only
these two types of PBW monomials in the expression of Vsing contribute to the
subleading term of the differential operator V.2.
A perfect improvement of lemma V.3 would give a closed formula for Fv
whenever v was a PBW monomial. This seems hopeless, as the diversity of PBW
monomials grows with the partition function. However it is still within reason to
ask for a theorem that expresses Fv as a sum over partitions. This could be useful
in determining more precise information about the form of Vsing. In NS(p, q) it has
been known since [9] that Vsing lies in V(p-l)(q-l) when p and q are odd, and in
2
V(P-l)(q-l)+l when p and q are even. But beyond this and theorem V.I, nothing is
2
universally known about Vsing'
39
VI. 2 Quasi-Bernoulli Numbers
In the process of examining NS(5, 3) we determined the values for E 1,3/Z
and E3,3/z. Recall that in general,
where the Cd,i are defined by (1 + z)d- 1 ln (1 + Z)-l = 2...= Cd,iZi. These numbers seem
to be some kind of generalization of the Bernoulli numbers
Em = t -.1_ i)-It (i) (n)m.
z+1 n
i=O n=O
At present this is just an observation, but there is undoubtedly more meaning to
it. The series Q~ (q) no doubt have modularity properties generalizing those of
GZk(q). The constant terms of GZk(q) are Bernoulli numbers (scaled by factorials),
and we believe there should be a meaningful way to relate the constant terms of
Q~(q) (the Em,d) to the Bernoulli numbers.
40
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