JHEP08(2019)007 ) is a N Springer Γ( / April 1, 2019 July 24, 2019 H : August 1, 2019 : ), labelled by : N Toda equations. ( ) = 1 1 Γ N − ( / N Received ˆ H Y Accepted A Published ) = N ( 1 Y is in correspondence with a one geometry of a degenerate class of ∗ N tt Published for SISSA by https://doi.org/10.1007/JHEP08(2019)007 equations turn out to be the well known ) and plays the role of spectral cover for the space of models. ∗ N ( tt 1 . 3 Y of 1803.00489v2 2. Each point of the space of level poles in the fundamental cell. The modular curve N = 4 Landau-Ginzburg theory, which is defined on an elliptic curve with The Authors. ≥ N c Extended Supersymmetry, Field Theories in Lower Dimensions
N Motivated by Vafa’s model, we study the , N [email protected] geometry of modular curves ∗ vacua and E-mail: SISSA, via Bonomea 265, I-34100 Trieste, Italy Open Access Article funded by SCOAP Keywords: ArXiv ePrint: cover of degree The presence of anvacuum abelian bundle symmetry and the allows toThe diagonalize underlying the structure Berry’s connectionand of number of the theory the emerge modular clearlycritical curves when limits and we of study these the the models. connection modular properties between and geometry classify the FQHE, this class ofmodels theories are has parametrized interesting by mathematicalan the properties. integer family We of find modulardimensional that curves these N Abstract: fractional quantum Hall effect (FQHE)transitively models on the with classical an vacua. abelian Despite group it of is symmetry not acting relevant for the phenomenology of the Riccardo Bergamin tt JHEP08(2019)007 30 ], the physics of the fractional 1 24 23 42 ) ) τ τ ; 21 ( z 2 ( ) 14 ,u 1 9 N,l – 1 – 23 u ( E W z = 2 level ∂ 37 N 9 4 32 11 9 16 32 38 40 5 37 14 geometry in 2 dimensions equations 4 1 ∗ ∗ 41 tt tt geometry ∗ 5.2.3 IR cusps 5.2.1 Boundary5.2.2 conditions UV cusps tt 3.3.1 3.3.2 3.3.3 Peculiarities of the 5.1 Classification of5.2 the cusps Local solutions 4.1 Modular4.2 transformations of Modular4.3 transformations of the physical Modular mass transformations of the ground state metric 3.2 Abelian3.3 universal cover 2.1 Derivation 2.2 Modular2.3 curves The role of number theory 3.1 The model on the target manifold quantum Hall effect (FQHE) isis not the yet nature fully of understood. thepossess charged anyonic One excitations statistics, of of but the the sayingsents if main Hall they a fluid. open are challenging questions These abelian problem quasi-holes or forpoint are non-abelian both of believed particles theorists view, still to repre- and many models experimentalists. have From been the developed theoretical to explain the observed filling fractions. 1 Introduction Despite its discovery dates back more than thirty years ago [ 6 Conclusions A Modular transformations of log 5 Physics of the cusps 4 Modular properties of the models 3 Geometry of the models Contents 1 Introduction 2 Classification of the models JHEP08(2019)007 ), z ( (1.1) W ) Chern- C , is interpreted ) = 1 units of geometry. To λ z ∗ ( e tt ], predict abelian anyonic ]. Each point of the curve 5 ]. 11 9 – 6 )[ , Z ) , ζ ) is the two dimensional coulombic − ζ z − z 2, also known as modular curves for the ) log( ≥ ζ equations can be derived. It turns out that ( ) log( ] a unifying model of FQHE which leads to ∗ e N ζ a set of positions of quasi-holes. The effect of – 2 – ( L tt 10 e ∈ ] and the idea of hierarchy states of Haldane and S = 4 supersymmetric Hamiltonian. More precisely, ζ X 2 ) are real numbers. The variable ζ ( N e ) = z ( ]. For a generic choice of the parameters defining and W 13 Λ. At this level the expression of the superpotential is just [ C geometry of these models is rather complicated, unless one ] ∈ ) of the modular group SL(2 ∗ z [ λ N tt ∂W C ( 1 = R ), labelled by an integer , with Λ a lattice and N S ( ], as well as Jain’s composite fermion theory [ 1 ∪ Γ 4 , / 3 H is a discrete set of = Λ L L ) = In this paper we study a particular class of theories of this type which have an abelian One of the main advantage of this model is that we do not need to compute the wave More recently, C.Vafa proposed in [ N ( 1 limit, since in thisof case eigenstates the of Berry’s such connectionthese symmetry can models and be are the completely parametrizedY diagonalized modulo in isomorphisms a by basis congruence the subgroup family Γ of Riemann surfaces vacua and studying the arranges the set ofsymmetry. quasi-holes In this in way we some canfor special construct phenomenological degenerate configuration purposes, models of to but FQHE at have which least are an not analitically enhancement realistic treatable. of subgroup of symmetry acting transitively on the set of vacua. This is the most convenient the classical vacua arering isolated, are i.e. identified the with theoryconnection is their of massive, set the and vacuum ofhave the bundle values an elements satisfies at answer in a about the theand set the critical chiral compute of statistics points. the equations of Berry’s known the Moreover, connection as quasi-holes the for one the Berry’s needs ground to states. solve However, these finding equations the classical valued. Therefore it requires aone more is precise considering. definition according to the class of models that functions to describe thering ground states, of since the these theory are labelled by operators in the chiral the standard setting ofto the be FQH systemsthe the lattice source is of tomagnetic reproduce electrostatic flux the interaction constant at is macroscopic a taken symbolic, magnetic point field since with the sum is taken over an infinite set of points and the function is multi- where as the electron coordinatepotential and which the describes term the interaction between the electron and an external charge. In of the FQH systemsSimons theory can in be themicroscopic realized 2 description + in in 1 terms a dimensional ofthe string bulk. a prototype of theoretical one-particle This supersymmetric context construction modelis as which motivates given is SL(2 in by relevant for the addition the Landau-Ginzburg FQHE a physics theory with superpotential Halperin [ statistics for the principal seriesbeen of explored FQHE. Also by the several possibility models of for non-abelian other statistics filling has fractionsnew [ predictions for the statistics of the quasi-holes. He shows that the effective theory Among these, the Laughlin’s proposal [ JHEP08(2019)007 5 the , models 4 ). More , 1 symmetry − N models are = 4, where Q = 3 ˆ N 1 ]. These are A − Z N N 21 N ˆ ). This apparent A ]. These appear in N all the 13 N we provide an explicit ) of level 3 N -Toda equations are related ( N Y Λ. By adding also the generators of / Toda equations [ we classify up to isomorphisms all the C , which is the number of vacua in the 1 2 − N N ˆ A ) is an ellitpic function with a z – 3 – ( equations simplify considerably on the spectral . At the level of superpotential, the action of the W ∗ z tt ∂ πi/N ) for the principal congruence subgroup Γ( ) of degree 2 e N N ( = Γ( 1 / Y N H ). ζ N ( symmetry group which is transitive on the vacua. The modular 0 ) = N ) with an abelian subgroup of symmetry acting transitively on the N ( Z ), with equations, neither the superpotential nor the ground state metric, and 1.1 Y N ∗ − ζ tt + ) is a cover of equations. On this space the symmetry group contains also the generators N N ζ ( ∗ ( identifies a theory where are embedded in this class of theories as critical limits, providing the regularity Y tt Q models appear. The beauty of the modular curves is that they possess various N N | 3 ˆ ). An interesting implication is that the solutions of the A Q The paper is organized as follows: in section Another interesting phenomenon appearing in this class of theories is that, despite the The theory of modular curves is also strictly related to number theory. These surfaces For each connected component of the space of models one can define its spectral cover N ( 0 and number theory arisedescription naturally of in these the models. derivation.to The pull-back In target the section manifold modelwrite is the on not the simply connected universalof and loops cover one in around needs the order poles to in define the the fundamental Hilbert cell. space The and symmetry algebra is non-abelian therefore the Berry’s connection,contraddiction are finds invariant a under consistent the explanationof in action the the model. of context Γ( of the abelian universal cover models of the typevacua. ( The underlying structure of the modular curves and the relation between geometry Γ by the action ofall the in Galois the group, same since orbit the of UV Γ cusps describedcovariance of by the the cusps are located at the vertices of platonic solidscan inscribed be in seen as the projective Riemannrationals algebraic sphere. curves defined over theGalois real group cyclotomic is reproduced extension of by a the third congruence subgroup which enters in this theory, i.e. to the congruence subgroups. Anwith outstandig fact is that forconditions a for given the solutionsonly to the equations.geometrical An structures. exception For is instance, the in the case modular of curves fundamental cell of thecurve and torus. can be The all normalized the in models the with formcurves a are of manifolds with cusps,theory. which represent These physically are the in RG correspondence flow fixed with points the of equivalence the classes of rationals with respect periodic physics. What is reallymathematical worth to structure. study in this class of theories is the rich underlying as the complex manifoldthe modular whose curves points identifyprecisley, a model and a vacuum [ generated by a torsionthe point lattice of one the obtains ellipticfrom a such curve properties basis that, for despitenot the they really are abelian inspired describe symmetry by that group thefundamental kind FQHE cell of setting, of have the these physics. to models model. do cancel Indeed It the the is charges flux of clear of the the quasi-holes magnetic inside field the in order to have a doubly of level JHEP08(2019)007 ) z ( W z ∂ satisfying we study 4 C ) cannot have z ( W z ∂ 5. The details of the ≤ N equations in the Toda form. ≤ ∗ , s tt ) s equations. In section − ( . Moreover, in this classification , e ∗ S z H tt S ∪ ∈ ∈ s X , τ + = Λ Z ζ L are instead given in appendix. The modular 1 πτ − 2 . z N – 4 – C ⊕ Λ ∈ Z X ζ ). We connect this phenomenon to the geometry of π −→ N S ) = z : ( Λ = 2 e W z ∂ ) with an abelian subgroup of symmetry acting transitively on the vacua. 1.1 The action of the abelian group is transitive on the vacua and, in the case of a non trivial kernel, can always beis a made copy faithful. of The theaccumulation abelian transitivity points, group. implies it that must In the be particular,this set given finitely property that of generated. are the vacua The zeroes lattices. abelian of subgroups Since of the group acts freely also on the set of poles and the We stress again thatfunction the with expression a above simplewe is pole allow just the at charges formal each ofsince point and the the of represents quasi-holes superpotential a to is be a meromorphic complex. complex function. This is mathematically consistent, where Our first aimtheories is ( to classifySince (up the to punctured isomorphisms) planemodels all is a the not superpotential models afrom on in simply the the the derivative connected target space, class manifold. one of cannot FQHE So, define we for have these to start the classification 2 Classification of the models 2.1 Derivation action and how thesection components we provide of a the classificationsuperpotentials ground of around state the cusps. these metric In points areFinally, particular and transformed. we we distinguish discuss study In between the the the behavour UV boundarythe of last Galois and conditions the group. IR of the critical solutions regions. and how they are related by This can be definedcostant as under holomorphic a function transformation only ofthe on Γ( modular the curves upper in half thecomputation plane simple of and cases the shifts of constant by genus fortransformations a 0, have a also i.e. generic the with effect 2 act of on changing the the states basis by of modifying the symmetry the generators representation and of the symmetry group. We study this the level of quantumWe states show by that considering this can trivialthe be representations modular done of properties consistently the of withunder these loop the the systems. generators. congruence First subgroups and wethe then consider critical of value the of the transformation one superpotential. the of vacua In which particular we use we to focus write on the on the universal cover and the abelian physics of the punctured plane can be recovered at JHEP08(2019)007 2, is a ≥ L N Λ. / = ) must be s C 2 ( e N . The fact that 2 which is periodic N e | must be a subgroup , 1 ) e = 1 and ζ N ( 1 e . N . ), the function ), where ) = z πi λ N 2 ( ,Q . But, according to the chinese e + 2 Λ W such that N ζ z E < n < N . 1 and a character ( 2 ∂ 2 ) = N , L ) Q Z ,N ≥ ( 1 , requires that N e ' ( C N 2 1 2 Λ for 1 Γ N ,N / 6∈ , e Z −→ 1 ) H it is immediate to conclude that also ,N z N ⊕ Λ ( – 5 – 1 N nQ 1 Z such that ) = L/ N W z : Z } N ' ∂ ( e ) Λ 1 ζ Λ Λ but Y ( ∈ e ∈ L/ in the elliptic curve. In particular, the torsion point must Λ, ) in a strict sense. N / NP z ) = , we can equivalently restrict the analysis to primitive char- ( Z C | ζ NQ e is non trivial. are fixed, the choice of the torsion point specifies an embedding W ' Λ ) has degree 0, + e z / z N for two positive integers z ( ∂ Λ C ( 2 W L/ ∈ N 2 W z . z Z ∂ P L ∂ Λ. By definition of homomorphism, the kernel of { ⊕ to be at least a pseudosymmetry for ∈ ⊂ ] is called the N-torsion subgroup of the additive torus group 1 L N λ N ] = [ , Z , i.e. such that N E L [ = 1 we obtain the trivial case in which there are no quasi-holes. N ' E ∈ is the elliptic curve Λ N ζ ∈ Λ L/ Q E The set Once Λ and the level In conclusion, the models are classified by couples ( Given the periodicity of Up to this point, we have a model for each lattice If we want . Up to a redefinition of the initial set of holes, this is represented by the sublattice Λ, This is true forWe a are compact going Riemann to surface, show as that it turns out to be the target manifold. • • 1 2 L is primitive implies the isomorphism equivalence classes of elliptic curves endowed with a N-torsion point and the space be of order 2.2 Modular curves Since we are classifyingare models related up by to an strict isomorphism. equivalence, we It have is to known identify those that which there is a bijection between the set of of the cyclic subgroup If we set where e remainder theorem, this can bebetween true the only two conditions if the onwith two the integers integers are coprime. The consistency of a sublattice Λ acters, i.e. with trivial kernel: for each of which is a symmetry for lattice, as well asour (the model. faithful representation of) the abelian subgroupextended to of a symmetries multiplicative of periodic character: principal divisor of JHEP08(2019)007 ) = ) for (2.1) Q τ ( ( 1 γ P , for some 0 = τ 0 ) as τ + Λ N ( . 1 mz ) N ), where r p −→ τ τ = ∼ mod + Λ 3 / ! ∞ + Λ } ∗ . z ) ! H 1 0 1 π/N ) and is finite. By adding these N ∈ ,
