Full Partition Function Research Articles

On accurate calculations of equilibrium H/D fractionations at super-cold conditions (≤200 K) I: Full Partition Function Ratio (FPFR) vs. Path Integral Monte Carlo (PIMC)

Accurate estimations of Hydrogen/Deuterium (H/D) isotope effects have been a long-standing problem since the beginning of stable isotope geochemistry, primarily due to their strong nuclear quantum effects (NQEs). Currently, the D/H ratio plays a key role in deciphering the origin and evolution of water ice and organics in the solar system. Additionally, the Big-bang origin of Deuterium necessitates that the observed D/H variations be linked to isotope exchange reactions, such as HD + H2O ↔ H2 + HDO. Therefore, an accurate benchmark dataset describing the equilibrium H/D isotope fractionations at low temperatures (≤ 200 K) is essential for interpreting rapidly accumulating astrochemical observation data. In this study, we implemented and compared two methods developed for evaluating the NQEs of H/D isotopes: 1) the Full Partition Function Ratio (FPFR) method and 2) the Path Integral Monte Carlo (PIMC) method. Both were assessed for their accuracy and efficiency in estimating equilibrium isotope effects (EIEs) using 10 common interstellar small molecules (H2, HF, HCl, HCN, HNC, H2O, H2S, NH3, HCHO, and CH4) as examples. Most of them are calculated at high precision for the first time. For the FPFR method, several higher-order corrections to the Urey-Bigeleisen-Mayer model were considered, including anharmonicity, quantum mechanical rotation with nuclear-spin statistics, and corrections to the Born-Oppenheimer approximation. For the PIMC method, four different actions related to the thermal density matrix are used. These include the primitive and three fourth-order actions (Takahashi and Imada, Suzuki-Chin and Chin approximations), based on the direct scaled-coordinates transformation. Three main conclusions are obtained: 1) Under the framework of Born-Oppenheimer approximation, the PIMC method holds the same and even slightly better accuracy (more comprehensive descriptions of quantum anharmonicities) in estimating EIEs for H/D isotopes as the FPFR method. However, both methods suffer from the neglection of the DBOC when using the Born-Oppenheimer potential energy surfaces. For the PIMC method, additional corrections for particle distinguishability must be considered when nuclear spin statistics exhibit its effects at low temperatures (e.g., < 160 K for H2). 2) Among the four actions, three fourth-order actions show better convergence and efficiency (2.2 to 26.5 times speedups at 50 K) than the primitive action, making them superior candidates for PIMC simulations at super-cold (≤ 200 K) conditions. 3) A combination of accurate potential energy surface with the PIMC method serves as an alternative way for theoretical estimations of H/D isotope effects.

Supersymmetry and trace formulas. Part I. Compact Lie groups

In the context of supersymmetric quantum mechanics we formulate new supersymmetric localization principle, with application to trace formulas for a full thermal partition function. Unlike the standard localization principle, this new principle allows to compute the supertrace of non-supersymmetric observables, and is based on the existence of fermionic zero modes. We describe corresponding new invariant supersymmetric deformations of the path integral; they differ from the standard deformations arising from the circle action and require higher derivatives terms. Consequently, we prove that the path integral localizes to periodic orbits and not only on constant ones. We illustrate the principle by deriving bosonic trace formulas on compact Lie groups, including classical Jacobi inversion formula.

Seven-dimensional super Yang-Mills at negative coupling

We consider the partition function for Euclidean SU(N)SU(N) super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six ``fixed" three-spheres on the \mathbb{S}^7𝕊7. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the {\mathbb S}^7𝕊7 is well-defined even when the square of the effective Yang-Mills coupling is negative. We show for an SU(2)SU(2) gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super Yang-Mills and, in fact, is gravitational.

Partition functions and fibering operators on the Coulomb branch of 5d SCFTs

We study 5d mathcal{N} = 1 supersymmetric field theories on closed five-manifolds mathcal{M} 5 which are principal circle bundles over simply-connected Kähler four-manifolds, mathcal{M} 4, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on mathcal{M} 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over mathcal{M} 4, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such mathcal{M} 5, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When mathcal{M} 4 is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.

Exact Toverline{T} deformation of two-dimensional Yang-Mills theory on the sphere

We study the Toverline{T} deformation of two-dimensional Yang-Mills theory at genus zero by carrying out the analysis at the level of its instanton representation. We first focus on the perturbative sector by considering its power expansion in the deformation parameter μ. By studying the resulting asymptotic series through resurgence theory, we determine the nonperturbative contributions that enter the result for μ < 0. We then extend this analysis to any flux sector by solving the relevant flow equation. Specifically, we impose boundary conditions corresponding to two distinct regimes: the quantum undeformed theory and the semiclassical limit of the deformed theory. The full partition function is obtained as a sum over all magnetic fluxes. For any μ > 0, only a finite portion of the quantum spectrum survives and the partition function reduces to a sum over a finite set of representations. For μ < 0, nonperturbative contributions regularize the partition function through an intriguing mechanism that generates nontrivial subtractions.

