Aspects of Two-Dimensional Conformal Field Theories Pos(TASI2017)003 1 Xi Yin Due to R C 12 − =

PoS(TASI2017)003 https://pos.sissa.it/ ∗ Speaker. In this set of lectureconformal notes, field theories, I including give some an oldin and introduction two new to dimensions. results the of the operator conformal approach bootstrap to program two-dimensional ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Theoretical Advanced Study Institute Summer School4 2017 June "Physics - at 1 the July Fundamental 2017 Boulder, Frontier" Colorado Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 USA [email protected] Xi Yin Aspects of Two-Dimensional Conformal Field Theories PoS(TASI2017)003 1 Xi Yin due to R c 12 − = . The traceless µ µν µ T T is the Ricci scalar of the 2D , but such an operator would not be R µν T , in the sense that the operator spectrum of local operators, which by the state/operator 1 compact H is the central charge of the CFT and c 0 is equivalent to conformal invariance, which holds in flat 2D spacetime as on the cylinder. Later we will consider examples of noncompact CFTs in which = i 0 µ | µ 3 T . For now, we will assume the CFT is 1 A piece of folklore states that all QFTs come from CFTs inBy the UV; “solving" this a is incorrect: CFT, typically, standard we counterexamples are mean determining the spectrumOf of course, local in operators this and case the one algebra may insist of on operator speaking of the traceless part of Furthermore, a 2D CFT has a conserved traceless stress-energy tensor Quantum field theory, a pillar of modern theoretical physics, was built over rather unsatis- Conformal field theories in two dimensions enjoy an infinite dimensional conformal symmetry In this set of lectures, I will outline the operator definition of 2D CFTs, discuss a number of A 2D unitary CFT admits a Hilbert space S 2 1 2 3 × gauge theories in 2 and 3 dimensions. product expansion. conformal anomaly, where spacetime. conserved. is discrete and in particularground state contains the identity operator 1, which is mapped to a normalizable condition this property is suitably relaxed. an operator equation. On a curved spacetime, this condition is replaced by mapping is canonically isomorphic toR the Hilbert space of the states of the CFT on the cylinder factory foundations. There are essentiallyHamiltonian three with approaches: UV (1) regularization, Lagrangian andful and (3) path for operator integral, formulating (2) algebra. effective theoriestheory The and for Lagrangian for symmetry approach perturbation protected is theory, observables, sometimes use- discretization and beyond of for perturbation the non-perturbative formulations Euclidean through spacetime.through lattice spatial The lattice Hamiltonian discretization or approach truncatedapproach is conformal is practically space the useful approach. most either intrinsic Theoperators formulation operator in of algebra QFTs, conformal and field is theoriesfield best (CFTs). theories understood in can Large the be classes context described of of as local Poincaré renormalization invariant local group quantum flows starting from a CFT in the UV. group. They are also subjectconsistency to the definition important of constraint a of 2D modularfar CFT invariance, from on which classifying any allows or for surface. solving a most Whileto 2D a plenty CFTs. superconformal of field For examples theory instance, are based everygenerally known, on Calabi-Yau not a we manifold known nonlinear gives are beyond sigma rise those model, whose protectedspace spectra of by of exactly supersymmetry, operators marginal at are deformations; a furthermore, genericcorner the point of Calabi-Yau in models the merely landscape their occupy of moduli a 2D tiny CFTs. standard examples, and sketch the bootstrap method one may use to classify2. and solve Defining CFTs. properties of a 2D CFT Aspects of 2D CFTs 1. Introduction Thus, the classification and “solution"ries. of CFTs are important in the study of quantum field theo- PoS(TASI2017)003 Xi Yin 2 forms a representation of the Virasoro algebra, which can be H There is a well defined correlation function of local operators at separated points on the plane Under operator product expansion, the stress-energy tensor generates the Virasoro algebra. Figure 3: Modular covariance of theon torus 2D CFTs. one point function is another key consistency condition (or the conformally equivalent Riemann sphere), whichexpansion. is entirely By specified the by completeness the of operator local product tion operators one as can states on construct the theconsistency circle, of correlation via the a functions CFT plumbing of on construc- local a general operators Riemann on surface any demands that Riemann the surface. correlation functions The com- Figure 2: The associativityfunctions. of the OPE is equivalent to the crossing symmetry of sphere 4-point decomposed into a direct summension of “primary" irreducible operator. representations, each of which contains a lowest di- The space of local operators Figure 1: A localcle/cylinder operator under at the the state/operator correspondence. origin of the plane is mapped to a state of the CFT on the cir- Aspects of 2D CFTs PoS(TASI2017)003 and (2.4) (2.3) (2.5) (2.6) (2.1) (2.2) ) Xi Yin z ( . Thus, } zz 0 T \{ ≡ with the line C ) ) ¯ z z ( , , z ) T ¯ ( z , term on the RHS is z ( 2 − O , z ) c ¯ ) w z ( itself, ( , e , ) T T ) z ) m z ( ¯ ∂ , − ) is the most general possibility ( w , ) ) T ( . n z T 0 as the anti-holomorphic Laurent is holomorphic on ε e δ ( 1 ( 2.1 )) i ) ε ¯ ¯ ) z n + w T n π z ( n ¯ L d ( is z ∂ 2 − e T − i z ) z T 1 ) ) 3 ¯ z ¯ z C π z dz ( , n ( I 2 z ( ε e T ) + ( 0 limit. The order ) c I 0 − z 12 ) + O ( , ( ¯ z . In particular, from this one can deduce the ) → = , z T z + z z 2 n ( ∂ε ( 2 m z 2 L T 3 + O ) n + ) − z at the origin, L ( , ) 4 w ) z ε z ( 2 ) c ( n 2 m 0 + T − ε ( n L ) 3 z − ∼ O w ∂ n ∞ ( ) − ε c 0 ) = ( ∞ 12 ¯ i z ∑ ( = j e n π T − , ] = ( dw 2 ) z j m z z term would be in conflict with the commutativity of the OPE). ( ( L C ) = term is determined by the Ward identity associated with translation 3 ) = , z I T z n 1 − ( . Assuming the absence of negative dimension operators, and that the ( z L − − ) T [ z T z ε ( δ T ) = generate the Virasoro algebra with central charge ¯ z n , z L ( , the stress-energy tensor has two independent components . ¯ z ) O ¯ z ε ( generate another copy of the Virasoro algebra, of possibly a different central charge δ dzd e n T ¯ L . The OPE of a pair of stress-energy tensors, say the holomorphic components, take = ) ¯ z OPE (note that a 2 ( , while the order is a counterclockwise contour encircling ¯ z ) ¯ z ds z z T TT ( C admits the Laurent series representation ≡ A general infinitesimal conformal transformation is generated by the Noether current On the Euclidean plane, which we parameterize with complex coordinates In the presence of a local operator zT ) ) ¯ z z ( ( . e e The corresponding transformation of a local operator infinitesimal conformal transformation of the stress-energy tensor c and similarly where where we have omitted non-singulardetermined terms by in the Ward the identity associated withrent dilatation symmetry generated by the Noether cur- symmetry generated by that is convergent (in the sense ofno correlation other functions) on local an operators annuluscoefficients centered are of at the inserted. origin where Likewise, we define only dimension zero operator being proportional to the identity, ( T the form Aspects of 2D CFTs puted from plumbing constructionsThis based is on different in pair-of-pants factfunctions) decompositions equivalent together must to with agree. modular the covariance associativity of of torus 1-point the functions OPE of primaries (crossing [1]. 2.1 invariance Virasoro of algebra sphere 4-point element for the T PoS(TASI2017)003 0 L (2.7) (2.9) (2.8) . The (2.10) (2.11) (2.13) (2.14) (2.12) Xi Yin . After n it − ¯ L − = = † n is transformed τ ¯ L ) z ( T , ) z ( are Hermitian operators. f ¯ at the origin. The primary w ¯ = w h e 0 T z , and likewise O n , − . } . and on the cylinder with the lowest L 0 . z . , to the complex plane parameterized e i c 0 24 c , is > h ww 24 z c i | . w 24 T { n n ) h + . | z + L , ¯ ) k c ( w 12 n z 0 h ) + in ( − inw z O − ε − ( = e L h e n ) zz i n + − z L T h ¯ ) ··· ( L z | 0 2 ∞ 1 n z ) ∞ T n z − z L 4 ∞ − ∑ − ∞ ∂ = w ∑ ) = = n L ∂ z n ) = ( , 0 0 − i z − z ( h ) = ( i ≡ 0 | 0 ) = ( z h h T 7→ | ) = ( ) = w 0 2 maps to a primary operator h N z w ¯ ( = w ( − i O ( i 0 h L ¯ are now two independent real parameters. Furthermore, in the ww z w h z | ww | ¯ T t w ∂ ∂ T 0 e T L in the presence of an operator at the origin now turns into a Fourier − is the Schwarzian derivative. ) σ 2 z ) ( 0 z = zz 2 z T ∂ 2 w ( ) 3 0 z z − 0 ∂ z on the cylinder, of the form ( and ¯ z 2 ∂ ) 0 , we can analytically continue to Lorentzian signature by taking τ z τ w 3 z i ( ∂ + 2 + (omitted anti-holormophic dependence) transforms under a general conformal trans- σ ww from the Euclidean cylinder parameterized by ) T = σ z , that obeys = ( } iw h h = z such that − w , 0 e O ) w z z ( { = 0 z We will restrict our attention to representations of the Virasoro algebra with the property that A particularly basic and important example of conformal transformation is the exponential We may obtain a finite conformal transformation by composing together infinitesimal ones. T , is bounded from below, which in particular is true for all unitary representations. Irreducible z 0 into Consequently, the Hermitian conjugation of the operator L representations of this type are generated by a primary state Now passing to the finite conformal transformation represented by The Laurent expansion of doing so, states of a unitary CFTthe on holomorphic the and (Lorentzian) anti-holomorphic Virasoro cylinder algebra. by definition form unitary2.2 representations of Representations of Virasoro algebra map expansion of Lorentzian signature, the stress-energy tensor components by Likewise, we have Writing eigenvalue A basis for the general Viraosro descendants is It is useful toinfinitesimal characterize form these is in terms of an associated holomorphic diffeomorphism, whose Aspects of 2D CFTs where Under the state/operator mapping, operator formation according to PoS(TASI2017)003 , 1 ≥ − 2 ) n k (2.20) (2.18) (2.19) (2.16) (2.17) (2.15) 25) or Xi Yin q ≥ − ≥ 1 , 1 c n , there will ( m 1 rs = ≤ ∞ k > Gram matrix is s ∏ n n ≤ = will be negative for 0 corresponds to the 1 . m ) , c q n = ( 1 admits a null Virasoro m − h P G s − , 0 r 0 ( 25 is a polynomial function of = m h ∞ m √ ) ≥ , 24 ≤ n = ∑ n NM ( ’s, det h ± r s √ . h , G c ) r that also have zero norm, giving = cannot have zeros on the positive ≤ h rs | − h − rs . 1. At a higher level n 1 M 2 ( | , say in descending order − , ) P √ } s = 1 , ) n r k = s } h − , with 1 n | r 1 = , and write , ν , known as the Kac determinant, which is , = h ) 1 sb N of weight 1 N ± | | ··· n , − ( − − h 1 α + ··· − b M basis states, { , rs L G ν h | 2 0. , 2 ( χ N ) + rb rs ) i n N n , ( 1 χ h b , > 2 ≤ 5 | 4 1 1 ms ) rs rs h + M with n ∏ ≡ are either non-positive real numbers (for − 0, but it does have zeros along the positive real axis = ≤ as the level of the state. The level − − | { − s ∑ 1 m α , rs r j Q N L r 2 ( s | n ≥ ) χ 1. The matrix element , 4 n h N Q 1 m K 2 h + 1 M = † − 4 = 1, Q = − + L + = j = being a real function of k rs | i 6 is not equal to one of the L ) α s h h ) χ m > , ∑ n | r r n h h ( + ( is the partition number, obeying ( c h (( h are purely imaginary. 1 = h G ) 1 4 G = | ± = m = ) ) is that a primary + α ( N s det n , | c NM ( r P 1 h G − 2.16 24 = 0 that a unitary representation must have c are normalized such that zero of the Kac determinant. h 25 and the integer partition = ) N rs = 1 χ , is non-negative for s 25), and det rs > N , ’s are positive real numbers; these give rise to null states in a representation , which may be written in the form , 1 r s ) c , ) h h − n r < rs , ( . 1 h n c b ( G + ··· 6 P < , m 3 ( descendants of the form 1, one can show that if , m 2 ) 1, det 1. We will refer to 1 Virasoro algebra. < rs − = c = ≥ is a positive constant, = are given by 1 . Of particular interest is the determinant of . A careful analysis leads to the following discrete possible unitarity representations with − k c n m s c h n , n n r K = The intuition behind ( If It follows from For ( h 1 [4, 5]: c P and < where be rise to an order complex (for 1 real axis, and therefore must be positive for all some c since some of the of the and Note that for real descendant at level where the coefficients identity/vacuum representation). If Aspects of 2D CFTs where we denote by Here It will be useful later to define ··· ≥ with the normalization convention c given by the formula [2, 3] defined as the matrix of the inner product of level PoS(TASI2017)003 . j > ˜ h | − , i , and ˜ ) h 34 ) (2.21) (2.23) (2.22) (2.24) Xi Yin 3 z − 5 ¯ | z k z ˜ , h ( 3 n ¯ λ z , and then j ( ) are entirely O h k in a suitable 4 − ) z φ i k ( h M 23 O m − z which enters the − k , labeled holomor- O ¯ h L c . By construction, 13 λ N . Under a rescaling z M |} | − k , k 35 L , determine the operator in terms of h N z ) ) | ; 5 3 j , ¯ 23 i jk z z | ¯ h z , C , , 3 25 i z z h 13 and is completely symmetric ( in terms of ¯ ( z k {| | i scales like N 4 k k O z ˜ B h ) φ k i j j ; , j max A 23 . i φ ¯ ˜ ˜ z i ) h h , 2 12 3 , > φ i ¯ ¯ z z i j h | i ˜ 13 h , δ h z ( 3 is chosen appropriately so that 12 2 15 ; ¯ z z z M ( | 4 23 k B z z ) = , O , expanding around i ) k 23 ) 13 2 z ¯ z O 2 z , j ¯ ( z , , 2 6 13 k i j O 2 z z A z , the coefficient | ( j k ( 3 i j k h ¯ z ∑ O , determined by the property that no other operator lie ; φ as well as the integer partitions ) j ¯ ) λ R 1 1 k h . This doesn’t mean that the OPE is not applicable at all ¯ z , provided that ¯ z , 3 φ , ) = ) i , → z 1 2 5 are structure constants. If we choose the basis of primaries , provided that 1 h ¯ z z z ) 3 z ( i ( , ( k ( 3 i j i z N i 2 n z , expanding around the point together with the structure constants z j C φ ( , ¯ B O i O ( h k 3 j i φ O M i , z O ∑ O N λ O ) k 1 i j with ¯ z C centered at , ) → with 1 1 k 3 z R z i ∑ ) ( ( z i i 4 z . O O ( ) = k i j m 2 ¯ A z O , 2 , or we can first take the OPE of z is the coefficient of the three-point function ( |} j k has a radius of convergence i j φ 24 C 3 ) z i | 1 z , ¯ , of the form z = | 3 , z 1 . 14 z i jk k z and ¯ ( , ) as an operator equation holds when inserted in a correlation function, where the expansion i C We can exploit the full conformal symmetry by organizing the OPE according to primaries Now let us consider the product of three operators, The set of primary operators The operator product expansion takes the general form j {| 3 φ i , to have normalization 2-point function (on the plane or Riemann sphere), namely z i k 2.21 ( where the summation is overaround all operators in the CFT, labeled by the index then take the OPE of these two expansion should leadon to the the coefficients same result, which implies a set of consistency conditions where the summation is overphic primaries and anti-holomorphic Virasoro descendants.determined The by conformal functions Ward identities and depend only on the central charge and descendants. For instance, the OPE of a pair of primaries take the form Virasoro algebra. The coefficients φ We can first take the OPE of max outside of the radius of convergence,way but to rather that make we the have expansion toprecisely well rearrange later defined the through in basis the a analytic correlation continuation function. of conformal We blocks. will characterize this more take the OPE of within the disc of radius then in product algebra, thereby all correlation functions onnontrivial the constraints plane due or to sphere. the They associativity are of subject the to OPE, highly which we will explore later. in Aspects of 2D CFTs 2.3 Associativity of OPE PoS(TASI2017)003 of the (2.25) Xi Yin 2 r = | 0 z | associated with each of the q . Modular invariance amounts to the map. A typical plumbing map takes ) i jk Riemann surface, based on the 3-point , C C , × g 2 ( C . The consistency of the CFT on the Riemann ∈ q PSL i jk C 7 , z / q 3 complex structure moduli of the Riemann surface. = 4 0 − z g on one of the two-holed discs to the boundary 1 Riemann surface can be constructed by plumbing together . There is one complex modulus 1 | r > q | = g | = z | 2 r 1 r Figure 4: Plumbing construction for a genus two Riemann surface. surface as a smooth manifold can be constructed by gluing pair-of-pants, or three- g 3 plumbing maps, giving rise to 3 2 three-holed spheres. The genus one case can be constructed by plumbing the inner and outer The equivalence of different plumbing constructions of the same Riemann surface would be A very interesting open question is the relationship between modular invariance and the notion of local Hilbert The plumbing construction gives a precise way to construct the partition function, and more Since every two-dimensional Riemannian manifold is conformally flat, a 2D CFT is canon- A genus 4 − − g g guaranteed provided that all sphereferent 4-point channels, functions and can that be all torus equivalentlyouter 1-point decomposed functions boundaries in constructed of two by a dif- plumbing 1-punctured togetherinto the annulus inner an are and independent annulus. of The choicelatter former of is amounts how the equivalent to to torus the is the modular cut associativity covariance open of of OPE the as torus already 1-point mentioned.space function in The the of sense all of the primaries. “split property" Namely, [6, 7]. function of arbitrarily Virasoro descendants onformal the Ward Riemann identities sphere, in which terms are of determinedstatement the by that con- structure all constants possible plumbingto constructions the of same the answer for same the (punctured) partition Riemann function surface (correlation lead function). generally, correlation functions, on an arbitrary genus which identifies the boundary ically defined on any Riemanndependence surface, on up the to metric a withinset a universal of conformal conformal local anomaly class. operators which inserted Furthermore,plumbing introduces on the construction a a correlation by general the function Riemann structure of surface constants any in a given CFT is determined via the Aspects of 2D CFTs 2.4 Modular invariance surface, which may be constructedconditions by known different as plumbings, modular leads invariance. to an important set of consistent the form other two-holed disc, with holed spheres. A general genus boundaries of an annulus. Theof plumbing three-holed construction spheres glues (or together two-holed a discs) pair by of an circular boundaries 2 3 PoS(TASI2017)003 4. , 3 (3.4) (3.3) (3.5) (3.2) (3.1) , (2.26) Xi Yin 2 , , 1 limit, by i 1 = ∞ φ | i ) ¯ , z → . ) , i ) z 4 ¯ z ¯ τ ( z , , 2 i z τ φ ( ( ˜ i is a modular form of M k f φ ) † − k ¯ ¯ τ in the L ˜ h , ) M ) τ ¯ τ 4 ( † − i ¯ z k ( L , f k | 4 h k z ) φ = ( τ h . 4 i ) . ) (∞) τ φ ) ∞ k ( 4 − 4 ˜ 2 h ¯ . z ϕ , T = ,

i c 4 4 ) ) ( z . z ¯ z ˜ ) = ( ( i ∞ M as ket and bra, and rewrite the 4-point (1) , ¯ 4 τ 1 ˜ ( z 3 N 0 , ) 4 / φ ( φ | 4 k G 1 φ 1 ∞ ˜ ) h ) ) φ ¯ ( z 2 4 k − h 0 1 ¯ = 4 , z , h ( 4 z φ 3 , 3 τ h ( z c 4 2 2 / φ (z)ϕ ( z 8 ) ’s in terms of sums of Virasoro descendants over 2 1 φ i ¯ z ) , ∞ and φ , − z 1 NM z → ( ( ) ( 4 k G 3 ¯ z = f 2 0 lim , i φ ) is equivalent to inserting a complete basis of states in ( 4 φ 2 (0)ϕ k | z 4 1 ) . Let us now carry out this computation concretely. Using z 1 4 φ , φ k φ | 0 of the Riemann sphere, we can put the four operators at φ ) 3 ϕ ( ˜ h N ¯ 34 , τ ) 1 φ ) = − , 0 C φ C ¯ ∞ , L τ

