arXiv subjects

Witold Skiba

Publications and source records attributed to Witold Skiba.

At least 19 recordsLinked to original sources

arXiv2026

Virasoro OPE and Conformal Blocks from the Inverse Shapovalov Form

We derive expressions for the Virasoro OPE and four-point conformal blocks on the sphere via the resolution of identity recently determined in [Phys. Rev. D 111, 085010 (2025), arXiv:2409.12224]. Even though the resolution of the identity depends on Virasoro singular vectors, our expression for the blocks does not depend on their precise form, but just on their well-known conformal weights. We verify that our expression is compatible with -- but differs from -- Zamolodchikov's $h$-recursion relation and we also examine the impact of various large central charge limits in our formula. A Mathematica notebook with a simple implementation of our expression for the Virasoro conformal blocks is provided as an ancillary file.

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arXiv2025

Conformal Bootstrap Equations from the Embedding Space Operator Product Expansion

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the $M$-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-$\ell$ exchange, with finite $\ell$-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars $S$ and vectors $V$, namely $\left\langle SSSV\right\rangle$.

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arXiv2025

Casimirs of the Virasoro Algebra

We explicitly solve a recurrence relation due to Feigin and Fuchs to obtain the Casimirs of the Virasoro algebra in terms of the inverse of the Shapovalov form. Combined with our recent result for the inverse Shapovalov form, this allows us to write the Casimir operators as linear combinations of products of singular vectors.

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arXiv2025

The Virasoro Completeness Relation and Inverse Shapovalov Form

In this work, we introduce an explicit expression for the inverse of the symmetric bilinear form of Virasoro Verma modules, the so-called Shapovalov form, in terms of singular vector operators and their conformal dimensions. Our proposed expression also determines the resolution of the identity for Verma modules of the Virasoro algebra, and can be thus employed in the computation of Virasoro conformal blocks via the sewing procedure.

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arXiv2023

One- and two-dimensional higher-point conformal blocks as free-particle wavefunctions in AdS$_3^{\otimes m}$

We establish that all of the one- and two-dimensional global conformal blocks are, up to some choice of prefactor, free-particle wavefunctions in tensor products of AdS$_3$ or limits thereof. Our first core observation is that the six-point comb-channel conformal blocks correspond to free-particle wavefunctions on an AdS$_3$ constructed directly in cross-ratio space. This construction generalizes to blocks for a special class of diagrams, which are determined as free-particle wavefunctions in tensor products of AdS$_3$. Conformal blocks for all the remaining topologies are obtained as limits of the free wavefunctions mentioned above. Our results show directly that the integrable models associated with all one- and two-dimensional conformal blocks can be seen as limits of free theory, and manifest a relation between AdS and CFT kinematics that lies outside of the standard AdS/CFT dictionary. We complete the discussion by providing explicit Feynman-like rules that can be used to work out blocks for all topologies, as well as a Mathematica notebook that allows simple computation of Casimir equations and series expansions for blocks, by requiring just an OPE diagram as input.

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arXiv2022

Conformal Blocks in Three Dimensions

We derive expressions for conformal blocks involving operators with arbitrary spins in 3-dimensional CFTs. We use previous results on the action of the OPE in the embedding space to derive the conformal blocks. The blocks are given as explicit power series in terms of the conformal cross ratios.

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arXiv2022

Feynman Rules for Scalar Conformal Blocks

We complete the proof of "Feynman rules" for constructing $M$-point conformal blocks with external and internal scalars in any topology for arbitrary $M$ in any spacetime dimension by combining the rules for the blocks (based on their Witten diagram interpretation) with the rules for the construction of conformal cross ratios (based on OPE flow diagrams). The full set of Feynman rules leads to blocks as power series of the hypergeometric type in the conformal cross ratios. We then provide a proof by recursion of the Feynman rules which relies heavily on the first Barnes lemma and the decomposition of the topology of interest in comb-like structures. Finally, we provide a nine-point example to illustrate the rules.

