Pub Date : 2024-11-14DOI: 10.1007/s11005-024-01879-9
Mamoru Ueda
We construct a homomorphism from the affine Yangian (Y_{hbar ,varepsilon +hbar }(widehat{mathfrak {sl}}(n))) to the affine Yangian (Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n+1))) which is different from the one in Ueda (A homomorphism from the affine Yangian (Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n))) to the affine Yangian (Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n+1))), 2023. arXiv:2312.09933). By using this homomorphism, we give a homomorphism from (Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n))otimes Y_{hbar ,varepsilon +nhbar }(widehat{mathfrak {sl}}(m))) to (Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(m+n))). As an application, we construct a homomorphism from the affine Yangian (Y_{hbar ,varepsilon +nhbar }(widehat{mathfrak {sl}}(m))) to the centralizer algebra of the pair of affine Lie algebras ((widehat{mathfrak {gl}}(m+n),widehat{mathfrak {sl}}(n))) and the coset vertex algebra of the pair of rectangular W-algebras (mathcal {W}^k(mathfrak {gl}(2m+2n),(2^{m+n}))) and (mathcal {W}^{k+m}(mathfrak {sl}(2n),(2^{n}))).
{"title":"Two homomorphisms from the affine Yangian associated with (widehat{mathfrak {sl}}(n)) to the affine Yangian associated with (widehat{mathfrak {sl}}(n+1))","authors":"Mamoru Ueda","doi":"10.1007/s11005-024-01879-9","DOIUrl":"10.1007/s11005-024-01879-9","url":null,"abstract":"<div><p>We construct a homomorphism from the affine Yangian <span>(Y_{hbar ,varepsilon +hbar }(widehat{mathfrak {sl}}(n)))</span> to the affine Yangian <span>(Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n+1)))</span> which is different from the one in Ueda (A homomorphism from the affine Yangian <span>(Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n)))</span> to the affine Yangian <span>(Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n+1)))</span>, 2023. arXiv:2312.09933). By using this homomorphism, we give a homomorphism from <span>(Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(n))otimes Y_{hbar ,varepsilon +nhbar }(widehat{mathfrak {sl}}(m)))</span> to <span>(Y_{hbar ,varepsilon }(widehat{mathfrak {sl}}(m+n)))</span>. As an application, we construct a homomorphism from the affine Yangian <span>(Y_{hbar ,varepsilon +nhbar }(widehat{mathfrak {sl}}(m)))</span> to the centralizer algebra of the pair of affine Lie algebras <span>((widehat{mathfrak {gl}}(m+n),widehat{mathfrak {sl}}(n)))</span> and the coset vertex algebra of the pair of rectangular <i>W</i>-algebras <span>(mathcal {W}^k(mathfrak {gl}(2m+2n),(2^{m+n})))</span> and <span>(mathcal {W}^{k+m}(mathfrak {sl}(2n),(2^{n})))</span>.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 6","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142636842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1007/s11005-024-01877-x
Bin Chen, Yujin Guo
We consider ground states of the N coupled fermionic nonlinear Schrödinger systems with the Coulomb potential V(x) in the (L^2)-subcritical case. By studying the associated constraint variational problem, we prove the existence of ground states for the system with any parameter (alpha >0), which represents the attractive strength of the non-relativistic quantum particles. The limiting behavior of ground states for the system is also analyzed as (alpha rightarrow infty ), where the mass concentrates at one of the singular points for the Coulomb potential V(x).
{"title":"Ground states of fermionic nonlinear Schrödinger systems with Coulomb potential I: the (L^2)-subcritical case","authors":"Bin Chen, Yujin Guo","doi":"10.1007/s11005-024-01877-x","DOIUrl":"10.1007/s11005-024-01877-x","url":null,"abstract":"<div><p>We consider ground states of the <i>N</i> coupled fermionic nonlinear Schrödinger systems with the Coulomb potential <i>V</i>(<i>x</i>) in the <span>(L^2)</span>-subcritical case. By studying the associated constraint variational problem, we prove the existence of ground states for the system with any parameter <span>(alpha >0)</span>, which represents the attractive strength of the non-relativistic quantum particles. The limiting behavior of ground states for the system is also analyzed as <span>(alpha rightarrow infty )</span>, where the mass concentrates at one of the singular points for the Coulomb potential <i>V</i>(<i>x</i>).\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 6","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142636846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1007/s11005-024-01869-x
Xavier Blot, Alexandr Buryak
The notion of a quantum tau-function for a natural quantization of the KdV hierarchy was introduced in a work of Dubrovin, Guéré, Rossi, and the second author. A certain natural choice of a quantum tau-function was then described by the first author, the coefficients of the logarithm of this series are called the quantum intersection numbers. Because of the Kontsevich–Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. In this paper, we relate the quantum intersection numbers to the stationary relative Gromov–Witten invariants of (({{{mathbb {C}}}{{mathbb {P}}}}^1,0,infty )) with an insertion of a Hodge class. Using the Okounkov–Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the “purely quantum” part of the quantum intersection numbers, found by the first author, which in particular relates these numbers to the one-part double Hurwitz numbers.
