Holomorphic Factorization Research Articles

Conformal quantum mechanics, holomorphic factorisation, and ultra-spinning black holes

We study a limit in which a relativistic CFT reduces to conformal quantum mechanics, and relate the partition functions of the two theories. When the initial CFT is holographic, our limit coincides with an ultra-spinning limit in the gravity dual. We therefore propose that ultra-spinning black holes are dual to an appropriate ensemble in finite-dimensional conformal quantum mechanics. The limit is studied in detail for SCFTs in four and six dimensions. These theories have a superconformal index which can be computed by gluing together two or more blocks. Applying our limit to the index effectively isolates a single such block. Our results therefore suggest that ultra-spinning black holes play the role of blocks in the gravitational dual of holomorphic factorisation.

Factorization of holomorphic matrices and Kazhdan's property (T)

In this article we deduce some algebraic properties for the group Sp2n(O(X)) of holomorphic symplectic matrices on a Stein space X: holomorphic factorization, exponential factorization, and Kazhdan's property (T). In holomorphic factorization we combine a recent result of the third author and K-theory tools to give explicit bounds for the case when X is one-dimensional or two-dimensional. Next we use them to find bounds for exponential factorization. As a further application, we show that the elementary symplectic group Ep2n(O(X)) admits Kazhdan's property (T).

Holomorphic factorization of mappings into Sp4(ℂ)

We prove that any null-homotopic holomorphic map from a Stein space $X$ to the symplectic group $\operatorname{Sp}_{4}(\mathbb{C})$ can be written as a finite product of elementary symplectic matrices with holomorphic entries.

RG flow between W3 minimal models by perturbation and domain wall approaches

We explore the RG flow between neighboring minimal CFT models with W3 symmetry. After computing several classes of OPE structure constants we were able to find the matrices of anomalous dimensions for three classes of RG invariant sets of local fields. Each set from the first class consists of a single primary field, the second one of three primaries, while sets in the third class contain six primary and four secondary fields. We diagonalize their matrices of anomalous dimensions and establish the explicit maps between UV and IR fields (mixing coefficients).While investigating the three point functions of secondary fields we have encountered an interesting phenomenon, namely violation of holomorphic anti-holomorphic factorization property, something that does not happen in ordinary minimal models with Virasoro symmetry solely.Furthermore, the perturbation under consideration preserves a non-trivial subgroup of W transformations. We have derived the corresponding conserved current explicitly. We used this current to define a notion of anomalous W-weights in perturbed theory: the analog for matrix of anomalous dimensions. For RG invariant sets with primary fields only we have derived a formula for this quantity in therms of structure constants. This allowed us to compute anomalous W -weights for the first and second classes explicitly.The same RG flow we investigate also with the domain wall approach for the second RG invariant class and find complete agreement with the perturbative approach.

Derived brackets in bosonic string sigma-model

We study the worldsheet theory of bosonic string from the point of view of the BV formalism. We explicitly describe the derived Poisson structure which arises when we expand the Master Action near a Lagrangian submanifold. The resulting higher Poisson brackets are all degenerate and essentially constant along their symplectic leaves. Deformations of the worldsheet complex structures define a family of Lagrangian submanifolds, parametrized by Beltrami differential. The worldsheet action depends nonlinearly on the Beltrami differential, but the structure of nonlinearity is governed by the BV Master Equation. This helps to clarify the mechanism of holomorphic factorization of string amplitudes.

Sequential warped product submanifolds having holomorphic, totally real and pointwise slant factors

Sequential warped product submanifolds having holomorphic, totally real and pointwise slant factors

The genus two free boson in Arakelov geometry

Using Arakelov geometry, we compute the partition function of the noncompact free boson at genus two. We begin by compiling a list of modular invariants which appear in the Arakelov theory of Riemann surfaces. Using these quantities, we express the genus two partition function as a product of modular forms, as in the well-known genus one case. We check that our result has the expected obstruction to holomorphic factorization and behavior under degeneration.

