Piotr Kucharski
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Piotr Kucharski has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Dmitry Noshchenko
School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin 4, Ireland
9 shared publications - Piotr Sułkowski
Faculty of Physics, University of Warsaw, Warsaw, Poland
9 shared publications - Tobias Ekholm
Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
6 shared publications - Hélder Larraguível
Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research, Jagiellonian University, Krakow, Poland
5 shared publications - Pietro Longhi
Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
5 shared publications - Sunghyuk Park
Department of Mathematics, Harvard University, Cambridge, USA
4 shared publications
Latest Publications
- Open Access
- 28 August 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (10) , 206
Quivers and BPS States in 3d and 4d
We propose a symmetrization relation between BPS quivers encoding 4d $$\mathcal {N}=2$$ theories and symmetric quivers associated to 3d $$\mathcal {N}=2$$ theories. We analyse in detail the symmetrization of BPS quivers for a series of $$A_m$$ Argyres-Douglas theories by engineering 3d-4d systems in geometric backgrounds involving appropriate 3-manifolds and Riemann surfaces. We discuss properties of these geometric backgrounds and derive the corresponding quiver partition functions from the perspective of skein modules, which forms the foundation of the symmetrization map for the minimal chamber. We also prove that the structure of wall-crossing in 4d $$A_m$$ Argyres-Douglas theories is isomorphic to the structure of unlinking of symmetric quivers encoding their partner 3d theories, which allows for a proper definition of the symmetrization map outside the minimal chamber. Finally, we show that the Schur indices of 4d theories are captured by symmetric quivers that include symmetrization of 4d BPS quivers.- 28 August 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (10) , 206
Open AccessQuivers and BPS States in 3d and 4d
We propose a symmetrization relation between BPS quivers encoding 4d $$\mathcal {N}=2$$ theories and symmetric quivers associated to 3d $$\mathcal {N}=2$$ theories. We analyse in detail the symmetrization of BPS quivers for a series of $$A_m$$ Argyres-Douglas theories by engineering 3d-4d systems in geometric backgrounds involving appropriate 3-manifolds and Riemann surfaces. We discuss properties of these geometric backgrounds and derive the corresponding quiver partition functions from the perspective of skein modules, which forms the foundation of the symmetrization map for the minimal chamber. We also prove that the structure of wall-crossing in 4d $$A_m$$ Argyres-Douglas theories is isomorphic to the structure of unlinking of symmetric quivers encoding their partner 3d theories, which allows for a proper definition of the symmetrization map outside the minimal chamber. Finally, we show that the Schur indices of 4d theories are captured by symmetric quivers that include symmetrization of 4d BPS quivers. - 1 April 2026
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 62 (2) , 379-396
Categorification of Quiver Diagonalization and Koszul Algebras
In earlier work by three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson–Thomas invariants of symmetric quivers was given using the so-called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver, such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.
- 1 April 2026
- Published by European Mathematical Society - EMS - Publishing House GmbH in Publications of the Research Institute for Mathematical Sciences
- Vol. 62 (2) , 379-396
Categorification of Quiver Diagonalization and Koszul Algebras
In earlier work by three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson–Thomas invariants of symmetric quivers was given using the so-called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver, such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.
- Top Cited PapersOpen Access
- 17 January 2026
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 116 (1) , 10
Unlinking symmetric quivers
We analyse the structure of equivalence classes of symmetric quivers whose generating series are equal. We consider such classes constructed using the basic operation of unlinking, which increases the size of a quiver. The existence and features of such classes do not depend on a particular quiver but follow from the properties of unlinking. We show that such classes include sets of quivers assembled into permutohedra, and all quivers in a given class are determined by one quiver of the largest size, which we call a universal quiver. These findings generalise the previous ones for permutohedra graphs for knots. We illustrate our results with generic examples, as well as specialisations related to the knots–quivers correspondence.
- 17 January 2026
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 116 (1) , 10
Top Cited PapersOpen AccessUnlinking symmetric quivers
We analyse the structure of equivalence classes of symmetric quivers whose generating series are equal. We consider such classes constructed using the basic operation of unlinking, which increases the size of a quiver. The existence and features of such classes do not depend on a particular quiver but follow from the properties of unlinking. We show that such classes include sets of quivers assembled into permutohedra, and all quivers in a given class are determined by one quiver of the largest size, which we call a universal quiver. These findings generalise the previous ones for permutohedra graphs for knots. We illustrate our results with generic examples, as well as specialisations related to the knots–quivers correspondence.
