Genus Two Modular Bootstrap
|
|
- Dylan Lindsey
- 5 years ago
- Views:
Transcription
1 Genus Two Modular Bootstrap arxiv: v5 [hep-th] 4 Dec 018 Minjae Cho, Scott Collier, Xi Yin Jefferson Physical Laboratory, Harvard University, Cambridge, MA 0138 USA minjaecho@fas.harvard.edu, scollier@g.harvard.edu, xiyin@fas.harvard.edu Abstract We study the Virasoro conformal block decomposition of the genus two partition function of a two-dimensional CFT by expanding around a Z 3 -invariant Riemann surface that is a three-fold cover of the Riemann sphere branched at four points, and explore constraints from genus two modular invariance and unitarity. In particular, we find critical surfaces that constrain the structure constants of a CFT beyond what is accessible via the crossing equation on the sphere.
2 Contents 1 Introduction 1 The genus two conformal block 4.1 OPE of Z 3 -twist fields in Sym 3 (CFT) The conformal block decomposition of σ 3 (0) σ 3 (z)σ 3 (1) σ 3 ( ) Recursive representation Mapping to the 3-fold-pillow The genus two modular crossing equation Some preliminary analysis Critical surfaces Beyond the Z 3 -invariant surface 18 5 Discussion 0 1 Introduction The conformal bootstrap program in two dimensions aims to classify and solve two-dimensional conformal field theories (CFTs) based on the associativity of the operator product expansion (OPE) and modular invariance [1 3]. A complete set of consistency conditions is given by the crossing equations for sphere 4-point functions and modular covariance of the torus 1-point function for all Virasoro primaries in the CFT [4,5]. In practice, while one may obtain nontrivial constraints on a specific OPE by analyzing a specific sphere 4-point function [6, 7], or on the entire operator spectrum of the CFT by analyzing the torus partition function [8 11], it has been generally difficult to implement these constraints simultaneously. In this paper, we analyze modular constraints on the genus two partition function of a general unitary CFT. The modular crossing equation for the Virasoro conformal block decomposition of the genus two partition function encodes both the modular covariance of torus 1-point functions for all primaries and the crossing equation for sphere 4-point functions of pairs of identical primaries. It in principle allows us to constrain the structure constants across the entire spectrum of the CFT. 1
3 A technical obstacle in carrying out the genus two modular bootstrap has been the difficulty in computing the genus two conformal blocks. Recently in [1] we found a computationally efficient recursive representation of arbitrary Virasoro conformal blocks in the plumbing frame, where the Riemann surface is constructed by gluing two-holed discs with SL(, C) maps. For a general genus two Riemann surface, however, it is rather cumbersome to map the plumbing parameters explicitly to the period matrix elements on which the modular group Sp(4, Z) acts naturally [13]. To circumvent this difficulty, let us recall a well-known reformulation of the modular invariance of the genus one partition function. A torus of complex modulus τ can be represented as the -fold cover of the Riemann sphere, branched over four points at 0, 1, z, and. τ and z are related by τ = i K(1 z), K(z) = F 1 ( 1 K(z), 1, 1 z). (1.1) The torus partition function Z(τ, τ) is equal, up to a conformal anomaly factor [14], to the sphere 4-point function of Z twist fields of the -fold symmetric product CFT, σ (0)σ (z, z)σ (1)σ ( ). The modular transformation τ 1/τ corresponds to the crossing transformation z 1 z. In this way, the modular invariance of the torus partition function takes a similar form as the crossing equation of the sphere 4-point function, except that the sphere 4-point conformal block is replaced by the torus Virasoro character. Usually in the numerical implementation, the crossing equation is rewritten in terms of its (z, z)-derivatives evaluated at z = z = 1. While a priori this requires computing the conformal block (the torus character in this example) at generic z, one could equivalently compute instead the conformal block at z = 1 with extra insertions of the stress-energy tensor, or more generally Virasoro descendants of the identity operator at a generic position (on either sheet of the -fold cover). Of course, the above reformulation is unnecessary for analyzing the modular invariance of the genus one partition function, as the torus Virasoro character itself is quite simple. However, it becomes very useful for analyzing genus two modular invariance. Let us begin by considering a 1-complex parameter family of Z 3 -invariant genus two Riemann surfaces that are 3-fold covers of the Riemann sphere, branched at 0, 1, z, and. Following [15], we will refer to them as Renyi surfaces ; such surfaces have been studied in the context of entanglement entropy [16, 17]. For instance, the period matrix of the surface is given by Ω = ( 1 1 ) i F 1 (, 1, 1 1 z) F 1 (,. (1.) 1, 1 z) 3 3 The genus two partition function of the CFT in question on the Renyi surface is given, up to a conformal anomaly factor, by the sphere 4-point function of Z 3 twist fields in the 3-fold
