Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces ()
1. Vertex Operator Super Algebras
In this paper (based on several conference talks of the author) we review our recent results [1-5] on construction and computation of correlation functions of vertex operator superalgebras with a formal parameter associated to local coordinates on a self-sewn Riemann surface of genus
which forms a genus
surface. In particular, we review result presented in the papers [1-5] accomplished in collaboration with M. P. Tuite (National University of Ireland, Galway, Ireland).
A Vertex Operator Superalgebra (VOSA) [6-10] is a quadruple
:
,
, is a superspace,
is a linear map
so that for any vector (state)
we have
,
,
![](https://www.scirp.org/html/6-7401581\50e653af-ef61-493c-a65a-72ef88b3bcf9.jpg)
,
-parity.
The linear operators (modes)
satisfy creativity
![](https://www.scirp.org/html/6-7401581\41d4de08-07ee-427d-9ad6-1e80e58d5e3c.jpg)
and lower truncation
![](https://www.scirp.org/html/6-7401581\16f7b828-570a-4a04-b056-27b2097a8bed.jpg)
conditions for
and
.
These axioms identity impy locality, associativity, commutation and skew-symmetry:
![](https://www.scirp.org/html/6-7401581\609945bc-206f-4788-b34a-f306bd71e3c3.jpg)
![](https://www.scirp.org/html/6-7401581\dbbf68e4-2fd6-4ba8-ac80-42ce84374a8a.jpg)
![](https://www.scirp.org/html/6-7401581\b3b25f3a-f2f5-43a4-8e1b-e0822a4e7166.jpg)
![](https://www.scirp.org/html/6-7401581\8195b82c-c950-47a3-8369-e76d09b98dbc.jpg)
for
and integers
,
.
The vacuum vector
is such that,
, and
the conformal vector satisfies
![](https://www.scirp.org/html/6-7401581\fcd44903-79a8-494b-844b-c8a7fe0f7941.jpg)
where
form a Virasoro algebra for a central charge ![](https://www.scirp.org/html/6-7401581\4c6b0bfb-bd45-419e-8c8f-fc1c48c7f3a5.jpg)
![](https://www.scirp.org/html/6-7401581\d514fa80-704d-4ecd-97fd-bb610bad9767.jpg)
satisfies the translation property
![](https://www.scirp.org/html/6-7401581\02491503-58f7-45ba-996b-0172d9e40df7.jpg)
describes a grading with
, and ![](https://www.scirp.org/html/6-7401581\7bbc5ea7-c99a-4970-adac-ece85ae45f33.jpg)
1.1. VOSA Modules
Definition 1 A
-module for a VOSA
is a pair
,
is a
-graded vector space
,
,
for all
and
.
,
![](https://www.scirp.org/html/6-7401581\12cb9776-ad7a-483b-affc-7588dead71a4.jpg)
for each
,
.
, and for the conformal vector
![](https://www.scirp.org/html/6-7401581\62743c93-a31e-467b-8780-cf28fd74c524.jpg)
where
,
. The module vertex operators satisfy the Jacobi identity:
![](https://www.scirp.org/html/6-7401581\e6a10a86-7e27-4ec4-8741-ac2f56320608.jpg)
Recall that
. The above axioms imply that
satisfies the Virasoro algebra for the same central charge
and that the translation property
![](https://www.scirp.org/html/6-7401581\8b84b8d3-f45a-4d7e-8e4c-f0e4570b4dcb.jpg)
1.2. Twisted Modules
We next define the notion of a twisted
-module [8,11]. Let
be a
-automorphism
, i.e., a linear map preserving
and
such that
![](https://www.scirp.org/html/6-7401581\481c65b7-0659-4638-9a7c-5e09b714392b.jpg)
for all
. We assume that
can be decomposed into
-eigenspaces
![](https://www.scirp.org/html/6-7401581\f912d746-55e8-4d8e-9228-42078b3c0f98.jpg)
where
denotes the eigenspace of
with eigenvalue
.
Definition 2 A
-twisted
-module for a VOSA ![](https://www.scirp.org/html/6-7401581\f291b544-bd78-4aeb-aef4-6e3e8f578ba1.jpg)
is a pair
,
,
,
, for all
, and
.
, the vector space of (
)-valued formal series in
with arbitrary complex powers of
. For ![](https://www.scirp.org/html/6-7401581\baa44556-0d63-441d-b86c-58efae07b24e.jpg)
![](https://www.scirp.org/html/6-7401581\d97aaa5c-6295-4ca7-a9a4-221fe3157033.jpg)
with
,
,
sufficiently large.
, ![](https://www.scirp.org/html/6-7401581\8ceb8eb8-fff4-4e73-9c73-5e0ff267b971.jpg)
where
,
. The
-twisted vertex operators satisfy the twisted Jacobi identity:
![](https://www.scirp.org/html/6-7401581\92cada63-1cc8-40b9-8a12-38d9607de935.jpg)
for
.
1.3. Creative Intertwining Operators
We define the notion of creative intertwining operators in [3]. Suppose we have a VOA
with a
-module
.
Definition 3 A Creative Intertwining Vertex Operator
for a VOA
-module
is defined by a linear map
![](https://www.scirp.org/html/6-7401581\13aa250b-2336-4fa5-85c5-271b110d791d.jpg)
for
with modes
; satisfies creativity
![](https://www.scirp.org/html/6-7401581\7994666b-dcfa-449d-89ac-77bb2d3c15f4.jpg)
for
and lower truncation
![](https://www.scirp.org/html/6-7401581\f1ae0048-dbaa-425c-97e1-d0f48b2577e5.jpg)
for
,
and
. The intertwining vertex operators satisfy the Jacobi identity:
![](https://www.scirp.org/html/6-7401581\c7e7aef6-75d1-4617-bc25-f99101ec6edc.jpg)
for all
and
.
These axioms imply that the intertwining vertex operators satisfy translation, locality, associativity, commutativity and skew-symmetry:
![](https://www.scirp.org/html/6-7401581\9b1107d2-3ee1-4a08-8623-77756e59c94f.jpg)
![](https://www.scirp.org/html/6-7401581\2135709b-5677-4afc-b964-859e1ce54d9e.jpg)
![](https://www.scirp.org/html/6-7401581\6342ded7-6589-4d0d-b9cf-781d8de11254.jpg)
![](https://www.scirp.org/html/6-7401581\5cb5cd26-b1cf-4a63-93c9-41c0238e95d2.jpg)
![](https://www.scirp.org/html/6-7401581\e682fea2-fd64-4d8d-840b-2bd55df9a171.jpg)
for
,
,
and integers
.