2 ( r t p q , b , N 1 d ) are the only ones which preserve ) ( τ
= , τ Γ . 1 + N Λ , + ) Q ]. It is worth to recall that a matrix / ) ( ) Γ ( ! τ E 1 H 1 / N 11 ) and can be shown to have the structure cτ aτ N ) ( P ( 1 c d a b Q N 1 + Λ ) to the rational projective line Y = ∪ ( ( ) =
, and state the bijection 1 1 X Z 0 τ H , 2 P N τ ∼ = ( ) ≥ ! −→ 1 – 6 – = π/N [ γ = N ) 2 N , X ( , ) H c d a b 1 N = ): τ tc qc N (
Λ : ( ) defined by the congruence condition Z 1 1 A , + −→ + Z E S N Γ −→ = = Γ ( , ) 0 { C ra τ pa H τ N ) ), we have ) is isogenous to ( ( SL(2 = ) = 1. Z 1 N ) = , ∈ ,Q ( Y c a 0 1 . These maps are the only bijections which preserve the group N τ a, c γ 0 ( ! Λ τ 1 ( SL(2 E S r t p q ∈ = Λ
) = τ N Λ ( r t p q 1 ), and that transformations of Γ m Γ ) is called modular curve for Γ Z ) is topologically a complex manifold with cusps and can be compactified. , ) one obtains N N if and only if Γ such that gcd( ( ( N 1 0 1 ) is a subgroup of SL(2 ( ) acts on the upper half plane by the usual fractional linear transformation Z 1 τ Y SL(2 X N Z Y ∈ ( . Given , ∼ ∈ 1 approaches the rational projective line, the cusps cannot be strictly considered as such that τ τ γ a, c C So, we learn that the whole space of models can be written as union of connected The set of cusps is given by SL(2 It will be clear later that the cusps can be interpreted as RG flow fixed points. ∈ 3 ∪ {∞} ∈ components labelled by the integer The space of a compactwhen Riemann surface.members Since of the this class two of dimensional theories, lattice but rather becomes as degenerate critical limits. with points to One first hasQ to extend the action of SL(2 where The space structure of the ellipticenhanced curve. elliptic curve With ( some these definitions itthe is choice immediate of the to torsion show point. that So, the we can define the moduli space for Γ and induces on them points of the elliptic curve the isogeny A complete proof ofγ this result can be found in [ where Γ JHEP08(2019)007 1 − such . It is τ τ Λ , one can P C/ + Λ ,...,N ∈ is coupled by ) is a primitive . = 0 P , with a modular P } πτ/N , τ P,Q 2 , k ( ] = 0 τ ] and can be shown ) − = 1 and simple zeroes at N ) which denote ellip- P , N [ + Λ N e [ ) N = τ . + Λ N z E ) Q ) | P Z z ,P,Q ) which preserves both the 1 for some ( + + Λ π/N mod ( kQ C τ 2 which has the same pairing ] is Z ]) = z P Λ − , ! ( 0 . We can also give an explicit ∈ M F . It follows from the derivation N Q E kQ [ k P = 2 [ P z ) as basis of generators for the ∈ N k F 1 0 0 1 E { τ ( Q γ ,
∈ = ]. These properties imply that, once ) = = + Λ ,...,N k , k e e , τ τ Q τ [ τ = k , µ – 7 – for some integer ) = γ N + Λ + Λ + Λ + Λ µ ] + ): kQ P kQ P,Q Z ], the matrix ( + , ([ π/N −→ between torsion points: N + π/N N πτ/N [ 2 P ] is picked, a torsion point of the vacua e 2 P E N " k W − ) parametrizes an elliptic curve and ) N z ( ): = ∂ N N 0 ( 2 = ( N 0 ! / 1 ζ Γ ( ). The cusps of Γ )] X C Q c d N a b ,...,N to the rational numbers. Such remarkable N − 0 is in Γ ( acts on the model as a parity transformation, ζ { 1 N ) is a normal subgroup of Γ I ζ I = + ) but inverts the sign of the torsion point. This − N – 9 – − γ ( N ):Γ N 1 ) which plays an important role in this classifica- ζ , for an integer ( ( Z ): 1 N , ( Q πd , these cusps represent again conformal fixed points N Z are actually two descriptions of the same model. So, X 0 2 , l N depending on an integer ∼ , we are dealing with a free massive field theory. We will − ) = 2, since SL(2 N d . Therefore, it is manifest that Γ πil N ) is [Γ N 2 + N ∈ N N e N cτ Z ( γ ( . However, a cyclic group has always two generators which are 1 -th root of unity and ). Moreover, Γ π ( Γ in N ) can be seen as a permutation of the co-levels. ) = l N Z / Z d , N Q . We know that a point on H ( ( = 2 ) = l I 0 e − N → Q ( SL(2 ∗ ) in the set of critical theories in section ) 0 γ Γ N ⊂ N ( ) does not leave the model invariant and maps the torsion point into an ( ) 0 0 N Γ N ( ( / 0 0 Γ H Γ has dimension exactly ) on the space of models is more clear when we choose is the inverse of ∈ ⊂ T a N ) ( has dimension between 1 and γ 0 We can be more precise about these statements by keeping into account that Γ N T ( 1 models. A fundamental property of the elliptic functions is that they are uniquely specified the orbits of Γ 3 Geometry of the3.1 models The modelNow on we the want target to manifold translate our abstract classification into an explicit description of these the cover totient function which count theexpect, elements this of is alsoactually the the degree real cyclotomic of extension theclasses number of field the defining Galois this group class described of by theories, the which set is a specific embedding of one the inverse of the other.since It it is does clear that not changemeans the that point the on co-levels except for the trivial case of co-level of the modular curve.co-level It corresponds is to clear that, pick for aa a different transformation fixed point of lattice on Λ Γ the space of models. Withcontains the this matrix definition, where of the Galois group ofby the adjoining cyclotomic a extension primitive [ connection reveals the algebraic natureto of define the a modular character curves. The above formula suggests Γ of Γ that inequivalent one the model corresponds to modify the character by the formula Another congruence subgroup of SL(2 tion is The three congruence subgroups that we have defined satisfy the chain of inclusions Γ( of on the spectral cover.under Instead, if theprovide a vacuum more is precise a description simple of zero the2.3 and physics around the the orbit cusps The of in role the section of cusp number theory at the leading order. If they are not simple zeroes of JHEP08(2019)007 . ) ), + τ τ ; N πlτ ; z 2 (3.2) (3.4) (3.1) criti- π ( N πlτ ( . This 2 ) − ζ N z − , τ N,l ( = → , are actually W 1 z ) z l η . ) = ( τ ∂ ; , − , , z Z ( ) ) P,Q N ) k τ τ πτ 1 ; ; , 2 N η z z N,l ” 1. From the property ( ( ( 2 and ⊕ ) ) − τ W l + Z ; z N,l N,l ( ( π ∂ π ) (3.3) z 1. By definition of elliptic and quasi-periods 1 τ 2 τ W W πil N ; 2 z z = 2 ); − R z e ∂ ∂ k # and choose ( ,...,N τ ( , so that the model is naturally ) ∈ l k l N N τ + ) N,l ) = ) = ) = = 0 ( with simple poles and simple zeroes − − lτ 1; Λ τ τ τ ( ; ; ; 1 2 1 2 α, β ,...,N S , k W − π π " z N 2 πτ ∂ πk π/N = 0 N + 2 − − 2 + 2 z z k log Θ ( z + 2 ) ( , ) = and co-level , . . . ., N z τ ) ζ τ ( N πilk – 10 – N,l ; ) is defined with the conventions symmetry. ) 2 ( N z N,l τ e ( = 0 ; . These transformations follows from the quasi- + Λ − N N,l 1 W N πilk z ( ( z A 2 W Z − ( ) =0 l ∂ z , k e πk ζ k X N N W − 2 ∂ τ 1 z = − . With this identification we can work with just ∂ =0 τ k X N N,N N ( Λ ; and ) = “ σ ) + Λ / τ ; k W τ S ) = ; z ; z τ ∂ + π/N π πτ ( ; = ) z + Λ lτ ( K ( ) + 2 + 2 + 2 N,l ) ( k π z z z N 2 N,l + ( W N ), we find that the superpotentials of co-levels lτ ( τ \ W π ]. This function is meromorphic on −→ −→ −→ ) is the theta function of character ( ; z 2 τ C z ∂ z z z 15 ( ; : : : ζ z = , defined by the actions )[ ( ) as torsion points. For convenience we invert the lattices of poles and vacua σ − A B τ τ ; β α ∈ S ) = πτ + Λ z ( τ σ, A, B ζ ; π We can also write for this class of theories a symbolic ‘primitive’ The algebra of the abelian symmetry group of this model is generated by three oper- The Weierstrass zeta function N 2 z = , − τ ( 2 where Θ function is multi-valued andIndeed, cannot the be target really manifold consideredon is this as not space. superpotential simply of connected the and model. we cannot find a true primitive and therefore, as pointedinverse out torsion in points the generating previous the section, they describe the same model with cal points, denoted by theζ equivalence classes related by a parity transformation with the additionalperiodicity relation properties of the Weierstrass function.allows to The double identity periodicity points of projected wich on differ the by torus an element of Λ ators respectively in function, the sum of thepoint residue of inside view the this fundamentalquasi-holes cell means charges. is that vanishing. Therefore, theclass From even a flux of though physical theories of with is the not beatiful magnetic interesting mathematical for field properties, the is FQHE this cancelled phenomenology. by that of the where η us focus on theΛ modular curve ofwith respect level to the previous derivation.The This derivative of can be the done superpotential with for the translation this class of theories is (up to multiplicative constants) by the positions and orders of their zeroes and poles. Let JHEP08(2019)007 ) ∗ to p Z ; z (3.6) (3.7) (3.5) S ( 1 H encircling 2 dz, dz, dz. ) − ) ) connecting τ x τ N τ ; . ; ∗ ; connecting ) z z z γ ( ( σ ( Z ) ) ) , γ 1 S N,l , . . . , ` N,l N,l ( ). The action on the ( ( − 0 ( 1 ` N 1 W W ` H W z z x z , ∂ ∂ ∂ a b ∼ k γ γ ` / Z Z (figure I dz b ) τ ) + ) + ) + , γ ; ∈ S} τ τ a τ z ; ; ; ( γ ∗ ) p p p p ( ( ( ). We choose a path ) ) ) N,l . We choose as local basis of ( N,l 3.5 N,l . We consider the smallest one, i.e. the N,l ( ( ( W S k (0) = ∈ S z ` W W ...... W ∂ z , p we need to specify a path p = = = Z x a , – 11 – H γ dz dz dz ) ) (1) ) ) = −→ S . Homology generators. on τ τ q τ = 0 in the homology group τ ; ; ; , and the anticlockwise loops ; 1 1] σ z z z z p ( , ( − ( , obtained by applying respectively the lattice vectors ( ) ) ) q b ) · (1) = : [0 N,l N,l N,l , γ in the definition ( ( ( N,l p p ( . a ( p ) on this space and give a formal definition of superpotential: Figure 1 ∗ γ ( { τ H W W W p 2 ; W z z z ` z = = ∂ ∂ ∂ ( ) p p q ...... p · · · H b a k γ γ N,l ` ∼ x ( Z Z Z p W z ) = ) = ) = to the base point 1 ∂ ), this is a well defined function which assigns a single value to each ` τ τ τ ; ; ; p x p 3.4 p πτ 1 poles contained in the cell with sides : ( ( ( ) ) ) ∼ − b . If the point is not orizontally aligned with the poles we can take for instance = 2 N,l N,l N,l ( γ ( ( π N N 2 W W W . Let us fix the point ∗ π, b ∗ ∗ k + π z N In order to define the action of In the covering model we have many more vacua than before, but at the same time 2 B L A ∗ p = 2 + Compatibly with the definition ofthe symmetry, action the of superpotential the shifts by homology a generators. constant under z to the one given bya the curves the first superpotential of the corresponding operators is Contrary to ( equivalence class of paths in a larger symmetry groupcover, to i.e. classify the them. homology. An Let evident us subgroup fix is a the point deck group of the We can pull-back pull-back the model onabelian the universal cover. universal This cover can of be defined in an abstract way as the space of curves 3.2 Abelian universalIn cover order to define a superpotential, as well as the Hilbert space of the theory, we need to JHEP08(2019)007 , b in γ ∗ (3.9) (3.8) γ · ) p ( σ . according to ) τ = k ; and generate a L p p , ( . We will discuss · 2 ) k , consistently with k ` σ a − L N,l and up to periodic iden- γ γ ( ) 1 , B p 1 W ( σ − πil N 2 . The vacua on the target ). Choosing the integration ,...,N e ) H 2 p ( such that = 0 σ ) = ) τ . σ ,...,N ∗ ; σ γ ∗ p γ (figure that the algebra is non-abelian on ( γ ) homologous to = 0 ( σ π σ, k 3 ) N 2 Z ; +1 , k N,l k dz ( S ) ( L πk N τ σ 1 dz 2 W ; ) γ = H do not commute with z commute. This is not inconsistent, since the τ away from the set of vacua and pick a curve ( , we define represent a complete basis for the symmetry ; = k ) σ ∗ ) + z H – 12 – 2 ( p k τ shifted by . Definition of ) N,l z ; − ( p p in ( N,l ( ) W p A, B, σ z . We see that the obstruction to abelianity on the ,...,N 1 ∗ ∂ W N,l ( z − k γ =0 ∗ k γ ∂ L · Figure 2 ) . W ) p differ by compositions with the loops 2 · k