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In this paper we extend previous work of Galleas and the author to elliptic SOS models. We demonstrate that the dynamical reflection algebra can be exploited to obtain a functional equation characterizing the partition function of an elliptic SOS model with domain-wall boundaries and one reflecting end. Special attention is paid to the structure of the functional equation. Through this approach we find a novel multiple-integral formula for that partition function., Comment: 31 pages, 3 figures; v2: minor improvements, reference added

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R^4/Z_r. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q -> -1, t -> -1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit., 38 pages, 4 figures; v2: minor changes, references added; v3: minor changes, a comment (Note added) and three references added, published version

In this manuscript, we study the logarithmic Penner type nonlinear interaction in the random matrix model for interacting RNA folding and structure combinatorics. The Penner interaction originally appeared in the studies of moduli space of punctured surfaces and has been applied here in the context of RNA folding for the first time. An exact analytic formula for the generating function is derived using the orthogonal polynomial method. The partition function for a given length L of the RNA chain, derived from the generating function enumerates all possible topological structures as well as the pairings. The partition function and the asymptotic large length distribution functions are found and show a change in the critical exponent of the secondary structure contribution from L − 3 / 2 for large N (size of matrix, N > L ) to L − 1 / 2 for small N ( N ≪ L ). The exact analytic results calculated in the proposed model allow evaluation of the specific heat versus T curve for large interaction strength. In particular, the second derivative of specific heat shows a striking behavior, changing from single peaked function for large N to a double peak for small N.

With the help of the F-basis provided by the Drinfeld twist or factorizing F-matrix for the open XXZ spin chain with non-diagonal boundary terms, we obtain the determinant representation of the partition function of the six-vertex model with a non-diagonal reflecting end under domain wall boundary condition., Comment: Latex file, 24 pages, 4 figures

We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential., Comment: Final version to be published in Nucl. Phys. B. Typos are corrected. 38 pages, 4 figures

Motivated by a careful analysis of the Laplacian on the supergroup SU(2|1) we formulate a proposal for the state space of the SU(2|1) WZNW model. We then use properties of b|1) characters to compute the partition function of the theory. In the special case of level k = 1 the latter is found to agree with the properly regularized partition function for the continuum limit of the integrable sl(2|1) 3 − ¯ super-spin chain. Some general conclusions applicable to other WZNW models (in particular the case k = −1/2) are also drawn.

We study the “paperclip” model of boundary interaction with the topological angle θ equal to π. We propose exact expression for the disk partition function in terms of solutions of certain ordinary differential equation. Large distance asymptotic form of the partition function which follows from this proposal makes it possible to identify the infrared fixed point of the paperclip boundary flow at θ=π.

We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${\cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type. Furthermore we derive the Poincar\'{e} polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the ${\cal N}=4$ partition function which is a modular form in agreement with the expectations of $SL(2,\Z)$ duality., Comment: 26 pages, few explanations added. version to appear in nucl.phys

We extend Zirnbauer's color-flavor transformation in the fermionic sector to the case of the special unitary group. The transformation allows a certain integral over SU(N_c) color matrices to be transformed into an integral over flavor matrices which parameterize the coset space U(2N_f)/U(N_f)xU(N_f). Integrals of the type considered appear, for example, in the partition function of lattice gauge theory., Comment: 19 pages, 2 figures; typos fixed, normalization corrected, some arguments improved, appendix with special cases added; version to appear in Nucl. Phys. B

We formulate the RNA folding problem as an $N\times N$ matrix field theory. This matrix formalism allows us to give a systematic classification of the terms in the partition function according to their topological character. The theory is set up in such a way that the limit $N\to \infty$ yields the so-called secondary structure (Hartree theory). Tertiary structure and pseudo-knots are obtained by calculating the $1/N^2$ corrections to the partition function. We propose a generalization of the Hartree recursion relation to generate the tertiary structure., Comment: 29 pages (LaTex), 13 figures (eps). Missing paragraph and figure added

It is known that the Seiberg-Witten invariants, derived from supersymmetric Yang-Mill theories in four-dimensions, do not distinguish smooth structure of certain non-simply-connected four manifolds. We propose generalizations of Donaldson-Witten and Vafa-Witten theories on a K\"{a}hler manifold based on Higgs Bundles. We showed, in particular, that the partition function of our generalized Vafa-Witten theory can be written as the sum of contributions our generalized Donaldson-Witten invariants and generalized Seiberg-Witten invariants. The resulting generalized Seiberg-Witten invariants might have, conjecturally, information on smooth structure beyond the original Seiberg-Witten invariants for non-simply-connected case., Comment: 29 pages, LaTeX2e

We consider the path-sum of Ponzano-Regge with additional boundary contributions in the context of the holographic principle of Quantum Gravity. We calculate an holographic projection in which the bulk partition function goes to a semi-classical limit while the boundary state functional remains quantum-mechanical. The properties of the resulting boundary theory are discussed., Comment: 20 pages, latex

The partition function of two dimensional QCD on a Riemann surface of area $A$ is expanded as a power series in $1/N$ and $A$. It is shown that the coefficients of this expansion are precisely determined by a sum over maps from a two dimensional surface onto the two dimensional target space. Thus two dimensional QCD has a simple interpretation as a closed string theory., 27 pages, PUPT-1376 LBL-33458

We derive a compact and explicit expression for the generating functional of all correlation functions of tachyon operators in 2D string theory. This expression makes manifest relations of the $c=1$ system to KP flow and $W_{1+\infty}$ constraints. Moreover we derive a Kontsevich-Penner integral representation of this generating functional.