1.4. Example: Heisenberg Intertwiners
Consider the Heisenberg vertex operator algebra
[10] generated by weight one normalized Heisenberg vector
with modes obeying
![](https://www.scirp.org/html/6-7401581\8cfc26db-0a16-4065-85c3-609890da8fb2.jpg)
.
In [3] we consider an extension
of
by its irreducible modules
generated by a
-valued continuous parameter
automorphism
.
We introduce an extra operator
which is canonically conjugate to the zero mode
, i.e.,
![](https://www.scirp.org/html/6-7401581\cf10f3bd-27a6-4d5a-a6a2-01bf809960eb.jpg)
The state
is created by the action of
on the state
. Using
-conjugation and associativity properties, we explicitly construct in [3] the creative intertwining operators
. We then prove:
Theorem 1 (Tuite-Z) The creative intertwining operators
for
are generated by
-conjugation of vertex operators of
. For a Heisenberg state
,
![](https://www.scirp.org/html/6-7401581\c45b40cd-49de-4467-9588-203f7955bb89.jpg)
The operators
with some extra cocycle structure satisfy a natural extension from rational to complex parameters of the notion of a Generalized VOA as described by Dong and Lepowsky [7,12]. We then prove in [3].
Theorem 2 (Tuite-Z)
satisfy the generalized Jacobi identity
![](https://www.scirp.org/html/6-7401581\1b0dd392-a7d5-469e-8840-2b1d4fb2db7b.jpg)
for all
.
1.5. Invariant Form for the Extended Heisenberg Algebra
The definitions of invariant forms [8,13] for a VOSA and its
-twisted modules were given by Scheithauer [14] and in [2] correspondingly. A bilinear form
on
is said to be invariant if for all
,
,
we have
![](https://www.scirp.org/html/6-7401581\0992fb0e-4e45-4980-a569-7b8551f36509.jpg)
![](https://www.scirp.org/html/6-7401581\037d4c92-26a6-4b0e-a680-b94b3ca22658.jpg)
We are interested in the Möbius map ![](https://www.scirp.org/html/6-7401581\a4b37d6a-5243-46f6-b3ed-622025af1765.jpg)
associated with the sewing condition so that
with
. We prove in [3]
Theorem 3 (Tuite-Z) The invariant form
on
is symmetric, unique and invertible with
![](https://www.scirp.org/html/6-7401581\e97302d5-f16e-45f5-97bd-3b90bab64f7d.jpg)
1.6. Rank Two Free Fermionic Vertex Operator Super Algebra
Consider the Vertex Operator Super Algebra (VOSA) generated by
![](https://www.scirp.org/html/6-7401581\e031bf61-9a58-4c8b-b4a0-f3de3da6df1d.jpg)
for two vectors
with modes satisfying anti-commutation relations
![](https://www.scirp.org/html/6-7401581\959988f1-9cae-4caf-906e-edea3de7bc72.jpg)
The VOSA vector space
is a Fock space with basis vectors
![](https://www.scirp.org/html/6-7401581\40a0800f-e49a-4360-b17c-a1169065da1b.jpg)
of weight
where
and
with
for all
.
1.7. Rank Two Fermionic Vertex Operator Super Algebra
The conformal vector is
![](https://www.scirp.org/html/6-7401581\eb98c258-791d-479c-aa84-c5527a257138.jpg)
whose modes generate a Virasoro algebra of central charge 1.
has
-weight
. The weight
subspace of
is
, for normalized Heisenberg bosonic vector
, the conformal vector, and the Virasoro grading operator are
![](https://www.scirp.org/html/6-7401581\eb80e0ba-f9a7-44b1-a558-1d6e0758d898.jpg)
![](https://www.scirp.org/html/6-7401581\d164fddf-c8d3-46cf-a57c-6d5a017a4488.jpg)
2. Sewing of Riemann Surfaces
2.1. Basic Notions
For standard homology basis
,
with
on a genus
Riemann surface [15,16] consider the normalized differential of the second kind which is a symmetric meromorphic form with
, has the form
![](https://www.scirp.org/html/6-7401581\175f9017-d52e-4bfc-a48d-b0ed2824f358.jpg)
A normalized basis of holomorphic 1-forms
, the period matrix
, and normalized differential of the third kind are given by
![](https://www.scirp.org/html/6-7401581\c5e614e7-2f74-4117-bf3f-77402cb83cea.jpg)
![](https://www.scirp.org/html/6-7401581\e3c41d61-5e93-4c8d-a469-23a62a03a149.jpg)
where
,
for
,
.
2.2. Period matrix
is symmetric with positive imaginary part i.e.
, the Siegel upper half plane. The canonical intersection form on cycles is preserved under the action of the symplectic group
where
![](https://www.scirp.org/html/6-7401581\0d0f8c39-8cb0-48db-b5ab-0bc8711507f1.jpg)
This induces the modular action on ![](https://www.scirp.org/html/6-7401581\62a12088-26a8-4f57-8006-dab6a017f785.jpg)
![](https://www.scirp.org/html/6-7401581\67c4ad66-29ee-4440-9d5a-e554c25a6f60.jpg)
2.3. Sewing Two Tori to Form a Genus Two Riemann Surface
Consider
two oriented tori
with
for
for
, the complex upper half plane. For
the closed disk
is contained in
provided ![](https://www.scirp.org/html/6-7401581\0f406409-00f8-49a4-a888-20fdef49180c.jpg)
where
![](https://www.scirp.org/html/6-7401581\737841a7-edbb-4378-a820-c6566ed47a12.jpg)
Introduce a sewing parameter
and excise the disks
and
where
![](https://www.scirp.org/html/6-7401581\5a2cf15e-15e7-4a4f-be31-d778106b40f8.jpg)
Identify the annular regions
and
via the sewing relation
![](https://www.scirp.org/html/6-7401581\3c9471b3-ed02-4aa9-8d95-5f7c4fc99f54.jpg)
gives a genus two Riemann surface
parameterized by the domain
![](https://www.scirp.org/html/6-7401581\0834292c-1b19-41b6-97c9-ea2728e675f8.jpg)
2.4. Torus Self-Sewing to Form a Genus Two Riemann Surface
In [1] we describe procedures of sewing Riemann surfaces [17]. Consider a self-sewing of the oriented torus
,
,
.