Tangle blocks in the theory of link invariants

The central discovery of 2d conformal theory was holomorphic factorization, which expressed correlation functions through bilinear combinations of conformal blocks, which are easily cut and joined without a need to sum over the entire huge Hilbert space of states. Somewhat similar, when a link diagram is glued from tangles, the link polynomial is a multilinear combination of tangle blocks summed over just a few representations of intermediate states. This turns to be a powerful approach because the same tangles appear as constituents of very different knots so that they can be extracted from simpler cases and used in more complicated ones. So far this method has been technically developed only in the case of arborescent knots, but, in fact, it is much more general. We begin a systematic study of tangle blocks by detailed consideration of some archetypical examples, which actually lead to non-trivial results, far beyond the reach of other techniques. At the next level, the tangle calculus is about gluing of tangles, and functorial mappings from Hom(tangles). Its main advantage is an explicit realization of multiplicative composition structure, which is partly obscured in traditional knot theory.

FRW and domain walls in higher spin gravity

We present exact solutions to Vasiliev’s bosonic higher spin gravity equations in four dimensions with positive and negative cosmological constant that admit an interpretation in terms of domain walls, quasi-instantons and Friedman-Robertson-Walker (FRW) backgrounds. Their isometry algebras are infinite dimensional higher-spin extensions of spacetime isometries generated by six Killing vectors. The solutions presented are obtained by using a method of holomorphic factorization in noncommutative twistor space and gauge functions. In interpreting the solutions in terms of Fronsdal-type fields in space-time, a field-dependent higher spin transformation is required, which is implemented at leading order. To this order, the scalar field solves Klein-Gordon equation with conformal mass in (A)dS4. We interpret the FRW solution with de Sitter asymptotics in the context of inflationary cosmology and we expect that the domain wall and FRW solutions are associated with spontaneously broken scaling symmetries in their holographic description. We observe that the factorization method provides a convenient framework for setting up a perturbation theory around the exact solutions, and we propose that the nonlinear completion of particle excitations over FRW and domain wall solutions requires black hole-like states.

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.

Symmetries in superintegrable deformations of oscillator and Coulomb systems: Holomorphic factorization

We propose a unified description for the constants of motion for superintegrable deformations of the oscillator and Coulomb systems on N-dimensional Euclidean space, sphere and hyperboloid. We also consider the duality between these generalized systems and present some example.

Invariant functionals in higher-spin theory

A new construction for gauge invariant functionals in the nonlinear higher-spin theory is proposed. Being supported by differential forms closed by virtue of the higher-spin equations, invariant functionals are associated with central elements of the higher-spin algebra. In the on-shell AdS4 higher-spin theory we identify a four-form conjectured to represent the generating functional for 3d boundary correlators and a two-form argued to support charges for black hole solutions. Two actions for 3d boundary conformal higher-spin theory are associated with the two parity-invariant higher-spin models in AdS4. The peculiarity of the spinorial formulation of the on-shell AdS3 higher-spin theory, where the invariant functional is supported by a two-form, is conjectured to be related to the holomorphic factorization at the boundary. The nonlinear part of the star-product function F⁎(B(x)) in the higher-spin equations is argued to lead to divergencies in the boundary limit representing singularities at coinciding boundary space–time points of the factors of B(x), which can be regularized by the point splitting. An interpretation of the RG flow in terms of proposed construction is briefly discussed.

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cn, λ ∈ C, in a domain Ω ⊂ Cn which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.

The moduli space of vacua of N = 2 $$ \mathcal{N}=2 $$ class S $$ \mathcal{S} $$ theories

We develop a systematic method to describe the moduli space of vacua of four dimensional $$ \mathcal{N}=2 $$ class $$ \mathcal{S} $$ theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin’s equations obtained from twisted compactification of 5d maximal Super-Yang-Mills, with local degrees of freedom at punctures given by (nilpotent) orbits. The crucial thing is the holomorphic factorization of the Seiberg-Witten curve and reduction of singularity at punctures. We illustrate our method by many examples including $$ \mathcal{N}=2 $$ SQCD, T N theory and Argyres-Douglas theories.