, ( k ( = N (or 0 2 4 k 12 f ( − 1 φ 2 z L C φ ) 1 1 φ PSL ) = ( 1 3 φ ), which may be organized into a sum over Virasoro descendants as , the torus 1-point function | + ) 4 ¯ ¯ τ z φ 3.4 i , , h 1 | z 1 ˜ ( φ M | k | + ) φ = ¯ z | τ , ˜ ( N z | k , . That is, ( | f ∑ ) 2 M k | φ ˜ h ) = , | 1 k N ( | h 3 ( φ k | ∑ 4 We will begin by considering the sphere 4-point function of primaries Performing the OPE of φ = h Figure 5: Inserting a complete basis of states (on the red dashed circle) in the four-point function. function equivalently in the form In radial quantization, we may regard defining Now we are ready to compute the correlator 3. Virasoro conformal blocks an entire basis of primaries, theterms 4-point of function the is structure determined constants by thethe conformal conformal Ward Killing identities group in To make the correlator finite and well-defined, we will rescale As already seen, writing the OPE of pairs of radial quantization in ( Aspects of 2D CFTs for every primary weight PoS(TASI2017)003 . ) ∞ ∞ C and 3.5 with (3.7) (3.6) (3.9) (3.8) = i ) i (3.10) Xi Yin h z 1 ξ 34 OPE ν , 2 → 1 to ν i i , . ) ) , k i = ) 0 0 1 ν z ( ( z φ ; ¯ k k M | k ν ν ) , − ˜ h ) ) ) 1 L , z z 1 ( ( 4 ( ( ˜ 2 ν ˜ h ρ , φ T T , i ) 2 ˜ ) ) 3 ) k M ˜ ν 1 1 ˜ h 0 . h † − . Now we can write ( , ( ( , ( , 1 k ¯ c 2 2 L k 2 c ˜ h ν ν ν ˜ ( ν h M ˜ ) ) , M + † − )] 1 ∞ ∞ − k NM ˜ L 1 h ( ( ¯ | 1. It is possible to analytically L h ( 0 0 ( 1 1 k G ( c 2 ν ν ) φ − = ρ h h k h , F ) | 1 1 2 is entirely fixed by conformal Ward ∂ν ν z ) ) 1 ˜ h . The radius of convergence of the + + M k z in the special case where the integer ) N | ν n z n involves a singular Virasoro generator ˜ ; − 1 ν , − + ) − k − − ˜ ) ξ 2 k N k N L z z h ) , ˜ h 1 ν , i i − ν , − 2 1 ( , + 3 4 ¯ L ξ N 2 π π L 2 k ν dz dz h , ˜ , h − 2 ν 2 − ν , , 3 3 ) − 4 1 0 n L ˜ 3 M ν ξ 1 1 ( , C C ν ˜ h h , ( − 2 2 ( , I I 4 − ρ + L and the central charge h ν 2 ˜ ¯ ν ρ z 9 ( , | | i h n ( = 1 , , and so N ξ ρ i − M | ρ i | − 1 ν ) } k ) = ) ( ( + 21 h + 0 n k k k 0 2 k ( 43 ( ρ h { ( ν h c h k C k , C + + ν 2 2 )[ ν 2 = F ) h = h n ν = z -plane with a branch cut that extends from ∞ , − 21 i − − N ( z ( i 1 k 1 1 1 0 k 1 h L h T ν C φ ) φ ν ) − | ( k − | 1 z ) as a series expansion in 1 z ˜ ρ | ( N 43 ( 1 denotes the three-point function of Virasoro descendants ] 2 c 2 ( M − C 1 = will be discussed in more detail later. | ν ) 2 i − h ¯ ν L h ) 1 F i = ) ) k ) φ ∑ | N 1 ∑ z ξ ˜ 0 ∞ N + ∞ M ( , φ | ( − has a (logarithmic) singularity at ( ( c | 2 k k 0 † − = 1 L 0 ) 1 ) ξ h ¯ ν ) ¯ L ν z i F z ν , ) h , 1 1 ( h ) = 3 M − 1 z ( c 1 z φ ξ ( ( † − 3 | 2 ; ( + 2 2 ) k φ L F h n ¯ ) = | | z ν ρ φ . Using the dilatation symmetry on the 3-point functions, we have h k k 4 , − ) ˜ , h − M z z ν are the components of the inverse Gram matrix of Virasoro descendants of a φ φ 4 ∞ ( i are small counterclockwise contours encircling 0 and 1 respectively, whereas ) denote holomorphic and anti-holomorphic Virasoro primaries of weight n h h † − h ( 2 ) to the entire complex 1 0 π − 1 i L , dz φ h ) ˜ 3 2 ) ν L ) C , M 1 − z , h c 1 ∞ ν † − ( , 2 n ( ( C n c contains only one element 2 ( L ν 3 I | 2 L h , and and φ k 1. We have F ( , N 1 h NM | i − h 0 φ 1 4 ν h G ν ≥ h ( C φ ( = [ = = h ρ n c As a simple example, let us compute So far, we have defined , n F respectively, and − i -expansion is 1, as ˜ the structure constants stripped off. In particular, where h identities in terms of the Virasoro descendatns where is a large clockwise contour. Further, we will write is the holomorphic Virasoro conformal block for the sphere 4-point function in the 12 partition L 3.1 Recursive representation channel. z continue The analytic structure of as primary of weight Aspects of 2D CFTs where where PoS(TASI2017)003 , 1 2 # . (3.12) (3.11) (3.13) (3.14) 2 2 Xi Yin h ≥ 4 n + h and/or in the , ) i 1 c . Whenever we )] ) 3 1 , 0 − ) ξ ξ ( 1 , 1 1 rs 1 ξ − ν ( , ξ ) m 2 4 . 1 L ξ k by repeatedly applying ( , fixed at a generic value, + m h 2 2 2 i 2 0 at the beginning of the L r h ν ξ ξ ) , ) + , s 3 z 3 − 2 > ( ξ ξ h − 1 ( ( n T r expansion. A far more efficient ) ρ − ρ ( z n 1 ∞ ) for ( i h p 1 1 0 ; it also allows us to trade a raising k ) − , n 3 4, the analytically continued block 0 ν − + in the central charge ) + , + ξ − ( h 1 ) 3 h 1 1 c L n m ξ 1 , 2 ν + , ) + ( 2 ) F n 2 , or 1 , + z + 1 1 2 ξ ξ 1 i ) ( n , − 1 ξ , n 2 ( π 3 , ( 2 = ∑ dz 2 l ν = 1 ξ 2 1. − ξ to lowering operators on m ) -plane. Again, generically we only encounter h ( ξ , i 0 z c 2 3 ≥ ρ rs ( C , ξ ξ i I s ) + " T )] h ) = m 10 , 1 ) for generic descendants , corresponding to simple poles in the inverse Gram 1 1 + 2 + ξ h ξ ∞ ν n ) i n ( ( , 1 ) = ) L 0 ≥ 0 L k 2 and ξ h ( 0 , L ν r ν , ( ( 1. Alternatively, we may hold 2 ρ otherwise, and h s c , 2 1 [ , ξ 1 k r − ξ ν , ≥ ∞ b ν , + ) ) for 3 n ( s 3 2 1 ξ ) z , m ξ ξ ( ρ s ( r , i , ( ( 2 ] r 2 ) = ρ 0 + π k on the complex ν c ) ρ dz n L 2 ) h ( 1 2 m ( c s z l , − ∞ , for r ( − + ) ) = − ) C h in 1 ) T 2 h c I n 3 ) , we can use the Virasoro algebra to reduce the length of the Virasoro ( ξ ( h c i s s , ξ , -expansion of the conformal block was discovered by Zamolodchikov . The locations of these poles are nothing but the zeroes of the Kac , − ∞ ξ r r 2 ( . For generic z ) 0 − F ( 0 b h 1, and ξ 0 h k h 1 = ∞ , L , ∑ with lowering operators on ν h m 3 ) = c = h 3 ξ 1 ( ) + 1 has only simple poles in − h n ξ ≥ − ν h + ) , ) , ( − ) = [ ) = n z n 1 NM s 2 1 z 1 1 , ; L ξ r i ν ( ξ ξ G h , whose locations b , + , , , π ρ c for ( k dz 2 2 coincides with 4 n 2 6 ν ξ ξ ( h n 1 h n 1 n , 2 . + C 3 − h − − ) I . This algorithm terminates after finitely many iterations and allows us to compute any h 1 L 1 L L i ) = , ( , , ξ ξ 2 n = [ = 3 3 , ρ ( h 2 ξ ξ l ) = , ( ( ξ 1 h , While such an algorithm can be implemented on a computer (using e.g. Weaver package [8]), it The idea is to consider the analytic continuation of Let us do another example, ρ ρ ( h 3 s ( , ξ r c ( c where More generally, we can compute the following conformal Ward identities ρ is highly time consuming and would besuch impractical as beyond the numerical first few bootstrap levels.block or For to some high Lorentzian purposes, accuracy, which continuation, requires going it to high is orders important in the to know the conformal chain of This set of identities allows us to trade aoperator at raising the operator beginning of theact Virasoro a chain in lowering operator on method of computing the matrix elements determinant, namely and examine the poles of simple poles in [9, 10], which we now describe. Virasoro chain in internal primary weight F are such that Aspects of 2D CFTs PoS(TASI2017)003 . rs χ limit, (3.19) (3.22) (3.15) (3.16) (3.18) (3.20) (3.21) (3.17) Xi Yin s , r h . . ) limit, in the → ) are nonzero. z 1 . ; s h ) , ) ν r 1 z , rs h , ; ν 2 , 1 h ν + 2 , − s → , ν ;2 r . rs , 4 h ) qb h χ h , s rs ). In the , ) , 4 M χ + r 1 − ( h − ( ν , 3 ρ L , pb , 3 2.19 h 2 ( 2 ) ( h which is relatively simple to ν 0 ρ , s + + , 5 , , r ) 2 , 2 and h h 0 h rs ) λ rs , ν ) , 2 χ is in fact finite and reduces to the 1 1 ( rs ( − h + c 6= ( h of is determined by the residues at its χ ρ s 1 ( , ) , SL ) r c − c F rs λ . 3 n 1 h 2 1 χ , 1 ν , ν F F h h − , , c − m . ) 4 ν 2 ( 4 3 ( + c ν c rs ν M h h qb ( , h A − NM , ( . Finally, the key property that leads to a ρ ( rs limit of + 1 2 1 1 L 2 rs i c G + h h c F − ) s P M λ rs , ≡ 2 ) r pb 1 4 rs 2 h χ h 1 i rs 1 2 c s χ ν . The contribution to the residue of the conformal + − , + rs h h h rs P r 2 11 − n N N 2 h = χ h | 2 nb − − | ∑ λ − rs ) c M 1 L h L rs − | 1 ) = + h P , ( + χ 1 h − 3 − c rs , ρ 1 h = ν b z ν . The large s λ , A c rs mb ) into , , r h c rs 2 ( 4 ) + h A s z ν χ ν ) = ) = b only receives contribution from Virasoro descendants of the ; − , → lim ( behaves like a Virasoro primary in the 1 ) = 1 s z ( h h 3.19 rs ν ) z step 2 ρ 1 ; ∏ 1 4 , = h | , ; rs χ h s h 4 n − 2 N ( h ∏ ( s χ , r | h − s for all positive , ν ≡ 4 , 1 , ρ , − + r , 4 h i n 3 1 1 r = c h rs , rs h h q L ∏ = − , as those of the null descendant 3 + , ] χ 3 s m = ) = , h 2 i r N h M , rs h h h rs 2 1 c rs , 2 − , χ + 2 step 2 χ χ 1 h 1 L 2 | , h , r = h − ( h 2 , h ∏ r 1 rs ( − − 1 ρ ν c rs c h 1 1 χ , h ( h h A is a null state, its 3-point function = 1 ( c [ F p − c ν ) z rs ( s F | , F = χ r vanishes, with ρ ∞ M ) h | s representations a normalized primary state of the same weight as the null state , h are defined by rs → r = lim ∑ | c , where − χ h rs 2 N h rs | h + λ ( − s χ , , s c rs r , r 1 h N h A h ( λ − ν s , r → lim L = h h This property has a suitable generalization beyond sphere 4-point Virasoro blocks, as we will discuss later. For now let us focus on the analytic continuation in 5 → lim h Note that while They are given by the “fusion polynomial" block is determined by form sense that it is annihilated by The residues at the poles has the same coefficients After rescaling by recursive formula for the residue is the factorization property Putting these together, we can turn ( poles together with its behavior at large corresponding conformal block of the global conformalcompute group and is in this case given by a hypergeometric function, the norm of where where Aspects of 2D CFTs There exists a closed form formula for PoS(TASI2017)003 . , 2 2 ∼ . + are Z ) n w z / + ; 2 ∞ h (3.25) (3.23) (3.26) (3.24) Xi Yin , + rs T 1 2 , we can , h + h z − . , 1 h ) h , z 4 − ; and h z , rs c 3 by h + z identification , h 2 2 , . In the next section, h 4 Z ) , h z 1 , ( h c 3 ( on the pillow is h s , r F , ) c for general w 2 z h ; n F . , is related to h to the pillow geometry + ) 1 , h z , with the 4 3 τ h ;2 ∞ ) + ( h h 4 . , 2 x s , τ h πτ r h 1 ;1; i s c , , 2 r − − π 2 1 rs z 1 c − e z , F h , P + 3 2 1 − )( π h = 4 3 z w , and the conformal block can be thought 1 2 ( x -expansion of ) h h 1 dx h h q z + = 2 τ F − ∼ s h . The four points on the plane 0 v s 2 , , r 1 r , 2 rs + respectively. rs c π ( c 1 P 2 Z 1 x P h s ( / , ) = r πτ 1 2 2 + z c rs p π − , ( h h , 12 T ) A 2 u w K 0 1 ) h s , r Z πτ h rs ∼ c on the Riemann sphere to , ( + + , we have 2 , h s P ) , πτ π s u w τ ) h r , ∂ r )) h , ( ( z c c rs ( ) 0 τ 1 s π = 2 1 ∂ z ( , A − F b b , r ( v 3 2 = c 1 − π θ h K ( ) b b , ( w = 0 + h = 0 2 K v ( h i h c s = ) = , = − r z ∂ 1 ; c = at w h w h ∂ τ c − , z 4 − h is one of the Jacobi theta functions. In the pillow frame, the four primary , 3 ) 2 ) = h n h z , ( ; q s 2 h , ∞ r h , 1 c , − 4 1 fixed points h = − h , ∞ n 2 ( 3 c c ∑ Z h 1 , F ≥ 2 ) s , h h ) = 2 ∑ ( , τ s , 1 ≥ r ( r c h 3 ( θ → c + c The explicit map from the coordinate To fully exhibit the analytic property of the sphere 4-point Virasoro conformal block, it is F . It is also conventional to define the “elliptic nome" Res w − operators are inserted at the corners Figure 6: The pillow geometry is the quotient Finally, we arrive at the recursion formula This allows for a relatively efficient computation of the mapped to the where useful to map the Riemann sphere with four marked points 0 Aspects of 2D CFTs Turning this into a residue in Suppose we have already computedsubstitute the this LHS result into up the to shifted blocks order on the RHS, and obtain results up towe order will describe an even more efficient approach. 3.2 The pillow geometry and the analytic continuation of the conformal block parameterized by the complex coordinate PoS(TASI2017)003 . disc. i (3.29) (3.27) (3.28) (3.30) Xi Yin q 12 ) differs ψ | rather than c 24 3.27 h are norms of , − 4 0 ) .( h n L q c a − q ; 3 | h h , 43 − 4 c ψ h 24 h -plane and more: the , . ) 4 z 3 h z h h . − , − 1 2 3 2 h 1 − h − ( ) , 2 n c 1 24 h 2 h ) -disc. Secondly, the matrix el- − q ( z 1 , q c h c − , the coefficients 24 − H − 3 1 1 2 ’s held fixed: is mapped to the eye-shaped region c − 1 h 24 i ( n ( − 1 ) z h 2 ) + ) ∞ h = = and the central charge 4 n h ∞ ∏ n h 2 , − ) exhibits several important properties. q 2 h 1 c q 1 n + h 24 and h , [ 3 i a h , − − c − h limit, the external operators inserted at the 0 4 h c + 3.28 24 1 ∞ ) = 2 h ∑ h z ( n h q ) 1 + 4 = ∞ = 1 h = 16 ∏ n h 1 + ( i 13 3 h 4 c 24 h 12 − = ( − + c 2 ψ i 2 h limit, with | h ) )) c 12 q 24 + τ ∞ 1 ψ ( − h | 16 ( 0 3 4 c ( L → 24 θ q − | − h × c 2 0 L 43 limit now allows us to derive a recursion formula in )) q ) = ( ) has unit radius of convergence, and is a holomorphic function ψ | τ h z h 1∞ ( ; 43 3 and are therefore non-negative (provided that the OPE is compatible h θ ψ , i 3.27 h 4 12 ∞ h are functions of the weights , ψ → -plane with a branch cut along 3 | lim ) = ( 0 h n z are the states created by a pair of primaries inserted at two corners of the z h a ; , i 2 h , h 43 4 , ) simplifies in the 1 ψ h | , h 3 ( z c h 3.27 , -disc away from the origin. This covers the entire complex disc. The Virasoro conformal block can be analytically continued to the entire -expansion of ( and F 2 ) has the property that, if we take q q h q i , -plane only maps to an eye-shaped region on the unit 1 z 12 h 3.27 ψ ( | c Thirdly, ( The knowledge of the large F , of the form c , and the coefficients on the unit Heuristically, the intuition here iscorners that are in unimportant, the and large coefficient) as the the pillow square root block of becomes the torus the character same of a (up primaryin to of weight a moduli independent Figure 7: The complex where pillow, projected onto the representation spaceh of the Virasoro descendants of a primary of weight The pillow representation ofFirstly, the the conformal block ( complex Virasoro descendants of with unitarity). on the unit Aspects of 2D CFTs of as the propagator on the cylinder [11] from the sphere 4-point block by a conformal anomaly factor, ement ( PoS(TASI2017)003 . k h and limit ∗ (3.33) (3.32) (3.34) (3.31) Xi Yin ∗ h , up to ˜ h n , -disc via . > ∗ ) q ≤ k h q h ; , whose con- rs 3 rs φ . . + n ) ¯ s z = q , r 2 h − φ , 4 , ;1 4 h k subject to , φ ˜ h 3 , s h 1 implies the convergence , 4 = ) , r ˜ h z 2 , 1 , < ; ¯ . i up to order -disc, in the large h φ 1 k , z q ˜ ˜ h k h h 12 1 24 , , ˜ h ψ 2 < 4 | ( ˜ . The equivalence between the two ˜ h h c → ˜ c 3 1, hence over the entire , ) are non-negative). The summand 24 , 3 H h 3 − ˜ ˜ h h < 0 ( , 4 3 ¯ : 13 L minimal model, also known as the 2D c 2 3.27 ¯ q h h q ) from operators of weights u ˜ | h 2 1 with , F < k rs 12 1 ) 1 c in ( , ˜ z ˜ h = h P ψ 3.32 n ( for a finite set of in three different channels: c 23 c − a ih 1 2 is kept in a finite range). This result amounts to ) i h h in the range 0 k 12 k q ) F ;1 → i ˜ ; ) h z k ¯ 14 ψ z z rs | -channel conformal block can be obtained from the h c , rs ; i t , c P or k 24 z 4 : 14 ( h c rs + k h − i , t 0 s h , A , φ 4 to emphasize its dependence on the internal weight L r 1 s in the interior of the unit 1 , h q h h r , in the range 0 i , | = , for generic internal weight , ∗ h 4 i 3 ˜ rs h 4 2 k in the large internal weight limit, and thus the convergence 12 k q 12 c 34 h q h h + ∏ − , k ψ ∗ , ψ H , h ˜ | h 2 while exchanging h h 3 h 3 → | h k + h z h 1 k q , , ( | h 2 12 on 1 channel. The ≥ c 2 − ) s ∑ h , C t h r k q : 12 ( 1 F , c k s k 1 + ∑ 16 h OPE in the pillow frame convergences exponentially fast as a sum over → F 23 ( and 1 ( k k z c 2 C s k 2 φ 34 12 H 1 14 ) = C C φ k q C ; 12 k h ∑ C , 4 k h = ∑ , 3 h , 2 close to 1 implies that the contribution to ( obeys h , c , and then use it to generate q ) behaves like would be suppressed by 1 n H h q ∗ ( ˜ With this understanding of the convergence of the conformal block expansion, we may de- Let us consider the sphere 4-point function in the special case Now we will discuss one of the simplest nontrivial solutions to the crossing equation, namely h c 3.32 H > -channel one by sending ˜ channels can be expressed as the crossing equation The equivalence of thesestraints different on conformal the block structure constants. decompositionsthe Up equivalence puts between to highly permutations nontrivial ofs external con- primaries, it suffices to inspect Cauchy-Schwarz inequality (since the coefficients (one can make a similar argument if either internal primaries. compose a general 4-point function the statement that the formal block decomposition in the pillow frame involves the sum of the conformal block sum for real h order 3.3 The convergence of OPE and the crossing equation The analyticity of the 4-point function for real the 4-point function of nontrivial primaries in the where we put the superscript for real 3.4 Solution to crossing equation: minimal model Aspects of 2D CFTs where In practice, we can implementequations) this recursion to formula solve efficiently for by first using it (as a set of linear in ( PoS(TASI2017)003 , ) . ¯ z . i , ) ¯ z z ( , (3.40) (3.38) (3.42) (3.39) (3.35) (3.36) (3.37) (3.41) Xi Yin ∂σ , z | σ ) ( ) 2 ¯ z f ¯ z , − ) , z . The con- z 2 L ( ) , ( ¯ 1 z f , h σ ) z ) 2 2 ( , 1 1 f ( + h z σ 2 | ∂ ) , i ≡ − σ ) ) 1 h z ∂ z ∞ 1 21 31 − z ( ¯ ¯ z z 0 z . , + − σ 0 i . ) 21 31 0 has a null descendant at level 1 σ z z . = ( = (( | ( ( ) 0 ) = 1 1 σ = f ¯ σ z ¯ z = = ) , i 2 , = , ¯ 3 3 z z ) 1 z z z , ( , , h ( z z z ∞ σ 4 f σ ( | ( ) 1 = = i 0 ) 2 2 2 , z z σ 31 1 σ , , σ − ) , | 2 z . ( 0 0 1 ) , | ¯ ) ) L 0 16 1 8 z 1 2 = = 1 z z 0 ( ) 1 1 3 4 ( ( z z z ∂σ = √ σ =