![](https://www.scirp.org/html/6-7401581\bcf6032e-8242-4b3d-a8d8-eb5fdfdfcb9e.jpg)
Define the annuli
,
centered at
and
of
with local coordinates
and
respectively. We use the convention
,
. Take the outer radius of
to be
.
Introduce a complex parameter
,
. Take inner radius to be
, with
.
,
must be sufficiently small to ensure that the disks do not intersect. Excise the disks
![](https://www.scirp.org/html/6-7401581\9048a72e-a73d-47b2-b488-301b00d7c6da.jpg)
to form a twice-punctured surface
![](https://www.scirp.org/html/6-7401581\70e7b094-5d2d-46a0-a201-8134fe51b0ea.jpg)
Identify the annular regions
,
![](https://www.scirp.org/html/6-7401581\a1e08a74-495b-41f2-935d-746605c8798a.jpg)
as a single region
via the sewing relation
![](https://www.scirp.org/html/6-7401581\fe78fade-5024-4365-aac8-343f839d4ee6.jpg)
to form a compact genus two Riemann surface
parameterized by
![](https://www.scirp.org/html/6-7401581\bca51a32-df20-4787-baf5-f01c9e36621d.jpg)
3. Elliptic Functions
3.1. Weierstrass Function
The Weierstrass
-function periodic in
with periods
and
is
![](https://www.scirp.org/html/6-7401581\6b474053-4f31-4576-93a2-3fa35a10e718.jpg)
for
,
. We define for
,
![](https://www.scirp.org/html/6-7401581\3940c0a6-a35d-446e-8bf8-089645c75d70.jpg)
Then
.
has periodicities
![](https://www.scirp.org/html/6-7401581\280ae0c2-8222-48c3-8134-ed1b83f27adf.jpg)
3.2. Eisenstein Series
The Eisenstein series
is equal to
for
odd, and for ![](https://www.scirp.org/html/6-7401581\801b8028-65ac-4779-8127-83fb227ee821.jpg)
![](https://www.scirp.org/html/6-7401581\fe50c796-8d14-49e1-b4fc-418d2d5da787.jpg)
where
is the
th Bernoulli number. If ![](https://www.scirp.org/html/6-7401581\bb04f687-a9ad-4173-a0b3-ac169b735f1b.jpg)
then
is a holomorphic modular form of weight
on ![](https://www.scirp.org/html/6-7401581\3c0f0da3-e240-4abd-954a-8e794aafef69.jpg)
![](https://www.scirp.org/html/6-7401581\092c4e90-7f50-4f33-a5ec-16d1ea028416.jpg)
for all
, where
.
is a quasimodular form
![](https://www.scirp.org/html/6-7401581\d691254f-ff50-426f-af0c-0663e7ba20f5.jpg)
having the exceptional transformation law.
3.3. The Theta Function
We recall the definition of the theta function with real characteristics [18]
![](https://www.scirp.org/html/6-7401581\9d9510a0-1eaf-4993-90b4-44c9f3ee7605.jpg)
for
,
![](https://www.scirp.org/html/6-7401581\8c865cc5-5496-4372-a104-ffc129822896.jpg)
![](https://www.scirp.org/html/6-7401581\f775b878-2174-4426-ab19-ffe97dd0eb82.jpg)
![](https://www.scirp.org/html/6-7401581\387f1fc8-405a-466d-902b-b37cfef5e7d3.jpg)
for
.
3.4. Twisted Elliptic Functions
Let
denote a pair of modulus one complex parameters with
for
. For
and
we define “twisted” Weierstrass functions for
[19,20]
![](https://www.scirp.org/html/6-7401581\7971180f-f3a1-4bae-beda-d63dc27829a3.jpg)
for
where
means we omit
if
. ![](https://www.scirp.org/html/6-7401581\a2c06ed8-6926-48f3-b254-c37c261be737.jpg)
converges absolutely and uniformly on compact subsets of the domain
[20].
Lemma 1 (Mason-Tuite-Z) For
,
![](https://www.scirp.org/html/6-7401581\020a6703-695d-4284-8c9b-45a3f5bc142a.jpg)
is periodic in
with periods
and
with multipliers
and
respectively.
3.5. Modular Properties of Twisted Weierstrass Functions
Define the standard left action of the modular group for
on
with
![](https://www.scirp.org/html/6-7401581\50c5d068-8d37-4567-a283-8251cb0d0364.jpg)
We also define a left action of
on ![](https://www.scirp.org/html/6-7401581\0177fdbc-e624-44b2-94f7-70eb8a20020e.jpg)
![](https://www.scirp.org/html/6-7401581\dbe8181c-e684-49e3-be34-58dddfbd7c54.jpg)
Then we obtain:
Theorem 4 (Mason-Tuite-Z) For
we have
![](https://www.scirp.org/html/6-7401581\cc646b72-9265-4f2a-b6d0-7851457ccaff.jpg)
3.6. Twisted Eisenstein Series
We introduce twisted Eisenstein series for
,
![](https://www.scirp.org/html/6-7401581\706a400c-1e84-4711-bba1-7b5bc89a3636.jpg)
where
means we omit
if
and where
is the Bernoulli polynomial defined by
![](https://www.scirp.org/html/6-7401581\f3297c50-1829-4c49-a4d7-9a0be078fd30.jpg)
In particular
.
Note that
the standard Eisenstein series for even
, whereas
![](https://www.scirp.org/html/6-7401581\1c6203f2-a537-44e9-a375-7db58585634b.jpg)
for
odd.
