20 research outputs found
Extended Seiberg-Witten Theory and Integrable Hierarchy
The prepotential of the effective N=2 super-Yang-Mills theory perturbed in
the ultraviolet by the descendents of the single-trace chiral operators is
shown to be a particular tau-function of the quasiclassical Toda hierarchy. In
the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental
hypermultiplets at the appropriate locus of the moduli space of vacua) or a
theory on a single fractional D3 brane at the ADE singularity the hierarchy is
the dispersionless Toda chain. We present its explicit solutions. Our results
generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support
the prior work hep-th/0302191 which established the equivalence of these N=2
theories with the topological A string on CP^1 and clarify the origin of the
Eguchi-Yang matrix integral. In the higher rank case we find an appropriate
variant of the quasiclassical tau-function, show how the Seiberg-Witten curve
is deformed by Toda flows, and fix the contact term ambiguity.Comment: 49 page
Integrability in SFT and new representation of KP tau-function
We are investigating the properties of vacuum and boundary states in the CFT
of free bosons under the conformal transformation. We show that transformed
vacuum (boundary state) is given in terms of tau-functions of dispersionless KP
(Toda) hierarchies. Applications of this approach to string field theory is
considered. We recognize in Neumann coefficients the matrix of second
derivatives of tau-function of dispersionless KP and identify surface states
with the conformally transformed vacuum of free field theory.Comment: 25 pp, LaTeX, reference added in the Section 3.
Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula
We solve the loop equations of the hermitian 2-matrix model to all orders in
the topological expansion, i.e. we obtain all non-mixed correlation
functions, in terms of residues on an algebraic curve. We give two
representations of those residues as Feynman-like graphs, one of them involving
only cubic vertices.Comment: 48 pages, LaTex, 68 figure
Conformal Mappings and Dispersionless Toda hierarchy
Let be the space consists of pairs , where is a
univalent function on the unit disc with , is a univalent function
on the exterior of the unit disc with and
. In this article, we define the time variables , on which are holomorphic with respect to the natural
complex structure on and can serve as local complex coordinates
for . We show that the evolutions of the pair with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting to the subspace consists
of pairs where , we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every homeomorphism of the unit circle corresponds uniquely to
an element of under the conformal welding
, the space can be naturally
identified as a subspace of characterized by . We
show that we can naturally define complexified vector fields \pa_n, n\in \Z
on so that the evolutions of on
with respect to \pa_n satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings . Moreover, in the latter case, the time
variables are Fourier coefficients of and .Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072
Rational Theories of 2D Gravity from the Two-Matrix Model
The correspondence claimed by M. Douglas, between the multicritical regimes
of the two-matrix model and 2D gravity coupled to (p,q) rational matter field,
is worked out explicitly. We found the minimal (p,q) multicritical potentials
U(X) and V(Y) which are polynomials of degree p and q, correspondingly. The
loop averages W(X) and \tilde W(Y) are shown to satisfy the Heisenberg
relations {W,X} =1 and {\tilde W,Y}=1 and essentially coincide with the
canonical momenta P and Q. The operators X and Y create the two kinds of
boundaries in the (p,q) model related by the duality (p,q) - (q,p). Finally, we
present a closed expression for the two two-loop correlators and interpret its
scaling limit.Comment: 24 pages, preprint CERN-TH.6834/9
Large N gauge theories and topological cigars
We analyze the conjectured duality between a class of double-scaling limits
of a one-matrix model and the topological twist of non-critical superstring
backgrounds that contain the N=2 Kazama-Suzuki SL(2)/U(1) supercoset model. The
untwisted backgrounds are holographically dual to double-scaled Little String
Theories in four dimensions and to the large N double-scaling limit of certain
supersymmetric gauge theories. The matrix model in question is the auxiliary
Dijkgraaf-Vafa matrix model that encodes the F-terms of the above
supersymmetric gauge theories. We evaluate matrix model loop correlators with
the goal of extracting information on the spectrum of operators in the dual
non-critical bosonic string. The twisted coset at level one, the topological
cigar, is known to be equivalent to the c=1 non-critical string at self-dual
radius and to the topological theory on a deformed conifold. The spectrum and
wavefunctions of the operators that can be deduced from the matrix model
double-scaling limit are consistent with these expectations.Comment: 34 page
Second and Third Order Observables of the Two-Matrix Model
In this paper we complement our recent result on the explicit formula for the
planar limit of the free energy of the two-matrix model by computing the second
and third order observables of the model in terms of canonical structures of
the underlying genus g spectral curve. In particular we provide explicit
formulas for any three-loop correlator of the model. Some explicit examples are
worked out.Comment: 22 pages, v2 with added references and minor correction
Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations
In this article, we classify the solutions of the dispersionless Toda
hierarchy into degenerate and non-degenerate cases. We show that every
non-degenerate solution is determined by a function of
two variables. We interpret these non-degenerate solutions as defining
evolutions on the space of pairs of conformal mappings ,
where is a univalent function on the exterior of the unit disc, is a
univalent function on the unit disc, normalized such that ,
and . For each solution, we show how to define the
natural time variables , as complex coordinates on the space
. We also find explicit formulas for the tau function of the
dispersionless Toda hierarchy in terms of . Imposing
some conditions on the function , we show that the
dispersionless Toda flows can be naturally restricted to the subspace
of defined by . This recovers
the result of Zabrodin.Comment: 25 page