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arXiv2022

Electroweak Constraints from the COHERENT Experiment

We compute bounds on coefficients of effective operators in the Standard Model that can be inferred from observations of neutrino scattering by the COHERENT experiment. While many operators are bound extremely well by past experiments the full future data set from COHERENT will provide modest improvements for some operators.

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arXiv2020

Conformal Conserved Currents in Embedding Space

We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator $\partial_μ$, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator $\mathcal{D}_{ijA}$ used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents $\langle JJJJ\rangle$ and four energy-momentum tensors $\langle TTTT\rangle$, we mostly focus on conservation conditions for $\langle JJ\mathcal{O}\rangle$ and $\langle TT\mathcal{O}\rangle$. Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.

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arXiv2020

Further Results on a Function Relevant for Conformal Blocks

We present further mathematical results on a function appearing in the conformal blocks of four-point correlation functions with arbitrary quasi-primary operators. The $H$-function was introduced in a previous article and it has several interesting properties. We prove explicitly the recurrence relation as well as the $D_6$-invariance presented previously. We also demonstrate the proper action of the differential operator used to construct the $H$-function.

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arXiv2020

Higher-Point Conformal Blocks in the Comb Channel

We compute $M$-point conformal blocks with scalar external and exchange operators in the so-called comb configuration for any $M$ in any dimension $d$. Our computation involves repeated use of the operator product expansion to increase the number of external fields. We check our results in several limits and compare with the expressions available in the literature when $M=5$ for any $d$, and also when $M$ is arbitrary while $d=1$.

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arXiv2020

Six-Point Conformal Blocks in the Snowflake Channel

We compute $d$-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two ${}_3F_2$-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

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arXiv2020

All Global One- and Two-Dimensional Higher-Point Conformal Blocks

We introduce a full set of rules to directly express all $M$-point conformal blocks in one- and two-dimensional conformal field theories, irrespective of the topology. The $M$-point conformal blocks are power series expansion in some carefully-chosen conformal cross-ratios. We then prove the rules for any topology constructively with the help of the known position space operator product expansion. To this end, we first compute the action of the position space operator product expansion on the most general function of position space coordinates relevant to conformal field theory. These results provide the complete knowledge of all $M$-point conformal blocks with arbitrary external and internal quasi-primary operators (including arbitrary spins in two dimensions) in any topology.

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arXiv2020

Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond

Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kampé de Fériet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.

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arXiv2020

Efficient Rules for All Conformal Blocks

We formulate a set of general rules for computing $d$-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism arXiv:1905.00434. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars $S$ and some specific irreducible representations $R$, namely $\langle SSSS\rangle$, $\langle SSSR\rangle$, $\langle SRSR\rangle$ and $\langle SSRR\rangle$ (where, when allowed, $R$ is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

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arXiv2019

New Methods for Conformal Correlation Functions

The most general operator product expansion in conformal field theory is obtained using the embedding space formalism and a new uplift for general quasi-primary operators. The uplift introduced here, based on quasi-primary operators with spinor indices only and standard projection operators, allows a unified treatment of all quasi-primary operators irrespective of their Lorentz group irreducible representations. This unified treatment works at the level of the operator product expansion and hence applies to all correlation functions. A very useful differential operator appearing in the operator product expansion is established and its action on appropriate products of embedding space coordinates is explicitly computed. This computation leads to tensorial generalizations of the usual Exton function for all correlation functions. Several important identities and contiguous relations are also demonstrated for these new tensorial functions. From the operator product expansion all correlation functions for all quasi-primary operators, irrespective of their Lorentz group irreducible representations, can be computed recursively in a systematic way. The resulting answer can be expressed in terms of tensor structures that carry all the Lorentz group information and linear combinations of the new tensorial functions. Finally, a summary of the well-defined rules allowing the computation of all correlation functions constructively is presented.

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arXiv2019

Conformal Differential Operator in Embedding Space and its Applications

We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal blocks. We show a number of relations that the components of conformal blocks satisfy and find invariance of our expressions under the dihedral group.

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arXiv2019

Conformal Four-Point Correlation Functions from the Operator Product Expansion

We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in arXiv:1905.00036 and arXiv:1905.00434, and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group.

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