Theorem 5 (Mason-Tuite-Z) We have
![](https://www.scirp.org/html/6-7401581\cec04c76-00d3-4cfc-90b1-f18f6c1a516b.jpg)
Theorem 6 (Mason-Tuite-Z) For
,
is a modular form of weight
where
![](https://www.scirp.org/html/6-7401581\e3993eed-ac1c-408e-ad6b-cbad9bbb9237.jpg)
3.7. Twisted Elliptic Functions
In particular,
![](https://www.scirp.org/html/6-7401581\d5fec9ef-3d19-4c90-8a30-dcc5fd134147.jpg)
![](https://www.scirp.org/html/6-7401581\528d5ca4-e2e5-42b2-abb6-590cf9558b12.jpg)
![](https://www.scirp.org/html/6-7401581\0c2258e7-b8f6-4cc0-94b0-decd3bd48c66.jpg)
where
![](https://www.scirp.org/html/6-7401581\0d01b469-8f0f-465c-982d-78f7b5224960.jpg)
![](https://www.scirp.org/html/6-7401581\c163800e-bf43-4124-9067-3a250a919287.jpg)
and
![](https://www.scirp.org/html/6-7401581\a0d69744-a234-4c4c-a0e8-8ea7b0596848.jpg)
4. The Prime Form
There exists a (nonsingular and odd) character
such that [18,21,22]
![](https://www.scirp.org/html/6-7401581\c43ed6a3-611b-4356-a963-3567d785fa3c.jpg)
Let
be a holomorphic 1-form, and let
denote the form of weight
on the double cover
of
.
We define the prime form
![](https://www.scirp.org/html/6-7401581\0f509e38-e197-48b6-922d-7ed5c506364a.jpg)
The prime form is anti-symmetic,
and a holomorphic differential form of weight
on
and has multipliers
and
along the ![](https://www.scirp.org/html/6-7401581\78df2281-a594-4ca3-b713-a7adfaa72b55.jpg)
and
cycles in
[21]. The normalized differentials of the second and third kind can be expressed in terms of the prime form [18]
![](https://www.scirp.org/html/6-7401581\befd031e-6bb0-48ea-b57f-cded130696cc.jpg)
![](https://www.scirp.org/html/6-7401581\8c0d0cb9-a87c-47e8-ba57-5108c58ce929.jpg)
Conversely, we can also express the prime form in terms of
by [22]
![](https://www.scirp.org/html/6-7401581\82be5d2e-5175-45cf-aa2d-d939b8a6ae4d.jpg)
Torus Prime Form
The prime form on torus [18]
![](https://www.scirp.org/html/6-7401581\bb5af253-15e2-4853-acb3-b5c5f985f643.jpg)
![](https://www.scirp.org/html/6-7401581\43b846cf-6691-42f4-b642-75b3dcc535fa.jpg)
for
and
and where
.
We have
![](https://www.scirp.org/html/6-7401581\6c21cabc-6703-43c5-9fdf-cec95b8c330f.jpg)
![](https://www.scirp.org/html/6-7401581\2ac5ed17-0ed3-45bc-baf8-119cd2a59e54.jpg)
![](https://www.scirp.org/html/6-7401581\c85aff86-d43b-475a-b154-e98eb466e427.jpg)
has periodicities
![](https://www.scirp.org/html/6-7401581\1348b222-a329-4427-abcb-2cf9657b8e89.jpg)
![](https://www.scirp.org/html/6-7401581\68c2b161-3335-4926-81b7-8ca9c6b98b8a.jpg)
5. The Szegö Kernel
The Szegö Kernel [18,21,22] is defined by
![](https://www.scirp.org/html/6-7401581\028f952d-ebef-4490-a2e4-a15a6a570a3e.jpg)
with
,
,
,
, where
is the genus
prime form. The Szegö kernel has multipliers along the
and
cycles in
given by
and
respectively and is a meromorphic
-form on ![](https://www.scirp.org/html/6-7401581\c68c4553-a4fb-460f-b3bf-1cd7acc5dc1d.jpg)
![](https://www.scirp.org/html/6-7401581\3cf1b252-a82f-407d-a536-4f118b3f1155.jpg)
where
and
.
Finally, we describe the modular invariance of the Szegö kernel under the symplectic group
where we find [21]
![](https://www.scirp.org/html/6-7401581\e841958f-f72c-4339-8437-0ea586c53cd4.jpg)
with
,
,
![](https://www.scirp.org/html/6-7401581\f75643e3-a09f-437e-8f42-a8fc6e0b604e.jpg)
![](https://www.scirp.org/html/6-7401581\5e17c920-10ca-45cd-8f83-3c181a963442.jpg)
where
denotes the diagonal elements of a matrix
.
5.1. Modular Properties of the Szegö Kernel
Finally, we describe the modular invariance of the Szegö kernel under the symplectic group
where we find [21]
![](https://www.scirp.org/html/6-7401581\d86ce2dc-8ec8-4ec6-a6f0-1497546b80aa.jpg)
where
,
for
![](https://www.scirp.org/html/6-7401581\948fcd62-1aa3-441e-b133-ae80031f10a1.jpg)
where
denotes the diagonal elements of a matrix
.
5.2. Torus Szegö Kernel
On the torus
the Szegö kernel for
is
![](https://www.scirp.org/html/6-7401581\8f2c8099-a09c-4185-81f5-10ff6591b9ee.jpg)
where
![](https://www.scirp.org/html/6-7401581\44afca7a-b6c0-41e0-8e82-a8c2c9567ad0.jpg)
for
,
and
for
.
6. Structures on
Constructed from Genus One Data
Yamada (1980) described how to compute the period matrix and other structures on a genus
Riemann surface in terms of lower genus data.
6.1.
on the Sewn Surface ![](https://www.scirp.org/html/6-7401581\26fd5bbf-fd0a-4807-9535-dda0e6b90cbb.jpg)
can be determined from
on each torus in Yamada’s sewing scheme [17,23]. For a torus
the differential is
![](https://www.scirp.org/html/6-7401581\9009a522-38b5-44a5-8eb6-4e88120703ee.jpg)
![](https://www.scirp.org/html/6-7401581\54fbbbc6-3f50-48ce-a1ac-508912dcfa2f.jpg)
for Weierstrass function
![](https://www.scirp.org/html/6-7401581\81574954-2ccf-4428-bcac-d0cf5aa3d28f.jpg)
and Eisenstein series for ![](https://www.scirp.org/html/6-7401581\1d7e4257-35d2-4397-aa1d-d2a3cfe7d4cd.jpg)
![](https://www.scirp.org/html/6-7401581\f8d2648b-72a7-4175-a22b-70da2c4e20d4.jpg)
vanishes for odd
and is a weight
modular form for
.
is a quasi-modular form. Expanding
![](https://www.scirp.org/html/6-7401581\b022ffa1-6de9-440a-8bac-5dd6a42548d2.jpg)
![](https://www.scirp.org/html/6-7401581\edf06771-5541-464e-aeab-72f69ba5a1d7.jpg)
we compute
in the sewing scheme in terms of the following genus one data, ![](https://www.scirp.org/html/6-7401581\48457ca7-8abf-4b61-9b03-32447bc98b2b.jpg)
![](https://www.scirp.org/html/6-7401581\7774f378-d84d-4bc0-af35-45f893de6e28.jpg)
6.2. A Determinant and the Period Matrix
Consider the infinite matrix
where
is the infinite identity matrix and define
by
![](https://www.scirp.org/html/6-7401581\b18e666f-933c-46b2-b0b4-4cdd19f5770e.jpg)
as a formal power series in
[23].