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $\tau$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $\tau$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the statement reduces to the early established \cite{MMM91b} relation between Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of $2d$ quantum gravity with $c, Comment: 67 pages (October 1991)

A general method is presented for deriving on-shell Ward-identities in (2D) string theory. It is shown that all tree-level Ward identities can be summarized in a quadratic differential equation for the generating function of tree-amplitudes. This result is extended to loop amplitudes and leads to a master equation {\it \`{a} la} Batalin-Vilkovisky for the complete partition function.

We present spectral density reweighting techniques adapted to the analysis of a time series of data with a continuous range of allowed values. In a first application we analyze action and Polyakov line data from a Monte Carlo simulation on $L_t L^3 (L_t=2,4)$ lattices for the SU(3) deconfining phase transition. We calculate partition function zeros, as well as maxima of the specific heat and of the order parameter susceptibility. Details and warnings are given concerning i) autocorrelations in computer time and ii) a reliable extraction of partition function zeros. The finite size scaling analysis of these data leads to precise results for the critical couplings $\beta_c$, for the critical exponent $\nu$ and for the latent heat $\triangle s$. In both cases ($L_t=2$ and 4), the first order nature of the transition is substantiated.

The character of an arbitrary Kac-Moody group is formally expressed as a partition function of an extension of the chiral WZNW model on a torus. Using a stationary phase approximation, I evaluate the path integral explicitly and obtain the Weyl-Kac formula for the character.

We study reductions of unitary one-matrix models. The unitary model admits an especially rich class of reductions of which the widely known symmetric model is only one. The partition function of a certain class of reduced models is shown to be a product of distinct Toda chain τ-functions. Virasoro constraints are also derived for the case of unitary models. It is claimed, in analogy with the reduced hermitian model, that in the continuous limit the Virasoro constraints must be imposed on a fractional power of the original partition function.

A complete set of recursion relations is obtained for correlation functions in two-dimensional topological gravity coupled with the ADE series of topological minimal models, by demanding the consistency of the contact algebra of physical operators. When formulated as linear constraints on the partition function, they coincide with the Virasoro constraints and the W-constraints that have been conjectured and partially verified in the multi-matrix models of two-dimensional quantum gravity. These recursion relations give a complete solution of the theory.

Using the framework of generalized Korteweg-de Vries flows, we compute some correlation functions in unitary minimal models coupled to gravity. A generalization to the (Z2 orbifold) D series case is given. We find the same fusion rules for the order parameters as in the underlying CFT, and derive the genus-one partition function. The results for irrelevant operators deviate from CFT.

In this paper we use the Coulomb-gas approach to construct the minimal-model conformal blocks of higher-genus Riemann surfaces. We define the higher-genus blocks by sewing, and write them in terms of the rational blocks of a compactified scalar field. We show that spurious states decouple, which implies that the blocks degenerate correctly. As an example, we compute the genus-two partition function, and verify modular invariance for the subset of minimal models which only require one type of screening charge.

We generalize the notion of orbifold to statistical mechanical models away from their critical points. In the simplest models, our non-critical orbifold construction reduces to Kramers-Wannier duality. More generally, it gives a new model whose toroidal partition function can be expressed in terms of that of the original model. The construction also preserves the star-triangle relations, so that the “orbifold” of an exactly solvable model remains exactly solvable. At a critical point, our construction reduces to the conventional orbifold construction of conformal field theory.

The partition function for two-dimensional scalar field theories which possess soliton states, is constructed in the semi-classical approximation. Divergences are handled by using zeta-function methods. The result is naturally interpretated after the appropriate mass renormalisations, as being that due to a massive particle confined in a box.

We give a formulation of the Wess-Zumino-Witten models on Riemann surfaces of arbitrary genus in which the Ward identities for the current algebras become complete. It requires twisting the models in a non-abelian way by Lie group elements. The Ward identities are written in terms of twisted Poincare series for Schottky groups and the zero modes of the currents are defined by a Lie derivation acting on the twists. Furthermore, we identify the denominator of the chiral partition function and we argue that both the numerator and the denominator of the partition function satisfy a heat equation on the moduli space.