{"title":"Quantum intersection numbers and the Gromov–Witten invariants of ({{{mathbb {C}}}{{mathbb {P}}}}^1)","authors":"Xavier Blot, Alexandr Buryak","doi":"10.1007/s11005-024-01869-x","DOIUrl":"10.1007/s11005-024-01869-x","url":null,"abstract":"<div><p>The notion of a quantum tau-function for a natural quantization of the KdV hierarchy was introduced in a work of Dubrovin, Guéré, Rossi, and the second author. A certain natural choice of a quantum tau-function was then described by the first author, the coefficients of the logarithm of this series are called the quantum intersection numbers. Because of the Kontsevich–Witten theorem, a part of the quantum intersection numbers coincides with the classical intersection numbers of psi-classes on the moduli spaces of stable algebraic curves. In this paper, we relate the quantum intersection numbers to the stationary relative Gromov–Witten invariants of <span>(({{{mathbb {C}}}{{mathbb {P}}}}^1,0,infty ))</span> with an insertion of a Hodge class. Using the Okounkov–Pandharipande approach to such invariants (with the trivial Hodge class) through the infinite wedge formalism, we then give a short proof of an explicit formula for the “purely quantum” part of the quantum intersection numbers, found by the first author, which in particular relates these numbers to the one-part double Hurwitz numbers.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 6","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142636891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-06DOI: 10.1007/s11005-023-01730-7
Fabio Deelan Cunden, Paolo Facchi, Marilena Ligabò
Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant (hbar rightarrow 0) and large quantum number (Nrightarrow infty ), with (hbar N) kept fixed. In a suitable topology, the limit is the discontinuous symbol (pchi _D(x,p)) where (chi _D) is the characteristic function of the classically permitted region D in phase space. A refined analysis shows that the symbol is asymptotically close to the function (pchi _D^{(N)}(x,p)), where (chi _D^{(N)}) is a smooth version of (chi _D) related to the integrated Airy function. We also discuss the limit from a dynamical point of view.
{"title":"The semiclassical limit of a quantum Zeno dynamics","authors":"Fabio Deelan Cunden, Paolo Facchi, Marilena Ligabò","doi":"10.1007/s11005-023-01730-7","DOIUrl":"10.1007/s11005-023-01730-7","url":null,"abstract":"<div><p>Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant <span>(hbar rightarrow 0)</span> and large quantum number <span>(Nrightarrow infty )</span>, with <span>(hbar N)</span> kept fixed. In a suitable topology, the limit is the discontinuous symbol <span>(pchi _D(x,p))</span> where <span>(chi _D)</span> is the characteristic function of the classically permitted region <i>D</i> in phase space. A refined analysis shows that the symbol is asymptotically close to the function <span>(pchi _D^{(N)}(x,p))</span>, where <span>(chi _D^{(N)})</span> is a smooth version of <span>(chi _D)</span> related to the integrated Airy function. We also discuss the limit from a dynamical point of view.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 6","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71908730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1007/s11005-023-01734-3
Davide Fermi, Daniele Ferretti, Alessandro Teta
We consider a system of three identical bosons in (mathbb {R}^3) with two-body zero-range interactions and a three-body hard-core repulsion of a given radius ( a > 0). Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues (E_n) accumulating at zero and fulfilling the asymptotic geometrical law (;E_{n+1} / E_n ; rightarrow ; e^{-frac{2pi }{s_0}},; ,text {for} ,; nrightarrow +infty ) holds, where (s_0approx 1.00624).