| − = = σ σ h ) ) i − 2 ) ) − 3 1 σ 1 ± , 1 ) ¯ , z 1 , ( h 1 w 31 31 21 21 1 2 1 , 1 2 ∞ 2 − ¯ ¯ ¯ ¯ z h ( z z z z ( z h h z ( ¯ L 1 8 ( 0 , , T p 1 − z = 15 ) = σ σ = 64 31 31 21 21 1 1 ) ) ˜ h z z z z ˜ ˜ = ( h invariance, h ( 2 3 − ( ( ¯ z ) − σ ) = f f = z , σ = = | z i − L ) 3 ∂ C 2 2 h ( . We need to compute the correlator involving , , z 2 h h σ , ) 3 4 1 1 ) z σ ( h ± | z h h − z 2 : ∂ : f ) 4 4 z ( 2 σ L − | | ¯ 1 1 z ) − ε 1 : σ , 1 3 4 − 1 2 − 31 31 z 1 ¯ z 2 − z z ( ), we derive ( 1 | | PSL , w − L z ( 2 1 3 σ i 1 z z 4 3 z ) )( π ( 2 ∂ ∂ − 1 0 dw 3.37 2 can be computed by going back to a slightly more general form of L ( + ( σ ( z ) = = 2 σ z σ C 1 | i i h ∂ ¯ z I ) ) ∂σ σ , h 1 ∞ ∞ = z ) into ( ( ( 0 0 ( i 2 1 σ σ σ z σ | ) ) h 3.40 ) 1 1 can be replaced with ¯ + z ( ( , ) 2 z z σ ) ( ( ) z ∂σ ¯ z σ σ ) 1 , ¯ ) and ( z 2 − z , ( − 1 z ( L ( σ ) ) 3.38 σ 1 0 2 ) ( , ( 0 1 σ ( h | acting on ∂σ σ 1 σ h h = h − Putting ( formal block in this case is2, particularly of simple, the due form to the fact that the 4-point function, as is ﬁxed by The correlators involving and so We will focus on the example of the 4-point function Aspects of 2D CFTs critical Ising model. There are only three primaries in this CFT, whose weights are Let us insert this into the 4-point function, L using This equation has two linear independent solutions, PoS(TASI2017)003 , 2 1 z εεε − = C 1 c (3.48) (3.45) (3.44) (3.46) (3.47) (3.43) Xi Yin → and OPE, as a z εεσ . C σσ internal primary 2

h , ··· 1 2 + F 4 z 2 ··· 1 √ + 4 613 − z . is invariant under 8192 ) 0 obey ) ¯ z ¯ z + ) ( F , z 1 2 3 2 117 z can appear in the ( minimal model, as z 8192 ( ± F √ f ε 2 1 f ,

) + . , ) z 25 , ¯ ) ) 256 3 τ ( = z z z ( 1 2 ∞ ( ( c 2 ˜ σσε h + 1 − − 64 C χ F 2 f f ) z 2 2 1 to + τ + 0, we have − 9 . ( 2 2 √ √ σσε 64 2 ) + ) h

= 1 2 z = z z C 1 2 χ z ( ( + ˜ z h 1 , = 64 z + + , is an extremely strong condition - in fact, this F h f f ) + ˜ h 1 4 16 d 2 ¯ z + ˜ ( h 1 , √ = 0 1 σσε + h ∑ 2 ) = ) = C 1 h F z z ) 1 8 + z ) = 1 and suppose there are only scalar Virasoro primaries. − − 3 8 − 0 ( ¯ z τ z 0 1 1 , F ( ( > τ = F 2 + − ( c f f 1 ) = ) Z √ z z

( ( ) = is the conformal block associated with a weight ¯ z − − f , f ) z . Expanding around z ) = ( ( 2 ¯ z − σ f h ) ) + − z z F ( ( 1 , + + f f z − 1 ) = ) = ( z z 1 so that there are no possible null Virasoro descendants. The assumption that f ( ( 1 2 0 > is the only nontrivial structure constant in the F F . We see here that only the identity operator and c i . Defined with a branch cut running from σσε ¯ z C − σσσσ Let us first inspect the torus partition function of such a CFT, which must be modular invariant. Now we will describe another nontrivial solution to the crossing equation. Let us consider There is still an extra set of consistency conditions that the CFT must satisfy, which are the 1 h → ¯ It takes the general form condition (conjecturally) fixes the CFT completely! We assume all Virasoro primaries are scalars, namely a unitary CFT with central charge z Thus, we have minimal model, but we will not verify it explicitly here. 3.5 Solution to crossing equation: Liouville theory modular covariance of the torus 1-point functions of primaries. This indeed holds for the Now we must inspect crossing symmetry, which demands that We see that crossing symmetry is obeyed if and only if In fact, can be shown to vanish by similar analysis. The CFT is thus solved completely. Aspects of 2D CFTs whose linear combinations givefinitely rise many to conformal blocks two (namely possible two)null in Virasoro conformal descendant this blocks. of case is an The accident fact due to that the there presence are of the As the notation suggests, for consequence of the null state differential equation. The 4-point function must then take the form PoS(TASI2017)003 (3.57) (3.58) (3.54) (3.55) (3.56) (3.49) (3.51) (3.52) (3.53) (3.50) Xi Yin . . 1 − 24 1 c − 24 , − c h − , modular invariance of the , namely . ) y ) h 1 x ¯ τ τ ( . π / − 24 1 2 . In the absence of null descen- 2 4 1 h and c − τη ν − i − x x πτ is the torus character of a Virasoro , − 4 − τ ∝ h h . − / √ χ n 1 , exp q ) 2 , . h , − . For the second equality, let us write 1 h c d 1 − 24 ( 1 2 ˜ h ) = ( | 2 πτ exp − 24 Z 1 − τ h τ ρ , and τ c 4 h − 24 h ∑ | = / 1 ) d c q ˜ − x 1 h 1 ∞ = h dh − h ∏ , n h = ) = V 1 − √ h − ∑ h 1 ∞ c − ( 24 y 24 ( c h Tr 17 = q η = − + Z exp h , ¯ h τ 2 x 0, we have q ∝ , d ) = τ → for all primaries, namely the spins are integers. This is π ) 1 τ ≥ h h 1 4 1 h ( ∑ = Z d ( ) = + h h − − 2 x − 24 τ 24 ρ χ | τ h ∈ , ( c c ) ( ∑ h h ˜ h τ Z 1 χ ( − − exp − 1 and h η h h | h , so that the anti-holomorphic Virasoro character is just the com- ) = d > c ¯ τ y 2 h , c , with the sum replaced by an integral with a certain spectral density | 2 ∑ π = ) = 1 τ τ τ ¯ 4 ( | τ − c 24 , c Z − π τ 4 ( is the space of Virasoro descendants of ≥ Z − h h exp V . Modular invariance amounts to h ˜ h d exp χ h h ∑ , and d τ y i h π ∑ √ 2 | is the number of primaries of weight e 1 τ ˜ | h , = h d q , ) h ( partition function can be written as where and write the above equation as Obviously, this is possible only if both sides are independent of This is in fact notof possible primaries for of a weight discrete spectrumρ of primaries. Rather, we need a continuous set of course satisfied if we only have scalar primaries, where The first equality is equivalent to and using the modular property of eta function To make things clearer, let us define plex conjugate of Aspects of 2D CFTs where representation of a primary of weight dants, which is guaranteed for For now, let us assume that ˜ PoS(TASI2017)003 (3.62) (3.63) (3.64) (3.60) (3.65) (3.59) (3.61) Xi Yin ’s? Re- i P . . 2

, ) z )) j − 1. Some additional    P 2 ) 2   j

;1 ≥ ) 2 i − z iP t 2 n P ; 3 2 , b t 2 2 P + − m P x ( 2 + b 4 + − 2 Q ϒ Virasoro descendants sinh P 2 2 Q ) 0 or 4 j , 2 = Q tb , and the OPE takes the form ) + ) ) + iP x h ≤ 1 h 3 ( 0 2 , = 2 P . b , n ( P ( 4 ( ) 3 ) , , h b sinh i 0 ϒ h P 2 x , . 1 , ϒ P m ( 4 V P + sinh 1 − 2 b , 3 h h x dP − P b 2 p 1 , Q 2 ϒ , 2 − 3 P ∞ ( 2 P b t ( bx + 0 h b is an entire analytic function on the complex ( ) h 2 2 2 − , Z , x C ϒ P ) δ 2 e − 3 1 2 x 1 1 h 2 ∝ , h ( − − , 3 = b = ) ( b 2 ) j 1 b 1 ∏ ) such that this equation is obeyed for all c x i P ( x h 18 ϒ h ( ) 2 ) ( γ ( , bx b F )) 3 0 − c − (

