7,078 research outputs found
Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I
We define the partition and -point correlation functions for a vertex
operator superalgebra on a genus two Riemann surface formed by sewing two tori
together. For the free fermion vertex operator superalgebra we obtain a closed
formula for the genus two continuous orbifold partition function in terms of an
infinite dimensional determinant with entries arising from torus Szeg\"o
kernels. We prove that the partition function is holomorphic in the sewing
parameters on a given suitable domain and describe its modular properties.
Using the bosonized formalism, a new genus two Jacobi product identity is
described for the Riemann theta series. We compute and discuss the modular
properties of the generating function for all -point functions in terms of a
genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one
point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun.
Math. Phy
Geometry of Spectral Curves and All Order Dispersive Integrable System
We propose a definition for a Tau function and a spinor kernel (closely
related to Baker-Akhiezer functions), where times parametrize slow (of order
1/N) deformations of an algebraic plane curve. This definition consists of a
formal asymptotic series in powers of 1/N, where the coefficients involve theta
functions whose phase is linear in N and therefore features generically fast
oscillations when N is large. The large N limit of this construction coincides
with the algebro-geometric solutions of the multi-KP equation, but where the
underlying algebraic curve evolves according to Whitham equations. We check
that our conjectural Tau function satisfies Hirota equations to the first two
orders, and we conjecture that they hold to all orders. The Hirota equations
are equivalent to a self-replication property for the spinor kernel. We analyze
its consequences, namely the possibility of reconstructing order by order in
1/N an isomonodromic problem given by a Lax pair, and the relation between
"correlators", the tau function and the spinor kernel. This construction is one
more step towards a unified framework relating integrable hierarchies,
topological recursion and enumerative geometry
Stateless HOL
We present a version of the HOL Light system that supports undoing
definitions in such a way that this does not compromise the soundness of the
logic. In our system the code that keeps track of the constants that have been
defined thus far has been moved out of the kernel. This means that the kernel
now is purely functional.
The changes to the system are small. All existing HOL Light developments can
be run by the stateless system with only minor changes.
The basic principle behind the system is not to name constants by strings,
but by pairs consisting of a string and a definition. This means that the data
structures for the terms are all merged into one big graph. OCaml - the
implementation language of the system - can use pointer equality to establish
equality of data structures fast. This allows the system to run at acceptable
speeds. Our system runs at about 85% of the speed of the stateful version of
HOL Light.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Macdonald's identities and the large N limit of on the cylinder
We give a rigorous calculation of the large N limit of the partition function
of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony
is a so-called special element of type . By MacDonald's identity, the
partition function factors in this case as a product over positive roots and it
is straightforward to calculate the large N asymptotics of the free energy. We
obtain the unexpected result that the free energy in these cases is asymptotic
to N times a functional of the limit densities of eigenvalues of the boundary
holonomies. This appears to contradict the predictions of Gross-Matysin and
Kazakov-Wynter that the free energy should have a limit governed by the complex
Burgers equation
D3-instantons, Mock Theta Series and Twistors
The D-instanton corrected hypermultiplet moduli space of type II string
theory compactified on a Calabi-Yau threefold is known in the type IIA picture
to be determined in terms of the generalized Donaldson-Thomas invariants,
through a twistorial construction. At the same time, in the mirror type IIB
picture, and in the limit where only D3-D1-D(-1)-instanton corrections are
retained, it should carry an isometric action of the S-duality group SL(2,Z).
We prove that this is the case in the one-instanton approximation, by
constructing a holomorphic action of SL(2,Z) on the linearized twistor space.
Using the modular invariance of the D4-D2-D0 black hole partition function, we
show that the standard Darboux coordinates in twistor space have modular
anomalies controlled by period integrals of a Siegel-Narain theta series, which
can be canceled by a contact transformation generated by a holomorphic mock
theta series.Comment: 42 pages; discussion of isometries is amended; misprints correcte
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