{"title":"Rigorous derivation of the Efimov effect in a simple model","authors":"Davide Fermi, Daniele Ferretti, Alessandro Teta","doi":"10.1007/s11005-023-01734-3","DOIUrl":"10.1007/s11005-023-01734-3","url":null,"abstract":"<div><p>We consider a system of three identical bosons in <span>(mathbb {R}^3)</span> with two-body zero-range interactions and a three-body hard-core repulsion of a given radius <span>( a > 0)</span>. Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of <i>a</i>. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues <span>(E_n)</span> accumulating at zero and fulfilling the asymptotic geometrical law <span>(;E_{n+1} / E_n ; rightarrow ; e^{-frac{2pi }{s_0}},; ,text {for} ,; nrightarrow +infty )</span> holds, where <span>(s_0approx 1.00624)</span>.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 6","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71909181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-30DOI: 10.1007/s11005-023-01725-4
David Corfield, Hisham Sati, Urs Schreiber
Weight systems on chord diagrams play a central role in knot theory and Chern–Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra and ask which weight systems are positive with respect to this structure; hence, we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental ({mathfrak {g}}{mathfrak {l}}(n))-weight systems on horizontal chord diagrams with N strands may be identified with the Cayley distance kernel at inverse temperature (beta = textrm{ln}(n)) on the symmetric group on N elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature (beta ); and we prove that the Cayley distance kernel is positive (semi-)definite at (beta = text {ln}(n)) for all (n = 1,2,3, ldots ). In particular, this proves that all fundamental ({mathfrak {g}}{mathfrak {l}}(n))-weight systems are quantum states, and hence, so are all their convex combinations. We close with briefly recalling how, under our “Hypothesis H”, this result impacts on the identification of bound states of multiple M5-branes.
{"title":"Fundamental weight systems are quantum states","authors":"David Corfield, Hisham Sati, Urs Schreiber","doi":"10.1007/s11005-023-01725-4","DOIUrl":"10.1007/s11005-023-01725-4","url":null,"abstract":"<div><p>Weight systems on chord diagrams play a central role in knot theory and Chern–Simons theory; and more recently in stringy quantum gravity. We highlight that the noncommutative algebra of horizontal chord diagrams is canonically a star-algebra and ask which weight systems are positive with respect to this structure; hence, we ask: Which weight systems are quantum states, if horizontal chord diagrams are quantum observables? We observe that the fundamental <span>({mathfrak {g}}{mathfrak {l}}(n))</span>-weight systems on horizontal chord diagrams with <i>N</i> strands may be identified with the Cayley distance kernel at inverse temperature <span>(beta = textrm{ln}(n))</span> on the symmetric group on <i>N</i> elements. In contrast to related kernels like the Mallows kernel, the positivity of the Cayley distance kernel had remained open. We characterize its phases of indefinite, semi-definite and definite positivity, in dependence of the inverse temperature <span>(beta )</span>; and we prove that the Cayley distance kernel is positive (semi-)definite at <span>(beta = text {ln}(n))</span> for all <span>(n = 1,2,3, ldots )</span>. In particular, this proves that all fundamental <span>({mathfrak {g}}{mathfrak {l}}(n))</span>-weight systems are quantum states, and hence, so are all their convex combinations. We close with briefly recalling how, under our “Hypothesis H”, this result impacts on the identification of bound states of multiple M5-branes.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 6","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71910588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-28DOI: 10.1007/s11005-023-01735-2
Lionel Mason
The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered (Lw_{1+infty }) celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real (Lw_{1+infty }) symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the analogous framework for Yang–Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.
{"title":"Gravity from holomorphic discs and celestial (Lw_{1+infty }) symmetries","authors":"Lionel Mason","doi":"10.1007/s11005-023-01735-2","DOIUrl":"10.1007/s11005-023-01735-2","url":null,"abstract":"<div><p>The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function <i>h</i> on the real twistor space. The recently discovered <span>(Lw_{1+infty })</span> celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, <i>h</i>, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by <i>h</i>. The real <span>(Lw_{1+infty })</span> symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the analogous framework for Yang–Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 6","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71910643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-11DOI: 10.1007/s11005-023-01716-5
Tommaso Maria Botta
There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver Q to the Yangian (Y^{Q}_textrm{MO}) by Maulik–Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver Q (framed CoHA), and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties ({mathcal {M}}_Q(text {v},text {w})) for all dimension vectors (text {v}) and framing vectors (text {w}) has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.