p ) in the transformation. Indipendently from the choice, by (1) A ( σ − πil L N p τ σ γ =0 ) = 0, we get the transformation of the superpotential σ 2 ( ; N k Z Z γ τ = e ∗ ; Q γ ∗ = = N ( Bσ, σL γ p ) ) = σ ( 0 we obtain an curve in = τ ) L ; N,l σ ( 1 σ, A, B, p ∗ N,l ) denotes the curve γ ( = ( − p ) p W ( N W σ N,l L ( σB . For these points there is an ambiguity in the definition of the cycle τ times W ∗ N σ and clearly the subgroup given by the homology is abelian. ), where A Z ; Now we can classify the vacua of the model defined on The generators S ( and 1 manifold are represented bytification the of points Λ since there is ahomology pole along path the which side bypassesthe of the critical the fundamental points pole cell. is on well We the choose defined, conventionally left. we the fix Now that the homology based on physics must be abelian onThe the abelianity target which manifold we butlevel have not by asked necessarily in projecting on the the thethis covering classification Hilbert point model. can in space be the on next recoverd theσ section. at trivial There the representation are quantum not of problems instead in the commutation between where we have universal cover is representedon by the the plane generators these of act trivially loops. and In the faithful representation group of the coveringthe model. space of It curves. isthe Indeed, clear algebraic the from relations generator figure Inequivalent definitions of different constant composing the operatorial relation a straight line.H Now, given a curve constant so that JHEP08(2019)007 . σ (3.13) (3.10) (3.11) (3.12) . ) based on N Z πilr ), we obtain 2 , 0 τ S 0 ; B ( 0 L 1 p πi e ( 2 H ) z x r and the residue for- N,l n ) ( 2 τ N 1 . . ; − W =0 +1 σ − 0 0 r 1 X k N p p N B L πilk ( = · − on 2 ) + k πi σ 2 1 A 2 γ N,l πil N − ( 2 · x πi e − e r πi W n r 2 ` πil N ,...,N k 2 , . By definition of abelian universal 2 ∗ with the cycles of ) + ) ) + 2 e =0 − Z σ τ =0 τ τ . All the other representatives in the 0 k Y r n r ; ; N ; ) p n 0 ∈ 0 H 0 k · σ ∗ r ), we easily get p p p L ( n b ( L ( ) + ) ( 2 ) in ) γ k 2 3.9 τ − · ; L 0 − N,l N,l N,l 0 N ( z ( ( N m a p ( ≤ γ – 13 – Y W ) W r W . By the relation 0 x 0 . Algebra of curves. ≤ = σ, A, B, B p 0 N,l ( 2 , . . . , n ) = n ) = ) = − 0 ∗ τ τ τ W N ; ; ; B 0 0 0 p m p N p πilk ( ∗ ( ( ,...,n 2 ) Figure 3 ) x ) 0 A e m, n, n N,l N,l N,l = ( ( ( ) = are labelled by the curves σ m,n,n τ W ; W W and k ; ∗ ∗ ∗ k H 2 , p as representative of B A L 1 − 0 N z − ,...,n to 0 σ B ∗ p ,...,N m,n,n ; k p = 0 ( ) k ). Let us take again the curve N,l ( We can also provide the corresponding critical values by computing the integrals 3.7 connecting W . Moreover, we can map fibers over different points to each other with the action of 0 0 the whole set of critical values Moreover, from the first algebraic identity in ( Finally, by acting with the generators mula we have where cover, each fiber is a copy of the homologyin group. ( corresponding fiber can be obtainedz by composing Therefore, the vacua on p JHEP08(2019)007 i X the i ]. In 0 k | ij 13 C respectively. i ¯ t and i , where the fields t ] i W X j [ ., , ∂ c C . k . k Φ ij = k ij + c C i R C = Φ i = k j z W. ) j 2 i geometry from the Kahler potential Φ C z δt θd ( 2 i ∗ 2 geometry in two dimensions [ d Φ tt θd . ∗ i 2 Z 0 d tt | i µ Z = = Φ −→ – 14 – depend holomorphically on we get . i c 2) supersymmetric theory in two dimensions can describing the F-term variations . i θ , k ij | , i 2 i / C , k 1 z W | + c k 2 − k ij Φ = (2 µ . generate a flow in the moduli space which has UV limit θd k and C ij = 4 model in one dimension and the other way around. 2 C N k µ d ij → = z δW N C 2 θ i = → ∞ Z j and ground states in the vacuum space. Denoting with j | θd µ i i 2 Φ from which the superpotential depends. The tangent space is i d and Φ i Φ t Z µz ]. Thus, the indipendence of → 24 , z . These operators and the anti-chiral ones form a commutative ring: W 23 i , ]. This is identical to the untwisted one on the flat space and contains as only t ∂ geometry in 2 dimensions 13 26 , ∗ geometry = ]. From 2) supersymmetric Landau-Ginzburg theories in two dimensions we can consider 0 and IR limit for i tt ∗ 25 , 13 tt → The geometry of vacua is naturally studied in the topological twisted version of the Another type of deformation that we can consider is the RG flow of the theory in the The F-term deformation space is a complex manifold with holomorphic coordinates µ = (2 vacuum bundle as In this basis of vacua the chiral operators are represented by the ring coefficients theory [ physical BRST observables the operatorsare in the fundamental the degrees chiral of ring freedom.between chiral In operators particular, Φ there isstate a corresponding one to to one the correspondence identity, we can define a frame of holomorphic sections on the Variations in the overall factor for where the structure constants space of couplings.theorems, Even though the the F-term superpotentialnates picks [ is up protected a by factor non-renormalization due to the rescaling of the superspace coordi- where Φ mechanics [ implies also indipendence from the dimensions. the complex couplings instead parametrized by chiral fields Φ known that the firstis ones do that not the affect computationsbe the for done vacuum a geometry. in An theThe important corresponding argument consequence is that weall can the choose a non-zero Lorentz-breaking modes Kahler arising potential which in suppresses the dimensional reduction to supersymmetric quantum 3.3.1 In this section webegin are by going recalling toN study some the known geometry results oftwo about vacua types of of this class deformations, of involving models. respectively We D-term and F-term variations. It is 3.3 JHEP08(2019)007 M, η (3.15) (3.16) (3.14) , ∗ ) k ij ¯ , = 0, and is compatible A . ] n j g i . = ( i Φ . i D dX j ¯ W A ¯ k i . n [Φ i ∧ = i − = 0 | i W j g k j i h ... ∂ ) i ¯ D ∧ = = A W . . . ∂ 1 ( i j ¯ I. ,A 2 i l ¯ = Res D k , i = dX i g , i.e. j l ¯ ) W ∂ j g. ∗ | g | 1 i ij ) 1 X h j ∂ i g − , A 1 η lk ¯ Φ( = M − ) = η 1 Γ = ij = – 15 – k ij Z − η )( i g g i n M i 1 | = ) , g ∂ , and represents an hermitian metric on the Hilbert − . i i ( 1 l ¯ πi η j ¯ t j A | ( ], or by using dimensional reduction to supersymmetric top i g (2 i h − j − ,A 27 i i = Φ and ∂ geometry is the Berry’s connection of the vacuum bundle. : j i = | t ij Φ i ∗ = M k ij η [Φ] = h ∂ tt i ]. One finds the formula | A W = k D h 29 ij , , its computation requires much more efforts. The 3 matrices is metric with respect to Res η = η 28 ¯ k , which implies the ‘reality constraint’ A plays the role of CPT operator of the theory, it must be satisfied the ij I A = M . Holomorphic and anti-holomorphic sections span the same vector space ∗ i ] is the Groothendieck residue symbol defined by i · | [ depends on both W MM g are related by g Now that the vacuum bundle is provided with an holomorphic structure, we can define is a symmetric pairing between chiral oparators and depends only on the holomorphic One can show that with the holomorphic structure ofconnection the of bundle. the Thus, vacuum the bundle Berry’s and connection in is the the holomorphic Chern’s basis we have with covariant derivatives The object studied byThis the describes how the groundthe states usual transform definition under variations of the couplings and has and Moreover, since condition Instead, space. Conversely to niques of the topological theoryquantum [ mechanics [ where Res two important quantities: η parameters. This is a purely topological object which can be easily computed with the tech- chiral fields and are related by a matrix which is also called real structure. Similarly, the anti-topological theory provide a distinct basis corresponding to the anti- JHEP08(2019)007 (3.20) (3.17) (3.18) (3.19) as complex . , 0 r i ,n 0 r p | n δ k 2 σ − r =0 equations become uni- Y n r r N 0 ∗ L #classical vacua , 2 tt j i C n,n − δ =0 δ Y r 0 N is n . ) = R i B j m,m , 1 δ } 1 m 0 − x . − A g { k j k,k † H ( δ ) δ i ) ) = i j Φ i X C } 2 is realized by the identification ( x ) = − Φ( { τ , g ). N ( j ; j } 2 x C W − X { h i ,...,n 3.16 0 N 0 ∂ – 16 – . = ) = ,...,n W 1 0 0 k j , m,n,n [Φ] = ; ) − i ,n i k 0 g j #classical vacua W | p j C ,n 0 j ( C i equations for this class of models. We first introduce the δ g∂ m ; ( Res ∗ 0 i = k −→ | tt ∂ and p i ( 2 defined by j 2 | − #classical vacua denote the classical vacuum configurations of R 2 − i N − N Φ N ,...,n ,..., is the Hessian of ,...,n 0 geometry prescribes a differential equation for the ground state met- 0 ,...,n 0 ∗ with the set of its values at the critical points. In particular, we can W = 1 tt j ]. These are usually called canonical coordinates. As well as being conve- m,n,n ∂ ; m,n,n i R , j ; 14 k m,n,n j ∂ k ; Φ } k )[ Φ i x equations } { ∗ x = det { ] tt ( H 13 W [ Another prerogative of massive theories is that they posses a natural system of coor- where If the theory is massive, i.e. all the classical vacua are isolated, it is possible to identify g = i The isomorphism between 3.3.2 Our next aim is topoint derive basis the Φ they represent an appropriateproperties choice of to the ground investigateversal state both and metric. different the physical In physical systems are these andon distinguished coordinates the mathematical by the solutions. the boundary conditions imposed dinates in the couplingw constant space, i.e. thenient set from of a practical critical point valuesthe of of soliton view, the masses these superpotential parameters of have the a theory clear are physical meaning, expressed since as differences of critical values. Therefore, Moreover, in the massive case the formula of the Groothendieck residue becomes: where the superpotential. In otheralgebra. words, Also for the a matrices massive of the theory F-term chiral deformations diagonalize: an operator in introduce a basis in‘point which basis’. the This chiral can ring be algebra normalized such diagonalizes that completely, usually called ric So, if we want to studyto how solve the this vacua equation transform under with deformations the of constraint the ( theory, we have In this frame the JHEP08(2019)007 . q t , (3.21) ) = 2 0 q − n . A point − N , q , S ) 2 ! s,n i − = 0 ). Since the ho- 0 N , . . . , t p n Z do not commute | 0 − , k . σ S , σ , i r,n ( i i r 2 1 = 2 2 n r 0 r, s, t , . . . , λ − ( − − H 0 L m ) N j ¯ r N 2 N − k, λ − =0 r g m Y r ) n N , i| q 2 2 λ n α, β, λ N − q , . . . , λ − ( =0 t , . . . , λ B , . . . , λ C j N r 0 ¯ 0 N 2 0 0 k, m − P matrix, since ⊗ =0 g N q A + ,...,n ) 0 0 P N 0 r the number of vacua on nβ × α, β, λ ,n i + λ + 0 α, β, λ α, β, λ ) have the Fourier expansion: ; 2 × ; ; 2 sβ N ,n k − − 0 | k k U(1) mα − + | | ( r N N , r m i ; N decomposes in a direct sum of irreducible λ rα k iα iβ iλ − ( ( +1 i p e e e e δ | S e N ) and Z 2 2 Z = = = ∈ Z − Z − ] label representations of =0 ∈ , . . . , λ r – 17 – i i i , Y N r q , . . . , λ π 0 N 2 2 2 equations, but the price to pay is to have more 0 S 2 ) X X − − − ( 0 , ∗ 1 r,s,t ,...,n β N N N m,n,n [0 0 ) Hom tt H k − ∈ α, β, λ − 2 α, β, λ ; j β 2 ( ( L ( k iαk m,n,n N j ¯ − | α ; δ , . . . , λ , . . . , λ , . . . , λ − j 2 , one finds that in the point basis the ground state metric k, i ) N ' 0 0 0 N 0 equations becomes an hopeless problem. Therefore, we have r p e g − e h is α S ∗ 0 N = V S tt − i V ) = ) = 2 α α, β, λ α, β, λ α, β, λ 2 2 , . . . , λ ( − ; ; ; 0 A, B, L − − δ N k k k , . . . , λ | | | N N 0 0 equations in such a way that the solutions are compatible with the = r A B ∗ L , λ 0 + 1 is the rank of α, β, λ tt , . . . , λ . The pull-back of the Landau-Ginzburg model on the universal cover is , β , . . . , t , . . . , λ N 0 r 0 0 0 α L ; j equations, but in the subclass reproducing the abelian physics of the punctured h r, s, t ∗ ( and α, β, λ j ¯ α, β, λ ; tt ( k, k j ¯ B | g The metric cannot be diagonalized also as k, g 0, since these are differential variables which enter in the equations. So, we have to check plane. Moreover, the importanceon the of vacua an is abelian thatproperty, even we symmetry writing can group the completely with diagonalizeto a the truncate transitive ground action state the metric.abelian Without representation this of the symmetry group. We cannot just set the loop angles to be with a necessary operationvacua to and write a the largerto space the of solutions. We are not interested in the most general solution where the matrix coefficients Using the symmetries diagonalizes with respect to the angles: where the angles mology is abelian, the angles are defined simultaneously in this common basis of eigenstates: basis of ‘theta-vacua’ for space. The vacuumrepresentations space of of the the homology theory group: on provided that where the operatorial action of the symmetry algebra is naturally transported on the Hilbert JHEP08(2019)007 as r (3.24) (3.22) (3.23) ). ˜ . B,L are simul- j,k 3.13 A, δ : σ S , and the over- ) ! corresponding V τ r τ ; k N ∂ πilk , 2 ) 2 λ ∂λ , − ! τ e ˜ N Bσ 1) ) ; σ 2 1 N N τ πilk πilr N πilr 2 − ; 2 = − − k 2 0 ,...,n e N e L k 0 σ p 2 2 ( 1 ) − − =0 − − =0 πi e ,...,n r X k Y + 1) N k 0 , 2 N N,l m,n,n − ( N ; 1 π r ( j,k N k N k 2 − n 2 δ p , W , k , L ( 2 − m,n,n − τ − ; i =0 − i 2 2 i − +1 k N 1 N k =0 1 k 1 πilk − ). Thus, if we choose − N k r =0 is consistent with all the equations. X p 2 ( µ∂ N − leaves invariant the superpotential. − − Y 0 N k ( 2 N L N e P g r πilk g g 2 0 r 0 r 1 ) 2 − 3.9 3 † + µ † =0 † µ λ τ − N − τ ,n ,n e k X + − N =0 ,...,n r r