- Open Access
Full twists and stability of knots and quivers
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region.Open AccessFull twists and stability of knots and quivers
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of $Sym^r$-coloured HOMFLY-PT polynomials under the addition of a~full twist to the knot. On the other hand, we show that symmetric quivers exhibit analogous stable growth under unlinking or linking of the quiver augmented with the extra node; in some cases this augmented quiver captures the spectrum of motivic Donaldson-Thomas invariants of all quivers in the sequence. Combining these two versions of the stable growth, we conjecture that performing a~full twist on any knot corresponds to appropriate unlinking or linking of the corresponding augmented quiver -- this statement is an important step towards a~direct definition of the knot-quiver correspondence based on the knot diagram. We confirm the conjecture for all twist knots, $(2,2p+1)$ torus knots, and all pretzel knots up to 15 crossings with an~odd number of twists in each twist region. - Open Access
- 17 July 2025
- Published by Atlantis Press SARL
- p. 19-33
Knot-quiver correspondence: a brief review
- 17 July 2025
- Published by Atlantis Press SARL
- p. 19-33
Open AccessKnot-quiver correspondence: a brief review
- Open Access
Knot-quiver correspondence: a brief review
This note is an overview of the knot-quiver correspondence, which relates symmetric quivers and their partition functions, a.k.a. motivic Donaldson-Thomas generating series, to quantum invariants of knots and links in $S^3$.Open AccessKnot-quiver correspondence: a brief review
This note is an overview of the knot-quiver correspondence, which relates symmetric quivers and their partition functions, a.k.a. motivic Donaldson-Thomas generating series, to quantum invariants of knots and links in $S^3$. - Open Access
Categorification of quiver diagonalization and Koszul algebras
In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.
Open AccessCategorification of quiver diagonalization and Koszul algebras
In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.
- Open Access
Unlinking symmetric quivers
We analyse the structure of equivalence classes of symmetric quivers whose generating series are equal. We consider such classes constructed using the basic operation of unlinking, which increases a size of a quiver. The existence and features of such classes do not depend on a particular quiver but follow from the properties of unlinking. We show that such classes include sets of quivers assembled into permutohedra, and all quivers in a given class are determined by one quiver of the largest size, which we call a universal quiver. These findings generalise the previous ones for permutohedra graphs for knots. We illustrate our results with generic examples, as well as specialisations related to the knots-quivers correspondence.
Open AccessUnlinking symmetric quivers
We analyse the structure of equivalence classes of symmetric quivers whose generating series are equal. We consider such classes constructed using the basic operation of unlinking, which increases a size of a quiver. The existence and features of such classes do not depend on a particular quiver but follow from the properties of unlinking. We show that such classes include sets of quivers assembled into permutohedra, and all quivers in a given class are determined by one quiver of the largest size, which we call a universal quiver. These findings generalise the previous ones for permutohedra graphs for knots. We illustrate our results with generic examples, as well as specialisations related to the knots-quivers correspondence.
- Open Access
- 13 November 2023
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 113 (6) , 1-76
Knot homologies and generalized quiver partition functions
We introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal $$L_K$$ , a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in $${\mathbb {C}}^3$$ .- 13 November 2023
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 113 (6) , 1-76
Open AccessKnot homologies and generalized quiver partition functions
We introduce generalized quiver partition functions of a knot K and conjecture a relation to generating functions of symmetrically colored HOMFLY-PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We interpret quiver nodes as certain basic holomorphic disks in the resolved conifold, with boundary on the knot conormal $$L_K$$ , a positive multiple of a unique closed geodesic, and with their (infinitesimal) boundary linking density measured by the adjacency matrix of the generalized quiver. The basic holomorphic disks that are quiver nodes appear in a certain U(1)-symmetric configuration. We propose an extension of the quiver partition function to arbitrary, not U(1)-symmetric, configurations as a function with values in chain complexes. The chain complex differential is trivial at the U(1)-symmetric configuration, under deformations the complex changes, but its homology remains invariant. We also study recursion relations for the partition functions connected to knot homologies. We show that, after a suitable change of variables, any (generalized) quiver partition function satisfies the recursion relation of a single toric brane in $${\mathbb {C}}^3$$ . - Open Access
- 21 June 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 402 (2) , 1551-1584
Quiver Diagonalization and Open BPS States
We show that motivic Donaldson–Thomas invariants of a symmetric quiver Q, captured by the generating function $$P_Q$$ , can be encoded in another quiver $$Q^{(\infty )}$$ of (almost always) infinite size, whose only arrows are loops, and whose generating function $$P_{Q^{(\infty )}}$$ is equal to $$P_Q$$ upon appropriate identification of generating parameters. Consequences of this statement include a generalization of the proof of integrality of Donaldson–Thomas and Labastida–Mariño–Ooguri–Vafa invariants that count open BPS states, as well as expressing motivic Donaldson–Thomas invariants of an arbitrary symmetric quiver in terms of invariants of m-loop quivers. In particular, this means that the already known combinatorial interpretation of invariants of m-loop quivers extends to arbitrary symmetric quivers.- 21 June 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 402 (2) , 1551-1584
Open AccessQuiver Diagonalization and Open BPS States
We show that motivic Donaldson–Thomas invariants of a symmetric quiver Q, captured by the generating function $$P_Q$$ , can be encoded in another quiver $$Q^{(\infty )}$$ of (almost always) infinite size, whose only arrows are loops, and whose generating function $$P_{Q^{(\infty )}}$$ is equal to $$P_Q$$ upon appropriate identification of generating parameters. Consequences of this statement include a generalization of the proof of integrality of Donaldson–Thomas and Labastida–Mariño–Ooguri–Vafa invariants that count open BPS states, as well as expressing motivic Donaldson–Thomas invariants of an arbitrary symmetric quiver in terms of invariants of m-loop quivers. In particular, this means that the already known combinatorial interpretation of invariants of m-loop quivers extends to arbitrary symmetric quivers.