The correlation function of the ZN super twist fields is calculated on the sphere. Also, the degeneration behavior of the higher genus partition function is used to derive the spectrum of dimensions for the non-abelian super twist fields.

We present a simple derivation of a grassmannian integral for the 3-dimensional Ising model partition function and study some elementary consequences.

We investigate the levels of the quartic oscillator ( − d 2 d q 2 + q 4 ) by means of the associated zeta function. By using several versions of the semiclassical approach, we prove exact results concerning the locations of the real zeros and of the poles of that function, the residues of which are essentially the coefficients of the semiclassical expansion of the Bohr-Sommerfeld quantization rule. We also compute the derivative of the zeta function at the origin, by relating it to the Fredholm determinant of the operator. These results translate as sum rules for the eigenvalues and thus are helpful for their precise computation, and a further analysis allows one to split them into separate rules for even and odd levels; the extension to higher anharmonic oscillators ( − d 2 d q 2 + q 2M ) is immediate. Finally we propose an asymptotic formula for large negative arguments of the zeta function, based on the nature and location of the complex singularities of the partition function of the operator ( − d 2 d q 2 + q 4 ) 3 4 .

The “fugacity” factor occuring in the partition function of the periodic CP 1 model is numerically analyzed. It is shown to behave as expected.

We use the duality and the algebra of the order-disorder variables to obtain a functional inversion relation for the Ashkin-Teller model in two dimensions. We then consider the completely integrable special cases to show that the algebra of the order-disorder variables, (self) duality, and a finite number of symmetries lead to the complete determination of the partition function of these theories solely in terms of the duality factor when the dynamics is specified by explicit analyticity assumptions. We also discuss the quantum nature of these operators and the duality.

The formulation of the effective action, in a way which is covariant under general field redefinitions, is considered. It is shown how Vilkovisky's ansatz for the effective action may be derived in the case of a flat field space. If the field space is curved, it is shown in a simple non-perturbative way why Vilkovisky's effective action is not correct. The same method is used to obtain DeWitt's result for the effective action. A generally covariant definition which includes both Vilkovisky's and DeWitt's definitions as special cases is given. We also discuss some of the features which arise in the case of a curved field space, including the analogue of the usual Legendre transformation which relates the effective action to the partition function.

A general non-perturbative approach to functional integrals describing systems possessing both thermal and quantum fluctuations is presented. The partition function is evaluated in the semiclassical limit by enlarging the class of paths in the functional integral to include complex-valued solutions of the classical equations of motion. Semiclassical energy eigenvalues, including tunnelling effects associated with decay of false vacua, can be obtained in this way for all states, not just the ground state. In this approach the imaginary part of an energy eigenvalue of an unstable state is directly related to physical boundary conditions at infinity; these results are compared to earlier attempts (fate of the false vacuum) to compute decay rates of metastable systems, involving an analytic continuation prescription for a negative mode of the second-order fluctuation operator. Use of complex-valued solutions obviates the need to employ the dilute-gas approximation which is generally invalid at finite temperature. The new method is checked by one-dimensional examples, and then extended to field theory, where a general expression for the partition function of a field theory is given in terms of stability angles about a complex classical solution.

The contribution of instantons to the partition function is calculated for the two-dimensional CP1 model defined on the torus.

The interrelations between exclusive and inclusive cross sections are investigated by means of efficiency matrices. A partition function Z is used as a generating functional for these cross sections which yields similarly also topological and other mixed cross sections. Regarding the partition function itself as a cross section, the general properties of Z and of statistical quantities derived from ln Z are investigated as functions of the chemical potentials. These statistical methods are used to analyze three simple examples of data for multiplicity distributions, in particular for the third distribution from the Echo Lake experiment a behaviour of Z is found which has similarities with a phase transition in statistical mechanics.

Applying the machinery of random matrix theory and Toeplitz determinants we study the level $k$, $U(N)$ Chern-Simons theory coupled with fundamental matter on $S^2\times S^1$ at finite temperature $T$. This theory admits a discrete matrix integral representation, i.e. a unitary discrete matrix model of two-dimensional Yang-Mills theory. In this study, the effective partition function and phase structure of the Chern-Simons matter theory, in a special case with an effective potential namely the Gross-Witten-Wadia potential, are investigated. We obtain an exact expression for the partition function of the Chern-Simons matter theory as a function of $k,N,T,$ for finite values and in the asymptotic regime. In the Gross-Witten-Wadia case, we show that ratio of the Chern-Simons matter partition function and the continuous two-dimensional Yang-Mills partition function, in the asymptotic regime, is the Tracy-Widom distribution. Consequently, using the explicit results for free energy of the theory, new second-order and third-order phase transitions are observed. Depending on the phase, in the asymptotic regime, Chern-Simons matter theory is represented either by a continuous or discrete two-dimensional Yang-Mills theory, separated by a third-order domain wall., 26 pages, 1 figure, references added, some parts rewritten and extended. To appear in Nucl. Phys. B