Theorem 7 (Mason-Tuite)
a) The infinite matrix
![](https://www.scirp.org/html/6-7401581\72e65b85-be4f-4392-bd06-0662aba39855.jpg)
is convergent for
.
b)
is non-vanishing and holomorphic on
.
Furthermore we may obtain an explicit formula for the genus two period matrix
on
[23].
Theorem 8 (Mason-Tuite)
is holomorphic on
and is given by
![](https://www.scirp.org/html/6-7401581\db024dff-715f-4a02-9115-5aefc5a173d0.jpg)
![](https://www.scirp.org/html/6-7401581\73da6edd-309d-4cb4-b1e3-a375d1bb2413.jpg)
![](https://www.scirp.org/html/6-7401581\57c286fd-59b3-4178-824e-871ce1b954f8.jpg)
Here
refers to the
-entry of a matrix.
6.3. Genus Two Szegö Kernel on
in the
-Formalism
We may compute
for
in the sewing scheme in terms of the genus one data
![](https://www.scirp.org/html/6-7401581\b40c9804-d375-4f07-83b8-0878e8d95dfc.jpg)
is described in terms of the infinite matrix
for
![](https://www.scirp.org/html/6-7401581\b2c6c17f-6064-401d-b65e-bde459b0d4d0.jpg)
Theorem 9 (Tuite-Z)
a) The infinite matrix
is convergent for
b)
is non-vanishing and holomorphic on
.
6.4. Genus Two Szegö Kernel in the
-Formalism
It is convenient to define
by
.
Then we prove [1] the following Theorem 10 (Tuite-Z)
is holomorphic in
for
with
![](https://www.scirp.org/html/6-7401581\051f40df-9237-41ff-8425-468ad848e262.jpg)
for
where
is defined for
, by
![](https://www.scirp.org/html/6-7401581\eeec822f-ef6d-4a59-82fa-14507083297b.jpg)
with similar expression for
for
.
Let
, for
and integer
. We introduce the moments for
:
![](https://www.scirp.org/html/6-7401581\9945e918-83b9-4889-a867-311cf5074052.jpg)
with associated infinite matrix
. We define also half-order differentials
![](https://www.scirp.org/html/6-7401581\bc0c17f4-b33b-4280-b67a-f954c1d679b3.jpg)
![](https://www.scirp.org/html/6-7401581\0f13981f-7b41-4a9f-8cc7-1e3c1964ec10.jpg)
and let
and
, denote the infinite row vectors indexed by
,
. From the sewing relation
we have
![](https://www.scirp.org/html/6-7401581\6ece94fa-45d9-4e44-8af0-a20769dca28f.jpg)
for
, depending on the branch of the double cover of
chosen. It is convenient to define
![](https://www.scirp.org/html/6-7401581\a2fde419-205e-4e81-8d5e-f1eff5edca0a.jpg)
with an infinite diagonal matrix
![](https://www.scirp.org/html/6-7401581\ba85fd3d-815d-47d1-b037-903ea769772d.jpg)
Defining
by the formal power series in ![](https://www.scirp.org/html/6-7401581\5b91694a-f5d9-4683-a422-774ca7d277b0.jpg)
![](https://www.scirp.org/html/6-7401581\4cb06f69-8763-40e9-b3ad-82e7b4ef1700.jpg)
we prove in [1].
Theorem 11 (Tuite-Z)
a)
is convergent for
b)
is non-vanishing and holomorphic in
on
.
Theorem 12 (Tuite-Z)
is given by
![](https://www.scirp.org/html/6-7401581\629fecdd-698b-4676-ab1c-8fbf6089bd50.jpg)
7. Genus One Partition and n-Point Functions
7.1. The Torus Partition Function for a Heisenberg VOA
For a VOA
of central charge
define the genus one partition (trace or characteristic) function by
![](https://www.scirp.org/html/6-7401581\3f9fe160-ad8b-4e3e-94fa-073a67ef4692.jpg)
for the Heisenberg VOA
commutation relations with modes
![](https://www.scirp.org/html/6-7401581\8cecbc60-78f6-4f8d-b5ac-4faaddaa86c9.jpg)
![](https://www.scirp.org/html/6-7401581\61ee122b-c425-4124-a776-82932db1b68c.jpg)
7.2. Genus One Twisted Graded Dimension
We define the genus one partition function for the VOSA by the supertrace
![](https://www.scirp.org/html/6-7401581\0f828336-120d-4fc8-8de6-a83056c05160.jpg)
where
.
More generally, we can construct a
-twisted module
for any automorphism
generated by the Heisenberg state
. We introduce the second automorphism
and define the orbifold
-twisted trace by
![](https://www.scirp.org/html/6-7401581\a8128d4c-0180-4fcb-8b32-2818c99958ed.jpg)
to find for
,
![](https://www.scirp.org/html/6-7401581\b649be15-0c14-486d-8525-87417d9bde5a.jpg)
7.3. Genus One Fermionic One-Point Functions
Each orbifold 1-point function can found from a generalized Zhu reduction formulas as a determinant.
Theorem 13 (Mason-Tuite-Z) For a Fock vector
![](https://www.scirp.org/html/6-7401581\05a62aff-1028-4ded-a17d-a589373009ff.jpg)
![](https://www.scirp.org/html/6-7401581\82dc4e33-0f77-45af-a8ac-e25756a11e66.jpg)
where for ![](https://www.scirp.org/html/6-7401581\d4ec9252-166f-43e6-9ca1-1b253184a1ce.jpg)
![](https://www.scirp.org/html/6-7401581\2fd4409f-1af6-4f06-97f5-f7631d31fc20.jpg)
7.4. Genus One n-Point Functions for VOA
In general, we can define the genus one orbifold n-point function for
by
![](https://www.scirp.org/html/6-7401581\3e244f11-aeca-4928-a53a-d050c8ddf704.jpg)
Every orbifold n-point function can be computed using generalized Zhu reduction formulas in terms of a determinant with entries arising from the basic 2-point function for
[19].
