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.
Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges. We study the Generalized Gibbs Ensemble with chemical potentials for these charges at high temperature. In a large central charge limit, the partition function can be computed in a saddle-point approximation. We compare the ensemble values of the KdV charges to the values in a microstate, and find that they match irrespective of the values of the chemical potentials. We study the partition function at finite central charge perturbatively in the chemical potentials, and find that this deg…
Following our previous works on extended higher spin symmetries on the torus we focus in the present contribution to make a setup of the integrable KdV hierarchies on $T^{2} = S^{1} \times S^{1}$. Actually two particular systems are considered, namely the KdV and the Burgers non linear integrable model associated to currents of conformal weights (2, 2) and (1, 1) respectively. One key steps towards proving the integrability of these systems is to find their Lax pair operators. This is explicitly done and a mapping between the two systems is discussed.
To every partition $n=n_1+n_2+\cdots+n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty}$ and $\hat{gl}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $τ$ is a $τ$--function of the $[n_1,n_2,\ldots,n_s]$--th reduced KP hierarchy and (2) $τ$ satisfies a `natural' string equation, we prove that $τ$ also satisfies the vacuum constraints of the $W_{1+\infty}$ algebra.
Two consistency conditions for partition functions established by Akemann and Dam-gaard in their studies of the fermionic mass dependence of the QCD partition function at low energy ({\it a la} Leutwiller-Smilga-Verbaarschot) are interpreted in terms of integrable hierarchies. Their algebraic relation is shown to be a consequence of Wick's theorem for 2d fermionic correlators (Hirota identities) in the special case of the 2-reductions of the KP hierarchy (that is KdV/mKdV). The consistency condition involving derivatives is an incarnation of the string equation associated with the particular m…
Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show tha…
We study the thermal partition function of Jackiw-Teitelboim (JT) gravity in asymptotically Euclidean $AdS_2$ background using the matrix model description recently found by Saad, Shenker and Stanford [arXiv:1903.11115]. We show that the partition function of JT gravity is written as the expectation value of a macroscopic loop operator in the old matrix model of 2d gravity in the background where infinitely many couplings are turned on in a specific way. Based on this expression we develop a very efficient method of computing the partition function in the genus expansion as well as in the low …
One-matrix model in $p$-critical point on torus is considered. The generating function of correlation numbers in genus one is evaluated and used for computation correlation numbers in KdV and CFT frames. It is shown that the correlation numbers in KdV frame in genus one satisfy the Witten topological gravity recurrence relations.
We present an unifying description of the graded $SL(p,q)$ KP-KdV hierarchies, including the Wronskian construction of their tau-functions as well as the coefficients of the pertinent Lax operators, obtained via successive applications of special Darboux-Bäcklund transformations. The emerging Darboux-Bäcklund structure is identified as a constrained generalized Toda lattice system relevant for the two-matrix string model. Also, the exact Wronskian solution for the two-matrix model partition function is found.
The Gauge/Bethe correspondence relates Omega-deformed N=2 supersymmetric gauge theories to some quantum integrable models, in simple cases the integrable models can be treated as solvable quantum mechanics models. For SU(2) gauge theory with an adjoint matter, or with 4 fundamental matters, the potential of corresponding quantum model is the elliptic function. If the mass of matter takes special value then the potential is an elliptic solution of KdV hierarchy. We show that the deformed prepotential of gauge theory can be obtained from the average densities of conserved charges of the classica…
In this paper, we investigate the partition functions of conformal field theories (CFTs) with the $T\bar{T}$ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtain…
Worldsheet (0,2) gauged linear sigma models are often used to study supersymmetric heterotic string compactifications with non-trivial vector bundles. We make use of supersymmetric localization techniques to determine their one-loop partition functions. In particular we derive conditions which ensure that the full partition function is modular invariant and we propose a method to determine the massless and massive target space matter spectrum.
In this paper we consider the interrelation between compactified string theories on torus and gauge fields on it. We start from open string theories with background gauge fields and derive partition functions by path integral. Since the effects of background fields and compactification correlate only through string zero modes, we investigate these zero modes. From this point of view, we discuss the Wilson loop mechanism at finite temperature. For the closed string, only a few comments are mentioned.