ρ 3 b , for integers ( P ϒ 2 0 2 γ ) P , b Q F 2 P Q 2 P

) = dh / − V + , P 1 ) n | 1 ) ) = 4 , , then 2 z ∞ − ) = − 24 P 1 | x 1 c 2 P   P + b b , ( ) , P ) Z P 4 P 3 1 ( − t 0 + P + + dt P V P ( + C , mb 1 , h ( x x Q 0 b . It is natural to normalize the two-point functions by 3 ∞ 2 2 P ( ( ( P 0 4 C ϒ P ( P b b b Q = ) i ( Z V , . For generic ϒ ϒ ϒ 1 P x C + = ,    Q P ) 2 ( 2 h P Q . P < , C ) ( , 3 2 exp ) x b 3 P x P ϒ , and ( dP − ( 1 2 ) = 1 ∞ P ( Re x 0 ( Q ) = ( Z Γ 6 dPC 3 b < / P ϒ ∞ ) ∼ + , 0 dPC x 2 ) 1 ( Z ∞ P 0 Γ , 0 ( = = is the Barnes double Gamma function, defined as the analytic continuation of 1 2 Z . P P c ≡ ) ( , which we denote by P V x ) ) ( P C ¯ x z b ( , z ϒ γ ( 1 It turns out that with this set of structure constants, the modular covariance of torus 1-point Let us suppose that there aren’t any further degeneracies, so that the scalar primaries are la- P V -plane, with simple zeroes at x markably, the answer is yes, andsolution appears to is be known unique as (up the togiven Dorn-Otto-Zamolodchikov-Zamolodchikov rescaling by (DOZZ) by [12–15] an structure overall constants, constant). The Are there any real valued functions where from the domain 0 useful properties are functions are also obeyed [16].ture The constants scalar-only define spectrum what of is primaries known together as the with Liouville DOZZ CFT struc- [15, 17]. The structure constants is now a symmetric function where beled by Aspects of 2D CFTs If we write Thus, we learn thatparameter the spectral density of primaries must be a flat distribution in terms of the The key consistency condition, as fartion as the sphere correlators are concerned, is the crossing equa- PoS(TASI2017)003 (4.2) (4.1) (3.68) (3.69) (3.66) (3.67) Xi Yin into its ) τ , σ ( . If we ignore 0 φ φ b 2 e πµ , 2 . z φ b ) = 2 ln ¯ 0 anti-holomorphic currents, e , φ i j 0 R ( δ n φ πµ ) V . iP 4 1 ) 2 . ¯ ( z − P ( + ∼ − . e U ` V R ) 1 2 φ σ X 0 ) in ( ¯ ∂ j . P e R QR ( ) 2 X S τ P ) + ( ¯ ) = z 2 n ¯ + z ( φ ( φ i 0 n R = ` φ ˜ ∞ j X iP − φ∂ c ∞ 2 12 ∑ 19 = m , e , n z ∂ ) 2 1 − z − ( ln ˜ mn h ) i L ) = g i j P X + τ ( δ , ∂ h S g σ ( √ ∼ z ∼ − ) = φ ) 2 z holomorphic currents and ) 0 ( d 0 i L φ j ( n ( j Z ) L P 1 X π ( Ψ 1 ) 4 z U ( in particular is subject to the potential i 0, which is an approximation that could be justified for lowest excitations L = ) X L τ 6= ( S n 0 φ with region, the energy eigenstates are scattering states described by asymptotic wave n ∞ φ is an arbitrary positive constant. On the cylinder, we may decompose → − 0 µ φ We begin with a set of Now we briefly recap some known constructions of unitary CFTs in two dimensions. A basic class of CFTs are constructed from the holomorphic and anti-holomorphic components The zero mode The Liouville CFT may be defined in more conventional terms by say canonical quantization, Of course, these aresingle-valued not and associative single OPE, such valued, that and the operator the spectrum idea is also is modular to invariant. build operators from these that obey 4.1 Narain lattice of free bosons, which may be defined by the OPEs whose energy is given by These are the states on the cylinder that map to the primaries 4. A brief survey of known 2D CFTs where Fourier modes, Figure 8: The scalar primary infield Liouville space. CFT describes a scattering state off the Liouville wall in the oscillators in the starting from the classical action functions of the form Aspects of 2D CFTs PoS(TASI2017)003 0 h 1 m 1, , L ⊕ 1 . In ≥ Γ ) (4.5) (4.6) (4.4) (4.3) , then ¯ τ lattice Xi Yin m ⊕ Γ / , ) k 1 n 1 ⊕ 8 , − Γ 1 , ( τ takes value in / of the orbifold 1 k -invariant states ] (but not always, g − [ G ( is a symmetry, the G g 1 H 2 case there are two 0 mod 8. If be a basis of Z G a = ≡ e k ) = m are called twisted sectors. ¯ τ channels of the four-point ] is equivalent to , the torus partition function − associated with an oriented t g [ τ n is the signature . Since h g g Γ ( g L 1 Z g 1 H , Z 1 and is the space of , commute. Γ s h ] g 1 h Z [ . g , 1, then ] C Z H and , g ∈ ∑ [ : and must obey ≥ h 8 ∈ R | g H E Γ 0 X R g m ] · 1 k , g R C [ · | k ik R , let us denote , + k k L = G X 8 , it is often possible to gauge · ˜ c − . . For the definite signature even self-dual lattice 24 L G 1 of 0 L M ik = 20 G . Consistency of OPE requires that − k e σ 0 h m , may meet at a point in cyclic order provided that · ¯ L . In particular, , : m in ¯ n n L q around the spatial circle and another line operator is even and self-dual [18]. Let G , = k g − c R 24 g conjugacy class R and Γ n L k − ≡ , ··· 0 = g L 0 , L is the only such lattice, and in the k k 1 q V 8 ] may intersect when , we can define a line operator ◦ g H [ is the root lattice of Γ = h k G . H i L 8 ∈ , 16 Γ i Tr g g Γ L and 1 case g and , such that the states propagating through the handle on the two sides L = Σ 8 k Γ ⊕ 8 is viewed as a vector in Γ with integral inner product are topological, in the sense that they can be continuously deformed (allowing ) , in the m R g k , k n L 8 , R L is an even unimodular matrix. Such a lattice = k b 1. In particular, twisted operator. By the crossing symmetry between ⊂ e n is the commutant (centralizer) of , and a given element 1 ◦ Γ = ( = g Σ − a k n C is of indefinite signature, say with e g are identified up to transformation by the symmetry action of g . Several line operators The twisted sectors may be formally constructed as follows. For the CFT on a Riemann Generally, the twisted sector spectrum can be determined by If a CFT admits a discrete symmetry group For a pair of commuting elements g Γ ≡ Σ L ··· of rank and ab 1 where the sum is over conjugacy classes of surface with pairing matrix given by the Pauli matrix of the original CFT, called the untwisted sectors, whereas the other loop around a handle of where of Γ splitting and joining) without changingon the partition function or correlation function of the CFT Modular invariance further demands that i.e. up to lattice isomorphic, where inequivalent ones, 4.2 Orbifolds due to ’t Hooft anomaly).operators This (or construction states is on known the as cylinder) orbifold of the [19]. orbifold The CFT Hilbert takes space the form of local line operators g CFT is related by a similar way, one can determineg the 3-point function between an untwisted operator and a pair of a lattice A of the original CFT witharound a the line Euclidean operator time circle. The partition function of the twisted sector Aspects of 2D CFTs Next, consider current algebra primaries of the form where PoS(TASI2017)003 3 g 2 g , (4.8) (4.7) 1 in the g Xi Yin , CFT is 4 i twisted g 1 ) − 24 C h 1 1 L φ − − 1 ) 1 g 3 − g 1 g 0 based on the 2 − 2 φ g ( g h . Now passing to , = , 3 twisted operators, 9 φ . g 2 c g , 1 g 2 φ g , − 2 , connected by topo- h ··· . 1 i g C g 4 h g , as illustrated in Figure g + ( 2 1 Z g φ g 2 3 24 and ˜ and q Z g 2 φ = 2 g g , with the special property that c φ ) 1 0 g , φ h 24 21493760 ( + channel involves , q t 720 twisted sector, i.e. − 1 ) 196884 g τ , the twisted field correlator channel conformal block decomposition involves ( + 21 j h s , g 744 ) = τ , whereas the + ( 2 1 g Z 1. The 1 − g , q 4 = g , 4 = 3 g g 3 24 C 3 g 1 ) 2 − )) τ ) g τ ( 2 1 ( g twisted sector operators, we can extract the 4 1 g E η g -invariant 1 ( j ( , 1 − 2 g g , , 1 is an even self-dual lattice of signature 1 g ) = -twisted torus character of the − 2 τ C 2 24 ( g g j L , 2 . g 24 , 1 L g is the elliptic ) τ ( j -channel and considering the identity channel in the OPE of the As a nontrivial example, consider the Narain lattice CFT with u . Through this procedure, all correlators of the orbifold CFT can in principle be determined channel OPE limit determines the most general twisted torus characters (the notation is schematic,twisted as sectors). we only labeled the primaries in the structure constants by their operator structure constants. This is a special case of the relation shown in Figure structure constants function of logical line operators with u Leech lattice it admits no lattice vectors of length squared equal to 2. The torus partition function the unambiguously. where 10 Figure 10: For a pair of commuting elements the Figure 9: Sphere 4-point function of twisted sector operators Aspects of 2D CFTs one can obtain the PoS(TASI2017)003 (4.9) (4.13) (4.12) (4.10) (4.11) Xi Yin , where ab and anti- ) kd 0 . , = ψ 1 to be invariant ( 12 ab ) τ )) κ / τ 1 ( 1 . In compact CFTs, 3 − G θ cannot be consistently ( ( , g 1 g Z − dim of weight 2 , ) G , . . One can define a conserved ) 12 z is proportional to an invariant G is expected to contain a set of z ) = G ( 12 )) ··· − T 2 for a long root τ a , ab τ G 1 dim . ( j 1 ( w ) κ 1 ) g k . = 4 ( τ 12 ) τ Z θ b ( = ( 0 ( − 2 ( ) η ψ ) a is positive definite, which implies that c ) θ a τ 2 j τ , and generated by + z ( ( ( ab c ψ G n 4 a η 12 j κ is thus θ ab z ab 2 Z = ) )) 2 d i f τ w 1 ) Z 24 ( ( ≡ τ ) / 2 a + ( n j , the reflection on all lattice elements, i.e. the 2 θ ) = g g 24 q g ( 1 2 τ ab ψ Z L ab z 22 / + κ d 1 , we expect that 1 12 ) + z ( n ∼ − . Furthermore, their OPE must take the form of current . The corresponding torus character is τ )) ( 1 ) → ) i ( L g 1 obeys τ lim ∞ 1 = w 0 1 g ( X ∏ n , Z ( ) Z 0 ab b 1 η h j ( ( d − ) q ) = + 11 z 1 ) + → − is a simple compact Lie group, we can write in fact admits a very large discrete symmetry group, that is ( τ of order generated by 2 k τ i ( a L ( ( j 2 1 G g g g 1 + 2 ) = X 24 as local primary operators, Z Z Z τ L . / ( a µ g current algebra for a simple Lie group 1 j 360 24 of weight ) = Z ) + k z L term is absent, indicating that there are no conserved spin-1 currents ) 744 ( τ − ¯ z ( 0 ) G ( − 1 1 a q τ ˜ T ) j Z ( j τ ( j 1 2 2 1 the level = = symmetry of k . If this fails, the orbifolding by the ) = b n G 2 τ ( Z + 2 τ Z / are the structure constants of the Lie algebra of 24 c → L Z ab τ f A unitary 2D CFT that admits a continuous symmetry group Denote by Note that given an element is normalized such that its inverse is semi-simple and compact. If ab conserved Noether currents Its modular S-transform is under constructed. This is an example194 of conjugacy discrete classes, ’t out Hooft of anomaly. which For 56 instance, cannot the be monster used4.3 group to has orbifold WZW due and to coset such models anomalies. such currents are necessarilyholomorphic operators given by holomorphicalgebra, operators say in the holomorphic case, Here spin-2 primary operator, known the Sugawara stress-energy tensor The torus partition function of the orbifold CFT In particular, the order in this CFT. Theisomorphic orbifold to the largestdiscovered sporadic by simple Frenkel, group, Lepowsky, alsoCFT. and known Meurman as [20], the is monster hence group. also referred This to CFT, as the monster bilinear form on the Lie algebra.G Unitarity demands that d reflection of all 24 chiral bosons Aspects of 2D CFTs Consider the PoS(TASI2017)003 , this is the j (4.14) (4.15) (4.16) Xi Yin θ . Thus, the n runs over all , one may try b R with a generic H . If it coincides , where ) G . Let us focus on k z ) c ( s a . The normalization , . minimal model with ) 0 i ≤ O ab k z ( , where ( WZW model produces θ for integer A , hd G associated with a pair of T ) b G 2 λ ˜ h ) h is labeled by a spin ¯ , z = , n z R c ( , ˜ i + ) i bd of the h 0 f φ obeys ( ( d c G λ ac O f c ⊂ h . − + ) H b 0 h ( − , with the OPE j . a , is equivalent to the h φ G is greater than or equal to G 1 h − a i j z , governed by a current algebra c z + t of k c + ) H dim form a representation of the holomorphic vertex k R ∼ ab 2 k 23 ( ˜ ) A h currents beyond the Virasoro algebra, these currents 0 c ( = or anti-holomorphic of weight s i SU ∑ , known as the Wess-Zumino-Witten (WZW) model ) φ / G ) ) ) ¯ c 0 z , where the representation z 1 , ( ) ) = ( ) s 2 G a 0 ( 2 ( j ( e ( T is built from a set of lowest weight current algebra primaries b k SU O SU ) OPE takes the form of that of a stress-energy tensor, with central b G z and = ) whose highest weight vector × ( of the left and right current algebra group 0 a k ) ( G ) z G ) O ( G 2 R ( G , T can only contain operators of weight ) T R ) z ( SU ˜ ( h . For ( , G G h T ( coset model [23, 24]. The construction of the full operator spectrum of the . . would also be the same as the CFT stress-energy tensor H k 2 ) 3 / G + ≤ G T k j 6 )( and produce a new CFT. Gauging a subgroup 2 of weight are non-negative integers. The OPE of a holomorphic operator + form a representation of the anti-holomorphic vertex algebra. The operator spectrum of representations H k φ is the dual Coxeter number, defined by the relation ( a WZW model contains precisely one set of primary fields , then h h h k is such that the G − c b If a unitary CFT admits conserved spin- Now given a CFT with continuous symmetry It is indeed possible to construct a fully consistent CFT whose holomorphic and anti-holomorphic G G 1 T that transform in an irreducible representation = i are necessarily holomorphic of weight 4.4 Rational CFTs identical condition is to gauge The central charge of thewith CFT stress-energy tensor The irreducible representations of highest root vector of c the holomorphic operators for now.giving rise The to OPEs a of holomorphic thegeneral (i.e. holomorphic form operators chiral) close vertex on algebra their that own, extends the Virasoro algebra, of the where operator stress-energy tensors are Aspects of 2D CFTs where of charge [21, 22]. To do so,ganized we not need only to as construct representationscurrent the algebra of spectrum (as the of the Virasoro local Virasoro algebra, generatorsresentation operators, can of but which be the as may built current representations now out algebra of of beφ the the or- currents). bigger Each irreducible rep- the so-called coset model is ratherfamily elaborate, of and coset models, we will not discuss it in detail here. Note that a particular set of operators with anti-holomorphic weight algebra, which may or mayweight not be irreducible. Likewise, the set of operators with holomorphic PoS(TASI2017)003 , ) Z , 2 (4.20) (4.17) (4.18) (4.19) (4.21) (4.22) Xi Yin ( transform into linear i χ , , 1 RCFTs admit higher spin τ ) . ¯ / τ ) > 1 ( ˜ j c 24 c χ − 1, and it follows that the central ˜ j i ˜ ∗ i h ( e → − S i = , π 6 τ ) 2 ¯ e τ T ) = ( i j , ˜ ¯ i τ δ c 24 . / χ det of the holomorphic vertex operator algebra . I ) 1 ˜ j − i = j τ 0 = − R ( = L n i j i ( 2 ˜ q i 3 T χ i ) ˜ ) i = χ R i 3 ˜ j ) n , ˜ ∗ i 24 Tr ) ST ˜ i , e S . In other words, all , i ) τ ) ST ∑ i j ( ( τ j Z S = ( ( ˜ ) = i , and that under , j i χ 2 τ 2 det n Z ) = χ , ( i j S ( ˜ ( i ¯ i τ T i j 3 , χ ∈ χ orbifold monster CFT we have discussed earlier. Even the S ) is a finite sum. Closure of OPE then requires all weights − τ j ) . The torus partition function of the CFT can be written as 2 ( i ˜ h 2 1 must involve a strictly larger holomorphic vertex operator ) = Z Z ) must form a finite dimensional representation of PSL e R S 1 ) = e S 4.18 . The modular invariance of the torus partition function demands − > τ ) i + of / det τ c h 1 ) τ ( ( ˜ j ¯ and ( is defined as τ i − ( i χ ˜ i ( χ e ˜ j T i ∗ R ˜ i χ χ e T CFT or its 24 ) = L 1 1, we have + + ¯ τ τ ( ˜ i χ → τ are non-negative integer degeneracies. A rational conformal field theory (RCFT) is one (and similarly for must be a rational number as well. ˜ i is nonzero only when i of the anti-holomorphic vertex algebra. c S ˜ n j ˜ j to be rational numbers. i n Under The torus character of There isn’t a complete classification of RCFTs, even though all explicitly known examples are Note that all RCFTs with e ) R ˜ h and , h combinations of themselves, and so do such that that and likewise the character where that contains finitely many representationsator of algebra the in holomorphic its and spectrum, anti-holomorphic( i.e. vertex oper- ( and similarly Aspects of 2D CFTs the CFT can thus be decomposed intoand representations symmetries beyond the Virasoro algebra. T algebra than Virasoro, because thefinite dimensional non-degenerate representations torus of Virasoro PSL characters do not transform in produced by coset models or Narainin lattice their CFT moduli and their space products ofmorphic and exactly orbifolds, CFTs marginal at whose special deformations. entire points Thealgebra, operator such simplest spectrum as class consists the of of RCFTs a are single mero- holomorphic vertex operator namely charge meromorphic CFTs are far from being fully classified. This in particular implies that PoS(TASI2017)003 . ) 1 )] ( X ( U (4.25) (4.23) (4.27) (4.26) (4.24) Xi Yin / Φ ) , 2 ) ( X ( SU i j G [ e f f S . . up to the leading two ) ) X Φ X ( ( β , Φ r Φ ) . 2 )] 2 X with a nontrivial 3-form flux , ( , X and ∇ j ( G ··· . This term appears for instance 2 1 gR ··· i 0. It turns out that the latter are G Φ X i j + b X √ coset model. β + β − 0 limit. In the end one arrives at an = ` ∂ [ i ) σ ) ) Φ 2 2 Φ 1 i jk j X ( X → ( d a ( R β R ∇ ` ε ∂ i U dimensions), and compute perturbatively Φ 1 4 Z / = ) i jk ab ∇ ε ) 2 π ( g 2 − R G 1 2 i j ) − 8 r ( R 4 1 β + 2 X 4 ε j ( + SL m + j ` = i j X + 2 a jk j ) X d 25 ∂ b i R gG X ∂ m i a X √ ` ∇Φ a ∂ k X i i σ ∂ a 2 R X ) + ( ∂ a d 1 2 X ab Φ ∂ ( ) 2 ε Z + G i j ) X ∇ π i j β 1 ( X 4 − ( R i j 1 2 i j -field, one can straightforwardly regularize the path integral using G = = = B B 2 ε = S G Φ σ i j a 2 − β β -field, is parity odd. It appears in the Lagrangian description of the a ) with a suitable regularizations scheme, or a Wilsonian effective action d B T = Z a a 4.23 π 1 T 4 stands for a suitable operator regularization, and r ] is a constant; this term vanishes in flat 2D spacetime and serves to modify the stress-energy . The second term involves the two-dimensional scalar curvature, and would be topological ··· [ ) X dB At least in the absence of A two-dimensional nonlinear sigma model (NLSM), defined by the Lagrangian ( = Φ equivalent to the target space equation of motion based on an effective action To turn this into aRHS meaningful in operator terms of equation renormalized in operatorsoperator the and equation quantum then of take theory, the the one form needs to rewrite the While these equations can be solved locally inespecially the when target the space, the target existence space of is the compact solution globally is far from obvious. is classically Weyl invariant.in It a would manner give that rise respects toits conformal a renormalization symmetry, but CFT group this if flow is thepath by far integral path analyzing from based integral on guaranteed. either ( can the We be need 1PI regularized to quantum study effective action defined the nontrivial perturbative orders are given by [25] Conformal invariance is equivalent to the condition the 1PI effective action, or more directly, the trace of the stress-energy tensor dimensional regularization (formally working in where including all possible relevant and marginal operators.as there The are Wilsonian infinitely picture many is relevant rather operatorsdeformations murky in are here, the of perturbation theory. the Other form classically marginal Aspects of 2D CFTs 4.5 NLSM H if in the NLSM description of cosetmodel, models, and the the simplest “cigar nontrivial CFT" examples which being is the equivalent to tensor hence modifying conformal transformations of the fields The first term above, the WZW model, which amounts to a NLSM on the group manifold PoS(TASI2017)003 M (4.29) (4.28) Xi Yin turned on, ) 1, 2, and 4. X ( = currents. There Φ 3 2 N 4 SCA which con- = 4 SCA which contains that is a Virasoro primary. N = of spin ) z ( N N G . ) 0 ( , 2 3 T . In the GSO projected theory, the half- r z 2 F + r G ) z 0 which amounts to the vanishing of 1-loop + − ν ( 3 = c + z 2 Z 3 ∑ i j ∈ 26 r R ∼ anti-commuting field . To do so consistently with modular invariance, one ) F 2 3 1 SCA, which contains in addition to the stress-energy 0 ) ) = ( z − ( = G ( ) G z ( N G , a single spin c , could the metric be deformed, with a dilaton profile ) X ( current algebra. i j ) “R-symmetry" current algebra, and the large G 1 ( ) 2 U ( × ) is solved to all order? I do not know the answer to this question, although, the SU ) 2 ( of central charge 4 SCA come in two qualitatively different types, the small 4.27 ) , which is an integer defined mod 2. The projection onto integer spin operators amounts z SU = ( F × T ) N CFTs with supersymmetries are interesting for a number of reasons, one of being that such For instance, the leading order condition Now we will discuss a few examples of SCA (by no means exhaustive) and CFTs that realize The superconformal algebra extends the bosonic Virasoro algebra, and so a superconformal 2 ( The only new independent OPE relation is Acting on the states at the origin it is useful to work with the Laurent series answer to the superconformal version of this question is yes, as4.6 we will Superconformal discuss symmetry next. theories are abundant and theirtry existence non-renormalization are properties. often easy Poincarésymmetry, to and supersymmetry establish together does due they not generate to a commute variousgebra superconformal supersymme- with (SCA) symmetry conformal are group. characterized The superconformal (but al- not uniquely) by the number Aspects of 2D CFTs beta function of the NLSMtopological demands restriction the on target the space to targetadmit be Ricci space. Ricci flat flat. A metrics are class Thiswith Calabi-Yau of Ricci puts manifolds. flat even highly metric One dimensional nontrivial can compactsuch ask, manifolds that given that a ( Calabi-Yau manifold are well known CFTs that admit holomorphic and anti-holomorphic SCAs of The tains a single must include twisted sector states with respect to tensor such symmetries. We begin with the SU field theory (SCFT) can be viewedwith as an an important subtlety: ordinary bosonic the superconformal CFT algebra withseems involves currents extra to of higher half-integer violate spin spin, the which symmetries, momentumThere quantization are two on ways the to make cylinder, the and theoryof also consistent: modular invariance spoils either slightly, we by modular somehow allowing relax invariance. ourselves the tospectrum, conventional work notion or with we half-integer spin insist operators on in thefrom the integer the spin spectrum. condition The and latter projectachieved is as out known follows. the as The half-integer half-integer the spin spin Gliozzi-Scherk-Olive operators operators (GSO)number are anti-commuting projection fields, [26], and and has odd is fermion to projection onto states invariant under integer spin superconformal currents arenonetheless be not used strictly to speaking organized the part spectrum of according the to representations spectrum, of but the they SCA. can PoS(TASI2017)003 , ) α ˜ = − 2 ψ , , 3 2 k 2 , has a ( and a ψ (4.32) (4.31) (4.30) Xi Yin 2 = k ) ) + 3 z z c k ( ( , where ± G ) = z G ( c A α is an extra index G A , 0 , n OPEs are non-singular. δ . c , and 6 − ) k 2 currents 0 G 6 η ( , which connects NS and R 3 2 − J = WZW model, but they are not η + G ∂ 1 minimal models can also be k n c z 1 ) J 1 minimal models with 2 SCA" [27, 28]. 