k 0 N ) ; Y k B 0 0 N n n 0 τ − N πilk , gC δ δ , gC p , gC ,n ; πi 2 N 0 N k N 0 = ( µ τ 2 2 0 τ 2 Nβ ,...,n e ) 0 0 ,n p L L − − σ ) C C C 0 =0 =0 σ ( Y Y 2 h r r ,n h τ h 1 = ) = N N N N,l 0 m ; ; ( − 0 0 − 0 =0 0 ,n B ˜ k β 0 N,l N k – 18 – p ( W N ) = ) = n,n n,n πilk ) = ( Φ m +1) = τ ; 2 1 ) 1 Q δ δ 1 ) α, 0 k 0 0 W ( 1 − k − − N µ∂ ke g N,l g − g − N,l Φ +1 ( ( k µ τ 1 τ ) B N k m,m m,m − − in the computation of the critical values in ( = = δ δ =0 W L W g∂ g∂ 0 0 g∂ N,l k X = N k N τ ( ( 3 ( ( ˜ B ¯ ¯ ¯ µ τ µ L j,k j,k ˜ − k,k k,k =0 B ) ) ∂ ∂ ∂ 2 Y πi W δ µ∂ δ k × is naturally projected on the theta-sectors of N N 2 2 2 − =0 1 − − µ Y = = = = k N − with N N = C 2 2 N 0 N − − 1 2 2 B 0 N 0 N − − − σB BL N N and πi , . . . , λ , . . . , λ τ ,...,n ,...,n = = 2 0 0 πil N 0 0 0 0 2 C ,...,n ,...,n ˜ ,n ,n 0 0 B e 0 0 σ ˜ ˜ β, λ β, λ ,n ,n 0 0 m m α, α, ; ; m,n,n m,n,n ; 0 ; 0 ( ( k k k k . The corresponding chiral ring coefficients in the point basis of the covering ) ) τ µ µ C C . ( ( ˜ B The action of The couplings we can vary in these models are the lattice parameter point basis point basis τ µ A, C C all scale model read to The ground state metric satisfy the following set of equations where we have replaced where we have usedgenerators the of commutation the relations homology,taneously in we defined can ( with find the a basis two angles in which the eigenvalues of Moreover, one can see that it commutes with From the equality we learn that the combination that the ansatz of a solution with constant vanishing JHEP08(2019)007 ). = 0 r 3.24 (3.30) (3.28) (3.29) (3.25) (3.26) (3.27) ): ∂g ∂λ τ ; equations 0 ∗ . p ( tt A ) = N,l ( = 0 and N 2 W σ − = N , λ w j,k = δ , equations and the F-term j,k N ∗ πilk ... , δ 2 i . tt , e . , = ) i i N i j,k πilk τ 1 in which the ground state metric 0 2 , δ i α, β ; ) − λ e ; 0 ) g α, β α, β k p N πilk τ | α,β † ; w ( 2 ( α, β ; ) j k | ; 0 e ϕ ) p k N N,l | + 1; e ( σ, A, B , gC ( πilkj = ) 2 πlj w k on the point basis | = − W k,j C j,k N,l +2 e τ ( i σ δ ) h N iα α 1 ˜ ( β e − µ∂ W i =0 can be obtained from this one by specifying the N – 19 – k X α, β ) = N α, ) = = = = e ( ; µ 1 i − = = N j,k j,k g α, β ) ) i i ( w ˜ ˜ + β β and α, β j ¯ ; k k, g∂ τ | k α, α, α, β α, β do not contribute to the g ( | ( ( ; ; ¯ point basis w w j j σ r | | ∂ ) as unique complex coordinate to parametrize models and C ∂ τ σ ∂λ ; 0 p ( ) -eigenstates as point basis point basis τ µ σ C C N,l ( . If the equation in the canonical variable is solved with the boundary w W = and ¯ . It is clear that with this ansatz the equations given above can be written w -basis the metric becomes w and we will see that this is the case. σ , the derivatives Nβ 5 r = ˜ β In the Now that we have recovered the abelian representation of the symmetry group, we and we note that the given definitions are consistent with the relation With this normalization we have the periodicity we can define a set of in section can construct a commondiagonalizes basis completely. of Given eigenstates the for action of conditions given by the cusps, theHence, solution automatically it satisfies is all the convenient equations toone in pull-back ( can the use equation onvacua. the By spectral consistency, cover the of cusps thetruncated should equations. model reproduce where We the are correct going regularity to conditions discuss for the the boundary conditions for the where The equations in the parameters variation of with universally as a unique equation with respect to the critical value for each deformations can be truncated at the non trivial part as well as the complex conjugates. If we demand that JHEP08(2019)007 ) 3.28 (3.33) (3.34) (3.35) (3.32) (3.31) , 2 ) = 0 and ( ) . r − ) λ α α, β . α, β ( ( k,m δ +1)+2 ) basis n ,...,N ) with − πl + , σ n,m ) for one chiral superfield. m η +2 point basis = 1 k,m ( 3.21 α , η ) m,n πl ; 0 ) (2 0 3.19 k β (2 β N ik . . . Φ N ik ) − − . . . − 0 0 0 0 0 1 0 0 − . e nβ β ) e 1 β ( . In particular, we will see that the + 1 ( δ − ...... δ ) − 5 =0 0 ) mα α, β k X 0 . . . ) N ( ( α i τ α 1 k . . . − , ,, i − − χ e − . . . 0 1 0 0 0 1 0 0 0 1 0 0 (0; α Z ) ) α ( ∈ = 0 = 0 = 0 N τ ( δ πilkj N,l δ 1 1 2 2 is understood. These equations appear tipically X ( – 20 – − − − m,n = − (0; i = ) = ) N N w e ϕ W ϕ ϕ 1 ) 2 z − 0 iαk N ) i N,l − − − 0 ∂ are real functions of the angles. It is straightforward =0 ( − 0 1 ϕ α, β k X , N , β ϕ e − e ( 0 , β 1 e W 0 N α 2 z − ϕ α ( − − ∂ e i ) = ( ) = ) = basis Toda equations: m ϕ 0 − m 1 − ϕ − Ψ σ w − , − 1 α, β α, β , χ α, β ) C +1 1 ( ( − ) = N ) ( i ϕ j k ˆ ϕ N symmetry group acting transitively on the vacua. We are going A e ,...,N e ϕ Ψ χ α, β − α, β α, β N + ( + 0 ( ( n Z 0 = 0 i ϕ k e ϕ Ψ ϕ χ point basis w , k k,m basis w + ) η ∂ ∂ − equations satisfied by these functions. Using W 1 W ¯ -eigenstates one gets ¯ w σ n,m w − ∗ σ ∂ η ∂ α, β N tt ( Res Res ϕ k w ϕ ∂ ¯ w ∂ We can also provide an expression for the symmetric pairing of the topological theory. where where In the basis of The topological metric canIn be the computed point with basis the the formula expression ( is are respectively to discuss the solutionRG to fixed these points equations in reproduceproviding section the a consistency appropriate check boundary for conditions the for abelian truncation. the TodaThe equations, operators in the chiral ring corresponding to the basis in ( where the dependence on thein angles and the models with a one finds the well known to derive the where the JHEP08(2019)007 ) 2 × α, β ( (3.36) η ), α, β ( g equations. A point , ) . , ∗ i τ 1 , 0 tt ; i η p z 0 | . In this case the Galois ( p − | τ σ ) (1) = 0 is left invariant by j j W τ p z L L ; ∂ + Λ n n π − . The simple poles and simple B B } − , π . , m m Z ∗ ∗ A A ) = πτ ) ) )] )] τ πτ β ; β jλ jλ 2 ]. − and 0 z ), but an explicit computation is hard + + in terms of the matrices − − z ⊕ 13 τ − ι, ( ι, ∗ ι. nβ nβ ( 1 α, 1 α, ζ 1 Z + + α, β − , η − − π − − ( W ∗ − + Λ ( ( z A B Lι, σ g g mα mα ) already discussed, the symmetry group of the η g ∂ ( ( π i i τ = 2 = = = = ; – 21 – − − = 1. models [ + τ e e l ισ ) = [ ) = [ ιL ιA πτ ιB Z Z N ∈ ∈ ;Λ − A τ α, β α, β z X X ( ( ( πτ, πτ σ, A, B, L ∗ ∗ = 2 level m,n,j m,n,j ζ + Λ g η = 2 and = = ) in terms of ; π N z i i β ) = + N τ − ; z α, ( −→ − − α, β, λ α, β, λ ) for W ( πτ, πτ z z g 0; 1; | | : ∂ 3.3 ι \{ C , as well as its algebraic relations, can be extended to the abelian universal . ι = ι ), where the complex conjugates ∈ S 3.16 z The presence of an extra symmetry, as well as the possibility of working with 2 This operator satisfies the following commutation relations with the generators of the In addition to the generators . Indipendently from the definition, the set of curves with basis of theta-vacua for this family of theories is cover. There are inequivalent choicesL which differ by compositions withthe the action loop of generator matrices, guarantees some semplifications in the derivation of the The action of which follows from ( abelian group: zeroes are located respectivelygroup in acts trivially and we have only one co-level. models of level 2 contains also the parity transformation We briefly discuss someof peculiar the aspects superpotential of is the class of models of level 2. The derivative where This condition gives in general. Insimplifies, the becoming case the of same a of trivial the representation3.3.3 of the homology, the Peculiarities of reality the constraint straint ( are respectively We can choose a basis and use one of these expressions to impose the reality con- JHEP08(2019)007 (3.40) (3.37) (3.38) (3.39) , the reality . ι . ) i . β β ! − , − 1 ) , i α, ) we find the Sinh- − α, − 1 0 0 − α, β ( ( ¯ 0 α, β 3.26 ¯ 0 1 1; | 1 1 g 1; g | − . , iα iα ) implied by , in ( . iα . iα − ) − e ! )] )] e e e τ . w ) ! α, β C ) ) = 1 )] ) ) = = ( ). The transpose and complex 2 (0; ) = , α, β α, β i i ) = ) , . α, β B ( ( ( ∗ † T W α, β α, β L α, β α, β L 2 z ( ( )] )] )] B α, β ( ( α, β α, β α, β α, β ( α, β ∂ ( ¯ 1 ¯ 1 β β β ) and L 2 A ( ( iα 0 1 ¯ 1 ) are real functions, while the positivity ¯ 0 B 0 1; 1; g g − − − e | | 1 B ) and sinh [ cosh [ ) α, β α, α, α, | | − ) = ( α, β ) ) ) ) ( ) g − − − 2 τ ) = τ α, β ( ( ( ) α, β ( B β g g g ( α, β α, β α, β A ( (0; and the topological metric in this basis are (0; − ( ( α, β 1 1 B – 22 – ( ¯ 0 ¯ 0 α, β 0 1 w ( ) = 2 sinh [2 W W iα α, 2 A ) = [ ) = [ ) = [ g g 2 z 2 z ) and C − A − ∂ ( ∂ e ( | | act on these states as: 2 , g , ι α, β | , g ) i α, β α, β α, β ( B ι ) ) α, β ( ( ( β ( τ L † ∗ β T ) = , σ ) = ) = g w g − A g i − α, β ) = ∂ (0; ( and ¯ α, w ¯ 1 α, α, β 1 α, β α, β σ ∂ , η ) and W ( α, β − ( ( g α, β − 2 z g ( ( ! A B ∂ 1; 0; g ¯ 0 | | | 1 imply respectively 0 α, β ) = 0. The matrix ( g − ι = = A 1 0 0 > i i α, β implies that ) ) = ( and g ¯ 0 0 α, β α, β ) = σ g α, β α, β ) = ) reduces to β ( 0; 0; ( read | | ¯ 0 − A ι 0 g σ g α, β 3.16 α, ( = 0, the operators w − ( C λ A Finally, by plugging theGordon given equation expressions of As a conquence, the metric can be parametrized in term of a single function of the angles We see that, byconstraint ( the parity properties of and therefore The hermiticity of requires that where conjugate of and Therefore the metric can be written as The symmetries The ground state metric in this basis is represented by the 2 x 2 hermitian matrix Setting JHEP08(2019)007 , ), → N 2.2 (4.2) (4.3) (4.1) πτ N Γ( 2 ∈ γ k k 2 2 cη cη , i.e. ) as + + N N τ N 1 1 ; z ( dη dη ) k N,l + 2 + 2 ( 1 . k N η ) 1 = 0 mod 2 2 W N and the modular properties η τ τ z b 2 cη ∂ . As discussed in section + , N )); ) )); + + N d τ d ; 1 τ + z + ). Now that we have provided an τ ); ). Given a modular transformation ( dη ζ k τ ) mod N ); cτ cτ ) )( ; ( ( k τ d d z 1 + ) acts on k . If we set k ( ; = 0 mod − ) + + + z , acting as the Galois group of the real bl N d c + + ( ( alτ π , l ) N,l al l lτ 0 N ( 2 ( cτ cτ ) = ) ( Γ b N b π a ) and Γ( π N,l N W 2 → N ( ∈ 2 = ( = ( + z + N l ∂ ( − , – 23 – − W , 1 aτ aτ b b z d d bl c d a b bl (( (( τ ∂ π + + τ + + ; π π N π 2 ; = 1 mod N N 2 N 2 2 bl = bl cτ cτ aτ aτ − π − ; − γ π − N 2 z a, d N 2 d z z z 1 ), the space of models and its spectral cover are described − η + − z ζ z ζ ζ ζ z 2.2 b b d cτ d ) ) + + + + N N N N πilk πilk πialk πilk ζ 2 2 2 2 N,l N,al cτ aτ ( cτ aτ e e ( e e ; ; 1 1 1 1 ). Let us first consider W d W − − d − − =0 =0 =0 =0 Z z ) preserves only one of the torsion point. The transformation z k k X X k k X X N N ), the derivative of the superpotential transforms as , N N ∂ + ∂ + z z ) ) ) N ) ) ) N ( d d d d d d ( 1 cτ cτ 1 + + + + + + Γ ) ) cτ cτ cτ ∈ cτ cτ cτ N,l N,l results in a shift of poles and vacua by ( ( = ( = ( = ( = ( = ( = ( ) b c d a b W W ) changes the co-level with the map + z z On the other hand, a matrix ∂ ∂ N N aτ ( ( = 0 π 2 where in the third lineΓ we used the factcyclotomic that extension. also the order of vacua is preserved modulo periodicity. We see that Γ where we have used the factof that the Weierstrass function: superpotential, as well as thesubgroups ground of SL(2 state metric, withγ respect to the relevant congruence 4.1 Modular transformationsThe of underlying structure ofparticular, this as class discussed of in modelsrespectively ( by is the the modular theoryexplicit curves of description for the of Γ these modular systems, curves. we want In to investigate the modular properties of the 4 Modular properties of the models JHEP08(2019)007 ) N ), but , where N + 1 acts dx τ vacua. On ) x ( → N , which is the C τ 3 ). In particular, , : S ) N x T ) = ( ( ). From the theory 0 x ( ) is connected to the dW N, l N dW πi 1 identify two decoupled N . A set of representatives 2 , 1 ), in addition to the effects − ) and Γ P e = 5, for these curves we have N N ( ( 0 1 N ) = x ), we learn that ); 0 N p ( ( ) and Γ 1 1 Γ(2) is isomorphic to ∞} N − / , ( ) σ 1 1 ( , Z , 0 { ) can be fixed up to a multiplicative constant dW x = ( – 24 – ) under Γ ). Let us give some examples. Among the curves ) with respect to some coordinate on the modular τ C + 1, is a UV cusp and 0 τ ) = ; N ; ) is the parameter that we have used to write the x z Γ(2) τ 0 ( ∗ τ ) C p ) changes the coordinate of the model on the spectral ; T ) must be a rational function with coefficients in the ( 0 → ) ; x N N,l p 0 ( ( τ ( p ) N,l C ( ( ) transforms non trivially under Γ( W z N,l τ ). Moreover, it is the projective coordinate which realizes the W ( ∂ ; 5 are the only ones with genus 0 and therefore the easiest to dW 0 ) and the Riemann sphere punctured with the cusps. Denoting N W p ≤ ( N ) ( ) = N x X N,l ( ( (2), we have 3 cusps which can be viewed as the 3 vertices of the X W ) is the chiral ring coefficient which describes variations of the critical dW x ∗ ; T 0 p ( ]. Genus 0 means in particular that the modular function field has a unique gen- W x 22 , being a fixed point of ∂ = 1 is understood. the coordinate on the sphere, we can define the one-form ∞ l x ), the cases with Starting from The fact that ) = equations. This means that Γ( N x ( ( ∗ equatorial triangle of a doublefor triangular the pyramid cusps inscribed is in given by where vacua. Moreover, the quotient group PSL(2 where cyclotomic field. Moreover, the polesical must mass be is located expected inby the to requiring free diverge. IR the cusps, Thus, correct where transformationfrom the properties the phys- under transformations Γ of on the one form as an automorphism of the sphere with the formula with C value with respect to theonly Hauptmodul. the Except co-level the 1 caseof (and of the projective inverse), algebraic so curves, we suspend the notation ( curve must be modular functionsX of Γ( describe [ erator, also called Hauptmodul. Thismodular function with is respect unique to up Γ( isomorphism to between Mobius transformations and geometry of the modularvalue can curves. be These definedplays spaces the only role are on of not deck theis group. simply universal defined This on cover, connected is the i.e. also and universal cover strictlythe the a of related other the upper to critical hand, target the half variations space, fact of where plane, that we the have where more superpotential Γ( than cover, with a consequent transformation ofshift the is ground state induced metric. on the Andiscussed analogous superpotential in constant also the by previous Γ and paragraph. solving the Before apparent giving contraddiction, an we want interpretation to of compute such these phenomena constants. It turns out thatit the shifts superpotential by is a not constant.since invariant On under a one a constant side, transformation it istime, of is irrilevant Γ( the not in wrong ‘physical determining to mass’ tt say the that physics the model of is a left system. invariant, But, at the same 4.2 Modular transformations of the physical mass JHEP08(2019)007 (4.4) . The by the k x ∗ x T = . Asking that 1 mod 3 which ∞ +1 − k acts on Γ(3) is isomorphic 3 x = / ) T x Z ) 3) , τ τ/ , with and (3 ( 3 η η ζ ∞ = 0 and ) as coordinate on the sphere, x 27 τ 1 ( , the one-form with values in the √ λ ) (4.5) 3 . i k n τ (1) = J Γ(3) acts on the Hauptmodul as 2) = 1 + q − x dx / = / 3 , x . k = ∞} − J 3 − ) ) (1 , − 3 2 3 (3) x x τ τ 3 . The set of cusps is 1 ζ ! x dx ζ . 