Differential equations and Minimal conformal field theories

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Differential equations and Minimal conformal
field theories

• Roberto Tateo

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OUTLINE
The Baxter TQ relation and the Bethe equations
The ODE/IM correspondence for the 6V model
The ODEs for minimal models
The Kac table from the ODE/IM correspondence
Conclusions
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Collaborators
Patrick Dorey
Clare Dunning
Ferdinando Gliozzi
Anna Lishmann
Davide Masoero
Adam Millican-Slater
Junji Suzuki
Mostly based on P. Dorey, C.Dunning, F.
Gliozzi, RT, On the ODE/IM correspondence for
minimal models,
J. Phys. A Math. Theor. 41 (2008) 132001
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Consider the 6-vertex model on a N x M lattice
with periodic BCs and N/2 even
(Spin reversal symmetry)
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To calculate the partition function Z, define the
transfer matrix T
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The partition function is
The free energy per site as is
Baxters TQ relation
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The BAE then emerge as a consequence of the fact
both t and q are entire. Setting
??
Twisted BCs
,
The conformal limit is achieved by sending
keeping
finite. Then
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The BAE reduces, in the region
to
The ODE/IM correspondence
CFT limit of the BA and the TQ relations
ODE/IM
Mgt0
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Q(E,l) spectral determinant for the radial
quantisation problem T(E,l) Stokes multiplier
spectral det. lateral quantisation problem
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The same BAE emerge from the study of c lt1 CFTs
in the framework developed by Bazhanov,
Lukyanov and Zamolodchikov
highest weight
c(M) central charge
p vacuum parameter
Take 2 co-prime integers altb.
The minimal model has central charge
and
conformal weights
lowest-possible conformal weight
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Then the minimal model is found by
setting
The term can be removed for
this value of by performing the
following transformation
the equation becomes
with the further rescaling
where
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and the change of variable has replaced the
singular potential
defined on a multi-sheeted Riemann surface, by
the simple potential
Any solution to this new equation is
automatically single-valued around z0!
To see which other primary states might have
similarly-trivial monodromy, keep real with
, and perform the same
transformation
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The Fuchsian singularity at z0 means that the
equation admits a pair of solutions which
generally have the power series expansions
where and are the
two roots of the indicial equation
We shall demand that the transformed ODE should
be such that, for arbitrary the monodromy of
a generic solution around z0 is
projectively trivial
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We shall show that the condition imposes the
following constraints
We shall call the integers in the asbt
sequence representable' and denote the set of
them by
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As a consequence, is quantised and runs
over its allowed values and the formula
precisely reproduce the set of conformal weights
of the primary states lying in the Kac table of
the minimal model
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Note that the requirement that a generic solution
y(z) be projectively trivial means that the two
independent solutions must have the same
monodromy
However in such a circumstance, while
keeps its power series expansion ,
generally acquires a logarithmic contribution
Unless D0 , this will spoil the projectively
trivial monodromy of y(z).
In fact the log term is absent iff the
recursion relations for the ds with D0
admits a solution
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Consider first the situation when
. Then, starting from the given initial
conditions, the recursion relation generates a
solution of the form
where the only non zero ds are those for which
the label n lies in Given that
, for these values of n the factor
on the LHS of the recursion relation is never
zero, and the procedure is well-defined.
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If instead , then equation
the recursion taken at
yields the additional condition
which is inconsistent for generic and
so the logarithmic term is required!! We can
define the set of non representable integers
To characterise these integers we first note that
given two coprime integers a and b any integer
n can be written as
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To see this use the Euler totient function j,
and the theorem
Then for some integer h and
satisfy the relation
But for any given n, this is not the only
possibility. We have
for some other pair of integers (s,t) if and
only if
for some integer k.
and
This is easily proved
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Write the relation as
Since (a,b)1, b must be a factor of
and so for some k. Dividing by b then
shows that , as required.
We have a line of representatives for n

with
For n to be a positive integer
or
and if none of these points has both
coordinates non-negative, then the corresponding
n will be a non representable integer..
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To keep t non-negative while making s as large as
possible, we shift t by a multiple of a so that
If s is still negative, then n is non
representable. Therefore, the numbers we want
are
Negating s, the trivial monodromy values are
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The allowed values for the conformal weights
precisely reproduce the Kac table
a3,b5
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The generating function
Another way to characterise the numbers
is through the generating function
It is straightforward to show that such a
rational function is actually a polynomial
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The description for the minimal models
There is another ODE that can be associated with
the same series of minimal models. This is
related to the , and
perturbations. For
Swapping a and b gives the related
ODE, while replacing a with 2a and b with b/2
yields the equation.
For a odd, one gets the same set of integers
as the su(2)-related case, while only a subset
is recovered for a even!!
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Other possible generalisations
The existence of a simpler version, of the basic
ODE for minimal CFTs is not restricted to clt1.
It generalises to the higher su(2) coset
CFTs and to the ABCD-related theories
at fractional level
with b-aKu, and u1,2,3,..
after simple changes of variable
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It is striking that when both L and K are
integer the CFT is unitary and W(z) simplifies
further to
K??L duality symmetry in the coset corresponds
to z?zE , E?-E
Maybe we should treat the points z0 and zE
more democratically also in the ODEs related to
primary fields!
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Conclusions
The zero monodromy property seems to play a
central rule in the ODE/IM framework both for
the selection of the set of primary fields and
the descendents (BLZ).
All the proposed ODE/IM equations for the
ground states have an equivalent zero-monodromy
version describing the corresponding minimal
series. It would be nice to reproduce the
associated generalised Kac tables using the
ODE/IM correspondence
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