2 = η . Some other examples are the free ( and anti-commuting fields 2 = ) + = + 3 2 + + 0 SU k N n 2 minimal models, with , with ( G ) ) L N N T 2 z + = ( ( z , 2 i G , which is not part of the superconformal → 2 J z c n SU N out 3 2 SCA admits an outer automorphism known ) + L / ) ) 0 2 2 ( ∼ ( = . The , ) J ) 0 ) 2 , z 2 27 0 n ( 2 SU ( z ( N δ J such that its OPE takes the form J c 3 + ) 2 SCA together with the spectral flow symmetry (with 4 SCA. It has four spin SU ) R-currents SCFT are the z z 3 η ( c k ( × = z = ) , the ) J 2 J k 3 n 1 + 2 ) J , ( n 2 N 1 , J N ∼ ( n , deformed to leading order by a marginal operator of the form 2 SCFT include ) SU L ˜ . 0 → ψ = ( , SU ¯ ( = , ϕ n ( ± r − β 2 SCA. It has a pair of spin J ψ 0. Such twisted sector operators are known as Ramond (R) sector G − N G N e = ) , = z − 1 with respect to η ( ˜ ν 2 SCFTs do admit spectral flow symmetry. An example is the ψ ± ± + N ± r − G ). This is known as the “extended G = ψ . Interestingly, the GSO projected η + → ··· N ) with . We may normalize , ) ± r 2 βϕ z , ( G − 1 J 4.29 e 1 Liouville CFT which at the Lagrangian level may be viewed as a linear dilaton + in the absence of branch cuts. In this case, the operator at the origin is said to be = carry charge ˜ ψ = 1 2 k ± + . One may also extend the spectral flow to half-integer 2 Liouville CFT [29], which at the Lagrangian level may be viewed as a compact boson , G ψ = ) Z N 4 . = . Note that these have the same central charge as the ν + theory consisting of a compact boson and a pair of chiral and anti-chiral free fermions ∈ is attached. This topological line operator creates a branch cut, and as a result R-current k βφ Certain “nice" Some other examples of Finally, let us discuss the small Next, let us consider Some basic examples of ··· 3 2 )( ) 12 F e η N , 2 ) 1 ˜ 2 ψ ( + = , k − ( integer or half-integer as the spectral flow, for sectors. However, the spectral flow need not be a symmetry of theNLSM SCFT. on a Calabi-Yau target space. Inthe this case, bigger we symmetry may generated organize by the the operator spectrum according that transforms under an outer automorphism c In terms of the Laurent modes is a doublet index with respect the realized as bosonic coset models and the CFT together with free fermion ψ and then The only new nontrivial OPE relation is 1 the same CFTs (in particular, theythe have different operator spectra).and Another a nontrivial linear example is dilaton, withthe a form pair of fermions, deformed to leading order by a marginal operator of Aspects of 2D CFTs where in the Neveu-Schwarz (NS) sector.projection In requires particular, the twisted identity sector operator( operator is on the NS which sector. aLaurent The topological series GSO line ( operator associatedoperators. with U PoS(TASI2017)003 ) 4 2 , = 2 and is (4.34) (4.33) (4.35) Xi Yin N 1 2 = ( + − G N , 0 . , n is an integer, and δ ) k 0 k 2 ( i η J ∂ + level z 1 3 n ) 2 SCA, the small J 2 αβ in a finite range [30]. In the i 4 SCA are supersymmetric 4 Liouville theory [31], of 1 spectral flow on the holo- η ( = is annihilated by 2 σ ` = = = Ω SU + , while the BPS primaries have N 1 η ) + O n 2 spin 0 − N L N k ( R , do not appear). T ) → ¯ i 2 z 2 . n ( d ··· ψ i L , αβ 1 2 ψ SU ε , . Further, , and 0 d η ··· ¯ i 2 1 z ) + i and = -fold with a suitable family of metric and pos- + WZW model. 0 ± n d ` h ψ ( 1 J i d i J − 28 k 2 ··· charge 4 SCA is relatively simple to describe: the unitary → 2 ) 1 z and i 2 R ± n ( = ) Ω J αβ i 2 SCA is relatively simple to describe [27,28]. Indepen- 1 (note that ¯ ( > ` σ ∼ SU = , N 2 SCA. In particular, the h Ω U 0 Ω + , n = O N δ αβ , from which one can construct the operator k and d ε i N η 2 3 d c z dz 2 + 3 corrected Ricci flatness condition) defines a family of = 3 n 0 J h would relate NS and R sectors). Rather nontrivially, here the spectral α AB η → ε ∧ ··· ∧ 4 SCA itself (in other words, unlike the . 3 n 1 ∼ i k 2 J = , ) , that may be constructed by marginally deforming the CFT consisting of a dz 0 k , and extended d ( i 6 η ··· of weight N B A d ··· , ± 1 β 3 ± r 2 1 i = Ω , G G c Ω O ) = 0 ! z (half-integer 1 d c ( → = must be an integer multiple of 6. The only new nontrivial OPE relation is A η 4 SCA also admits a spectral flow map ` A = α c ± r G = Ω -field (that solve the G 4 SCA and seeing that it is invariant under spectral flow. 2 SCA is quite delicate [32,33], including a variety of different shortened representations, the and B The representation theory of small The supersymmetric NLSM on a Calabi-Yau A class of nontrivial examples of SCFTs that admit small Our knowledge of the Calabi-Yau models as SCFTs so far is largely limited to the moduli space Let us briefly discuss the operator spectrum. While the general representation theory of the N ` = = = -form for integer SCA is automatically “extended").N This can be shown by studying the vacuum character of the NS sector, the non-BPS primaries have flow is part of the representations are labeled by primaries of weight sibly a chiral primary. It hasd a natural interpretation in the NLSM description: there is a holomorphic linear dilaton, four free fermions, and an 4.7 Calabi-Yau models The NLSM on hyperKähler manifolds.central charge There is also a family of morphic NS sector maps themary identity operator operator (NS sector vacuum state) to a superconformal pri- Aspects of 2D CFTs symmetry and generally not a symmetrytherefore of the SCFT. Note that the h SCFTs with In the interacting NLSM the RHSrenormalized a due priori to should the be holomorphy regularized, of but in fact such an operator is not of exactly marginal deformations andthe the non-BPS spectrum operator of spectrum BPS obtained operators, from(called and conformal Gepner some bootstrap. points) crude Exceptions in bounds aredescription the on special as moduli points an space orbifold of where product the of SCFT minimal models. happens toN be rational and admit a representation theory of the extended dently on the holomorphic and anti-holomorphic side, there are non-BPS (massive) representations PoS(TASI2017)003 , - 1 . 4 1 , 1 2 2 , − 1 φ (4.38) (4.39) (4.40) (4.36) (4.37) − Xi Yin and , φ 1 , 1 2 0 odd), and φ , , . and d 0 6 ) 3 2 1 2 E , . > (if transformation 0 , ) − 1 4 φ n ) 2 1 Q 2 − E q Z − d . 1 n , , , ) − q 2 ) y ) y ( 2 1 is given by 1 1 2 2 1 . ) − ) 0 q n − + y 2 J z , n q ) y , 1 q c − q n + τ ( c 1 2 d 24 − )( q ( 1 Q . The space of weak Jacobi − 2 1 b 1 − χ ( 2 y even), and saturates the BPS d 2 − 0 ( − )( has PSL − 2 ) c 6 NS L τ n z 1 2 ± 1 n d + q , χ 2 , d ( χ − q 0 i 1 ¯ yq n J π i (if − 2 BPS )( π ··· + yq 2 d ) = e 1 2 , e 1 1 , y Tr ( 2 c − ( + , 24 c n q 1 ± 1 1 ≡ − ) = ( , 0 2 ( ∞ − = yq ) τ ¯ L ∏ 1 R n d Q 1 y ¯ / q χ + 1 , ∞ . Due to spectral flow symmetry (by integer = z ± 0 ∏ n 0 J , , q + ¯ 1 ±b J ( y 0 τ ( ) m , Q , ) c / 1 2 1 24 + 1 ) NS . Furthermore, Q = 1 are two weak Jacobi forms of index 1. For half- ∞ = + y m − − ··· τ ∏ ) χ n 0 1 2 d m 1 , − 29 1 Q , ( L ( q 2 Q 2 + − y q , yq χ d + − z m ± ( q φ m , RR ( y + + ) , 2 1 , 1 Q 1 1 → 1 2 ) Tr , m dependence drops out, due to cancelation between excited − mQ − ± z 0 1 z + )( c ¯ 6 d , , 0 2 τ + − φ ( , 1 2 m J 2 d 0 τ y 2 Q y , ) = ( m − q z yq 1 0 c − ) 24 m = χ , 2 − mQ 1 2 in the range c d 24 τ − + + < 1 and yq q 0 Q ( -BPS, while an operator that is BPS on one side and non-BPS on 2 q + 1 Q L 2 1 NS h -BPS. m ) = χ Q ∞ + + − q 1 z 1 4 − z 2 2 1 , − 1 ch ∞ . The entire weight and charge content of the non-BPS representation d ∑ ( , = 1 | − BPS ) q 2 m Q q → | 1 − + spectral flow on a representation of shifted primary weight and R-charge, ( Z c ) = 24 − z . Note that the τ ∈ y 1 2 Z y > ∑ iz non ( − , m , ∈ is a weak Jacobi form of weight 0 and index h π q ∑ h χ q 2 m = c Tr q ( 24 ) ( e in the range z c Q 24 Q − η , , ≡ = R h 2 NS = Q − − Q τ ) ( χ y q q y ch , , χ τ . The corresponding NS character q i are Eisenstein series and | ( π ) = ) = ) = 2 Q 6 2 Q | y y y , e , , , E NS h , q q q = 4 = ( ( ( ch h E q is invariant under NS NS NS 0 Q Q There is a supersymmetric index that is very useful in constraining the spectrum of χ χ χ χ ), integer index, one can in addition make use of two other basic weak Jacobi forms η BPS operators, known as the ellipticsector genus [34]. whose To ground define state this index preservessector we supersymmetry. character must by work The in R the Ramond sector character is related to the NS The elliptic genus is defined as a character in the (R,R) sector, of the form where states with even and odd anti-holomorphic R-charge property That is to say, where A BPS representation oninteger the R-charge other hand contains a chiral or anti-chiral primary that carries an the other side is referred to as forms of integer index is spanned by polynomials in four basic Jacobi forms Aspects of 2D CFTs and BPS (massless) representations. A non-BPSthat representation carries contains a an superconformal integer primary R-charge obeys the BPS bound is summarized by the NS character bound An operator that belongssectors to is BPS loosely representations referred to in as both the holomorphic and anti-holomorphic PoS(TASI2017)003 X · i ik . e -point (4.43) (4.42) (4.41) Xi Yin O n . While ··· ) -th order ¯ = z n + i , z V q ( ) 2 O − y ’s). The , it is generally not 20 µ λ that break conformal + 0 limit of the X 1 O ’s. − i , → y k 4 SCA. i + k 1, one may deform the the- ) = 128 ∞ = ( − 0 ˜ . h n N ) -BPS multiplets must pair up and V z 1 4 = ) , 216 1 τ h ( ( + 1 3 2 . This requires the vanishing of the y , − 0 λ n 2 characters in the RR sector. The result φ V ) -BPS states in the K3 CFT, assuming the 128 ) gives rise to the vertex operator 1 1 4 = , 0 2 − ( O h 2 2 y − N − n and with weight 1 20 V , 30 2 1 1 # O h ) + ( OPE contains another marginal primary, a nontrivial i ¯ z 1 ) , i − 0 . There is a natural notion of metric on this deforma- z y ) = ( ( n into correlation functions. In a theory with Lagrangian ( z 2 i , O V ) τ + i . ) ¯ z ¯ ( z z 2 into extended , 3 2 , z λ 20 z ) d ( ( z CY , + O χ Z O ) for all τ -BPS multiplets that correspond to the deformation moduli of the y z 3 ( 1 1 2 2 2 3 2 − = , i ∏ d n 0 4.43 2) . This result has an interesting implication on the BPS spectrum: while R φ ) = 1 z *" λ R = − , . By matching the Witten index with the Euler characteristic of K3, one χ − τ 1 , ( 0 1 N − , φ ghosts, where a marginal primary 0 R 1 φ c χ , 2 . Matching the Witten index with the Euler characteristic gives the result 3 case, which includes Calabi-Yau 3-fold models, the elliptic genus must be pro- 2 case, where an example is the SCFT with K3 target space, the elliptic genus must 2 3 b = , 0 = = 2 3 ) = -BPS multiplets must contain either extra conserved higher spin currents (expected to φ , -integrals may be regularized by analytic continuation in the z i d 0 1 4 and d , z φ µ τ ( X K3 A useful way to understand the obstructions to conformal invariance at higher orders is to view Exactly marginal deformations are marginal deformations that lead to a continuous family When a CFT admits a marginal primary In the In the χ preserved at higher orders, due to thesymmetry. regularization of For coinciding operators instance, ifbeta the function is generated at order where the zero-momentum limit of ( the CFT as part ofbosons the worldsheet theory of a compactified bosonic string theory, along with free of CFTs, parameterized by the deformation parameter conformal invariance is preserved to first order in the deformation parameter of a massless scalar fieldobstruction in to Minkowskian spacetime conformal (parameterized invariance by is the captured by the simultaneous description, this would amount to deforming the Lagrangian density by the operator ory by inserting exp string tree level scattering amplitude SCFT, the tion space, given by two-point function of the marginal operators, known as the Zamolodchikov One can then decompose be absent at a generic pointcan on potentially the be conformal lifted manifold), as or non-BPS the multiplets under marginal deformations. 4.8 Exactly marginal deformations there are (extended is simply Aspects of 2D CFTs be proportional to derives [35] absence of extra conserved higher spin currents beyond the small portional to This can be used to fix the degeneracy of all PoS(TASI2017)003 , L R ) n ) z ) are are 1 ( ) 1 ( v ¯ 5 ¯ b z ( w ˜ j (4.45) (4.44) Xi Yin , U Q cannot a U z w j ( . λ is a chiral 2 1 + , and + . To see that + , 1 , regularized , ) . + φ 1 Q + c 2 1 ≥ − , . Φ − − c SCA and central r − − , (chiral, chiral) or ¯ φ ) G 1 + 2 1 4 ) 1 2 ( , 1 2 − − + − − , 4 ¯ , G . When + G 1 2 for all , where ) 2 1 ). In particular, the case . ( φ 4 on K3 (corresponding to 5 = ( + − c + r 1 2 1 2 . T 5 Q G − − c G ( + − 1 5 Q e N G G + Q Q 1 2 , the deformation by 1 1 )+ + up to a total derivative. Let us ¯ − b , then the D1-D5 CFTs labeled z Q − − ¯ , − ˜ j = 1 θ 0 5 ) ) , to a chiral super-parameter (also + ¯ k Q Q w w zG θ φ 0 1 , ’s among themselves that 2 λ , 1 2 ¯ w − z d + Q , + − ( , z ¯ R z + or Sym G = ( O 2 1 φ SCFT, which preserve the full supercon- λ ) 5 + + ( , 3 ) − − 1 2 Q + + 2 K G 1 , is annihilated by ( G Φ 2 ) 1 Q 2 + ( λ ¯ , + w − or 5 + , ) + Q w θ -dimensional conformal manifold isormorphic to φ 1 w 31 ( 2 R − Q , (corresponding to + n − , zd − L , then promoting z 4 + 2 ( n φ + T d , φ 2 1 1 2 + R ) + − . More generally, whenever a CFT admits a set of z − − ¯ φ on can be moved past ( G anti-holomorphic currents ¯ )) G 1 v 1 2 R ) . In some limits on the conformal manifold, the D1-D4 CFT SCFTs is that marginal deformations are necessarily exactly ] i R z + − k Q n 1 n ) ( π ( ) dz − , G 2 2 + 1 can only be renormalized holomorphically. This follows from O , L i ( G + 2 ) 2 I ) × ( λ ) ¯ U w + z ) , are both coprime with , w ( L ( + v ) w n ) = v ( 0 5 φ ( ¯ 1 2 w lie on the same conformal manifold. This implies that conformal NLSMs + Q , , and , O , which admits an , ) − − a represents a (chiral, anti-chiral) primary with R-charge z ) + ( + R w 0 1 0 5 j ¯ 0 n / ( G φ , − J ) Q Q , 1 2 L 1 2 ( , O R n + − − + 0 1 ) − − n Γ z φ ¯ , 0 G Q G ( L L ( ) and + n 2 = for a positive integer w ( ) G ( 5 )( represents a (chiral, chiral) primary with holomorphic and anti-holomorphic ) k O , and 1) or instanton number v O and z ) w 6 \ Q h ( ) ( ) + , 1 + self-dual Yang-Mills equation with instanton number , v w 5 v 1 R , = 5 i + n ∂ ) 1 , ∂ Q Q 5 c π 2 ( L Q φ , ( dz n 2 1 1 corresponds to the orbifold SCFT Sym 1 Q = = [ Γ Let us now discuss marginal deformations of a An interesting example that exhibits some of the rich phenomena that may occur on the confor- An important property of ( Q Q ( = I ( U 5 = inspect this in the (chiral, chiral) case: formal symmetry to first order. Such a deformation is based on weight this preserves superconformal symmetry, we needthe to superconformal show that current for any holomorphic function in a supersymmetric manner, writing the deformation in superspace as superfield whose lowest component is known as the “spurion"), and consider all possibleunder local RG. superspace integrals It that then could be follows generated from the non-singular OPE of mal manifold is the D1-D5 CFT [39–41], which is a SCFT with small holomorphic currents exactly marginal. charge be renormalized at all. marginal [37, 38], thereby giving risebasic to reason abundant is examples of that (super-)conformal when manifolds. we The deform the theory by say where we have made use of the assumption that coprime, the D1-D5 CFT isthat expected if to be compact. Argumentsby based on string dualities suggest Aspects of 2D CFTs metric [36]. Therefore, this moduliifold, space of sometimes exactly called marginal the deformationsthe conformal is a Narain manifold. Riemannian lattice man- Basic examples are free boson CFTs based on (chiral, anti-chiral) operators, through their descendants of the form where charge can be described by the supersymmetric NLSMto whose target space is the modulik space of solutions Q Aut PoS(TASI2017)003 Xi Yin respectively, critical three- vector models . This can be 3 ) / 5 4 ) i 2 π N + 2 = ( n − 6 )( c O e 1 through the truncated + n ( n and . Provided that there are ) − i 3 / X 1 i ). ( 3 π 2 = W e -fold product CFT has only two c AdS copies of the N symmetry, under which the weight N 3 Z , and for small n CFT deformed by an operator of the form φ 32 minimal model. The three-state Potts model contains is uncharged. The ) , with a superpotential ε i 4 transform with a phase X D ∗ , 4 supersymmetric version of LG models [53], describe by the σ A ) ( 2 , Virasoro algebra. It has a 2 ( 5 4 = c “energy operator" ) 2 5 and its conjugate , 2 5 ( σ primary ) 1 Under better control are the The third example is a class of “short" RG flows, analogous to the critical The first class of RG flows are integrable flows [42–47]. The simplest examples involve an The second class of RG flows are that of Landau-Ginzburg (LG) models. In the bosonic A CFT perturbed by a relevant operator will flow to a different CFT of smaller central charge 15 generates a fine tuned RG flow to the minimal model with , n under the renormalization group. This is an important tool for exploring the relations between 1 2 15 while the weight ( checked using conformal perturbationconformal theory space for approach by large directly diagonalizingmethod the [50–52]. deformed Hamiltonian with Rayleigh-Ritz Lagrangian of a set of chiral superfields no mass terms (i.e. quadraticof the terms superpotential in ensures the nontriviality fields) ofuseful to the in infrared begin fixed determining with, point. the the The conformal non-renormalization LGin manifold property model the as is IR well particularly CFT. as the structure constants of BPS operators in three dimensions. Let usstate Potts begin model, with also the known tensor as the product of RG flow to a massive theorya in factorized the S-matrix. IR, with In athe this finite UV case, number CFT of the is species QFT fully of resulting determinedthese elementary from by data, particles one the an and can IR integrable in particle relevant principle spectrumnamic perturbation determine and Bethe of the 2-body ansatz spectrum S-matrix (TBA), of and elements. the recover QFT From correlation the on functions UV a of CFT circle local from using operators the the may smallto thermody- be radius constructed a limit. from set the Furthermore, of form the factors, analyticityS-matrix the constraints [48, latter 49]. and subject crossing symmetry properties determined by the factorized case, it is understood that the noncompact free boson CFTs as well as constructingfixed potentially points new are CFTs. generally StartingRG beyond from flow the is the known reach UV to of CFT, be thedifficult conformal integrable infrared or to perturbation protected RG determine theory, by the symmetry and (typically infraredflows so supersymmetry), in CFT it unless two is precisely. dimensions the quite that are Here under we control. briefly mention a few classes of RG φ c Aspects of 2D CFTs on seemingly very different targetdeformations space through geometries a are strong somehow connected couplingFurthermore, by regime there exactly (where are marginal the points target inspectrum space above the becomes a moduli highly dimension space gap curved). where (thesestrings continuum that the can states D1-D5 move are CFT to the infinity develops holographic at a duals finite of continuous cost long of closed energy in global 4.9 RG flows nonzero spin Virasoro primaries thatdegenerate primaries are of built the from different holomorphic and anti-holomorphic PoS(TASI2017)003 I ∈ (5.2) (5.1) (4.46) Xi Yin ) ˜ h , h ( . I acting on the space 0 for all implies α ≥ The simplest example ) ¯ ] τ . Indeed, it appears that 6 ˜ h . / N χ 0 1 / h being the gauge coupling). expansion, and the operator − χ , g [ ( N τ )] = α / / ¯ g τ 1 / / 1 1 − . ( 0 − and the strictly inequality holds for = Z ( = ¯ z ˜ h 0 R in the dimension spectrum of Virasoro = . χ ∆ z j such that ) 0