7.5. Zhu Reduction Formula
To reduce an
-point function to a sum of
-point functions we need:
The supertrace property
![](https://www.scirp.org/html/6-7401581\2f8760f5-ccdc-41f3-b3f8-7a3c21f89e6e.jpg)
Borcherds commutation formula:
![](https://www.scirp.org/html/6-7401581\cda2885f-0665-4161-8ae6-0ce0e9843e7f.jpg)
expansions for
-functions:
![](https://www.scirp.org/html/6-7401581\4fd03c30-d3f6-4a20-8655-9225553350b8.jpg)
![](https://www.scirp.org/html/6-7401581\5e29cc24-5bb4-4d10-aba9-c9a53612d44c.jpg)
Theorem 14 (Mason-Tuite-Z) For any
we have
![](https://www.scirp.org/html/6-7401581\22d81144-1bb3-464a-9d9a-b14816d55c80.jpg)
where
is given by
![](https://www.scirp.org/html/6-7401581\ab3029a3-4700-432a-a764-f4247c9e9d35.jpg)
7.6. General Genus One Fermionic n-Point Functions
The generating two-point function (for
) is given by
![](https://www.scirp.org/html/6-7401581\eebf068d-abf4-4b31-9a0e-c0d03d7eb1bd.jpg)
Theorem 15 (Mason-Tuite-Z)
![](https://www.scirp.org/html/6-7401581\13849600-01db-47d5-8a83-2d0de663b5d8.jpg)
Theorem 16 (Mason-Tuite-Z) For
Fock vectors
![](https://www.scirp.org/html/6-7401581\6fec0473-55d5-4e68-8471-08e2a454dc40.jpg)
and
![](https://www.scirp.org/html/6-7401581\7730b7e7-205c-42ba-a314-327167914794.jpg)
for
and
with
. Then for
the corresponding
-point functions are non-vanishing provided
and
![](https://www.scirp.org/html/6-7401581\c6c77956-5ec3-4806-8bf9-564fb05bf830.jpg)
where
is certain parity factor. Here
is the block matrix
![](https://www.scirp.org/html/6-7401581\9e497765-c908-44f8-a447-7c6cc2fb7079.jpg)
with
![](https://www.scirp.org/html/6-7401581\5ceff498-a6c1-4a2b-a19b-e4d5fb1ccca5.jpg)
for
with
and
![](https://www.scirp.org/html/6-7401581\f1c0043c-835a-49bb-b7bd-802f5cb8cf14.jpg)
for
with
and
.
is the sign of the permutation associated with the reordering of
to the alternating ordering.
Furthermore, the
-point function is an analytic function in
and converges absolutely and uniformly on compact subsets of the domain
.
7.7. Torus Intertwined n-Point Functions
As in ordinary (non-intertwined) case [2,19,20,24-29] we construct in [4] the partition and
-point functions [30-39] for vertex operator algebra modules.
Let
,
,
be VOSA
automorphisms commuting with
. For
and the states
we define the intertwined
-point function [4] on the torus by
![](https://www.scirp.org/html/6-7401581\da23f917-ad6a-491e-ba24-f51dbb9280c2.jpg)
where
,
,
,
;
, for variables
associated to the local coordinates on the torus, and
is dual for
with respect to the invariant form on
. The supertrace over a
-module
is defined by
![](https://www.scirp.org/html/6-7401581\32488f2f-0d70-4213-ba84-378ce591564b.jpg)
For an element
of a VOSA
-twisted
-module we introduce also the differential form
![](https://www.scirp.org/html/6-7401581\5b5b487c-6841-47b2-afab-37ef86878ce8.jpg)
associated to the torus intertwined
-point function.
7.8. Torus Intertwined Two-Point Function
The rank two free fermionic VOSA
, [10]
is generated by
with
![](https://www.scirp.org/html/6-7401581\caa34323-4074-4dfe-b35e-600318fdafe1.jpg)
The rank two free fermion VOSA intertwined torus
-point function is parameterized by
,
, and
, [2, 4] where
![](https://www.scirp.org/html/6-7401581\907d4693-c4f2-4f4b-8eee-397286cbfa0f.jpg)
for real valued
,
,
,
.
For
and
, ![](https://www.scirp.org/html/6-7401581\1c349f2b-a150-4224-9057-e3812cf7e65f.jpg)
we obtain [4] the basic intertwined two-point function on the torus
![](https://www.scirp.org/html/6-7401581\456f6981-1a19-4d24-9125-ee0e37ccd106.jpg)
We then consider the differential form
![](https://www.scirp.org/html/6-7401581\80bad698-2419-4de8-9f7d-f272fc9eb2bd.jpg)
associated to the torus intertwined
-point function
![](https://www.scirp.org/html/6-7401581\b88d2c0f-8bdb-4561-9f1d-920515fce1be.jpg)
with alternatively inserted
states
and
states
distributed on the resulting genus two Riemann surface
at points
. We then prove in [4].
Theorem 17 (Tuite-Z) For the rank two free fermion vertex operator superalgebra
and for
the generating form is given by
![](https://www.scirp.org/html/6-7401581\7d6fc76d-b336-4cd0-bdae-76d91ac1e546.jpg)
![](https://www.scirp.org/html/6-7401581\08ebc0cb-3361-4fb9-9601-11657c6826d1.jpg)
is the basic intertwined two-point function on the torus, and
-matrix
with elements given by parts of the Szegö kernel.
8. Genus Two Partition and n-Point Functions
8.1. Genus Two Partition Function in
-Formalism
We define the genus two partition function in the earlier sewing scheme in terms of data coming from the two tori, namely the set of 1-point functions
for all
. We assume that
has a nondegenerate invariant bilinear form—the Li-Zamolodchikov metric. Define
![](https://www.scirp.org/html/6-7401581\eb3bc912-99a2-494c-bf8a-f0f955dcc205.jpg)
The inner sum is taken over any basis and
is dual to
wrt to the Li-Zamolodchikov metric.