1 (1 4 , J 3 ) = 1 α n 1 x 2 A + (0; (0; 3 2 τ =1 =0 τ / ∞ n/ τ = 4 4 2 3 Y q n ( k X 3 1 q = θ θ 2 ζ (0) = 0 x , ) = α − 24 x 0 x − x / = – 25 – ∗ { , x 1 ( 1 τ 1 on the sphere. It is straightforward to check that ) = q 3 T ) = = τ J and x ( is the modular lambda function defined by =1 . Denoting ∗ ∞ ). dW ∞ ( Y n λ 3 x ) = T ζ ) = 1 ( Γ(3) = τ πi/k ( ∞} 1 3 dW C 2 ∞ x , η e ( − τ 1 dW x q , 1 = (1) = 0 − 3 ) must have a simple pole in 3 { 27 ζ k x 1 J / ζ ) ( √ 1 i C = C P , ) = 1 x ( ∼ ) = = 3 the cusps are located at the 4 vertices of a regular tetrahedron , x −→ τ ), we find is the only UV cusp, while the other rationals are in the same orbit of ( N x dW 3 ( ∗ J and ∞ T Γ(2) is instead sent to / dW (0) = 0 πiτ 3 H 2 − ∞ J e ): = = = τ is a complex constant. We see that there is a unique generator of Γ(2) which acts ) = τ ( 0 τ x q λ x α ( ) satisfies x In the case of ( dW and identify free theories. The integers coprime with 3 are 1 and 2 = ) and produces the constant shift of the superpotential. ∗ where cusp at dW where we use thechiral notation ring is where is the Dedekind eta function. The generator of Γ T describe the same theory. The generator of the function field is inscribed in the Riemann sphere.to It the is symmetry not group a of coincidence this that solid PSL(2 figure, i.e. Also in this case where non trivially on the critical∞ value. This generates loops around the cusp in 0 (as well as in transformation of PSL(2 From what we have said T where The 3 cusps fall in the point symmetry group of a triangle. We can use JHEP08(2019)007 . . , } 4 4 1 2 3 J S , / 2 (4.6) = ∼ ∞ (1) = . The , 4 8 T x acts on = / ,J 5 τ , N Γ(4) ∞ 2 / / ) 1 Z ), with , , 3 x ( / (0) = 1 4 , , their residue must J ∞} 7 = 0 mod dW , γ / 1 b 2 dx, , { 4 , ) ) / ) ) ) 3 1 3 2 n 4 − − = 4. In this curve the cusps , 2 , which are the fixed points τ ζ and n n n/ τ 3 q 5 5 ) with τ τ q / } . ∞ N q q 2 1 , − N − 4 , , ( Γ(5) acts on the Riemann sphere − − (1 + ζ 8 4 0 / (1 ∞} / / / ) (1 2 Γ = 0 , 1 5 3 2 ) 1 Z )(1 )(1 ) , , 4 x , , 2 4 ∈ − 2 n 4 . 3 − − . 4 n/ τ τ / / x / 4 γ 5 n n q q 1 1 2, which are both fixed points of 4 2 5 5 τ τ J J , , / , J q q + 1 − − 5 9 and fall in the points 4 2 − with symmetry group PSL(2 − / / 5 4 ζ / − − ζ 2 ζ 2 (1 (1 1 , which correspond respectively to 1 1 T , , 4 , 1 P (1 (1 3 0 ζ + = and 1 3 = =1 ∞ / { – 26 – / Y 5 n x 1 4 − =1 1 ∞ J ∞ 1. The critical points are given by 4 , J Y , n ∗ / 7 − ∗ 0 1 τ 5 / respectively to T { / q T 1 2 1 8 5 , − 1 τ , while = − ). J 4 √ q / x 3 8 5 x ∞ ( 1 = 8. Imposing also the correct transformation property ζ ζ , 3) = 1. These must be simple poles for Γ(4) c 9 ) the Hauptmodul / C α / dW C ) = ) = 1 2 (1 ( τ 4 as as , τ − 1 5 and 4 ( ) that a transformation ( 0 ζ / ) = P 4 5 2 and the Galois group acts on the set of cusps by exchanging the ,J { T T x J , ) J 4.3 ( are sent by 4 = ζ ) = = 5 and = 1 x ∞ dW ( l b , Γ(5) (1+ 5 / C / dW ∗ T 3) = 1 / (2 8 if we set (5) is the last case of genus 0. This modular curve has 12 cusps which identify the 4 ). It follows that if two IR cusps are in the same orbit of such / , the symmetry group of this platonic solid. Among the 12 cusps x 5 . We know from ( X Another spherical version of a platonic solid appears for ,J ( A 4 T ζ of the cyclotomic units throughdW the Galois groupbe and related permutes by the the residueand of correspondent 5 the Galois poles transformation. in It is the case for instance of 0 which transforms under The UV cusps 2 represent the 5 decoupled vacuaalent in co-levels, two i.e. inequivalent IR limits.two This UV curve limits has and two inequiv- theprojective two coordinate groups on of IR theories. As in the previous cases, we can use as vertices of a regular icosahedron.as The quotient PSL(2 the UV fixed points are 2 which reads and satisfies The fixed points of thisThe map 4 are IR 0 cusps and− are 1 all in the same orbit of remaining ones represent instead free IR critical points. The Hauptmodul of level 4 is which transforms under are the 6 vertices of aAlso regular in octrahedron in this case the co-levels are just In this curve we have two UV cusps, i.e. JHEP08(2019)007 ) ), τ N and ( (4.7) (4.8) (0; 0 2 5 i ζ 1 2 u u + = 0. The − − 5 1 2 1 2 , z ζ . ) h z # ) /q 1 2 n τ − q k N − )( 1 2 (0) = 1 + )(1 − 5 z l J N q dx. ( n τ k 5 q ζ πi , 2 = 0 and determine its critical − − πiz k e z 2 (1 ). Since the constants generated ) 2; ). This implies in particular that e τ x x 8), with =1 1 ∞ ( 3.4 / Y = n − (0; ) z (5 ]. The theta functions Θ z x 5 # dW q J l k N N + k 5 22 − , ζ k − − (1 1; 1 2 1 2 17 = 2 x , / 1 " k 1) , − – 27 – 16 2 2; Θ − x 1 u x " u ( , q + 2 ), we find , a representative of 8) are both normalized to 1. log 4 5 τ 4 is a convenient normalization constant. This function / =0 τ πiτ ζ 1 πiu k X 2 ) which act non trivially on the critical value. Once the ) 2 (5 e u N − 1 2 πilk α e 5 N 2 2 3 J (0) and − 5 = = / e 5 ζ ) k N 1 τ z 1 J ) = q u − − k =0 5 is the second Bernoulli polynomial. Because of their modular )( ( x k X ζ 2 ( 1 2 N 5 1 6 ζ B τ − ) reads q = l N (0) and τ + dW ) = − 5 ( k τ x J (1 + πi 1; (0; 2 / ) x − ) ) = . Then, we recall the definition of Siegel functions: − (0; 4 ) 5 τ 2 e N,l ( ζ ( x 2 /N N,l ( − ,u W Z = 1, one can repeat the procedure and fix all the coefficients up to an overall 1 2 5 l u W ) = ∈ ζ g x 2 ( for 2 ), and their residue are related by Galois transformations. ) and the Dedekind eta function [ , u B x 1 T τ ; ( 8) = ( u 0 2 / A similar procedure could be carried on in principle also for modular curves of higher By summarizing, the generators of loops around the IR cusps on the sphere correspond p ,u ( (5 1 . In particular, all the Hauptmoduls defined previously can be expressed in terms of 5 u where properties, these objects areN a sort of ‘buildingg blocks’ for the modular functions of level with is equivalent to choose,value. for We a first fixed set the notations where the phase remains ill defined as long as we do not specify the determination of the logarithm. This expression of the criticalproperties. value as Let function us on take theby the upper the multi-valued half modular function plane transformations inexpression and are of ( study indipendent its from modular the point, we can set with the residue formula.which Moreover, acts the by free multiplication IR onall cusps the the are coefficients poles all of in haveW the the same same orbit order, of which Γ must be 1genus, from but the it non is trivial more complicated. monodromy of It is instead much more convenient to find a general where the residue of to the subset of generatorsnormalization is of fixed, Γ( the constant generated by modular transformations can be computed constant. Setting J under JHEP08(2019)007 ] (4.9) 16 (4.12) (4.13) (4.14) (4.15) (4.16) (4.10) (4.11) ). The 4.12 . ) ) τ z ( q 2 , , , ,u − without the root of , , 1 1) u ) (1 k . N − E z 2 = 0 6= 0 2) , / / d ) is odd /q is odd = c c 1 ( 1) τ n τ 2 , − ) = . − c d ( 2 q ) 2 2 u 2 0 2 τ u for for u τ ( if if , u ,u ( − ( − 0 1 1 du . Provided the above relations, l 2)( N k u +1 , N b / , H 2 , , ) 1 + 1 E πiu )(1 6 ) l τ ,u N u 2 / z = − 1 ∈ τ ( 1 , 1 e q ( 0 2 u 6 E ) 1 u − τ 1 n log τ bu / ( ,u τ u q +3)) 0 1 ( bu πi c − cd 3)) = u log η 2 + − − 2 2 ) − 2 e 0 2 b E )) u d ( ) as τ ,u d τ 24 ( N + τ πilk 1 ( / )(1 πiδ 2 + a 2 ( u 1 τ ( )+ e n γ τ η ,u e c 2 q ) ( q E 1 bc d 1 2 ) 2 u 2 / )+ 1 − − ) − =0 ,u 2 – 28 – u u 1 ig c 1 k , u X ( (1 N 1 u ), they transform with a phase: u 2 2 − (1 ,E ( u ac ) and ) 2 Z E ( (1 cu τ , τ B ) = =1 τ a, b, c, d ∞ ) = ( iπ bd ( Y ( πibB q n τ + 2 ( 2 τ 2 + e ε ie ) on the upper half plane. Therefore, the critical value ,u iπ ,u 1 × − 1 τ ab 1 e − (0; SL(2 (0; u 2 1 u ) au g ) = # ( ) = log u ∈ E )) = (0; b 1 2 ) = γ 1. This can always be achieved by the property ( 2 N,l τ = ( u u i τ = ( ( + 2 1 2 0 1 < ) = δ γ πiu − − u u ,u τ c d a b u ( ) as W 2 (0; 1 0 2 ( − − 2 1 2 1 2 τ u − 2 1 2 1 2 # ,u " e , u χ 1 2 ,u 1 h = 0 1 1 u u (0; u − Θ u ) γ E , u a, b, c, d − − E 2 ( . Under an integer shift of the characters, these functions satisfy [ N,l 1 2 1 2 2 ε ( ) = / , u τ " 1 ( 1) W ). Here we assume the characters of the Siegel functions to be normalized ) is the Siegel function of characters 2 Θ − Z τ 1 ,u ( , u < u ( k N +1 2 -product representation , 1 q l u N πiu SL(2 E 2 E e ∈ γ evaluate the difference for such that 0 and In order to compute the constants generated by the modular transformations, we need to where Moreover, being the Siegel functionsular up properties. to a For multiplicative constant, they have good mod- where unity valued branch of log Θ can be consistently defined as holomorphicwe function can of rewrite and can be written in terms of Since the Siegel and Dedekind functions have neither zeroes nor poles, there is a single- have the JHEP08(2019)007 (4.18) (4.23) (4.20) (4.21) (4.22) (4.17) (4.19) , , we find the ) , γ ) N ( . πilk ) ) γ 2 i . ) ( c N e 1 N ) ( 0 2 ) . u 0 N,l N 1 u ( ( Γ 1 u N,l = ). Using the fact that ( − , Γ W i ) = . 0 1 N c = 0 the transformations ]. In appendix we follow W γ ( ) 0 2 i ( . 0 c c ) du ) u ) 17 Γ h 0 2 = 0 mod τ N ( 0 2 ) + ∆ represents the fractional part u − N,l b 1 u πix/c ( ∈ τ Γ( i ) + ∆ 2 0 1 + B − (0; e x τ ) ) γ 0 1 W h 0 1 . (0; 0 1 du ) du ) − h u N,l (0; 1 du ( ( ) i− with the residue h u 1 0 2 ( N,al c )(1 W ) + ∆ B ( 2 k N,l cu ( 2 τ c 2 − , B 0 1 i W N 1 2 W 1 b , πibl − du (0; b 2 2 i h πixd/c ) 2 and we obtain 1 2 πi bu c d N − N − ) e πial πil N,l bu N , 0 1 2 2 + e ( πix u Z 2 2 – 29 – − − + ( d,c − e e W e ) = 2 ] ) = 2 2 0 2 ' du γ b (1 h B ( u ) h 2 , u + 0 1 + = ,u N )) = )) = )) = = τ = 1 τ τ τ 0 2 c a u ( ( ( Γ( 0 2 ) has already been done in [ ) denotes d,c , we have to plug in the above expression: χ x, u / γ γ γ ] (0; 1 [ ) , and summing over ) 0 2 N N u Z d,c ( N ] (0; (0; (0; /c , u 6=0 2 k N,l N ( ) ) ) 0 2 Z ( 0 1 x 1 , u X ∈ B i , u x , u N,l N,l N,l W 1 1 = ( ( ( 6= 0, we get the formula 0 1 x, u u [ 1 2 c πi 2 c au W W W 2 h u = 2 is the first Bernoulli polynomial, x, u = 1 mod πi ) follows from this by requiring further / , 0 1 − = 1 l u N N ad 0 1 − u Γ( ): ): ): ) = 2 = x γ N N N ∈ ( and 1 ( ( 2 u 0 1 γ Γ( ), these becomes ) = ,u N Γ Γ 1 x N ∈ u ( ( ∈ ∈ 1 χ 1 γ . This is consistent with the fact that the modular shift of the critical value is γ γ B Γ τ and the symbol [ Setting ∈ x γ = 0 mod The case of modular transformations of the physical mass. In sequence c If This result turns out tofrom be indipendent from theindipendent branch of from the the logarithm vacuum and thatthe in we cases particular choose. of From the the general congruence formula subgroups. we can Let reduce us to consider where of Indeed, these transformations simply translate the vacua: On the other hand, for closely that derivation, adaptingbelong to it the to coset the group general Γ case. For It is clear that in this case we cannot appreciate a modular shift of the critical value. computation for the case of Γ( JHEP08(2019)007 = 0. (4.27) (4.26) (4.24) (4.25) τ , ) ) implies that , ) N ( 1 πix/c 2 i πix/c e 2 e − , al/N ) h d c − d = 1, one finds c − , πi. )(1 − 1 . , l )) l )(1 N πix/c l/N τ # 2 ( , . These formulas are coherent e λ = 2 πixd/c )) )) + 2 k 2 1 πix 2 d − πix τ τ 1 0 2 πixd/c − c 2 ( ( − − N − e 2 1 e e λ λ " − l )(1 N e − − − = log(1 − 2 (1 πix 1 4 ) with 2 πixd/c (1 N e 2 to derive the modular transformations of = N − , 4.24 e ) ) Z = log(1 , mod τ τ Z = 2 that the formulas above give the results /c Γ(2) − ). But, if we assume the perspective of the Z mod /c ), ( X al ∈ Z – 30 – l N W (0; (0; x ) that we found in the previous paragraph. X N (1 − ∈ 4 3 − x τ τ = + 2)) = log(1 is the Hauptmodul of level 2 that we have defined ; N = 2 4.23 x Θ Θ z τ x τ 4 ,T , ( ( − b ) Z b λ 2 # )) 1 2 mod /c τ N Z N,l N l − X ( ∈ πil πial 1 2 0 1 − 2 2 x (0; λ " ) = log = W ) is invariant under transformation of Γ(2), the logarithm − − 3 τ x z e e τ − = Θ ∂ ( log(1 ) and the ( / 1 1 follow from the summation over λ (0; ) T τ c c c x − W πiN πiN πiN 4.18 with another one of the same fiber in the universal cover. Therefore, 2 2 2 log (0; 0 4 p − − − is the generator of anticlockwise loops around the IR cusp in 2 ) = ) = ) = T γ γ γ ( ( ( ) = (Θ ) ) ) ) ) ) τ N N N equations for these class of theories are manifestly covariant under transforma- ( ( ( N,l )). Γ(2) is freely generated by the matrices λ 1 0 ( N,l N,l ∗ Γ( τ ( ( Γ Γ ( tt − W λ W W ∆ ∆ ∆ − The We can check in the simple case of not only models, butchoice also of the vacua. vacuum Thethe modular new transformation coordinate on simply the changes spectral the curve describes initial the same model,tions but of a the different congruence vacuum. subgroups. In particular, the covariance under Γ Indeed, the constant is the same we obtain4.3 with the residue Modular formula. transformationsThe of modular the shift of ground the stateis superpotential metric seems invariant to under contraddict the transformationsuniversal statement that cover, of the there model Γ( is no contraddiction at all. Indeed, the physical mass parametrizes It is clear that Using respectively the ( where 1 previously. Although 1 is not. Welog(1 can use the expression for ∆ where the constraints on with the transformations of obained with the geometrical approach. The physical mass has the expression with JHEP08(2019)007 , ) al N ( 0 ) and is left N k i as a generator ) under Γ 0 i τ p 0 i | ; i p B | z bl ( bl ) − /d, β − σ without changing the /d, β ) k N,l ) preserves the torsion ) kd 0 , ( c ) N c p σ βc c N N βc n W , the fiber index ) changes also the torsion z − ˜ σB B . B . − AB N ∂ d i N i α m ( 0 α = ˜ σ = 0 . ( p ;( A | i ;( c c n ) k N | . The fact that the generator of bl /d, β bl B B ) B nβ − d N d m + − σ σ ) A /d, β ) βc σ N c ) πialkj = 1 mod βc on the point basis is kd 2 = ˜ = − B = d | − βc d are left invariant by the transformation − i 7→ d γ ˜ α α σ A A . e 0 A ( − ∗ ∗ k , which implies the vacuum transformation p ( 1 ;( N 3 m γ γ ) πilkj L α bl − ( 2 =0 i aj and ( π k X nβ | N ;( − N 2 − ⇒ | + ) ) e e bl N ⇒ − 1 – 31 – Z βc βc ) = z d d − mα − ∈ =0 − − ( 0 k i X ) = α α N p X d → − ( ) + + kd m,n | e bl . Therefore, we can consider again z ) βc + ) Z ), we can write − d − ∈ πalj πalj change in relation to the transformation of the torsion . The action of βc πτ σ βc α 2 2 d N cτ d k − − ( ( ( ( X ( = 0 mod α α A 0 L the operator associated to the torsion point of co-level m,n bl bl bl N N N ( π ( b i i i Γ N −→ 2 d N bl k kd N − − − iαk N i i σ ∈ 0 e e e B, L and − p − − γ e e e = = = = −→ = = σ i k σ = = = π B N L i 2 ∗ ∗ γ α, β γ ; the homology cycle satisfying j α, β | symmetry, resulting in a permutation of the metric components. Let us d ; k γ A | N γ Z = -eigenstates transform consequently as σ ) in the new model. The operators ˜ . Differently, equation that we discussed in section A Z al ∗ ; tt S ( ) change the basis of lattice generators and consequently the representation of the 1 We have also to include the shift N H ( 1 We note that, ininvariant. the This case follows of frompoint the up fact to that a a shiftfiber. of transformation The a of lattice Γ( vector, which moves the base point Let us study what theseagain transformations trivial mean representations at of the level of vacuum states. We consider where we denote withand ˜ with loops are not involvedof in the the modular transformations is consistence with the truncation one can read hownew the function generators is of still theof periodic symmetry group of are 2 as modified. well, We since note the that correspondingco-level the homology cycles have thepoint same of definition the in vacua. the For a model of modular shift, the groundΓ state metric is not lefthomology invariant. group. Indeed, matrices Besides ofpoint this Γ( of effect, the a transformationconsider of this Γ more general case. From the transformation of the equation naturally descends on the space of models. However, as a consequence of the JHEP08(2019)007 (5.1) (4.28) . ), since the . First of all, N ( 0 πk N 2 , i ∪ {∞} − . . Q z ) /d, β τ τ , ) ; ; i 1 2 ) in βc, β βc N πk πk N N n . − − nz, 2 2 i /d, β 2 ) α α − − ; ;( βc sin 2 z z w . In the case of Γ /d, β ( sin aj n τ ) − | 1 2 1 2 ) changes the representation of the q aj = 0, leave the metric completely in- n τ βc on the transformed states is given by ) n τ α q βc ϕ c N q n − τ 0 1 1 βc − ;( q − − − 1 Θ Θ πk N α α ˜ aj 1 ) = 2 ( α A, B | =1 i ) ∞ N N ;( πilk X n − σ, =1 → 2 ∞ e βc X n α, β e z aj − ; | α 1 N α + 4 πilj – 32 – ( w − 2 =0 i iβ + 4 ), i.e. with ∗ 1 2 z k X e e e N γ N ( 1 2 = = = j Γ( i i i ϕ cot / = cot ) ) = ) ) N ) = τ τ /d, β /d, β /d, β ( N ; πilk ; ) ) ) 1 2 z z z, τ ( ), i.e. the equivalence classes of Γ( e ( ( α, β βc βc βc ) . Moreover, we rewrite the derivative of the superpotential in 1 0 1 1 ; N − − − − Θ Θ =0 w N,l ( k N X πlτ ( N Γ( α α α 2 j / 1 2 W ϕ ;( ;( ;( ] H z ∗ + ∂ 2, where we have a non trivial co-level structure. We note further that γ aj aj aj z 15 are inequivalent and have a different set of eigenstates, we appreciate . It is convenient to come back to the initial lattices of poles and vacua ) = | | | τ ˜ ˜ σ σ A ∞ ; B → i ) as z , the expression above becomes N > ( τ z ) ; = z and πiτ τ ( N,l 2 ( 1 e σ W = z τ ∂ q Using the relation [ with terms of Θ behaviour around the boundary regionsof of the the modular domain. curve Thesewe are have represented by to the understand cusps whichthe cusp type at of modelwith each the cusp shift corresponds to. Let us begin with 5 Physics of the5.1 cusps Classification ofNow the that cusps we have discussed the modular properties of the solution, we want to describe its homology, resulting in theoperators character ˜ shift also the exchange ofspecifically the for metric componentstransformations along of the the diagonal. cosetvariant. Γ This effect takes place in the following way: As anticipated, we see that a transformation of Γ( From the eigenvalues of the new basis, we learn that the ground state metric transforms The action of the symmetry group generators ˜ JHEP08(2019)007 . c a , γ ∞ and i k (5.2) (5.3) 2 ) = 1. . One → N cη ∞ z . r, j . Let us i + N ) with 1 τ = ; mod dη z τ l inz . ( π, πτ ) − ! e ) = 1 a modular = + 2 are integers such ) N,l n τ . ( n , q n . In particular, the + a, c n τ τ l l W ( ! z q b, d − ) of the zeta function, ) z ); l z ) 1 ∂ d l − l N πilk 4.2 2 − N N + ( − e N i ( 1 i − cτ N − e =0 − ( with gcd( mod =1 ∞ k X e , we have also to take N ) n k l X l c . Clearly we have gcd( a τ ) − ). Let us introduce the integers π − l c Q N d 2 = inz − N n ( τ − N + = q − e πk ( τ N by acting on − 2 q r z z. model of co-level − − c cτ a πk N ( ) + − 1 2 inz satisfies respectively n π ζ ilz . In this definition e cos = − N z 2 and e − c a l n l ilz − τ τ n N ( τ e q l τ q ˆ Q z N n τ A q n = 0 corresponds to take again N to q πilk l N τ πik − ∼ 2 2 q are truncated at the leading order. Moreover, c = 1 ) e 1 2 e – 33 – ∞ j τ τ 1 1 i iN N sin q , ; − − =0 =0 N } z k X n τ k X N ( N q cot ) =1 n τ ∞ mod ∞ d n W X . Manipulating the expression, we get q i − c, N inz l ∞ n N e + { = i πilk 1 → . Using the modular properties ( 2 n τ escape to infinity for large −−−−→− → cτ e ∞ τ τ =1 ) i ∞ n −−−−→− τ 1 min X n q τ − ) =0 n τ = ( ; which sends k X τ → ≤ iN N q +Λ z − ; N ( πilk , the series in τ 1 2 z ) 1 2 − Q b πk d N ( e 2 ) ∞ c d a b N,l are not 0 if and only if i 1 ≤ + =1 + ∞ ( + X n − ) = N,l =0 k ( i k X τ N W → cτ aτ = ; z N πlτ ; ) and clearly the case of W − z 2 τ ∂ c a ( d Z ) γ − = = 2 , . Therefore, the derivative of the superpotential becomes the infinite sum in + z N,l S ), with 1 N ( S cτ SL(2 W = 2 corresponds to the Sinh-Gordon model z c, N ) ∈ ∂ mod N c a l N,l ( γ − where the torsion point of the poles is now W = gcd( = z ∂ Q transformation that can study the behaviourand of then the taking the model limit around we have wich we recognize ascase the of superpotential of a Now let us consider the rationals. We associate to a cusp Integrating this expression, we find Taking the limit since the vacua Thus, we obtain The two sums over n denote with The theta function in these expressions is normalized with quasi-periods JHEP08(2019)007 1 1. − (5.4) (5.5) Q > , . . . , j ∞ i τ → ; = 0 τ dp m ) correspond to π m Q 2 2 j c, N . rη − τ 2 , with , m j πr + jp 2 . So, we learn from this πi , + ∞ ∞ 2 Q i i = gcd( τ ∞ πd m N i ; − → → 2 Q τ τ π ) → = 2 τ τ + τm p p , ( k 2 1 1 τ 2 p ) j πr 1 Q Q d dη dη 2 η j πr , τG Q = 1 mod 2 2 2 2 1 + − = (2) + + + ad z ζ 2 cτ τ. 2 ( − η with divisor τ τ 2 by writing } τ ζ z ; ; 0 ; ∞ c a ∞ k i . p i \{ dp dp ζ → Z π → Q πilp p τ Q π π X ∈ τ 2 −−−−→ 2 – 34 – Q Q −−−−→ 1 2 2 e 2 1 ) c,d Q − πilp 1 Q dη η η τ − − 2 − ( z 2 =0 e z z 2 p X Q ) = 1 + G τ ζ − , =0 ] ( ζ ζ p X ) Q N 2 m πilm τ 30 6= 0 the torsion point approaches 2 G π ( e Q 2 2 Q Q N 1 πilp πialp πilp m πilm ) that the cusp 1 2 2 2 2 G N =1 e e e − dη b e d b j X d 5.2 m 1 1 1 = 1 ) + − − − + + =0 =0 =0 =0 + + 1 − d p p p X X X Q Q Q j X 2 η m ) ) ) cτ aτ j 2 ) + cτ aτ 1. Let us consider for the moment the cusps with divisor d d d rη ; d ; d d cτ + + + − + + z + z cτ cτ cτ + ( cτ + 2 ( cτ cτ = ( = ( = ( ∞ ) is the Eisenstein series i ) ,...,Q ) τ ) is the Riemann zeta function, one has → ( z τ N,l 2 −−−−→ N,l ( ( ( = 0 ζ G becomes very large, for W W p z Moreover, by the formulas [ z τ ∂ ∂ where in the lastexpression line and we the have limit used ( the fact that where Thus, we get the limit and the asymptotic behaviour where As and One obtains By these definitions, we can split the sum over JHEP08(2019)007 c a τ ). ), = 0 N ) as 2.3 ( (5.8) (5.6) (5.7) c 1 N , or by ( X ), ( N X → 2.2 ≤ ) N . Thus, for the ( < Q Q N X , = ! ). The cusps which j z and count the number ) , or equivalently N ) for 1 al ( = al N N 1 − h 5.6 − Q = ( ), one can write the equal- i Q of Γ − Q N . e ( π 1 , ) c a , )” al . are all pushed to infinity when N. + 2 models ( z, − in Γ N. ∞ τ Q z 1 i ( !+ τ 2) ” ) we can write symbolically − sin q h ∼ Q 1 8 → mod a/c τ z/ 1 0 1 ˆ + 5.1 z mod q A τ ) + Λ 2 # d # c a 2; al ialz c a " ∞ e + decoupled theories which appear on " i z/ * ( − j → 1 al τ = cτ – 35 – = τ q −−−−→ ( = # c . a ) # c πk a N γ τ 2 ch satisfying this condition is ) = “ log sin ( ; c γ a Q ch ) are ramification points of the cover c ) z z “ log Θ Γ + h ( c ( 1 N + N 1 a ( ∼ ( c a − c W a a 1 Θ " 1 γ " Γ → X W 1 τ −−−→− : c a vacua split in − ) ) with the mignus sign. In this case the stabilizer of the cusp γ = 0. Thus, using the ( al = 1. In particular, the cusps with τ N vacua up to periodicity ; k 5.8 in z Q ( Q ) = 1, the poles h , the ) if ). The absolute value can be equal or less than N,l 1 0 1 ( Q is called width of the cusp and represents the minimal integer such that the stability group of the cusp Q ) and we have N 5.7 h W c a N ( 1 in Γ( Γ − a/c model of co-level 1 , are the UV limits, since the vacua tend to a unique point on the spectral curve. ∼ ] − N h Q The exceptions to this picture are represented by the so called irregular cusps, i.e. The number We know that the cusps of In the case of 11 ˆ + A -degenerate points. These are described by the those which satisfy thebelongs ( to It is clear that thecusps minimal with integer divisor Q the free theories ( mod satisfy the relation withthe the stability plus condition sign can are be called written as: regular. From the theorems ( which is satisfied with oneis of conjugated the to two signs. This relation means thata/c the generator of Γ of degenerate vacua of the model labelled by the cusp Denoting with Γ ity [ one finds that these cusps are decribed by the multi-valued superpotential This model represents the freeof version poles of and our one class of of vacua. theories, with a 1 dimensional lattice becomes large except for and using the asymptotics which is a theory with an JHEP08(2019)007 is , = 2, ∞ ) ck i c a N + γ al → ( k N/Q N τ ; 2 model with l B = N 3 1 2 τ χ to be coprime, ˆ q Q A ) for c τ N ( πilk log 2 2 e ,u and 1 1 N u πilk a . − 2 =0 ) E k X N e ) 1 mc ck 2 for the curve of level 4. − + =0 N + l / ) + k X ( N al 2 ( , ∞ ) as for the cusps with divisor m N πik i . ∞ 2 i . ) 2 e πim i 5.3 → 2 → )) 1 1 πimx ). Despite we have m τ e 2 −−−−→ − τ u =0 1 and require . ∞ e ( h k N X ) i N π ( ( τ −∞ 6=0 1 ck 2 ( bl ∞ N al = −∞ X → 6=0 ∼ − m + ∞ N B 2 m + 4 = N + X ). m bl τ dk 2 1 m z ( m + πim N al , c 2 a 2 N πilk dk ) )) e ∼ 2 γ 4.19 ( ) = , e ck z 2! πi ∞ τ k N + 1 N i −∞ ( 2 – 36 – 6=0 , (2 ∞ al − ck 2 = =0 X l m B N N + k X m N ,u → − 1 al E E 2 2 = 2 where 1 u τ N ) ) πilk ( 2 E c E a ) = 1 1 N πi πi e log log i γ ∞ 1 i x (2 (2 h log − , =0 N N ( πilk πilk ) (0; k X N 2 2 2 − − ) ord e e N N,l 1 2 B πilk c a ( = = 1 1 2 N,l ( K = 1, and the corresponding theory is actually a τ ) e − − =0 =0 = q 1 k k X X N N ) h W N,l − =0 c a ( k X = = = N N,l c a ( K = log = K c a c a ) for the critical value, we have ) is given by the formula in ( → c a τ γ → ( 4.11 0; τ k N ; 0; ) l N ) χ N,l ( Let us consider the limit of the first piece. The leading order of Now we want to determine the positions of the cusps on the W-plane. Using the If we exclude the trivial case of N,l ( W W we get where Let us develop this expression. Usingpolynomial the Fourier expansion of the second Bernoulli periodic Therefore, we find where expression ( we find that theThis stability is condition known is to satisfied bethe only the width unique of by irregular the cusp4 cusp cusp 1 for vacua. is Γ The superpotential2, is but the we Sinh-Gordon have one to in impose ( the identification JHEP08(2019)007 N ≤ = 0 to a < Q τ ). In general rescaling the , N , which admits µ ( 0 N ) = 1 . As we said, the ), these models are | , µ | N ) = 1 to another cusp ( c, N equations and describe 1 ). Instead, the other IR ] shows that for a given ∗ N a, Q = 0 mod 11 tt ( 2. From this point of view, . 