) = ε n ¯ ¯ τ i z τ ∆ > α , ε / ∂ j -function is equal to that of free matter fields. However ˜ τ 1 h m z c ( N < ∑ i ∂ − Z + , ( n , h N 3 h m χ 33 ), hence ruling out the spectrum α and . Given any linear functional − ˜ h n Z , on the derivative basis 5.1 ) , h i odd N ¯ τ d ε S α ( n ∑ 1 ˜ h + N = ∑ χ i m ) τ = , where one may consider an 1 , if we can find an ( h N α I χ 0 holds for all [ α ≥ ˜ h , ] h ˜ h d χ ˜ h , h , we may expand h ∑ χ ¯ z gauge theory in 2D, whose energy spectrum on the circle is isomorphic to [ limit, this spectrum does not organize into a representation of any Virasoro ie ) α 1 0 (assuming the identity operator is part of the operator spectrum), then we ∞ ( ≡ , the modular crossing equation ¯ ¯ > τ τ U → , -theorem is perfectly applicable and the UV , ] c z 0 τ R ), based on the positivity of 3 case gives a very short RG flow [54]. ie χ 0 = ≡ χ [ 3.48 may be thought of as a double trace deformation. This leads to a short RG flow, under τ 0, we would rule out the possibility of a gap N α j ε = i ε ˜ Note that the For instance, if Finally, let us mention that another important class of RG flows in two dimensions are gauge We now return to the bootstrap approach to 2D CFTs and explore general constraints based on To begin with, let us inspect constraints from the modular invariance of the torus partition h j 6 < = N i Given any hypothetical spectrum encounter an immediate contradiction with ( of functions in Defining h primaries. the UV limit is not thewell same as as the the holonomy theory degree of of free freedom matter when fields, the due theory to is put the on gauge a singlet spatial constraint circle on (necessary local for operators, modular as invariance). and say Aspects of 2D CFTs relevant operators that are invariant under that of a free non-relativistic particle on a circle of radius The former would generate apoint, at massive least RG for flow. sufficiently large The∑ latter generates an RG flow to a neweven fixed the is the pure (free) theories, including gauged linear sigmaas models Calabi-Yau (GLSM) models) in that the often IR flowby [55]. to a These nontrivial relevant do CFTs operator, not (such as fit the in gauge the theories standard are paradigm not of conformal CFT in perturbed the UV. unitarity, crossing, and modular invariance. 5.1 Bounding the gap function ( which the central charge is expected to change by an amount of order 1 algebra, and the spectral density is not scale invariant (unlike in a5. noncompact CFT). Carving out the 2D CFT landscape Clearly, in the PoS(TASI2017)003 , 0 c is is c ∆ . It ) 0 c c (5.3) (5.4) (5.8) (5.7) (5.6) (5.5) ∆ ( jumps Xi Yin can be to find ) ) c ∞ mod is strictly ) with the N ( ( mod ∆ ) ∆ → c in the linear 5.2 mod ( in the large N . z ∆ α ) mod 3 ∆ c and ¯ . , or equivalently, the c z 0, and the value of c . The result is shown in < c 14328 3 . of the form ( a 0 . Numerically, . α ) ) , + 2 . The slope of 55 or so; we then numerically N ] ( mod )] 2 . Let us further simplify things − . 4 c ) c ∆ ∆ = 0 , ( q ( , , whose coefficients depend on 1 = ∞ . 3 ¯ [ N z 0 ˜ ) h O − → R = ∆ τ z ∈ 3 ) 1 N ( + +