Differential equation for partition functions and a duality pseudo-forest

We consider finite quantum systems defined by a mixed set of commutation and anti-commutation relations between components of the Hamiltonian operator. These relations are represented by an anti-commutativity graph that contains necessary and sufficient information for computing the full quantum partition function. We derive a second-order differential equation for an extended partition function z[β, β′, J] that describes a transformation from a “parent” partition function z[0, β′, J] (or anti-commutativity graph) to a “child” partition function z[β, 0, J] (or anti-commutativity graph). The procedure can be iterated, and then one forms a pseudo-forest of duality transformations between quantum systems, i.e., a directed graph in which every vertex (or quantum system) has at most one incoming edge (from its parent system). The pseudo-forest has a single tree connected to a constant partition function, many pseudo-trees connected to self-dual systems, and all other pseudo-trees connected to closed cycles of transformations between mutually dual systems. We also show how the differential equation for the extended partition function can be used to study disordered systems.

Higher genera Catalan numbers and Hirota equations for extended nonlinear Schr\xf6dinger hierarchy

We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin–Frobenius manifold is a tau-function of the extended nonlinear Schrödinger hierarchy, an extension of a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental–Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation.

Resurgent analysis for some 3-manifold invariants

We study resurgence for some 3-manifold invariants when Gℂ = SL(2, ℂ). We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in S3. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds M3. In particular, this directly indicates that the homological block for the torus knot complement in S3 is an analytic continuation of the full G = SU(2) partition function, i.e. the colored Jones polynomial.

Dual path integral: a non-perturbative approach to strong coupling

We develop a non-perturbative method for calculating partition functions of strongly coupled quantum mechanical systems with interactions between subsystems described by a path integral of a dual system. The dual path integral is derived starting from non-interacting subsystems at zeroth order and then by introducing couplings of increasing complexity at each order of an iterative procedure. These orders of interactions play the role of a dual time and the full quantum partition function is expressed as a transition amplitude in the dual system. More precisely, it is expressed as a path integral from a deformation-operators dependent initial state at zero time/order to the inverse-temperature dependent final state at later time/order. We provide examples of strongly coupled systems with up to first-order interactions (e.g. Ising model) and arbitrary high-order interactions (e.g. 1+1hbox {D} QFT). We also discuss a possible emergence of space-time, quantum field theories and general relativity in context of the dual path integral.

Partition functions of the tensionless string

We consider string theory on AdS3× S3× \U0001d54b4 in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal AdS3, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal AdS3 and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal AdS3 and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal AdS3. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately.

Topological correlators and surface defects from equivariant cohomology

We find a one-dimensional protected subsector of mathcal{N} = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general mathcal{N} = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

Twisting with a Flip (The Art of Pestunization)

We construct mathcal{N}=2 supersymmetric Yang–Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. It turns out that for every fixed point one can allocate either instanton or anti-instanton contributions to the partition function, and that this is compatible with supersymmetry. The equivariant Donaldson–Witten theory is a special case of our construction. We present a unified treatment of Pestun’s calculation on S^4 and equivariant Donaldson–Witten theory by generalizing the notion of self-duality on manifolds with a vector field. We conjecture the full partition function for a mathcal{N}=2 theory on any 4D manifold with a Killing vector. Using this new notion of self-duality to localize a supersymmetric theory is what we call “Pestunization”.

Liouville quantum gravity on the annulus

In this work, we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists initiated by Polyakov in 1981. It is also a very important example of a conformal field theory (CFT). Results have already been obtained on the Riemann sphere and on the unit disk, so this paper will follow the same approach. The case of the annulus contains two difficulties: it is a surface with two boundaries and it has a non-trivial moduli space. We recover the Weyl anomaly—a formula verified by all CFT—and deduce from it the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We also show that the full partition function of Liouville quantum gravity integrated over the moduli space is finite. This allows us to give the joint law of the Liouville measures and of the random modulus and to write the conjectured link with random planar maps.

Resurgence analysis of 2d Yang-Mills theory on a 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 imaginary. We find that the non-perturbative correction obtained from the resurgence is reproduced from a certain analytic continuation of the grand partition function of a system of non-relativistic fermions on a circle. Our analytic continuation is different from that considered in hep-th/0504221.