0 ) / . ) if gcd( otherwise c a ) mc γ c a N ( + ( , γ l k ( N φ ; ) and gcd( ) l 6= 0 N N ( 0 ) = 1 the cusp order is not vanishing χ N,l 2 Q ( 1 Γ m N ). W πilk N 2 c, N ) = N e ( 1 0 = ∆ − mod c, N =0 −∞ k ∞ X N P = lr – 37 – UV m R-symmetry, which is broken off-criticality by the − = = . Thus, we have V 0. It is known that at the critical point a generic c a ) = 1 with a matrix of Γ m N ) = 1. A computation in [ c a r (1) 0; 2 → with gcd( 0; ,Q U c, N N 0 ) µ πial )) cusps of Γ a 2 ) N,l a/c is located at the origin and provides the boundary condition − ( can split in different equivalence classes of Γ N,l 2 ( ) this becomes W ∞ and their number is equal to is not vanishing if and only if N πi i W l ) (2 0. N Q, N/Q , = and gcd( 0 − mc τ → . So, we learn that if gcd( + l is coprime with Q ( µ N < Q < N such that gcd( (gcd( c = φ N πik ) 2 ) = e a/c = 0 mod mod N,l 1 c a ( c ,N − 0 1 =0 c K − N k c we have P = Q r It is clear from the last formula that the UV cusps are all in the same orbit of the Galois with gcd( 0 0 c a in the space ofsuperpotential. couplings Since near an theseof overall points phase the regard can superspace, the alwaysUV the overall be parameter fixed solution absorbed point must in is dependsLandau-Ginzburg the theory reached only fermionic gains when the on measure the module 5.2 Local solutions 5.2.1 Boundary conditions In this last section wethe solution provide around the the boundary cusps. conditionsasymptotic behavour for Approaching required a the by critical the point, physics the of solution the has corresponding to cusp. match The the deformations cusps with divisor 1 the Galois group maps a cusp divisor group of the modularlabelled curve. by Counting the the co-levels equivalencechoosing classes the of co-level Γ isfree equivalent IR to cusps pick are whichgeneric all UV rational in cusp the to same put orbit in of the the origin. Galois group, Also since the we can map In particular, the cusp for the critical limit In the case of where and therefore the critical valuefixed is points divergent. described Coherently by with free our theories.connecting analysis, them In these this become are limits the infinitely allhave IR the massive. a vacua decouple finite On and coordinate the the on solitons contrary, the the W-plane: cusps with 1 solution only when The sum JHEP08(2019)007 ]. → 19 z (5.9) : 2 [ σ c/ 2 and ˆ 0 we have ˆ = 1. In the UV c/ ]. Denoting with → − t Nz 20 , it is clear that the , e . The symmetry acts N equations in canonical πi Z z 19 ∗ . These are known to be z , 1) 1 tt 1) − + 2 13 are the scaling dimensions l − [ l z CP − , ( (1) spectrum of the UV and Q − N Q ∼ ( , e U z symmetry generated by e j ¯ l ) in the chiral ring from the vacua i z l ]. The . δ . Indeed, in this limit the unique N z − − ) . 2 n 19 symmetry. The Ramond charges of l β Z N 2 ( − N − Q e R − n/ N ]. They are Landau-Ginzburg theories ( j ¯ ˆ c − 2 + e generator = 1 [ i, R i 19 ) q c = 1 lz = V − l − ) − R g lz e ¯ µ t (1) − µ e U ( and the U(1) charges defining the solution can µ – 38 – 0 gt∂ ( 0 N → is the central charge ofthe CFT. It is shown that t 1 2 −−→Q σ → ) = c µ ) with vanishing ) ····· ····· t −−−→ t ; = of the i ( ¯ i ) = z j ¯ i t ( i g A ) 3.31 ( q are symmetrically distributed between z j ¯ Q z i i 2 2 q N,l − ( e Q . Given that e l − -models over abelian orbifolds of 3, where − W 2 1 σ . Since the superpotential must have R-charge 1, these have c/ equations near the critical point is given in terms of the charges N z c/ z ∗ z connection is related to the matrix = , e − = ˆ e tt z = 0. A basis of U(1) eigenstates in the chiral ring can be generated e ∗ c and − R i tt α e l q 1 = ˆ . Since U(1) is broken by the superpotential to α is the generator of the Ramond U(1) max vacua, provided the periodic identification eigenstates. In what follows we study the q R ) = 1. These are integrable models with a Q N V models are well studied in literature [ 1 is the complex dimension of the target manifold. In the limit l, N − and n , one can define N | ˆ πi = 0 and A N µ 2 | + = min respectively charge condition, we find that the set of eigenstates split in two ‘towers’ form are the Todanon equations trivial in homology ( operatordepend is only on generators of the twoconsider symmetry the groups case have of awith common the basis operators of eigenstates. Let us first and gcd( z transitively on the criticallimit points, these which theories tend are to givenasymptotically by free the theories with condition central charge ˆ 5.2.2 UV cusps The with superpotential So, the solution to the of the U(1) IR cusps. the chiral primaries In the sense given bycan the be above seen formula, as thethe Berry’s an ground connection off-criticality associated state definition metric to of the can the RG be U(1) flow diagonalized generator. in Near a the basis UV of fixed vacua point with definite charge: where The matrix of the chiral primaryq operators of thein CFT. the UV These limit aret the real numbers distributed among superpotential. The eigenvalues JHEP08(2019)007 3 ˆ → A z for a : z ) σ k 6= 0. The − . N α ( e α/d = equations are the → when kz ∗ α . − N tt z e 1 Z ) , shows that the set of ) does not change the l π α 1 − 2 , N N l l − 1 − ( − 2 0 l − ( − N . This model belongs to − l , z e N mod 1 2 in the chiral ring. So, one can − − symmetry generated by − e ak 2 N by multiplying the basis above by orbifold. The Z = 1 and write the action of the Galois = z 2 . z 2 α 0 z ) Z − − 2 k π , . . . , l ,...,N / πl e α α e 2 and puts in relation the solutions of the 1 2 is the unique generator of the homology at the level of corresponding charges. In − + 1 = 1 = 1 al − A z ( l CP k 2 − − 1 − e , with e → ,...,N 0 N N l al k µ – 39 – ∼ = 0 are periodic of , k , k , ∼ l ) l k k ·········· l ) = , z al ak = 1. The operatorial equality z π π ) l k α α ( − 2 2 π = 0 a transformation of Γ α is vanishing, the UV cusps turn out to be fixed points 2 = . This theory has a α l Nz W = N + − β l e ( β k πi -model on the 1+ k − l 2 2 k k π ( q σ e 2 N + 2 l l ]. We point out that in this language the Galois group acts 1 z − 19 1 is compensated by the rescaling of the angle 2 of the modular curve of level 4 is described by the superpotential ∼ l / N z z al and the co-level − l π 1 1 ) that for α 2 k e → N , which have respectively charge 0 and 1. Near the critical point these = 4 and a basis of U(1) eigenstates is given by l 4.28 lz of the metric components. This can be seen at the level of charges by the . The relation N − k al models. To see that the set of charges is invariant under this map we have α e implies the equivalence 1 − → Z charges N and k ∈ V ˆ q A I k ). This is consistent with the fact that , but 4 vacua determined by the condition N 2 iπ . In this way the operators have the correct eigenvalues under The irregular cusp 1 We note further that, since We can include the dependence from the angle z π + α 2 with the identification z family and is asymptotically a Toda ones with in this regime. It is cleardependence from on ( fact that the map of Γ( group as charges is left invariant by this map. e U(1) charges as functions of the angle are different to use the chiralgeneric ring condition general, the integer recast all the charges above as on the other one accordingidentity to how we take thetwo operators limit. correspond We to complete a thecorrection unique basis to marginal by degree adding the of the scalingdirectly freedom [ which on gets the a U(1) logarithmic charges with the map Approximately, we can say that the theory splits in two, with a set of operators dominant with U(1) JHEP08(2019)007 ) = 1. φ, ψ ( (5.16) (5.15) (5.10) (5.11) (5.12) (5.13) (5.14) Q µ C ) one gets . z i ( W z φ, ψ | ∂ . π α ∂ 2 ∂ψ − − 2 ∂ , ∂φ i 1 2 , t. 0 ) | π n 0 log t,φ,ψ . B ( . . The chiral ring operator = 2 L ) m 1 2 πiµ, πiµ. e µ i 2 z B m 2 0 π ) − | α is pretty much the same of the previous equation is singular in the UV cusps: = 2 ) − − n i 0 ∗ nψ ) ) + 2 α n − + cot ( tt B p p ( i ( ( 1 µ q m φ, ψ mφ | ( W W B i πiµ 1 2 ) 2 − ) = – 40 – e m < Q < N τ φ, ψ ) = ) = = 2 Z h ; p p ∈ − z 0 ( ( ( n → X n,m t ) = ( n,m −−→− W W W ∗ z 0∗ ) W πi π = ∂ α and has simple poles and simple zeroes respectively in B 2 α 2 B ) i ; t π t, φ, ψ ( nψ ( i 1 + models for 1 + g φ, ψ some vacuum state of the covering model. Setting to 0 the ϕ | 1 equation in the parameter i mφ − 1 2 . Moreover, it is odd with respect to the parity transformation ( 0 ∗ i | Q Z tt − ˆ A ) we can construct the unique theta-vacua of this theory: e ∈ Z ∈ κ 3.21 , π X α . From the residue formula and the parity properties of 4 n,m kπ 3 = i + 2 . Since the target space is not simply connected we need to pull-back the model π z φ, ψ | in figure µ 0 and → − C z : kπ We define the ground state metric We want to derive the acts on the theta-vacuum as differential operator in the angles where we denote with corresponding critical value, the whole set is simply the transformations of the superpotential Proceeding as in ( This function is2 periodic of 2 ι on the abelian universalB,B cover. A natural basis for the homology is given by the cycles The discussion for the paragraph. So, we focusThese on IR the cusps free are massive Landau-Ginzburg theories models corresponding described to by the the derivative case of A solution in termswhich is of a regular simply trascendents connected space. can be5.2.3 given only on IR cusps the upper half plane, We conclude by saying that the solution of the with charges respectively JHEP08(2019)007 . × R (5.17) (5.18) (5.19) (5.21) (5.22) (5.23) (5.24) (5.25) (5.20) 2 T , )) ψ 2 m , . We can normalize | − R µ i | φ 1 ∈ ) implies . = m ) t ( 2 i 5.17 ) = 0 , m 1 m ) exp ( . ( . . | ) ) ) . 2 t, φ, ψ 2 ) ( m ]. The solution can be expanded in L , m φ, ψ . 1 + ( φ, ψ ι, 31 t, φ, ψ ( 2 1 ι, , L m 1 ( 1 ,A L ( m ) L − 0− − 21 | A 2 ) = 1 ∂ − B B ∂ψ πt ) = reality constraint ( , in order to have the abelianity of the solution ) = , m φ = = (4 ) = 1 − – 41 – ψ ∗ 0 ) = 1 0 − t, φ, φ m tt − ( ιB geometry of the modular curves is rich of interesting ∂ ( K ιB ∂φ g ψ, , m ) ∗ 3.3.2 A φ, 2 2 − tt − ( − t, ψ, φ 2 m ( ( ( , m L π L 1 L A ) can be determined by imposing appropriate boundary ) = ) requires 2 m 2 + 4 , we simply find the trivial solution ( ¯ ψ ) reads µ m A , m 5.19 ∂ 1 } − = 0 µ , m ∂ 1 \{ ( φ Z t, φ, ψ m A ∈ ( 2 − g X ( ,m A 1 m ) is a real function of the angles and the RG scale ) = monopoles have been studied in [ ∗ t, φ, ψ tt equation for ( t, φ, ψ L ∗ ( L tt the usual setting degenerates in a doubly periodic physics on the complex plane. In the 6 Conclusions In this paper we have shown howphenomena the and outstanding connectionsThese between Riemann geometry, surfaces number parametrize theory a and family physics. of supersymmetric FQHE models in which According to the discussion inone section should consider trivialangle representations to of vanish, the namely loop generator. If we demand the loop Combining these two conditions one gets the further constraint conditions. One can easily see that the while the parity condition ( Abelian Bessel-MacDonald functions as where the coefficients The We recognize in this expression the equation of a U(1) Bogomolnyi monopole on we have also the state so that the topological metric is 1. Thus, the reality constraint implies Moreover, by the commutation relations where JHEP08(2019)007 Toda 1 − N ˆ A , 2 u + τ 1 equations, which can be u ∗ = tt models play the role of UV critical ) 1 τ ( − 2 equations. This requires to pull-back the N ˆ ,u ∗ , z A 1 tt u πiz 2 E e – 42 – = z , the elliptic functions playing the role of superpotentials N , q πiτ 2 poles in the fundamental cell, with the corresponding residues which e N = τ q we have setted the notations 4 vacua and N Our investigation has also revealed the algebraic properties of the modular curves. As The known results and theorems about the cusps counting and classification have been Studying the modular properties of these models, we have underlined that the non A Modular transformations ofIn log section limits and belong to the same orbit of the GaloisAcknowledgments group. I want to thank my PhD advisorof Sergio this Cecotti work. for his useful guide and constant supervision IR nature of the corresponding RG fixed point. we pointed out, thegroup of most the remarkable real connection cyclotomicequations. extensions with acts number This on theory the follows regularity from is conditions the that of the fact the that Galois the components of the ground statesymmetry metric, group. since they change the representation of the abelian recoverd in a physical languagetheories. when One we of have the classified main the point critical is limits that of the this width family of of a cusp allows to determine the UV or be defined only on thein upper projective coordinates half on plane, the sinceof modular the the curves. F-term platonic This variations solids has areis inscribed been rational more studied in functions in convenient the the to Riemann easiestthe parametrize sphere, cases modular the but function critical for field. value surfaces The in of congruence terms higher subgroups of genus have it the a fundamental not trivial