( a c h c 3 ≥ 1 . We then take the limit h 1 811209 ) χ . − − ) ¯ 24 lim z x ) c c 0 = N ∂ ) + ∀ for τ ( mod 0. The result can be expanded in the large − = ( ∆ ∆ + q ∆ 1 3 + ( 0 η 1 4 z c 1 522621 1 4 . ∂ τ ) = + R mod ≥ , and thus the asymptotic slope of ( τ x 0 1 03374 c 6 1 ∆ 3 ) ( . a x − 3 a 12 [ ) is taken to zero, with 0 + c ) = ( 34 ) R 41425 ) = 3 τ 1 3 − . ( τ − 5.6 12 R ) + a 0 c 0 ( ) is positive and ¯ 3 z ˆ h χ . − ∂ a becomes slower with increasing − ˆ ) χ ∆ 5.6 ) + ( c ) + x ( π z N ) + 0118212 ( 2 as a function of the central charge → polynomials in ( mod . 471824 ∂ x 1 − . ) 0 ( ( ∆ n mod R e 0, and 0 1 1 τ 1 − ∆ ( mod a 245875 R ( a h + . > 1 ∆ 1 ] = χ 0 a c = h 6 a ˜ h is exactly equal to such that ( − χ 1 α ( ) ≈ 3 h − 3 12 for c c a for the dimension gap bound receives more contribution from higher 0 χ a ( , [ 1 π to find ∆ 1 2 − 24 are degree increases, suggesting that the derivative basis is inefficient at large c α + α would give an upper bound on the gap in the operator spectrum. e a mod ) ∞ − c 0 h ∆ ∆ ∆ ( = q , and find the strongest bound on the dimension gap within this subspace of ] = n 1 4 0 N 4, and then decreases smoothly and monotonically with N R τ χ 0 ≤ to = χ [ c n ) = ) . The convergence of τ α N 1 ( mod 12 ( + at . Such a h ∆ [57, 58]. For instance, we may consider linear functionals 0 ˆ 1 8 m χ 0, where ∆ derivatives as . α ¯ 1, z to > 11 , ˜ z 6 1 h > , It is observed that This result can be improved by including higher order derivatives in As a simple example, consider linear functionals of the form [56] It is also of interest to bound the dimension of the first nontrivial scalar Virasoro primary in c h limit as greater than the upper bound on the twist gap We may simplify the modular crossing equation slightly by replacing For function no less than optimal linear functional order linear functionals; the resulting bound will be denoted by assuming that there are no conservedto Virasoro compute primary currents of any spin. It is straightforward for some optimized (i.e. minimized in this case)is when then ( given by the largest root of truncation the optimal modular bound on the dimension gap, from remains an open question to determine the form of the optimal linear functional Aspects of 2D CFTs and on the vacuum character We would like to choose c limit, and the corresponding bound a compact unitary CFT. Such a bound can be obtained by working with a different hypothesis on for computed using SDPB package with reasonable efficiencyextrapolate up to Figure PoS(TASI2017)003 , ) 8, 8 25. ( = (5.9) < Xi Yin c SO c , at 2 ) G c 8, the scalar , ( , ) 0 4 3 , ( = s mod 5 exists for 55 results with a 14 ∆ (red) as a function , SU 0 25. The numerical ≤ , 2 0 (blue) [58]. = ) , s mod = = s mod 2 1 N ∆ ( only, together with the ∆ c mod = ≤ 0 ∆ provided that they satisfy c SU 8 must admit a nontrivial ∆ | ˜ h < ≥ − c ∆ ), normalized by h | 5.2 = s (in red) as a function of central charge (by fitting 31 4, where the bound is less than or equal , and diverges at ∞ ≤ . ) mod c 1 c = ∆ ( − N for mod ) ] = ∆ to c 0 35 ( ) χ 0 N . ( mod mod χ [ ∆ 12 ∆ α ) [58]. coincides with N is strictly greater than ) 0. A nontrivial upper bound on the dimension gap / c ) ( ≥ c 0 ( , compared with the all-spin dimension gap bound ˜ h = c 0 , s mod = h ∆ s mod ∆ is saturated by level 1 WZW models with gauge groups , namely allowing scalar primaries of dimension 0 4, I = s mod > ∆ c . 8 We may also bound, say, the scalar degeneracy at the gap, by considering a slightly different By definition, E , obtained by numerical extrapolation of optimization problem. Consider linear functionals of the form ( to 1. For results exhibits a number of interesting features. There is a kink of the bound and Figure 12: Modular bounds on the dimension of the lightest scalar operator c of the central charge where the bound isrelevant equal scalar Virasoro to primary. It 2. is alsogap seen This bound that implies at the that special all values CFTs with Figure 11: The modular bound on the dimension gap Aspects of 2D CFTs the spectrum quadratic polynomial in 1 vacuum, and arbitrary weights forthe primaries unitarity of bound nonzero spin The numerical results are shown in Figure PoS(TASI2017)003 (5.11) (5.10) Xi Yin gives a rigorous 1 . − , subject to Z ∗ i ∈ 4. The black dots cor- ∆ 0 2 1 2 ∆ ˜ , we show the numerical h ≤ χ − 13 = c 0 2 ∆ h ˜ # h . ˜ ≤ h χ c , h 0 = χ ∗ h h ≥ χ α for 1 ˜ h , ) h c ∑ . In Figure ( , 0 the rest 0 ˜ h = s mod = + s mod , 6= ∆ 0 2 ∆ . Thus, ∆ i h 0 over a range of 0 2 χ ∆ ∆ 0 0 2 ∆ or WZW models, where the degeneracy bound is χ = s mod 0 χ 2 36 1 ∆ , ∆ ) ∗ χ 8 gap ∆ h ( = d , then applying it to the modular crossing equation, we 1 2 ∗ 0 ∗ α + SO ∆ α ≥ 0 ˜ gap h χ d 0 and = χ should not exceed for the first scalar primary at + when " 1 h ] 0 1 ) ˜ ∗ h 2 ∆ α gap χ , G h d ( 0 ≥ − = χ [ , 0 ≥ 1 α ) ] ˜ h 3 ( χ h χ SU [ . Of course, , α 1 gap ) d 2 ( is the scalar degeneracy at dimension SU gap d 2. The dotted lines highlight the gap and the presence of marginal scalar primaries in the spin-0 = case, as well as conserved spin-1 currents in the spin-1 case [58]. Figure 14: Logarithmic plots of thethe optimal functional degeneracy acting of on operators spin-0 at andc spin-1 the characters gap when is maximized and the dimension gap bound is saturated at saturated [58]. Figure 13: Upper bound on the scalar degeneracy at where upper bound of Aspects of 2D CFTs We now seek to maximize results for the upper bound on respond to Suppose we find such a linearhave functional PoS(TASI2017)003 i ) ∞ ( 0 φ ) 0, 1 φφ ( > φ 0 for , as a ) ) (5.13) (5.12) (5.16) (5.14) (5.15) ) Xi Yin ¯ z ˜ ∗ h , , ∆ z ] = h in the ( ˜ ( h ( f ) φ χ 0 , ˜ ) h h 0 , 0 χ ( [ h F ( ∗ φ h α , ) = ¯ I z , z ∈ (we have suppressed the , ( ) 0 g . ˜ h h , = h odd ( ) ∀ . ¯ z = ) 1 2 ). The crossing equation takes the , n − , ( c 0 1 ˜ ) h + ¯ ( z ˜ h ( ≥ is always positive, i.e. m F ˜ h ) ) ) 2 1 , F ˜ 2 1 ˜ h h ) 0 F , ( , ) z ) = h h h 2 1 z . This is known as the “extremal functional , where the zeroes of the extremal functional ≥ ( ¯ ( z ( ) − , , n F 0 h ˜ , h n 1 2 1 2 , 14 ˜ 1 h , , 0 , = m ( ( = 2 F h h ¯ z h g , of the form F z m ˜ h F ( , C φ = 2 n h 37 z ∗ F , ∆ , ∆ C m ) 0 ˜ − h < y ¯ z , ˜ h ) ( h ∑ ¯ z ˜ + h odd ( h ) = ˜ h n ˜ h F ∑ ∑ and the central charge ) = , ) + ¯ z F z h m φ , ) ( on the spin 0 and spin 1 characters are seen to agree with the ( z z h n ) = ( − ( , 2 3 ∗ h g 0 m F , ∆ as a function of 0 = F ( F n y ] ˜ ¯ h z f 0 , ˜ h ∂ ˜ 2 h h to the correlator. ∆ , − χ m 2 h z C ∗ ) h ˜ h ∂ 2 case is shown in Figure C , ∆ ∆ χ ˜ happens to be saturated by a CFT, then we must have h h [ ∑ , ∗ ≡ = h − ∑ α ) WZW model of the corresponding spins above the scalar gap. c ∗ gap ˜ h ) ∆ d , ( 3 h ( θ ( n is the Virasoro conformal block with internal weight , SU m ) z . To do so, consider the inequality F ( ∗ h ∆ represents the sum of OPE coefficients squared of primaries of weight F 0. We define the spectral function as ˜ h , 2 h ≥ C in the rest of the CFT spectrum. In other words, the rest of the spectrum of primaries is a ˜ h , Now we turn to the spectral function method, which allows us to extractConsider the from conformal block the decomposition of crossing the 4-point function Using the crossing equation, one can obtain nontrivial upper and lower bounds on If the bound on An example in the ) h ˜ that maximizes the gap at h , ∗ h or in terms of of four identical scalar primaries of dimension where OPE, and explicit dependence on the weight of form for equation not just bounds onacross gap the in entire the spectrum [61]. operator spectrum, but on the contribution from operators ( subset of the zeroes of α spectrum of the 5.2 Bounding the spectral function Note that the conformal block itself evaluated at By its definition, theCFT spectral is function a of non-decreasing a functionoperators that 4-point up ranges correlator to between dimension of 0 identical and 1. operators in It measures a the unitary function contribution of from Aspects of 2D CFTs method" [59, 60] and can becharacter applied or to conformal all block sorts decomposition of is bootstrapof known analysis interest. or whenever conjectured a to coefficient be of maximized the by the CFT PoS(TASI2017)003 , 1 ≈ − 12 0 , c only 0 . The (5.18) (5.17) Xi Yin F 7 12 / 0 , = 1 φ F ), we would ∆ and summing . At any given ) R ˜ 5.16 h . , 1 that admits ∈ 0 h n ( , 8 with > 0 ≥ m , c 0 y ) = 0 , F c ˜ 0 h * , y Δ 2 h C subject to ( − ) 0 , ∗ 0 ∆ OPE, and y ( f ) with φφ subject to the opposite inequality of )( 0 1 2 , 5.16 , 0 . y 2 1 φ ( x ∆ g 13 2 = − ) = ,N ˜ h 0 1 , we would recover all the OPE coefficients. An , Δ ∗ ) is simultaneous satisfied for all possible weights 38 (a) h = . By maximizing . , resulting in a lower bound on the spectral function, ≈ ∆ ( 1, i.e. using first order derivatives only. If we could ) ) ϕ ) 0 N ∗ ∗ Δ , x 0 = ∆ ∆ 5.16 ( 8, ≤ ( ( 0 F = , ) f N n ] c 0 N 0 y ( , + + 0 f . Numerically, we can do this via semidefinite programming, y m ) spectral function bounds in a CFT with , then by multiplying ( ∗ − i ∆ ) I ( ∆ 0.2 0.4 0.6 0.8 1.0 1.2 f , sometimes called the “scaling block", we would write h ’s such that ( − . Similarly, by minimizing n + , ) ∗ φφφφ φ ∗ m h h ∆ y 2 ( ∆ ( − ) Θ 0.6 0.4 0.2 1.0 0.8 z [ N ˜ h ( − , f by h ) C z (from green to red), assuming only scalar primaries for ˜ h , ; h ∑ h N ( F , and deduce a lower bound on the spectral function ) is a rigorous lower bound on ). It is rather weak in the large weight limit however, where we already knew that the h is a hypothetical spectrum for operators appearing in 0 , + 0 Virasoro primaries (i.e. no primaries of nonzero spin). We have seen that such an assumption φ I y 5.16 obeying the hypothesis on , by taking its derivative with respect to h ), we would find an upper bound ) Let us now inspect the As a toy example, consider the case ) 2 ∗ ˜ h , if we can find a set of ∆ − , ∗ ( ( h 5.16 f 2 and its spectral density is equal to that of Liouville CFT. Now if we know the spectral function Figure 15: Upper and lowerderivative bounds order on the spectral function from linear functionals of increasing ( over operators in the OPE, using the crossing equation, we have find an optimal lower bound on provided that we make the truncation which we denote by ( Of course, this result isfrom only ( approximate, but istail qualitatively contribution similar from high to dimension the primaries exact must bound be exponentially derived suppressed. shaded regions are excluded andLiouville theory the [61]. black curve denotes the corresponding spectral function of scalar combined with modular invariance implies that the CFT is noncompact, has a dimension gap approximate Aspects of 2D CFTs where ∆ Thus, PoS(TASI2017)003 . i * ∆ ) ) Δ m 2 , ∆ are q n as a , with i 12 p 2 ∆ ) ) 16 m (5.22) (5.20) (5.21) (5.19) Xi Yin , v , 2 ∆ ( 2 ( ∆ n , m , 0 25 (red) , , 2 p ∆ 2 ∆ n 0 ( ( , p n 0 = , , F m h 0 , 10 m . 2 ∆ n F N F N C p decays at least ) h and ∆ 2 ∆ , we may directly ( ) = i C 8 v ∞ ρ ) ∆ and , ( ∆ ∆ v . The bounds narrow i n = ( h , d v grows like polynomial , m ρ . 15 N R v p 33 in solid red) with the 1 (blue) to ) 0 6 h (b) ∆ = ( ≥ = n , ∆ N N m ∀ p 4 . . , . Plots taken from [61]. ∆ ) ) ∆ odd 12 55 π 2 ∆ 2 ∆ d − , n , ) . e = 2 2 ∆ 2 i ∆ takes the form ∆ + ) ( ( φ ( n so that 1 2 0 ∆ , . Now taking the inner product of , ∆ m µ ranges from 0 ) m ) ) π , * − 2 ∆ = 2, F ∆ F , 0 Δ ) N ∗ ¯ ( ( 13 is shown in Figure 2 z = ∆ n ∆ , N be the spectral density, so that the conformal y ∆ ( = ( f limit, but this is hard to see numerically as the ≥ 0.2 0.4 1.0 0.8 0.6 0.4 0.2 0 ) ( m , λ µ = = ) c 0 - - ) y n ∆ , as θ 1 2 ∞ F ∆ z , ( 27 in solid blue and , N , 39 * 2 ∆ 7 ( v ∗ 12 1 2 odd m C h Δ x ρ ( ) = → g n ∆ ∑ ∞ = ( , + 0 N , N 2.0 ρ over the positive real axis defined with the inner product ( 0 φ up to ) = m Z 1 to ensure the convergence of both ∗ ∆ ) ) ∆ + = ∗ ∗ ∆ = . Let ∆ = 0 amounts to the completeness of ( ) 0 ) = i λ ∆ ∆ 8, i , i f n ( ( 0 ∆ y ∆ 0 − , , ≥ ) . Recall that the Virasoro conformal block grows like ( , y ( N e = m 0 f N x v ∆ ∗ decays at least as fast as µ ( p p ± h c f ∆ ∼ , , 1.5 ) v v ) = ) ∆ h h ∆ ( for should be chosen so that the inner products ∆ v ( ) ) ) − ∗ µ ∆ ∗ ∆ ( ∆ ( ∆ ( µ ( µ N (a) 1.0 f θ . In practice we can choose π is the elliptic nome, convergence of OPE demands that 2 q < , and so λ ∆ 0.5 − < , where ∆ ) * ), we see that the spectral function is given by Δ The measure factor If we make the assumption that the upper and lower bounds do agree at . Thus we can choose ( N ∆ f 0.2 0.6 0.4 0.2 1.0 0.8 5.19 - ( in provided 0 defined by convergent integrals in as fast as 16 at large block decomposition of the 4-point function at for some suitable measure factor Figure 16: (a) Plot of basis of the Hilbert space of functions of We would like to define the vector exact DOZZ spectral function (dashed, black) for Aspects of 2D CFTs example of numerical results for down an allowed window,bounds in could which conceivably converge the in Liouville the spectral function lies. The upper and lower with step of 2. (b) Comparison of computation of the bounds becomes very time consuming at higher truncation order seek solutions to the equality The existence of solution for all PoS(TASI2017)003 , , * 2 ) Δ c I ( = , the c (5.24) (5.23) Xi Yin mod mod ∆ ∆ 43 = converge slower ¯ τ , and Euclidean BTZ 12, N τ = 3 , the uniqueness of the simply from the linear i 1 v (b) φ AdS , N . Two examples are shown 2 ) i P φ i , ) 2 ∆ − and converges to the Liouville N φ 1 − φ = N ≤ h ∗ ¯ 100, dimension gap τ n ∆ derivatives on ( = , the bounds pin down the spectral ( ˜ h c + ) 2 3 i 5 10 15 20 θ N χ , m ) v i = = N ¯ τ is the orthogonal projection onto the sub- = P 100 case, assuming a gap close to odd h gap − τ N to converge and pin down the spectrum. In , ( ∆ P = h 0 = ) c ∗ ) = χ τ ˜ ∗ 1.0 0.8 0.6 0.4 0.2 h ≥ ∆ , ( ( ∆ h * 40 n Z ( Δ d , N ∗ f mod m ∆ , where odd. Now we can solve f v < ˜ h , N , m + 0 P to be the modular bound on the dimension gap, 1 h stabilizes with increasing + 0 = ∑ ) = by n 100. The dotted black curve represents the modular spectral ∆ ∗ i i n v ∆ 0 , , = . (b) Upper and lower bounds on the modular spectral function ( 31 0 m ) = for 2 3 N c ∗ = obtained using up to order p p f , , ∆ m ) pure gravity computed from the thermal , = at ( WZW model. In the N N ∗ n 3, N / v v 3 1 2 p ∆ h h ) ( = mod mod mod 3 f ± , ( ∆ ∆ AdS 1 CFT with only scalar Virasoro primaries as suggested by the numerics ) . If we take and (a) N 0 ( mod SU > 0 ∆ f , 0 c p . 2 case, assuming the maximal gap 2, dimension gap c = = c , where we see that c 16 at large . N ) Two examples of the numerical bounds on the modular spectral function are shown in Figure In a similar way, we can define the modular spectral function Along with similar results for mixed correlators of the form * 3 Δ ( . In the 0.0 0.5 1.0 1.5 2.0 mod f 0.80 0.75 0.95 0.90 0.85 1.00 numerical bounds converge slowly toward the answerAdS expected from semi-classical pure gravity in and bound it from thesay modular a crossing equation, dimension with gap a suitable assumption on the spectrum function as that of the would imply that the only CFT with such property is Liouville theory. spectral function in Figure 17: (a) The upperassuming (blue) a and gap lower equal (red) to bounds on the modular spectral function at with 17 assuming a gap close to function of perturbative saddles in the gravitational path integral. Plots taken from [61]. we expect the upper and lower bounds on practice, the bounds Aspects of 2D CFTs Numerically, we may approximate space spanned by equations and obtain an approximate spectral function spectral function. in Figure PoS(TASI2017)003 , RR i φ (5.26) (5.25) Xi Yin . When- ) R spin for the , except for ,` ) L 4 SCA. In the 2 ` M ( -BPS primaries ( which may take = 4 1 ` SU = N h of the superconformal ` 2 lies within the positive , gives a singular K3 CFT, = . maybe decomposed into its 2 R ] spin Z , ` 4 M 0 , R . , ) ∈ − s 20 )) ··· 2 ; 2 L ˜ h , Γ ( 4 ` 1 2 ( 2 − , ∈ ≡ SU 1 h ` -BPS primaries are determined by the SO . Combining the holomorphic and the 2 , which we denote by i , ` 1 4 i ⊕ h 4 = , ` 0 × 2 1 h and s ) 20 > 4 descendants have integer spin and would h 0; , R and with 20 ] , ˜ h ( s 0 1 2 = , ` -BPS representations give rise to 80 exactly [ , h 1 2 s SO ; , and the only allow valued of s , ( N ] 2 1 n / 41 0 h ) 1 , > ` 4 , ∞ = ˜ h SCA, corresponding to the tangent directions of the , i s h , M may be equivalently described as the moduli space of i 1 2 ) 0; . A lattice vector , 20 1 2 , 4 i 4 ( , i ⊕ , ; has h 0; [ M , 4 SCFT defined by the supersymmetric NLSM on K3. The ] 1 2 2 1 20 s ( SO ; [ h 0;0 ) ,` R h \ 2 1 4 h ) , [ h 4 , 4 ( BPS : 20 20 6 1540, etc. The 20 BPS : BPS : Γ − ( i ⊕ − − = = c 4 2 1 1 3 vacuum : non Aut 0. The BPS representations on the other hand has 0;0 n h ' embedded in = 4 ` , 462, M R-symmetry. It is related by half unit spectral flow to the (R,R) primary 0, the corresponding point in the moduli space 20 ) Γ -BPS (NS,NS) primaries transforming in the doublets of holomorphic and anti- = = 2 2 1 ( 2 R n ` SU 4 SCA is ; we will simply denote the representation by 90, 1 2 be the = = ± , 1 N ± i n An exact result that captures the moduli dependence of the K3 CFT is the following [62, 63]. We will briefly describe some results on bounding theFirst, gap we in consider the the OPE of BPS operators in While the BPS operator spectrum is invariant along the conformal manifold 6 φ sector, the representations are labeled by the weight = primary. A non-BPS representation c value 0 or holomorphic NS have half-integer spin, but their half-integersurvive level the GSO projection. The degeneracies of the operator spectrum can be organized according to representations of the small Let elliptic genus. The result can be summarized by the following BPS operator content with marginal deformations preserving the conformal manifold special points where some non-BPS primariesspin become currents, BPS the and non-BPS give operator rise spectrum to hasdeformations. highly extra nontrivial conserved The dependence higher on conformal exactly manifold marginal the Narain lattice in that the CFT develops aadmit continuous an spectrum ADE above a classification, certain havingIIA dimension to string gap. do theory Such with compactified singularities the on the enhancement K3. of 6D gauge symmetry for type SCFTs in two nontrivial examples. Aspects of 2D CFTs 5.3 Some results on superconformal theories anti-holomorphic sector, we have the following types of representations: projection onto the positive and negativeever the subspaces lattice of embedding is such thatsubspace, a lattice i.e. vector Here we assume tono be extra working conserved at currents a beyond the generic superconformal point algebra. on the Note conformal that the manifold, where there are PoS(TASI2017)003 ,

0 ) RR 1 and = ∞ φ ( s t 0 (5.29) (5.27) (5.28) Xi Yin R turns to = φ ] ) s stands for 0 1 , spins). The ( , Λ h ) R [ R 2 φ ) u ) 2 , z ( 0, corresponding 2 ( t R , → 2 SU φ s . Now consider the ) 2 R ( ) 0 1 4 ( ,` O , , 2 L R ) 4 1 ` + ¯ φ τ ) can be understood from ( 24

, -BPS primaries are the vac- u ` 1 2 τ )) . j | y , τ · y ) has 5.28 ( ik L ( 2 R ` i B Λ η ` π ( 2 Θ + = + t τ

2 L 2 R 2 ) ` ` k j ¯ τ d i `, ∞ i π ( , whose components are in correspond- 0 F B − Z (i.e. 2 L 20 RR ` 0 ` + ` τ R supergravity, which we will not review here, i φ s = y ` -integrand). In the context of type IIB string ) π )

k z 1 e , 0 acting on the upper half plane, and ` ( as exact functions on the conformal manifold of i j , Λ y ) , the latter encoding dependence on the conformal ` 2 ∈ B ∑ ∂ RR 42 ` k k -BPS 4-point function in terms of super-Virasoro ( Z , k exists as an integral over the moduli of a genus two 2 , 2 1 φ + y i j y ) 2 ` 1111 ` 4 2 ¯ z B ∂ k ( τ π , j A , 2 ∂ i jk i j z y e ( A B ∂ 2 lies in the positive subspace, corresponding to the K3 i and RR y i + ` ) = = φ takes non-negative real values, and diverges in two different SCA. By a simple analysis of Ward identities one can show ∂ ¯ τ ) i jk 2 ) , ` 0 2 is deﬁned as ` j , parameterizing a point on the conformal manifold of the K3 A 4 τ ( 4 π | , δ , Λ 1 1111 u ) has the interpretation as the string tree level scattering amplitude y 4 RR ik i ( 20 A Θ 16 δ φ Λ R with

= ( 5.27 = Θ + 1 ` ∈ ` − i j t 4 , N δ i jk 4 superconformal block for the BPS correlator − t ` | lies in the positive subspace i 20 A z = OPE as a function of δ Γ y − + RR 1 1 N -BPS multiplets of the K3 CFT. The formula ( | ` φ singularity, or (2) a null lattice vector k 2 1 , and the LHS is defined by analytic continuation from a suitable domain of 1 6 t δ 1 − s external operators in the R sector) in a non-BPS channel labeled by RR 1 s = i j A − φ 1 4 − δ . It turns out that c | s z | z is the fundamental domain of PSL 1111 π 2 , for instance, is given by ≡ − ` 2 A d F u i jk = Z We will simply make use of the integrated 4-point function of a single primary, say A To proceed, we need to decompose the where the integral converges (it is also possible to expand the integral directly around by subtracting appropriate counter terms from the ing with the 20 theory compactified on K3, ( The auxiliary vector uum/identity and non-BPS representations (whichholomorphic necessarily have vanishing that the only representations that could appear in the OPE of a pair of where heterotic/type II duality asmoduli a of heterotic a string torus. one-loop A similar amplitude, formula hence for the integration over the conformal blocks of the (with weight Riemann surface. of four tensor multiplets infollow from six analysis dimensions. of superamplitudes By inone 6D supersymmetry can determine non-renormalization the results coefficients that Aspects of 2D CFTs which is a singletintegrated under four-point function the R-symmetry and has conformal weight the K3 CFT. where the embedded lattice CFT. The lattice theta function namely limits: (1) a latticedeveloping an vector manifold [64]. to the large volumecontent limit of of the the K3 surface. We would like to constrain the non-BPS operator t PoS(TASI2017)003 ) 1 1 A ( 1 up U -BPS (5.31) (5.30) Xi Yin / 1 2 + 2 h ) R , which is ˜ , h , 2 2 h ( C 12) and (2) the . SL ) ≥ ¯ z with a rectangular ( c 2 R , (the orange dots are Z 4 / = . 4 limit, while not obvious ) N h 1111 z T free orbifold: twisted sector ∞ A 2 1; F . This is in fact saturated by /Z ) 4 1 4 z → + ( Bound T h R , , = 4 4 superconformal block turns out 1 2 cigar CFT. The vacuum channel 1111 , = gap 1 = A = , N ∆ 2 SCA, which puts highly nontrivial h 1 , N F = 1 N ˜ h , ( 1111 6 2 h A C N = ˜ h , c 28 h ∑ . = Virasoro c = 100 18 ). F 43