LOOP VARIABLES AND GAUGE INVARIANT EXACT RENORMALIZATION GROUP EQUATIONS FOR CLOSED STRING THEORY

We formulate the Exact Renormalization Group on the string worldsheet for closed string backgrounds. The same techniques that were used for open strings are used here. There are some subtleties. One is that holomorphic factorization of the closed string vertex operators does not hold in the presence of a cutoff on the Euclidean worldsheet. This introduces extra terms in the Lagrangian at the cutoff scale and they turn out to be crucial for implementing gauge invariance. This naive generalization from open string to closed string requires a massive graviton and the gauge symmetry is Abelian, just as in open string theory. Interestingly, it turns out that if one introduces a nondynamical background metric (as in background field formalism) and combines a gauge transformation on the field with a transformation on the coordinates and background metric, the graviton can be massless. Some examples of background coordinate covariant equations are worked out explicitly. A preliminary discussion of massive modes, massive gauge transformations and the role of worldsheet regulator terms is given. Some of the gauge transformations can be given a geometric meaning if space–time is assumed to be complex at some level.

Fluxed M5-instantons in F-theory

We analyse the non-perturbative superpotential due to M5-brane instantons in F-theory compactifications on Calabi–Yau fourfolds. The M5 partition function is obtained via holomorphic factorisation by explicitly performing the sum over chiral 3-form fluxes. Comparison with the partition function of fluxed Euclidean D3-brane instantons in Type IIB orientifolds allows us to fix the spin structure on the intermediate Jacobian of the M5-instanton. We furthermore analyse the contribution of the M5-instanton to the superpotential in the presence of G4 gauge flux, where the superpotential is dressed with matter fields. We explicitly evaluate the pullback of G4 onto the M5-brane as a measure for the presence of charged instanton zero modes. This accounts for the M5 charge both under massless U(1)s, if present, and under what corresponds in Type II language to geometrically massive U(1)s.

Holomorphic factorization of mappings into SL_n(C)

We solve Gromov’s Vaserstein problem. Namely, we show that a nullhomotopic holomorphic mapping from a nite dimensional reduced Stein space into SLn(C) can be factored into a nite product of unipotent matrices with holomorphic entries.

A universal inequality for CFT and quantum gravity

We prove that every unitary two-dimensional conformal field theory (with no extended chiral algebra, and with central charges c_L, c_R > 1) contains a primary operator with dimension Delta_1 that satisfies 0 < Delta_1 < (c_L + c_R)/12 + 0.473695. Translated into gravitational language using the AdS_3 /CFT_2 dictionary, our result proves rigorously that the lightest massive excitation in any theory of 3D gravity with cosmological constant Lambda < 0 can be no heavier than 1/(4 G_N) + o(|Lambda|^(1/2)). In the flat-space approximation, this limiting mass is twice that of the lightest BTZ black hole. The derivation of the bound applies at finite central charge for the CFT, and does not rely on an asymptotic expansion at large central charge. Neither does our proof rely on any special property of the CFT such as supersymmetry or holomorphic factorization, nor on any bulk interpretation in terms of string theory or semiclassical gravity. Our only assumptions are unitarity and modular invariance of the dual CFT. Our proof demonstrates for the first time that there exists a universal center-of-mass energy beyond which a theory of "pure" quantum gravity can never consistently be extended.

Instantons beyond topological theory. I

AbstractMany quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyse the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kähler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.In Part I we describe the class of models under study and present our results in the case of one-dimensional (quantum mechanical) models, which is quite representative and at the same time simple enough to analyse explicitly. Part II will be devoted to supersymmetric two-dimensional sigma models and four-dimensional Yang–Mills theory. In Part III we will discuss non-supersymmetric models.

Holomorphic Factorization of Superstring Scattering Amplitudes

The holomorphic factorization of the superstring partition function is verified at arbitrary genus. The evaluation of scattering amplitudes and the implications of genus-dependent estimates on the string coupling are given.