We study the free energy of the pure glue QCD string with a torus target space and the gauge groups $SU(N)$ and (chiral) $U(N)$. It is highly constrained by a strong/weak gauge coupling duality which results in modular covariance. The string free energy is computed exactly in terms of modular forms for worldsheet genera 1 - 8. It has a surprisingly mild singularity in the weak gauge coupling/small area limit.
We compute the partition function of five-dimensional abelian gauge theory on a five-torus T5 with a general flat metric using the Dirac method of quantizing with constraints. We compare this with the partition function of a single fivebrane compactified on S1 times T5, which is obtained from the six-torus calculation of Dolan and Nappi. The radius R1 of the circle S1 is set to the dimensionful gauge coupling constant g^2= 4π^2 R1. We find the two partition functions are equal only in the limit where R1 is small relative to T5, a limit which removes the Kaluza-Klein modes from the 6d sum. This…
In view of the recent interest in formulating a quantum theory of Ramond-Ramond p-forms, we exhibit an SL(10,Z) invariant partition function for the chiral four-form of Type IIB string theory on the ten-torus. We follow the strategy used to derive a modular invariant partition function for the chiral two-form of the M-theory fivebrane. We also generalize the calculation to self-dual quantum fields in spacetime dimension 2p=2+4k, and display the SL(2p,Z) automorphic forms for odd p>1. We relate our explicit calculation to a computation of the B-cycle periods, which are discussed in the work of …
We revisit the problem of quantizing a chiral boson on a torus. The conventional approach is to extract the partition function of a chiral boson from the path integral of a non-chiral boson. Instead we compute it directly from the chiral boson Lagrangian of Floreanini and Jackiw modified by topological terms involving auxiliary fields. A careful analysis of the gauge-fixing condition for the extra gauge symmetry reproduces the correct results for the free chiral boson, and has the advantage of being applicable to a wider class of interacting chiral boson theories.
Two different families of abelian chiral gauge theories on the torus are investigated: the aim is to test the consistency of two-dimensional anomalous gauge theories in the presence of global degrees of freedom for the gauge field. An explicit computation of the partition functions shows that unitarity is recovered in particular regions of parameter space and that the effective dynamics is described in terms of fermionic interacting models. For the first family, this connection with fermionic models uncovers an exact duality which is conjectured to hold in the nonabelian case as well.
We construct black hole geometries in AdS$_3$ with non-trivial values of KdV charges. The black holes are holographically dual to quantum KdV Generalized Gibbs Ensemble in 2d CFT. They satisfy thermodynamic identity and thus are saddle point configurations of the Euclidean gravity path integral. We discuss holographic calculation of the KdV generalized partition function and show that for a certain value of chemical potentials new geometries, not the conventional BTZ ones, are the leading saddles.
We compute the partition function of four-dimensional abelian gauge theory on a general four-torus T4 with flat metric using Dirac quantization. In addition to an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagnetic S-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abelian gauge theory has its origin in symmetries of the 6d (2,0) tensor theory, by computing the partition function of a single fivebrane compactified on T2 x T4, which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of the abelian gauge theory τ= {θ\over 2π} + i{4π\over e^2} with …
We study the large $N$ 't Hooft expansion of the chiral partition function of 2d $U(N)$ Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deform…
Recently, Okuyama and Sakai proposed a novel holomorphic anomaly equation for the partition function of 2d Yang-Mills theory on a torus, based on an anholomorphic deformation of the propagator in the bosonic formulation. Using the boson-fermion correspondence, we derive the formula for the deformed partition function in fermionic description and give a proof of the holomorphic anomaly equation.
We consider the partition function of a general vertex operator algebra $V$ on a genus two Riemann surface formed by sewing together two tori. We consider the non-trivial degeneration limit where one torus is pinched down to a Riemann sphere and show that the genus one partition function on the degenerate torus is recovered up to an explicit universal $V$-independent multiplicative factor raised to the power of the central charge.