8.2. Genus Two Partition Function for the Heisenberg VOA
We can compute
using a combinatorial-graphical technique based on the explicit Fock basis and recalling the infinite matrices
.
Theorem 18 (Mason-Tuite) a) The genus two partition function for the rank one Heisenberg VOA is
;
b)
is holomorphic on the domain
;
c)
is automorphic of weight
;
d)
has an infinite product formula.
8.3. Genus Two Fermionic Partition Function
Following the definition for the bosonic VOA we define for
, ![](https://www.scirp.org/html/6-7401581\572433f0-9c41-484c-93bf-be4b19a54bfe.jpg)
![](https://www.scirp.org/html/6-7401581\244600ee-e34b-4ce8-a23d-8e85a4286903.jpg)
The inner sum is taken over any
basis and
is dual to
with respect to the Li-Zamolodchikov square bracket metric.
is the genus one orbifold 1-point function. Recall that the non-zero 1-point functions arise for Fock vectors
![](https://www.scirp.org/html/6-7401581\680237e9-52b2-4955-a127-006ea84b6e48.jpg)
such that
,
![](https://www.scirp.org/html/6-7401581\fb7e7ed4-06f2-4069-aa83-1a7ccb8c8ba1.jpg)
The Li-Zamolodchikov metric dual to the Fock vector is
![](https://www.scirp.org/html/6-7401581\aada9ea0-03d4-4419-90d0-3917df276490.jpg)
Recalling the infinite matrix
we find
Theorem 19 (Tuite-Z)
a) The genus two orbifold partition function is
![](https://www.scirp.org/html/6-7401581\edea1025-da80-4137-a87b-5c667cbbe9fe.jpg)
b)
is holomorphic on the domain
;
c)
has natural modular properties under the action of
.
8.4. Genus Two Partition and
-Point Functions in
-Formalism
Let
be automorphisms, and
be twisted
-modules of a vertex operator superalgebra
. For
with
and
,
, we define the genus two
-point function [4] in the
-formalism by
![](https://www.scirp.org/html/6-7401581\bf675bd7-6c07-45c6-87b8-58e215cf7ad0.jpg)
where
, where
(respectively
)
denotes the pair
,
(respectively
,
). The sum is taken over any
-basis.
In particular, we introduce the genus two partition function
![](https://www.scirp.org/html/6-7401581\33701555-f5e6-4dcb-9dfb-f8f9d106f77d.jpg)
where
is the genus one intertwined two point function.
Remark 1 We can generalize the genus two
-point function by introducing and computing the differential form associated to the torus
-point function containing several intertwining operators in the supertrace as well as corresponding genus two
-point functions.
Similar to the ordinary genus two case [2], we define the differential form [4] associated to the
-point function on a sewn genus two Riemann surface for
and
,
with
,
,
![](https://www.scirp.org/html/6-7401581\e51b861e-7d40-455b-891d-3fb89da06b7a.jpg)
9. Generalizations of Classical Identities
9.1. Bosonization
The genus one orbifold partition function can be alternatively computed by decomposing the VOSA into Heisenberg modules
indexed by
integer eigenvalues
, i.e., a
lattice [26]. Let
be lattice elements of the rank one even lattice,
, and
-cocycle. Then
Theorem 20 (Tuite-Mason)
![](https://www.scirp.org/html/6-7401581\6f65d593-17ba-4661-9d99-f574756e65bb.jpg)
Then ther genus one twisted partition function is given by
![](https://www.scirp.org/html/6-7401581\bc216445-82c3-412d-8df2-1a748663461e.jpg)
Comparing to the fermionic product formula we obtain the classical Jacobi triple product formula:
![](https://www.scirp.org/html/6-7401581\397be951-15fc-4512-8a21-71bbd2d51577.jpg)
9.2. Genus Two Jacobi Triple Product Formula
The genus two partition function can similarly be computed in the bosonized formalism to obtain a genus two version of the Jacobi triple product formula for the genus two Riemann theta function [19]
![](https://www.scirp.org/html/6-7401581\fe7b62fd-f6ee-4ee7-8673-12890640076a.jpg)
for an appropriate character valued genus two Riemann theta function
![](https://www.scirp.org/html/6-7401581\cfbeadfa-b735-45b7-9eae-49716642fe3f.jpg)
Comparing with the fermionic result we thus find that on ![](https://www.scirp.org/html/6-7401581\56f1834b-ec9c-4ef8-a77f-0b8259e11104.jpg)
![](https://www.scirp.org/html/6-7401581\6ef7e47f-5a25-4c7d-aca4-70fd501a1742.jpg)
9.3. Fay’s Trisecant Identity
Recall Fay’s trisecant identity [21]
![](https://www.scirp.org/html/6-7401581\2487d788-2f40-4e0a-8423-b055b389f8cc.jpg)
for
,
, where
is the Jacobian of the curve.
9.4. Bosonized Generating Function and Trisecant Identity
In a similar fashion we can compute the general
- generating function
in the bosonic setting to obtain:
Theorem 21 (Mason-Tuite-Z)
![](https://www.scirp.org/html/6-7401581\e3d313e3-80f6-4a5c-a247-40c92a2d2662.jpg)
Comparing this to fermionic expressions for
we obtain the classical Frobenius elliptic function version of generalized Fay’s trisecant identity
[21]:
Corollary 1 (Mason-Tuite-Z) For
we have
![](https://www.scirp.org/html/6-7401581\123da698-24f1-4329-ae00-66f10c2c3e4a.jpg)
9.5. Generalized Fay’s Trisecant Identity
We may generalize these identities using [26]. Consider the general lattice
-point function. We have [19], For integers
satisfying
, we have
![](https://www.scirp.org/html/6-7401581\b2e6b650-2ef2-4e3e-8a20-ad50c0ee9f7b.jpg)
![](https://www.scirp.org/html/6-7401581\7eb9c3f6-db84-4786-ade4-550cd5b24c91.jpg)
Comparing this to the expression for
-point functions we obtain a new elliptic generalization of Fay’s trisecant identity:
Corollary 2 (Mason-Tuite-Z) For
we have
![](https://www.scirp.org/html/6-7401581\cf6afa2b-7a28-4c33-a826-a48a1eb86605.jpg)
Here
is the block matrix
![](https://www.scirp.org/html/6-7401581\18248251-cf52-461a-8315-78bb020f52a6.jpg)
with
the
matrix
![](https://www.scirp.org/html/6-7401581\880d224a-4e5a-4c28-98f3-14248f54a5a2.jpg)
for
and
, and
-functions are given by the expansion
![](https://www.scirp.org/html/6-7401581\bba3679a-eff0-464d-bad7-2c6e39672103.jpg)
10. Genus Two Intertwined Partition and n-Point Functions
In [4] we then prove:
Theorem 22 (Tuite-Z) Let
be
twisted
-modules for the rank two free fermion vertex operator superalgebra
. Let
. Then the partition function on a genus two Riemann surface obtained in the
-self-sewing formalism of the torus is a non-vanishing holomorphic function on
given by
![](https://www.scirp.org/html/6-7401581\c81ad364-ba77-4088-8f68-e242d3e8b498.jpg)
where
is the intertwined
module
torus basic two-point function, and
is some function.