1 2 ) ) ∞ z 5.30 ( 0 − RR 80 1 1 ( φ 2 1 ) -BPS twist ﬁelds at a ﬁxed point of z 1 1 2 ( OPE in Figure RR 0 limit of ( 1 ) = 2 subalgebra (which would not be the case for z 1 , is found numerically to be =1 φ i 60 ( ) φ = → ¯ R z 1 18 , , 4 φ h z = ( can be expressed simply as an extra linear relation on N withR SCFTs [65], which include supersymmetric NLSM on Calabi-Yau 3- N RR 4 h 1 ) φ 2 40 1111 F ) , 0 A 2 ( ( 28 bosonic Virasoro block with external weight 1 and internal weight RR 1 9 free orbifold: untwisted sector = SquareT φ 2 c

= cigar CFT, which admits an exact description as the supersymmetric 20 /Z c 4 1 T A 4 OPE, as a function of the integrated BPS four-point function = 1 2 blocks with external BPS operators are related to Virasoro conformal blocks in a simple gap φ Another example we will discuss very brieﬂy is the constraint on the OPE content of It is now straightforward to implement the constraints from the crossing equation on the cor- The asymptotic value of the dimension gap bound in the Δ 1 = N φ . Plot taken from [64]. 0.5 1.5 1.0 2.0 4 The constraint from on the same footingreproduced as here. We the simply summarize crossing the equation.ﬁrst result non-BPS for primary the The in upper the bound details on of the dimension the gap of analysis the is too lengthy to be from the plot range of Figure folds. As already mentioned, the latter has extended numerical data points which havered already represents stabilized the at gap 20 in the derivative OPE order). of The shaded region in The reason behind this simple relationto is coincide that with (1) that the of the singularity. operators in relator T be equal to the to a simple prefactor, N way; this can be argued byblock studying can the be correlators obtained of as the the the coset, and is conjectured to capture the singular limit of the K3 CFT where the K3 develops an Figure 18: The solidin blue curve shows the upper bound on the gap of the non-BPS primaries Aspects of 2D CFTs PoS(TASI2017)003 (5.32) Xi Yin of an R-charge λ , applied to quintic λ OPE channel (CC) and in the its conjugate (anti-chiral, anti-

+ , ) over the Kähler moduli space of − , + ∞ λ − ( φ φ + , −0 , -BPS operators have been determined + contours. The conifold point is labeled − 2 1 φ φ λ ) 1 , we show an example of numerical bounds ( + , 19 + φ ) 44 ¯ z , z ( , computed using linear functionals up to 20 derivative − , λ − 2 cross section is expected to shrink to zero width as the φ ) and = 0 OPE. In Figure ( crossing equation. + gap CC gap + CC , , z (chiral, chiral) primary, and ∆ ∆ + + 1.5 1.6 1.7 1.8 2.0 2.2 2.6 3.2 ) − Conifold φ 0.0 φ 2 1

1 + , , 1 2 + → ( is the BPS primary associated with the Kähler deformation and the BPS φ -0.2 z + , 1.7 + 1.6 φ -BPS four-point functions, through the OPE coefficients of the BPS operators, -0.4 1.56 1 2 is a known function over the Kähler moduli space. Some points on the bounding as a function of λ -0.6 Gepner be a weight gap CA ∆ 2 superconformal block decomposition in the + , OPE channel (CA), and constrain the gap of the first non-BPS primary in both channels -0.8 + = φ − , − N BPS primary in the -1.0 φ 1.8 2.0 2.2 2.6 3.2 Let So far, we have analyzed constraints on individual OPEs from crossing symmetry and con- The moduli dependence can be incorporated through the OPE coefficient ) Conifold t-plane + 2 , , + 2 0.8 1.6 1.4 1.2 1.0 1.8 the quintic 3-fold [66]. The blackby curves the are black the dots, constant whileCA the channel Gepner point is shown in red. Right: Upper bounds on the gap in the Figure 19: Left: The contour plot for the chiral ring coefficient as an exact function on theinto conformal the analysis manifold. of This allows usthereby to allowing incorporate moduli for dependence constrainingequation. the non-BPS operator content of the OPE using the crossing orders. The narrow peak on the 5.4 Genus two modular bootstrap straints on the spectrum from the modular invariance of the partition function. One may extend on the non-BPS spectral gap in the CC versus CA channel, as a function of derivative order is taken to infinity. Plots taken from [65]. OPE coefficient surface are seen to be saturated by OPEs in free theories. Aspects of 2D CFTs constraints on the 3-pointquintic function 3-fold, of the BPS 3-point operators. function of In say some (chiral, cases, chiral) such as the NLSM on the chiral) primary. We can analyze the four-point function by its 3-fold model, where simultaneously using the φ ( PoS(TASI2017)003 , = i j j b C α (5.34) (5.35) (5.36) (5.37) (5.38) (5.33) Xi Yin · a over the α . Here we ) 0 for some ikk Ω 2, such that , , C = 1 Ω i j j ( 12 as a set of linear = C )) τ a / or Ω , , 1 ( a ) ¯ γ 2 τ i jk ω ( ; C j ˜ h → − , , are defined as follows. Pick j τ b . ˜ h β ab , ! i ab . Ω ˜ h 1 A ( 2 − AB . ) CD T c + b D

ab F α ) Ω + ≡ τ ab , ; , with the intersection numbers γ = Ω j B 2 ! b C h β , , , ω 12 22 → j )( 1 a a h B Ω Ω β β ! , β , i I I 0 + 2 h 11 21 I ( α 45 , Ω 2 0 , Ω Ω , b − T 1 A c β ab

( transformations, of the form

α δ ab F ) = ≡ = C = Z i j j ) , Ω T b + C . However, there are no a priori positivity property for 4 Ω γ b ω of a CFT may be written as a function of ( j ( into torus 1-point Virasoro conformal blocks, a i j j ∑ γ α ! 2 α ) is positive deﬁnite. The entries I C have integer entries and obey Sp 0 ¯ Z ab z I = , 7→ I 0, possibly with singularities at the loci D Ω ) D z 0 τ , ( − ( Ω i 2 C → φ

, T a i γ B Ω ) , α ¯ z , A . There are two independent holomorphic 1-forms z ( i ab and that Im φ δ h T is defined up to = Ω } b a β = β · 2 matrices , a Ω a × α α { 0, = b The first parameterization is through the period matrix As for the crossing equations for the sphere 4-point functions, ultimately one would like to This is partially achieved by consideration of the modular constraints on the partition function β · a which obeys Siegel upper half space Im will restrict our attention tocomplex the dimension former. 3, and The may moduli bethree space parameterized parameterizations of below. in the a genus number two of Riemann different surface ways. has We will describe a basis of the dimension 1 homology group The basis where the 2 Correspondingly, the period matrix transforms by The genus two partition function combine the crossing equation for allbeyond primaries a in few the primaries. CFT So simultaneously, while but theof this sphere torus is 4-point 1-point very crossing functions difficult equations in and principle the give modularCFT, the we covariance complete need set of to consistency organize relations these for crossingset a equations 2D of in unitary primaries a and way satisfies that certain takes positivity into property. account OPEs of a large of the CFT on agenus genus two two Virasoro conformal Riemann blocks, surface. whose coefficients Such involving either a partition function may be decomposed into Aspects of 2D CFTs the latter to the modular1-point covariance function of of torus a 1-point primary functions. Indeed, by decomposing the torus and it is not easy to implement such constraints using semidefinite programming. β it is straightforward to writerelations down on the the modular structure crossing constants equation for PoS(TASI2017)003 . . , ) ) 3 c c 2 ( ) rs C , X SL + 2 AdS Z G ( j (5.39) (5.40) Xi Yin ( h / 2 Z , ) ) of genus two } i ∞ , that identifies a q q are the plumbing { shifted to ; i j → q rs h c , that is invariant under + in the summand is the ) j and its adjacent weights ) h Ω j j , h ( h → Ω rs c ( j ) G is the free (non-Abelian) group h } ; i i } q β { , ext a ; h } α 1 CFT, say a free boson, in the same i , h { h i ( = ) { j β ; c h , ( } is the limit set of the Schottky group action. , for a complex parameter α rs , and z c h ) ext a Λ / with the internal weight / G h are the internal weights, transformations. We can choose the PSL C ) q ) { ) , j i ) j ( Λ 2 ) h h h ( C 7→ 2 ( 46 s ( , ( , z j r rs 2 is a known function of rs } − SL ( c c Q s ∞ G , j r ) − } Q c j ∪ { limit of the vacuum block in the plumbing frame, , 1 q j C ≥ { ∞ s ( , 2 ∑ conformal block, the summation in the second line runs over = ≥ r 0;0; c ) 3, are moduli of the resulting Riemann surface. ( j 2 , ∞ ∑ ( 2 G , + SL 1 would then be a well defined function of is the c = ) = ) ) i } } i X , j i q Z q q ( { , known as the Schottky group. { ; / β 2 } , we must specify a conformal frame, or equivalently a choice of metric within i 2 Z h Z that identifies an inner circular boundary to an outer boundary. The three complex 0;0; { and ( ; } ∞ qz α , up to a universal conformal anomaly factor that depends only on the central charge G are two loxodromic elements of PSL ext a stand for possible external weights, ) 7→ h is the genus two partition function of a given Z β { z , , ext a ( X 4 limits with fixed primary weights are finite, thus permitting a recursive representation in c h Z is straightforward to compute as an expansion in the plumbing parameters. The vacuum α ( G ). ) ∞ The plumbing construction also specifies a conformal frame, which we refer to as the plumbing The second parameterization is the Schottky parameterization, where we realize the genus two The third parameterization, which is closely related to Schottky parameterization and useful 2 Sp ( → ∈ SL , generalizing Zamolodchikov’s recursion formula for sphere 4-point blocks. Schematically, the 5.38 on the trivalent graphconformal associated block with evaluated at the central conformalG charge block, and generated by frame. The Virasoro conformal blocksc in the plumbing frame havec the special property thatgeneral their recursion formula takes the form Riemann surface as the quotient internal weights labeled by the index for the computation of general Virasorowhere conformal we blocks, form the is genus based two onplane Riemann the along surface their plumbing by boundaries, construction, gluing using a three pair PSL of two-holed discs on the complex Aspects of 2D CFTs γ Thus, to specify its conformal equivalence class.Alternatively, We we will may discuss avoid the two conformalwhere particularly anomaly useful altogether conformal by considering frames the later. conformal ratio frame. ( where There is a useful formulaof that expresses the the Schottky period group, matrix throughgravity, which the infinite we Schottky products parameterization will over is not elements useful(and explain in higher) writing in Virasoro the conformal detail classical blocks. here. limit ( Due to its connection to gluing maps to be an inversion mappair of of the inner form circular boundaries centered atthe the form origin (or shifted as necessary), or a scaling map of where is the corresponding global plumbing parameters parameters, PoS(TASI2017)003 , with (5.43) (5.41) (5.42) (5.44) Xi Yin ! 2 / 1 0 − γ and a pair of q 3 , 2 ) σ / by directly sum- z 0 γ 1 ; ¯ , written in affine z q k 1 ˜ h P

, is deﬁned as follows: j × ˜ γ h , 1 q i P ˜ h ( twist ﬁelds c 3 3 1 2 F Z OPE. The Renyi frame confor- h h h ) , z 2 1 3 ; , k − σ

) h 3 ) n γ , σ j − 2 q it is conjugate to . x h ) ) ) ( , − i 3 − + 2 2 C 1 h x x , σ ( ( ) 2 2 c − − ( ∞ for the Renyi surface (still parameterized by = − 1 ∏ F n x x x c ( )( )( 2 i jk G P 3 ∈ − + 1 1 ∏ C σ γ x x 47 k ) , j + 2 , ∑ − − i x ) = x x ( can be computed as an expansion in ( ( } = 3 OPE channel. j descendants in the ) σ

z 3 q ) = 3 ) ; ¯ { σ k + ⊗ 1 3 ∞ 3 x h y ( ( , σ 0 3 j 3 3 3 0;0; σ h σ σ σ ( ) ,

i ∞ 1 h ( G , as an element of PSL ( 3 γ c σ ) F ¯ z 3 3 , σ σ z ( 3 twist ﬁelds in the 3-fold product CFT on the sphere. Right: The genus two σ ) 3 0 , ( as 3 Z 3 ) σ σ y

, x ( is the set of primitive conjugacy classes of the Schottky group. P 1. The partition function computed in the plumbing frame is not manifestly modular invariant We begin by considering the “Renyi surface": a complex 1-dimensional locus in the moduli < | γ q however, due to the conformalcomputable anomaly manner, factor. we To will formulateRenyi work the frame, in modular as follows a crossing [68]. equation yet in different a conformal frame, which we refer to as the | in the 3-fold symmetricdecomposition product takes the CFT form on the Riemann sphere. Its genus two conformal block mal block is related to the plumbing frame block ming over contributions from Virasoro where pick any element of the class Figure 20: Left: Thesurface. 3-fold cover The of partition the function Riemannpoint of sphere function with the of four CFT branch on points the is covering a surface genus-two can be regarded as the four- The genus two partition functionby on the covering the map, Renyi is surface, equal in to the a four particular point conformal function frame of specified a pair of where the conformal block conformal block associated with the Aspects of 2D CFTs block admits a closed form expression in terms of the Schottky parameterization [67], anti-twist fields space of the genus two Riemannfour surface, points. that is Explicitly, a we 3-fold may cover realize of the the Riemann Renyi sphere surface branched as at a curve in coordinates PoS(TASI2017)003 , ··· as set + are 1 2 6 2 D h (5.47) (5.45) (5.46) (5.48) Xi Yin = z ¯ z 27 given by . and = 1 z not h ··· . , + n 21 z q ) 2356039 3 . 2 h ) + , 2 2 i jk 2 3 5 h C h h , . − outside the domain in the space of dimension 1 ) z 54 1 z 27 h ) are positive valued provided ; ( h k D 3 ( n ) ∆ h 3 A , , j h + 0 is special, and is 0 2 , ∆ ∞ = 2 h 2 ∑ , n ) = , h i 1 1 96552 c 8 ), by expanding around , 3 2 ∆ h h 1 h − h ( ( + , h 3 c − h ( 4 cl = 54 , 5.44 n G 3 + 3 2 2 h i A F h h ( with ) z h , 27 z + of ( 2 1 ( + z = h i jk h cl 0 limit of the generic block with q , 2 1 C − 1 ) 5 h 18 F 3 h 1 → 1 c − h . An example is shown in Figure 48 3975 h 3 h )) → D h − + 54 τ z 2 3 ( exp h 3 ( or more general Virasoro descendants of the identity, -expansion of θ z ( z ) 27 + c ) = z 7 72 z ( 3 ; − h 3 T )) h + z , 162 2 2 2 − h h + , 1 2 1 + ( h z 1 , for generic weights ( c h z c 27 G , and is given by the vanishes, corresponding to the vacuum channel, then the conformal block F 2 ) = ( + 3 h 6 z h ; 1 ) by a conformal anomaly factor, , the crossing equation implies a critical domain 3 z = z h ) + 1 , such that the OPE coefficients 3 , z limit of h ) h 2 ( is the elliptic nome, and the coefficients ) 3 + h z ) and ¯ 2 ∞ , z ∆ ; ’s, say h ( i 1 log z , 3 1, on any of the three sheets covering the Riemann sphere. This allows us to write τ + 2 h i h · h 2 9 → 1 ( π , k ∆ h c 2 e n c , − 1 h − 0 limit of the generic block, as the latter would incorrectly include contributions from null F , = ∆ z L 1 27 ( q ) = h → z ( ··· i The genus two conformal block for the Renyi surface may also be mapped to the 3-fold-pillow A good amount of knowledge of 2D CFTs has been accumulated over the lastEven four the decades. simplest class of rational CFTs, namely the meromorphic CFTs, are far from being To deform away from the Renyi surface, rather than changing the geometry we can sim- We may analyze the crossing symmetry ( ∞ 1 h cl = n G − F the states. frame, to be equal and positive. The vacuum block with bounded by those with weights that lie within exists only when Yet, the solvable 2D CFTs arecoset largely models, limited and to orbifolds free thereof. or rational theories, a handful of noncompact classified: it is easy totion, constrain but their it spectrum is from quite the difficult to modular show invariance that of there the exists partition a func- set of structure constants that obey the entire If one of the in the analysis of sphere 4-point functions.derivatives in As a simple but nontrivial example,triples just using first order down the expansion offormulating the the genus complete two set of block genus of two modular general crossing moduli equations for around6. the Renyi Summary locus, thereby L ply insert the stress-energy tensor Here we record the first few terms in the and of the where Aspects of 2D CFTs the same cross ratio that the unitary bound is obeyed. PoS(TASI2017)003 Xi Yin 2 Δ 1.5 =0.2 =0.55 =0.9 =1.25 =1.6 1 1 1 1 1 Δ Δ Δ Δ Δ Commun. Math. Phys. , superconformal theories. 1.0 ) 4. Right: Cross-sections of 2 c=4 , 2 = ( c 0.5 for ) 3 ( ∆ D 3 Δ 1.5 1.0 0.5 So either we are missing important constraints 7 49 Structure of representations with highest weight of infinite dimensional Classical and Quantum Conformal Field Theory (1979) 97–108. 34 . Plot taken from [68]. 1 ∆ Adv. Math. , really the optimal upper bound on the twist gap [58]?), or we have no clue what the 1 − (1989) 177. 12 c 2D CFT looks like. Clearly, we are only beginning to carve out the landscape of 2D CFTs, Lie algebras 123 The critical coupled Potts model of [54] is a potential candidate, but to the best of my knowledge the possibility of I would like to thank the organizers of TASI 2017 for the invitation and the participating Our knowledge of interacting, compact irrational CFTs is embarrassingly limited: the only 7 [2] V. G. Kac and D. A. Kazhdan, [1] G. W. Moore and N. Seiberg, it being equivalent to one of the known rational coset models is not yet ruled out. students for their enthusiasm and stimulatingCho, questions. Scott Collier, I Petr Kravchuk, am Ying-Hsuan grateful Lin, Shu-Heng toWang Shao, Chi-Ming for David Chang, Simmons-Duffin, collaborations, Yifan Minjae and ThomasKeller, Dumitrescu, Zohar Daniel Komargodski, Harlow, Simeon Alex Hellerman, Maloney,Strominger, Christoph Eric Cumrun Perlmutter, Vafa, Slava for Rychkov, illuminatingInvestigator Ashoke Award conversations. from Sen, the Andy Simons This Foundation work and by is DOE supported grant by DE-FG02-91ER40654. aReferences Simons generic and much is to be learned in the decades to come. Acknowledgements this domain at constant There isn’t a single example ofconserved a unitary, currents compact beyond CFT that Virasoro is symmetry. 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