Wavefunctions of AdS$_3$ Universes and $T\bar{T}$-deformed Torus Partition Functions
School of Science, Huzhou Normal University（湖州师范学院理学院）；Center for Gravitational Physics and Quantum Information (CGPQI)（引力物理与量子信息中心）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
We evaluate quasi-classically the Ramond partition function of Euclidean D=10 U(N) super-Yang--Mills theory reduced to a two-dimensional torus. The result can be interpreted in terms of free strings wrapping the space-time torus, as expected from the point of view of Matrix string theory. We demonstrate that, when extrapolated to the ultraviolet limit (small area of the torus), the quasi-classical expressions reproduce exactly the recently obtained expression for the partition of the completely reduced SYM theory, including the overall numerical factor. This is an evidence that our quasi-class…
We have studied the free compact boson on a $n$-sheeted covering of the torus gluing alone $m$ branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the $n$-th Rényi entanglement entropy of $m$ disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.
Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter-Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and W_{3} cases.
We compute an SL(6,Z) invariant partition function for the chiral two-form of the M theory fivebrane compactified on the six-torus. From a manifestly SL(5,Z) invariant formalism, we prove that the partition function has an additional SL(2,Z) symmetry. The combination of these two symmetries ensures SL(6,Z) invariance. Thus, whether or not a fully covariant Lagrangian is available, the fivebrane on the six-torus has a consistent quantum theory.
We compute the partition function of five-dimension Abelian sYM including a graviphoton term on a five-torus. The result agrees with Abelian M5 brane partition function with zero characteristics on an associated six-torus.
We study the large $N$ 't Hooft expansion of the partition function of 2d $U(N)$ Yang-Mills theory on a torus. We compute the $1/N$ genus expansion of both the chiral and the full partition function of 2d Yang-Mills using the recursion relation found by Kaneko and Zagier with a slight modification. Then we study the large order behavior of this genus expansion, from which we extract the non-perturbative correction using the resurgence relation. It turns out that the genus expansion is not Borel summable and the coefficient of 1-instanton correction, the so-called Stokes parameter, is pure imag…
We study five-dimensional minimally supersymmetric gauge theory compactified on a torus down to three dimensions, and its embedding into string/M-theory using geometric engineering. The moduli space on the Coulomb branch is hyperkaehler equipped with a metric with modular transformation properties. We determine the one-loop corrections to the metric and show that they can be interpreted as worldsheet and D1-brane instantons in type IIB string theory. Furthermore, we analyze instanton corrections coming from the solitonic BPS magnetic string wrapped over the torus. In particular, we show how to…
Recursion relations for the sphere $4$-point and torus $1$-point ${\cal W}_3$ conformal blocks, generalizing Alexei Zamolodchikov's famous relation for the Virasoro conformal blocks are proposed. One of these relations is valid for any 4-point conformal block with two arbitrary and two special primaries with charge parameters proportional to the highest weight of the fundamental irrep of $SU(3)$. The other relation is designed for the torus conformal block with a special (in above mentioned sense) primary field insertion. AGT relation maps the sphere conformal block and the torus block to the …
We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a defi…
We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions o…
Computation of superstring partition function for the non-linear sigma model on the product of a two-torus and its dual within the scope of the doubled formalism is presented. We verify that it reproduces the partition functions of the toroidally compactified type--IIA and type--IIB theories for appropriate choices of the GSO projection.
Study of the matrix-level affine algebra $U_{m,K}$ is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of $U_{m,K}$ modular-invariant partition functions. Here we connect the algebra $U_{2,K}$ to strings on 2-tori describable by rational conformal field theories. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra $…
The instanton contributions to the partition function and to homologically trivial Wilson loops for a U(N) Yang-Mills theory on a torus $T^2$ are analyzed. An exact expression for the partition function is obtained as a sum of contributions localized around the classical solutions of Yang-Mills equations, that appear according to the general classification of Atiyah and Bott. Explicit expressions for the exact Wilson loop averages are obtained when N=2, N=3. For general $N$ the contribution of the zero-instanton sector has been carefully derived in the decompactification limit, reproducing the…
We show that the solitonic contribution of compactified strings corresponds to the quantum statistical partition function of a free particle living on higher dimensional spaces. In the simplest case of a compactification in a circle, the Hamiltonian corresponds to the Laplacian on the 2g-dimensional Jacobian torus associated to the genus g Riemann surface corresponding to the string worldsheet. T-duality leads to a symmetry of the partition function mixing time and temperature. Such a classical/quantum correspondence and T-duality shed some light on the well-known interplay between time and te…
We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background…
The $T{\bar T}$ deformation of a relativistic two-dimensional theory results in a solvable gravitational theory. Deformed scattering amplitudes can be obtained from coupling the undeformed theory to the flat space Jackiw--Teitelboim (JT) gravity. We show that the JT description is applicable and useful also in finite volume. Namely, we calculate the torus partition function of an arbitrary matter theory coupled to the JT gravity, formulated in the first order (vierbein) formalism. The first order description provides a natural set of dynamical clocks and rods for this theory, analogous to the …
ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等理工学院）；Instituto Superior de Agronomia, Universidade de Lisboa（里斯本大学高等农业学院）；CEMS.UL, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院数学系）
For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This tau-function acts as a partition function for the conserved densities, which fits its potential interpretation as the effective action of some quantum system. The class consists of multi-component generalizations of the Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The relationship between the former and the approach of Feigin, Frenk…
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 use holomorphic factorization to find the partition functions of an abelian two-form chiral gauge-field on a flat six-torus. We prove that exactly one of these partition functions is modular invariant. It turns out to be the one that previously has been found in a hamiltonian formulation.