We may similarly compute the genus two partition function in the
-formalism for the original rank one fermion VOSA
in which case we can only construct a
-twisted module. Then we have [4] the following:
Corollary 3 (Tuite-Z) Let
be the rank one free fermion vertex operator superalgebra and
,
, be automorphisms. Then the partition function for
-module
on a genus two Riemann surface obtained from
formalism of a self-sewn torus
is given by
![](https://www.scirp.org/html/6-7401581\ff0824c8-d584-4835-85fb-afe4adb026eb.jpg)
where
is the rank one fermion intertwined partition function on the original torus.
10.1. Genus Two Generating Form
In [4] we define matrices
![](https://www.scirp.org/html/6-7401581\5aedc3ed-15d7-44ef-8293-416eafa90b59.jpg)
![](https://www.scirp.org/html/6-7401581\4798e8d0-f9d0-453d-9802-cb9cba281d89.jpg)
and
are finite matrices indexed by
,
for
;
is semi-infinite with
rows indexed by
and columns indexed by
and
and
is semi-infinite with rows indexed by
and
and with
columns indexed by
. We then prove
Lemma 2 (Tuite-Z)
![](https://www.scirp.org/html/6-7401581\161ca212-c600-4fdc-89f4-c02eebb88a28.jpg)
with
,
.
Introduce the differential form
![](https://www.scirp.org/html/6-7401581\50f0e5a7-66f1-4b66-b03a-d9088630eb72.jpg)
associated to the rank two free fermion VOSA genus two
-point function
![](https://www.scirp.org/html/6-7401581\b73183e3-8812-4f0b-9567-ab2ec6f73592.jpg)
with alternatively inserted
states
and
states
. The states are distributed on the genus two Riemann surface
at points
. Then we have Theorem 23 (Tuite-Z) All
-point functions for rank two free fermion VOSA twisted modules
on self-sewn torus are generated by the differential form
![](https://www.scirp.org/html/6-7401581\59645e25-88f2-43aa-9d23-f0bb1c4b83fc.jpg)
where the elements of the matrix
![](https://www.scirp.org/html/6-7401581\bc2b7d47-8915-44f3-8224-16fef82b6dfc.jpg)
and
is the genus two partition function.
10.2. Modular Invariance Properties of Intertwined Functions
Following the ordinary case [20,27,40] we would like to describe modular properties of genus two “intertwined” partition and
-point generating functions. As in [27], consider
with elements
![](https://www.scirp.org/html/6-7401581\4729c142-35b4-4cb0-bbea-1d1ce5c63e85.jpg)
is generated by
,
and
with relations
.
We also define
where
with elements
![](https://www.scirp.org/html/6-7401581\1d885471-c512-449e-ad2f-cd5ff3b03307.jpg)
Together these groups generate
.
From [27] we find that
acts on the domain
of as follows:
![](https://www.scirp.org/html/6-7401581\8224a322-bb11-4613-bdf2-58a41453f025.jpg)
![](https://www.scirp.org/html/6-7401581\8a9818df-46e9-4128-b213-6f57152dbf9c.jpg)
We then define [4] a group action of
on the torus intertwined two-point function
![](https://www.scirp.org/html/6-7401581\decdedcd-0d15-4a88-9ace-3873dd04ae09.jpg)
for
:
![](https://www.scirp.org/html/6-7401581\de7d7a35-d330-451a-8bb5-6bced35bffb7.jpg)
with the standard action
and
, and
and the torus multiplier
, [1,19]. Then we have [4]
Theorem 24 (Tuite-Z) The torus intertwined twopoint function for the rank two free fermion VOSA is a modular form (up to multiplier) with respect to ![](https://www.scirp.org/html/6-7401581\c8ad3ccd-95b5-4f0d-a737-b9cc494848db.jpg)
![](https://www.scirp.org/html/6-7401581\b193ea7b-1662-496f-9c3c-3f4cc5b7d923.jpg)
where
.
The action of the generators
,
and
is given by [1]
.
In a similar way we may introduce the action of
on the genus two partition function [4]
![](https://www.scirp.org/html/6-7401581\1e45d873-09f8-482e-af24-de6afff07bda.jpg)
![](https://www.scirp.org/html/6-7401581\bca616d8-9459-4cd7-96c9-329774ba49ca.jpg)
We may now describe the modular invariance of the genus two partition function for the rank two free fermion VOSA under the action of
. Define a genus two multiplier
for
in terms of the genus one multiplier as follows
![](https://www.scirp.org/html/6-7401581\6c0150e5-2722-4ce7-b5de-f871ff2786ab.jpg)
for the generator
. We then find [4].
Theorem 25 (Tuite-Z) The genus two partition function for the rank two VOSA is modular invariant with respect to
with the multiplier system, i.e.,
![](https://www.scirp.org/html/6-7401581\478cce93-07b6-49d0-abf0-6a4f90a5c514.jpg)
Finally, we can also obtain modular invariance for the generating form
![](https://www.scirp.org/html/6-7401581\20b0a3b1-2395-48f9-bb38-72b955cf3906.jpg)
for all genus two
-point functions [4].
Theorem 26 (Tuite-Z) ![](https://www.scirp.org/html/6-7401581\f6d29358-70b8-47f8-913c-4aa12610f4a8.jpg)
is modular invariant with respect to
with a multiplier.
11. Acknowledgements
The author would like to express his deep gratitude to the organizers of the Conference “Algebra, Combinatorics, Dynamics and Applications”, Belfast, UK, August 27-30, 2012.