{"title":"Framed cohomological Hall algebras and cohomological stable envelopes","authors":"Tommaso Maria Botta","doi":"10.1007/s11005-023-01716-5","DOIUrl":"10.1007/s11005-023-01716-5","url":null,"abstract":"<div><p>There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver <i>Q</i> to the Yangian <span>(Y^{Q}_textrm{MO})</span> by Maulik–Okounkov, whose construction is based on the notion of stable envelopes of Nakajima varieties. In this article, we introduce the cohomological Hall algebra of the moduli stack of framed representations of a quiver <i>Q</i> (framed CoHA), and we show that the equivariant cohomology of the disjoint union of the Nakajima varieties <span>({mathcal {M}}_Q(text {v},text {w}))</span> for all dimension vectors <span>(text {v})</span> and framing vectors <span>(text {w})</span> has a canonical structure of subalgebra of the framed CoHA. Restricted to this subalgebra, the algebra multiplication is identified with the stable envelope map. As a corollary, we deduce an explicit inductive formula to compute stable envelopes in terms of tautological classes.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 5","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10495305/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10235533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-05DOI: 10.1007/s11005-023-01701-y
Paul Gustafson, Mee Seong Im, Remy Kaldawy, Mikhail Khovanov, Zachary Lihn
This paper explains how any nondeterministic automaton for a regular language L gives rise to a one-dimensional oriented topological quantum field theory (TQFT) with inner endpoints and zero-dimensional defects labeled by letters of the alphabet for L. The TQFT is defined over the Boolean semiring (mathbb {B}). Different automata for a fixed language L produce TQFTs that differ by their values on decorated circles, while the values on decorated intervals are described by the language L. The language L and the TQFT associated with an automaton can be given a path integral interpretation. In this TQFT, the state space of a one-point 0-manifold is a free module over (mathbb {B}) with the basis of states of the automaton. Replacing a free module by a finite projective (mathbb {B})-module P allows to generalize automata and this type of TQFT to a structure where defects act on open subsets of a finite topological space. Intersection of open subsets induces a multiplication on P allowing to extend the TQFT to a TQFT for one-dimensional foams (oriented graphs with defects modulo a suitable equivalence relation). A linear version of these constructions is also explained, with the Boolean semiring replaced by a commutative ring.
{"title":"Automata and one-dimensional TQFTs with defects","authors":"Paul Gustafson, Mee Seong Im, Remy Kaldawy, Mikhail Khovanov, Zachary Lihn","doi":"10.1007/s11005-023-01701-y","DOIUrl":"10.1007/s11005-023-01701-y","url":null,"abstract":"<div><p>This paper explains how any nondeterministic automaton for a regular language <i>L</i> gives rise to a one-dimensional oriented topological quantum field theory (TQFT) with inner endpoints and zero-dimensional defects labeled by letters of the alphabet for <i>L</i>. The TQFT is defined over the Boolean semiring <span>(mathbb {B})</span>. Different automata for a fixed language <i>L</i> produce TQFTs that differ by their values on decorated circles, while the values on decorated intervals are described by the language <i>L</i>. The language <i>L</i> and the TQFT associated with an automaton can be given a path integral interpretation. In this TQFT, the state space of a one-point 0-manifold is a free module over <span>(mathbb {B})</span> with the basis of states of the automaton. Replacing a free module by a finite projective <span>(mathbb {B})</span>-module <i>P</i> allows to generalize automata and this type of TQFT to a structure where defects act on open subsets of a finite topological space. Intersection of open subsets induces a multiplication on <i>P</i> allowing to extend the TQFT to a TQFT for one-dimensional foams (oriented graphs with defects modulo a suitable equivalence relation). A linear version of these constructions is also explained, with the Boolean semiring replaced by a commutative ring.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 5","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47376727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-05DOI: 10.1007/s11005-023-01714-7
Belal Nazzal, Anton Nedelin
We study superconformal indices of 4d compactifications of the 6d minimal ((D_{N+3},D_{N+3})) conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with (A_N, C_N) and (left( A_1right) ^N) global symmetries. We mainly concentrate on (C_2) case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our (C_2) operator and previously derived (A_2) generalization of van Diejen model. Finally, we also consider compactifications with (A_N)-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.
{"title":"(C_2) generalization of the van Diejen model from the minimal ((D_5,D_5)) conformal matter","authors":"Belal Nazzal, Anton Nedelin","doi":"10.1007/s11005-023-01714-7","DOIUrl":"10.1007/s11005-023-01714-7","url":null,"abstract":"<div><p>We study superconformal indices of 4<i>d</i> compactifications of the 6<i>d</i> minimal <span>((D_{N+3},D_{N+3}))</span> conformal matter theories on a punctured Riemann surface. Introduction of supersymmetric surface defect in these theories is done at the level of the index by the action of the finite difference operators on the corresponding indices. There exist at least three different types of such operators according to three types of punctures with <span>(A_N, C_N)</span> and <span>(left( A_1right) ^N)</span> global symmetries. We mainly concentrate on <span>(C_2)</span> case and derive explicit expression for an infinite tower of difference operators generalizing the van Diejen model. We check various properties of these operators originating from the geometry of compactifications. We also provide an expression for the kernel function of both our <span>(C_2)</span> operator and previously derived <span>(A_2)</span> generalization of van Diejen model. Finally, we also consider compactifications with <span>(A_N)</span>-type punctures and derive the full tower of commuting difference operators corresponding to this root system generalizing the result of our previous paper.\u0000</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"113 5","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10480275/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10191033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}