We rewrite the worldsheet torus partition function of the Thermal AdS CFT by isolating the boundary parameters. Using this, we show that the spectrum of the Euclidean BTZ black hole and Lorentzian AdS3 can be extracted -- the latter as a zero temperature limit. A similar procedure recovers the Lorentzian BTZ spectrum proposed in an earlier work. We then use our expression to construct a boundary modular invariant expression as a Poincaré series.
Duality relations for the 2D nonhomogeneous Ising model on the finite square lattice wrapped on the torus are obtained. The partition function of the model on the dual lattice with arbitrary combinations of the periodical and antiperiodical boundary conditions along the cycles of the torus is expressed through some specific combination of the partition functions of the model on the original lattice with corresponding boundary conditions. It is shown that the structure of the duality relations is connected with the topological peculiarities of the dual transformation of the model on the torus.
Motivated by string theory connection, a covariant procedure for perturbative calculation of the partition function of the two-dimensional generalized $σ$-model is considered. The importance of a consistent regularization of the measure in the path integral is emphasized. The partition function is computed for a number of specific 2-manifolds: sphere, disk and torus.
We consider the set of partition functions that result from the insertion of twist operators compatible with conformal invariance in a given 2D Conformal Field Theory (CFT). A consistency equation, which gives a classification of twists, is written and solved in particular cases. This generalises old results on twisted torus boundary conditions, gives a physical interpretation of Ocneanu's algebraic construction, and might offer a new route to the study of properties of CFT.
Starting from the recent action proposed by Sen [1,2], we evaluate the partition function of the compact chiral boson on a two-dimensional torus using a path-integral formulation. Crucially, we use a Wick-rotation procedure obtained from a complex deformation of the physical spacetime metric. This directly reproduces the expected result including general characteristics for the theta functions. We also present results for the chiral 2-form potential in six dimensions which can be readily extended to 4k+2 dimensions.
We analyze the modular properties of the effective CFT description for paired states, proposed in cond-mat/0003453, corresponding to the non-standard filling nu =1/(p+1). We construct its characters for the twisted and the untwisted sector and the diagonal partition function. We show that the degrees of freedom entering our partition function naturally go to complete a Z_2-orbifold construction of the CFT for the Halperin state. Different behaviours for the p even and p odd cases are also studied. Finally it is shown that the tunneling phenomenon selects out a twist invariant CFT which is iden…
We study the modular transformation properties of Euclidean solutions of 3D gravity whose asymptotic geometry has the topology of a torus. These solutions represent saddle points of the grand canonical partition function with an important example being the BTZ black hole, and their properties under modular transformations are inherited from the boundary conformal field theory encoding the asymptotic dynamics. Within the Chern Simons formulation, classical solutions are characterised by specific holonomies describing the wrapping of the gauge field around cycles of the torus. We find that provi…
We provide quantitative evidence for our previous conjecture which states an equivalence of the partition function of a 3d N=2 gauge theory on a duality wall and that of the SL(2,R) Chern-Simons theory on a mapping torus, for a class of examples associated with once-punctured torus. In particular, we demonstrate that a limit of the 3d N=2 partition function reproduces the hyperbolic volume and the Chern-Simons invariant of the mapping torus. This is shown by analyzing the classical limit of the trace of an element of the mapping class group in the Hilbert space of the quantum Teichmuller theor…
