We compute the exact partition function of the isotropic 6-vertex model on a cylinder geometry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz and algebraic geometry. We perform the computations in both the open and closed channels. We also consider the partial thermodynamic limits, whereby in the open (closed) channel, the open (closed) direction is kept small while the other direction becomes large. We compute the zeros of the partition function in the two partial thermodynamic limits, and compare with the condensation curves.
We compute off-critical local height probabilities in regime-III restricted solid-on-solid models in a $4 N$-quadrant spiral geometry, with periodic boundary conditions in the angular direction, and fixed boundary conditions in the radial direction, as a function of $N$, the winding number of the spiral, and $τ$, the departure from criticality of the model, and observe that the result depends only on the product $N \, τ$. In the limit $N \rightarrow 1$, $τ\rightarrow τ_0$, such that $τ_0$ is finite, we recover the off-critical local height probability on a plane, $τ_0$-away from criticality. I…
We show that the determinant of the co-exact $p$-form on spheres and anti-deSitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact $p$-form contains characters of anti-symmetric tensors of rank lower to $p$ all the way to the zero-form. Using this result we evaluate the partition function of $p$-forms and demonstrate that they obey known properties under Hodge duality. We show that partition function of conformal forms in even $d+1$ dimensions, on hyperbolic cylinders can be written as integral transforms involving onl…
We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $I \times Σ$, with $I$ an interval and $\dim Σ= 4l+2$, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $\dim Σ= 0$ is considered as a toy example. We show that one can identify - for certain choices of residual fields - the "physical part" (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper [arXiv:2012.13270], without any quantum correction…
The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space $S^a\times AdS_b$ can be mapped to $S^{a+b}$ provided $S^a$ and $AdS_b$ are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on $S^a\times AdS_b$ and $S^{a+b}$. We then demonstrate that the partition function of higher derivative conformal scalars and fermio…
We show that any 2D scalar field theory compactified on a cylinder and with a Fourier expandable potential $V$ is equivalent, in the small coupling limit, to a 1D theory involving a massless particle in a potential $V$ and an infinite tower of free massive Kaluza-Klein (KK) modes. Moving slightly away from the deep IR region has the effect of switching on interactions between the zero mode and the KK modes, whose strength is controlled by powers of the coupling, hence making the interactions increasingly suppressed. We take the notable example of Liouville field theory and, starting from its w…
Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
利用二维Yang-Mills与布朗运动之间的关系，证明了在一类透镜空间中，柱体上的二维Yang-Mills与Chern-Simons理论之间的关系.或者，这可以被看作是计算共形场论中的某些标量的2dYM。我们发现，对于特定的YM耦合值但有限的k和N，2dYM的配分函数约化为Chern-Simons理论中的U=ST^pS型算子. U是一个算子，从中可以得到S^3/Z_p上的陈-西蒙斯配分函数，以及威尔逊圈的期望值。对应关系涉及杨-米尔斯耦合的虚部是一个有理数，可以看作是陈-西蒙斯/WZW理论和拓扑2dYM的Witten，和Blau ant Thompson之间的关系的推广。本重新制定的一些性质的2dYM的圆柱明确。特别是，我们表明，模变换性质的配分函数是密切相关的仿射字符。
We establish the relation of partition functions of conformal higher spin fields on Weyl equivalent spaces in $d=4$ dimension. We express the partition function of Weyl graviton and conformal higher spin fields as an integral over characters on $S^1\times AdS_3$, $S^4$, and $AdS_4$. We observe that the partition function of conformal higher spins on hyperbolic cylinders differs from the partition function on $S^4$ by the `edge' contribution. The logarithmic coefficient obtained from the character integral of the partition function of conformal higher spins on $AdS_4$ is the half of that obtain…
We consider KdV currents in a quantum field theory obtained by deforming a two dimensional conformal field theory on a cylinder via the irrelevant operator $T{\bar T}$. In this paper we determine their one-point functions modular properties. We find that the one-point functions factorize into two components each with a definite modular weight. We also obtain a general differential equation that the generalized torus partition sum satisfies.
We analyse the computation of the partition function of 5d $T_N$ theories in Higgs branches using the topological vertex. The theories are realised by a web of $(p,q)$ 5-branes whose dual description may be given by an M-theory compactification on a certain local non-toric Calabi-Yau threefold. We explicitly show how it is possible to directly apply the topological vertex to the non-toric geometry. Using this novel technique, which considerably simplifies the computation by the existing method, we are able to compute the partition function of the higher rank $E_6$, $E_7$ and $E_8$ theories. Mo…
A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of…
We study the thermodynamic limit of the six-vertex model with domain wall boundary and reflecting end. We evaluated the partition function explicitly in special cases. We calculated the homogeneous limit of the Tsuchiya determinant formula for the partition function. We evaluated the thermodynamic limit and obtain the free energy of the six-vertex model with reflecting end. We determined the free energy in the disordered regime.
We compute the partition function for 6d $\mathcal{N}=1$ $SO(2N)$ gauge theories compactified on a circle with $\mathbb{Z}_2$ outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d $SO(8)$ and $SU(3)$ gauge theories with $\mathbb{Z}_2$ twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.
We develop an efficient method to compute the torus partition function of the six-vertex model exactly for finite lattice size. The method is based on the algebro-geometric approach to the resolution of Bethe ansatz equations initiated in a previous work, and on further ingredients introduced in the present paper. The latter include rational $Q$-system, primary decomposition, algebraic extension and Galois theory. Using this approach, we probe new structures in the solution space of the Bethe ansatz equations which enable us to boost the efficiency of the computation. As an application, we stu…
We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of arXiv:1709.01928. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO($N$) gauge theories and the pure $G_2$ gauge theory. The obtained results agree with the results known in the literature. We also compute …
We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the invariants in terms of the topological vertex. Utilizing identities for the topological vertex proved in arXiv:1603.05271, we derive product formulas for the partition functions. The connected version of the partition function is written in terms of Jacobi forms. In the special case where the elliptic surface is a K3 surface, we get a derivation of the Katz-Klemm-Va…