units effect of also on the solution with vanishing looprecasted angles as is Toda consistent equations with in all the canonical the coordinates. trivial modular transformationsgeometry of of the the superpotential modular curves. are a A critical natural value consequence as of coordinate on the the spectral cover can find the necessary topological data tomodel write on the the abelianphysics universal is cover non-abelian. of the Onof target this loops space manifold, around the where the symmetry we poles,between group have which the is seen are enlarged that generators responsible with the of for thethe the the generators classification algebra. non can trivial Hovewer, commutation be the relations recovered abelianity at that we the have quantum required level. in In particular, the ansatz of a have add up to zero byand definition. the The quasi-holes cancellation guarantees of theanalitically the these enhancement total models. of flux symmetry In between thattransitive particular, the makes action the magnetic possible presence on field of to the an face vacua abelian allows symmetry to group diagonalize with the a ground state metric, as well as to subclass of theories of level JHEP08(2019)007 ) τ ( 2 (A.7) (A.1) (A.2) (A.3) (A.5) (A.6) (A.4) ) and ,u 1 u N with the E ) must be C γ ( 2 for Γ( , ,u ) 1 τ u N ( , χ 2 , , ,u , , ) , 1 z . These objects sat- u 2 1) E / /q 6= 0 . = 0 is odd is odd n , τ − 1) c c ) cd q ) is single-valued on the ) we know that there is a c d − d τ + τ 1 1 ) = ( . ( − τ ( u 2 if if τ 2 0 2 for for 2 ( 2 ( A.3 c u 2 ,u ,u 0 1 du 1 )(1 +1 − u 1. − z u πiu 2 , + 1. Using the expansion of the + 2 b q , E ) c a E . ,u , 1 e < ) n τ τ 1 τ ) 1 6 q u τ ( 1 u 0 2 / = ( 0 2 , log bu u 1 6 . From ( − − b ( d ,u , u ) = / 2 0 1 − 0 1 bu = + + +3)) τ u (1 c cd + ( B /N 3)) 0 2 cτ aτ )) 2 2 2 , u E − γ 2 1 − Z 2 =1 b τ u ∞ ,u d ( Y ( n 1 d πiδ πi ∈ + , u γ u ) + ). Therefore, the difference e ) = a ( 1 z 2 ( )+ ) E Z τ 2 c q 2 ) bc ( , d ,u 1 , u – 43 – 2 = 0. These transformations belong to the coset γ )+ 1 ) as , u − u − 1 < u ) = 2 u 2 ( u 2 ,E c Z c u 1 2 γ (1 ) , (1 ( E − u cu τ ac 2 2 ( ( (1 / a, b, c, d ,u ) 2 ( πibB + iπ 1 1 bd + 2 e ε ( ,u u u 1 SL(2 . Moreover, since log 1 ( ie χ iπ 2 u τ ab ∈ e − au ) = log B τ 2 1 ) which is equal to one. This number is 12 E ) = q γ )) = 2 ( u b τ = ( ( τ 2 0 2 ( 0 1 + = c d a b πiu 6= 0 and write ,u and are generated by γ ,u 2 u ) = ) = τ 1 δ ( 0 1 c τ − ( u 2 u ( N 2 e = χ 2 ,u ) we easily obtain Z ,u 1 ) and the whole SL(2 − ,u /E γ 1 u 1 u N ' )) u E ( A.1 E τ a, b, c, d ) -expansion of the Siegel functions, is the principal branch on , and defined the modular units ) = 0 E ( ( q τ Γ ε γ N ( , ( /N 2 ) and generic characters ) 2 Γ( Z ,u Z ,u / N times a rational number. Given that the upper half plane is simply connected, , 1 ) ( ∈ +1 u 1 1 2 N πi E u ( 1 SL(2 E , u ] 1 for Γ ∈ u 16 2 Let us first consider the case with γ N Siegel function in ( From now on we assume negative real axis deleted. Fromchanges now by on a we phase will undergenerality use an the this integer canonical determination. shift normalization of Because 0 the characters, we can assumegroup without Γ loss of power of 12 equal to 2 this number isupper independent half of plane, it issuggested also by indipendent the from the branch of the logarithm. A natural choice, With these definitions, we want to compute the difference for and where and transform under which are the Siegelisfy functions [ up to the root of unity with JHEP08(2019)007 and (A.8) c d . ) )) τ z ; 0 → − /q z n τ τ ( q m − Q 1 m . 0, i.e. y . i m 2 | → c c X ) approach 0, therefore y , τ + 0 2 ) + log(1 ; γ Im 0 z u z , c a z q ) c d ( n τ /q + τ ) q ; . Q τ τ z ( ) = 0 1 →− ( m − γ τ τ ) u Im c d q ( . z ) ) is pure immaginary, we have to − = 0 1 m m τ z /q γ − Q 0 m u q q ( τ ) in the limit ( )) and τ ) + lim 2 q (log(1 2 ) γ τ τ − ,u )) = 0 γ ( ( 1 =1 ; B 1 ∞ z 2 0 τ 1 γ u X =1 + ( u m ∞ q ( z ( ; ,u X n + ( χ , 2 γ 1 γ m − z ; u z c m a q B ( γ ∞ χ ) ) + i 1 2 z Q 1 1 z Q ( m ) = , γ q u 1 πi z m – 44 – 0 ( → Q q Im 2 =1 − ) ∞ , z m X B 2 m τ Im − - c d ( u X y > c ) = 2 c γ d τ 1 2 lim ) = →− ( , τ ) + Im 2 τ 00 lim →− log(1 πi ; τ + log(1 ,u τ with L ( c d − z 1 τ ( γ u ) )). As + 1 1 Q E τ 0 iy, →− ). We can decompose it in two pieces: u ) = 2 ( u τ τ L , we can calculate ( γ γ + ; = 2 ( τ log ; 0 2 ) we have = c γ d z γ ( z ,u z ) = lim ( 1 − πiB 4.8 τ Q u ; Q 0 χ = z ( τ ) = 2 Q τ ( 2 Im ,u 1 c d , by applying the Abel limit formula. Setting u ∞ E in the upper half plane and the characters canonically normalized, the conditions i lim →− τ τ log → Using again the ( ) τ ( Now it is the turn of Let us start with Im and keeping only theevaluate immaginary the expression parts, since Since it is indipendent of γ Then, let us put where Therefore, using series expansions like for the logarithms in the expression, we obtain With of absolute convergence for the standard series of the principal logarithm are satisfied. JHEP08(2019)007 . 00 L . (A.9) . (A.10) ) i 0 2 c u 0 2 , + u 0 1 u − c m d ) , we define | 0 1 − c | Z ( ) 0 2 du πi u m h M | 2 . We introduce + c e 0 1 1 6∈ m u −| ) = , B | λ x c ) . ( c ) 0 1 − dε m . , m ( u − ) m ( m − | ϕ i ) and 2 c and t λ 1 | πi h 1 2 m λ m B ( Z − M M. ( − m ζ 1 ζ/λ . m e - c πi M m m - − 2 m | /M , λ c X λ − 1 ) c + ( c ( 1 0 Z | πiB ) is pure immaginary and an odd − mod mod z m 2 2 ) m ) m 1 ∈ | m ) mod m τ x x = λ πid/c /q − c 2 ( ) = 0 1 | q 2 τ x 0 2 x − τ u ϕ q m − m u ; = m, x 2 − ) |i 0 e ( + 0 1 = 6= = ) = c z 1 | 0 1 a u − ( + ( 0 2 = u d/c − ) m m m 0 m m u c − m z (1 =1 (1 ( r πimt ∞ q = Q 2 r X + m τ dε =1 – 45 – − − | ∞ 1 0 1 m − ) X − e m ( 1 u τ ) = 1 if ) m ) = πi ; m − = c 0 τ 2 0 1 - x 1− if 0 if ( ; z u e ( X 0 c ( m r z f dε , ζ ( m − | πimt ), we note that m 1 2 c Im λ m | m . Taking the limit under the summation sign, we obtain Q | h− 0 e c m X ) = m ( c Q d 2 − L 1 ( N 1 =1 . Now, for each class 1 ∞ m ϕ | → X ) = m m B = c r →− | | λ m, x | m τ c ( ) c =1 | 1 ( ∞ = lim | 1 N 0 1 a X 2 m denote the sum respectively for 2 ϕ = lim ) = u 00 is real and not integer, it holds the Fourier expansion ) ) 0 = 2 00 0 1 L 0 1 t is the first Bernoulli polynomial. Thus m L u ( L u − 1 2 2 ] that the partial sums of these series are uniformly bounded. Therefore, 2 M ϕ . If (1 − − − − 18 and , m /c x | | (1 0 ,M c mod X c 17 | L πi πy ) = = = (1 2 2 m ) = m 00 − x − ) = ( e L − c 1 ( ( = = B ε ϕ 00 r L where Since function of Now we turn to the last piece where where where Taking the limit under the summation sign, one gets we are allowed toUsing take the the notation limit above and under taking the the sign immaginary of part, we summation. have Let us consider first The symbols It is shown in [ where JHEP08(2019)007 (A.11) (A.12) in terms of . One gets ω 0 L . . and ) ) λ x x − ω ω . − ) x x ) x − ) . . ω | )(1 1 c | x ζ/λ )(1 − k x x ζ/λ ( ζ − ) x ) ( − r πix/M ζ . 2 − ω )(1 N 1 ) 1 − − x ζ/λ λω N. x e (1 − of the first term in ( ( ζ 6 ( 1 (1 x 1 f k − k − − =0 ) - k X 1 N − x X − | c mod 6 ( (1 c | ) x ry − 0 , ϕ k + x | 0 2 , + | ω + c − Z ) c | | rω c 1 y , ) x Z ω k − k Z 1 x ω ω λ M − Ncu 1 ω =0 − + /M becomes πix/M 6∈ − r X N Z − 1 2 – 46 – x X x 0 − y − ∈ = 1. Using the definitions of 1 2 e y x λ x 6=0 mod L 0 1 )(1 x | ) 1 ζ x c x = x y )(1 | λ − − )(1 x =0 ζ λ ) x λ − x 1 k X x r 1 2 ζ N ζ , ζ/λ | ζ 1 Ndu − y c − | − = λω ζ − 1 = y (1 + ( 0 1 iπ M λ (1 M − r = 1 if and only if L x − (1 − | − x λ 1 c | - x x ) = = < y < X c - - r ) = X X 0 S c c x ( λω f πi πi πi M M M the partial sum in the variable 2 2 − − − S in the last two terms, we find = = = 0 x 0 . Using this expression L | L c − | to πi/N x , we see that ( 2 can be rewritten as e 0 L = ] is shown that , d, c 0 2 ω 17 , u 0 1 The sum on the rightu is 0 unless ( Let us denote with This expression canvariable be further simplified. We decompose the sum by introducing the Changing Let In [ Then, JHEP08(2019)007 (A.14) (A.15) (A.13) ) sN ) . i + c i 0 2 0 2 u d,c cu ] − 0 2 − 0 1 u . 0 1 1, we have ) du − h , du N 0 1 h ( | − 0 2 ( u c u πix/c 1 y 2 Nc . + − B e 0 1 ) , one obtains further ≤ | πi 1 d,c 0 1 2 ] du − s e u 0 2 d,c ] x, ) ( i− [ πiys/c 0 2 , 1 0 2 ≤ , u 2 Z c )(1 ry 0 1 cu B sN , u /c 6=0 − c Z 0 1 2 0 1 x X rω ∈ + sNe x x, u du − [ i . 1 N h πixd/c , πiy/c 0 2 x, u ) Z 2 with 0 + 2 c d =0 |− 1 − e /c c s ) X cu | 6=0 Z e . 0 1 = [ d,c πix x X mod 0 2 i , ∈ ] − sN 2 u – 47 – 0 2 1 x πiy/c − u 0 2 − 0 2 ( d,c e 2 − 1 ] d,c + 0 1 2 cu e + ] i 0 2 N 0 1 , u c πi 0 2 − (1 0 2 B i ≤ 0 1 Ncu 0 1 2 du X − du u 0 2 r , u cu h − ≤ 0 1 + = du − i− − d − 0 h cu 0 2 N 0 1 , 0 1 x, u c c can be obtained from the first one with the substitution a y )(1 c ( [ 0 1 = , cu d,c 1 ) 0 − du ] x, u Z u πi 0 ), which permits any use, distribution and reproduction in Nu h 1 − [ 0 2 , 2 0 1 L =0 |− 0 1 c /c y = u L Z 6=0 c s e X Z − r | ( du x X , u ∈ /c πi 6=0 2 du πiyd/c h 1 x Z 0 1 y y y 2 h 2 x X ∈ B e ζ ζ ζ − y y y x x, N y e λ λ λ πiy − − − ζ x, u c − [ 2 πi y 2 1 c πi = − , we get 1 1 1 e λ 2 − 0 2 | | | − r πi CC-BY 4.0 c c c 1 1 1 u 1 (1 | | | − = − − N N − 0 This article is distributed under the terms of the Creative Commons L → − ) = 2 = = = = = γ 0 2 ( S 2 , u ,u 0 1 1 u u χ − 1 From the property [ → 0 1 Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. Putting all the pieces together, we finally have Noting that the second sumu in In order to write the final result in a more compact way, we introduce the symbol Letting consequently JHEP08(2019)007 B 66 ]. Nucl. (2014) , ]. 05 ]. Phys. Rev. (1991) 359 Nucl. Phys. SPIRE , , IN [ SPIRE SPIRE Commun. Math. JHEP Bull. London ]. IN , Harvard , (1999) 8084 , IN , Phys. Rev. Lett. [ B 367 , , Bachelor thesis, SPIRE B 59 ]. IN , Cambridge University [ functions ]. , Springer, Germany (2005). dimensions η SPIRE 4 Nucl. Phys. (1984) 2390] [ (2008) 155109 IN , [ SPIRE arXiv:1511.03372 52 Phys. Rev. and IN , A new supersymmetric index , ]. [ 3 (1983) 1395 B 78 ]. 50 supersymmetric theories supersymmetric theories , Springer, Germany (1986). SPIRE Two-dimensional magnetotransport in the IN (1982) 1559 SPIRE = 2 = 2 ][ (1983) 605 IN 48 N N , Springer, Germany (1981). Erratum ibid. Phys. Rev. geometry in ][ – 48 – , Springer, Germany (1977). [ 51 , ∗ tt A course of modern analysis ]. Phys. Rev. Lett. A first course in modular forms , ]. ]. Nonabelions in the fractional quantum Hall effect (1984) 1583 SPIRE Modular units hep-th/9204102 Topological properties of Abelian and non-Abelian quantum Hall IN ]. Beyond paired quantum Hall states: parafermions and [ Ising model and Topological antitopological fusion Ising model and Phys. Rev. Lett. 52 SPIRE hep-th/9209085 ][ , SPIRE IN Phys. Rev. Lett. [ IN , ]. ][ [ SPIRE Fractional quantization of the Hall effect: a hierarchy of incompressible IN Anomalous quantum Hall effect: an incompressible quantum fluid with [ The arithmetic of elliptic curves (2004) 671. Statistics of quasiparticles and the hierarchy of fractional quantized Hall SPIRE (1992) 405 IN 36 NonAbelian statistics in the fractional quantum Hall states Riemann surfaces and modular function field extensions (1993) 139 Composite fermion approach for the fractional quantum Hall effect [ Transformation formulas for generalized Dedekind Introduction to modular forms Fractional quantum Hall effect and M-theory ]. (1989) 199 Phys. Rev. Lett. 157 B 386 , 63 arXiv:1312.1008 (1991) 362 [ SPIRE IN cond-mat/9809384 Phys. 055 Harward University Cambridge, Massachusetts, U.S.A. (2002). Phys. HUTP-92/A044 (1992) [SISSA-167/92/EP]. Press, Cambridge U.K. (1902). Math. Soc. [ incompressible states in the[ first excited Landau level states classified using patterns of zeros (1991) 802 360 quantum fluid states states Lett. extreme quantum limit fractionallycharged excitations S. Cecotti, D. Gaiotto and C. Vafa, B.R. Cais, D.S. Kubert and S. Lang, S. Lang, S. Cecotti and C. Vafa, S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, E.T. Whittaker and G.N. Watson, Y. Yang, F. Diamond and J. Shurman, J. Silvermann, S. Cecotti and C. Vafa, S. Cecotti and C. Vafa, C. Vafa, X.G. Wen and Z. Wang, X.G. Wen, G.W. Moore and N. Read, N. Read and E. Rezayi, B.I. Halperin, J.K. Jain, D.C. Tsui, H.L. Stormer and A.C. Gossard, R.B. Laughlin, F.D.M. Haldane, [9] [6] [7] [8] [4] [5] [1] [2] [3] [21] [22] [17] [18] [19] [20] [15] [16] [11] [12] [13] [14] [10] References JHEP08(2019)007 , ]. ]. , SPIRE SPIRE IN ]. IN ][ Nucl. Phys. [ , (1991) 755 ]. supersymmetry geometry for SPIRE ∗ IN , [ tt = 2 B 355 SPIRE (1991) 337 N IN [ function A 6 ζ arXiv:1412.4793 (1988) 411 [ Nucl. Phys. , 118 (1989) 701 -models: nonperturbative aspects σ (2016) 193 B 328 20 Mod. Phys. Lett. Nonperturbative aspects and exact results for Singularity theory and ]. ]. , – 49 – SPIRE SPIRE IN IN Nucl. Phys. Twistorial topological strings and a [ [ , Commun. Math. Phys. , Landau-Ginzburg families ]. = 2 -models N (1991) 1749 (1991) 2427 σ Adv. Theor. Math. Phys. SPIRE , d IN A 6 A 6 Landau-Ginzburg versus Calabi-Yau 4 . [ = 2 Topological On the structure of the topological phase of two-dimensional gravity N Geometry of Landau-Ginzburg models A direct evaluation of the periods of the Weierstrass Topological Landau-Ginzburg models ]. (1990) 281 theories in = 2 N = 2 SPIRE IN [ arXiv:1304.7194 N B 340 Int. J. Mod. Phys. the Int. J. Mod. Phys. S. Zemel, S. Cecotti, A. Neitzke and C. Vafa, C. Vafa, S. Cecotti, S. Cecotti, S. Cecotti, L. Girardello and A. Pasquinucci, E. Witten, E. Witten, S. Cecotti, L. Girardello and A. Pasquinucci, [30] [31] [27] [28] [29] [24] [25] [26] [23]