Q2DTor: A program to treat torsional anharmonicity through coupled pair torsions in flexible molecules

The Q 2DTor program (Quantum 2-Dimensional Torsions) is designed to calculate accurate rotational–vibrational partition functions (also called rovibrational partition functions) and thermodynamic functions for molecular systems having two [1] or more torsions. Systems with more than two torsions can also be studied by treating the torsions by pairs. The program searches for all the torsional conformers and evaluates the rovibrational partition function using the multi-structural harmonic oscillator (MS-HO) approximation and the extended two-dimensional torsion (E2DT) approximation. The latter incorporates full coupling of the two torsions by means of the two-dimensional non-separable (2D-NS) approximation [2], and it also includes their influence on the remaining degrees of freedom. The program also calculates the ideal gas-phase standard-state thermodynamic functions at the requested temperatures. Twenty molecules have been used to test Q 2DTor. Program summaryProgram Title:Q2DTorProgram Files doi:http://dx.doi.org/10.17632/wbgchgc2kp.1Licensing provisions: GNU GPL v3Programming language: Python 2.7Nature of problem: Calculation of accurate partition functions and thermodynamic functions in molecular systems involving two torsional modes. Torsional anharmonicity is treated quantically and includes full coupling in the kinetic and potential energies between the torsions and between the torsions and the rest of the degrees of freedom.Solution method: The program uses the variational method to solve the Schrödinger equation of a two-dimensional torsional potential using Fourier series. All of the remaining degrees of freedom (non-torsional) are incorporated through a projected (the torsional modes are removed) rigid-rotator harmonic-oscillator partition function which is calculated at every torsional stationary point and that is allowed to vary with the torsional motion. The integration of the rovibrational partition function over the torsional space leads to a mixed quantum-classical vibrational partition function, which is transformed into a full quantum partition function by including the quantum contribution due to the torsions. For the evaluation of the integral, the rovibrational partition function at nonstationary points is carried out through a Delaunay triangulation procedure using the calculated rovibrational partition functions at the stationary points as nodes.Additional comments including Restrictions and Unusual features: The program is limited to two coupled torsional modes.

Exact deconstruction of the 6D (2,0) theory

The dimensional-deconstruction prescription of Arkani-Hamed, Cohen, Kaplan, Karch and Motl provides a mechanism for recovering the A-type (2,0) theories on T2, starting from a four-dimensional mathcal{N}=2 circular-quiver theory. We put this conjecture to the test using two exact-counting arguments: in the decompactification limit, we compare the Higgs-branch Hilbert series of the 4D mathcal{N}=2 quiver to the “half-BPS” limit of the (2,0) superconformal index. We also compare the full partition function for the 4D quiver on S4 to the (2,0) partition function on S4× T2. In both cases we find exact agreement. The partition function calculation sets up a dictionary between exact results in 4D and 6D.

Modular properties of full 5D SYM partition function

We study properties of the full partition function for the $U(1)$ 5D $\mathcal{N}=2^*$ gauge theory with adjoint hypermultiplet of mass $M$. This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function $G_2^C$ associated with a certain moment map cone $C$. The answer exhibits a curious $SL(4,\mathbb{Z})$ modular property. Finally, we propose a set of rules to construct the partition function that resembles the calculation of 5D supersymmetric partition function with the insertion of defects of various co-dimensions.

Elliptic CY3folds and non-perturbative modular transformation

We study the refined topological string partition function of a class of toric elliptically fibered Calabi-Yau threefolds. These Calabi-Yau threefolds give rise to five dimensional quiver gauge theories and are dual to configurations of M5-M2-branes. We determine the Gopakumar-Vafa invariants for these threefolds and show that the genus $g$ free energy is given by the weight $2g$ Eisenstein series. We also show that although the free energy at all genera are modular invariant the full partition function satisfies the non-perturbative modular transformation property discussed by Lockhart and Vafa in arXiv:1210.5909 and therefore the modularity of free energy is up to non-perturbative corrections.

Large volume susy breaking with a solution to the decompactification problem

We study heterotic ground states in which supersymmetry is broken by coupling the momentum and winding charges of two large extra dimensions to the R-charges of the supersymmetry generators. The large dimensions give rise to towers of heavy string thresholds that contribute to the running of the gauge couplings. In the general case, these contributions are proportional to the volume of the two large dimensions and invalidate the perturbative string expansion. The problem is evaded if the susy breaking sectors arise as a spontaneously broken phase of N=4→N=2→N=0 supersymmetry, provided that N=4 supersymmetry is restored on the boundary of the moduli space. We discuss the mechanism in the case of Z2×Z2 orbifolds, which requires that the twisted sector that contains the large extra dimensions has no fixed points. We analyze the full string partition function and show that the twisted sectors distribute themselves in non-aligned N=2 orbits, hence preserving the solution to the string decompactification problem. Remarkably, we find that the contribution to the vacuum energy from the N=2→N=0 sectors is suppressed, and the only substantial contribution arises from the breaking of the N=4 sector to N=0.

Tao probing the end of the world

We introduce a new type IIB 5-brane description for the E-string theory which is the world-volume theory on the M5-brane probing the end of the world M9-brane. The E-string in the new realization is depicted as spiral 5-branes web equipped with the cyclic structure which is key to uplifting to six dimensions. Utilizing the topological vertex to the 5-brane web configuration enables us to write down a combinatorial formula for the generating function of the E-string elliptic genera, namely the full partition function of topological strings on the local 1/2 K3 surface.