4 symmetric product CFT, whose conformal block decomposition takes the form σ 3 (0) σ 3 (z, z)σ 3 (1) σ 3 ( ) = CijkF c (h i, h j, h k z)f c ( h i, h j, h k z). i,j,k I (1.3) Here I is the index set that labels all Virasoro primaries of the CFT, C ijk are the structure constants, and F c (h 1, h, h 3 z) is the holomorphic genus two Virasoro conformal block in a particular conformal frame, with central charge c and three internal conformal weights h 1, h, h 3. We will see that F c can be put in the form F c (h 1, h, h 3 z) = exp [ cf cl (z) ] G c (h 1, h, h 3 z), (1.4) where the factor exp [ cf cl (z) ] captures the large c behavior of the conformal block, essentially due to the conformal anomaly. G c is the genus two conformal block in the plumbing frame of [1] (with a different parameterization of the moduli) whose c limit is finite. It admits a recursive representation 1 G c (h 1, h, h 3 z) = G (h 1, h, h 3 z) + 3 i=1 r,s 1 A rs i (h 1, h, h 3 ) G crs(h c c rs (h i ) i )(h i h i + rs z), (1.5) where c rs (h) is a value of the central charge at which a primary of weight h has a null descendant at level rs, and A rs i are explicitly known functions of the weights. The Z 3 cyclic permutations of the three sheets are themselves elements of the Sp(4, Z) modular group. A nontrivial Sp(4, Z) involution that commutes with the Z 3 is the transformation z 1 z. This gives rise to a genus two modular crossing equation, [ F c (h i, h j, h k z)f c ( h i, h j, h k z) F c (h i, h j, h k 1 z)f c ( h i, h j, h ] k 1 z) = 0. i,j,k I C ijk (1.6) Together with the non-negativity of Cijk for unitary theories, this crossing equation now puts nontrivial constraints on the possible sets of structure constants. For instance, we will find examples of critical surfaces S that bound a (typically compact) domain D in the space of triples of conformal weights (h 1, h, h 3 ; h 1, h, h 3 ), such that the structure constants C ijk with (h i, h j, h k ; h i, h j, h k ) outside the domain D are bounded by those within the domain D. In particular, applying this to noncompact unitarity CFTs, one concludes that there must be triples of primaries in the domain D whose structure constants are nonzero. We emphasize that the existence of a compact critical surface for the structure constants is a genuinely nontrivial consequence of genus two modular invariance, which does not follow simply from 1 In contrast to the form of the recursion formulae presented in [1], here we include the factor z h1+h+h3 in the definition of the blocks, so that the residue coefficients do not depend on z. 3
5 a combination of bounds on spectral gaps in the OPEs (from analyzing the crossing equation of individual sphere 4-point functions) and modular invariance of the torus partition function (which does not know about the structure constants). The crossing equation for (1.3) does not capture the entirety of genus two modular invariance, since the Renyi surfaces lie on a 1 complex dimensional locus (1.) in the 3 complex dimensional moduli space of genus two Riemann surfaces. Instead of considering general deformations of the geometry, equivalently we can again insert stress-energy tensors on the Renyi surface, or more generally insert Virasoro descendants of the identity operator in the twist field correlator (1.3) (on any of the three sheets). This will allow us to access the complete set of genus two modular crossing equations, through the conformal block decomposition of (1.3) with extra stress-energy tensor insertions, which is computable explicitly as an expansion in z (or better, in terms of the elliptic nome q = e πiτ, where τ is related to z by (1.1)). Explicit computation of the genus two Virasoro conformal block of the Renyi surface in the twist-field frame will be given in section. The genus two modular crossing equation will be analyzed in section 3. In particular, we will find critical surfaces for structure constants simply by taking first order derivatives of the modular crossing equation with respect to the moduli around the crossing invariant point. In section 4, we formulate the crossing equation beyond the Z 3 -invariant locus in the moduli space of genus two Riemann surfaces. We conclude with some future prospectives in section 5. Note added: This paper is submitted in coordination with [18] and [19], which explore related aspects of two-dimensional conformal bootstrap at genus two. The genus two conformal block In this section, we will study the genus two Virasoro conformal block with no external operators, focusing on the Z 3 -invariant Renyi surface that is a 3-fold branched cover of the Riemann sphere with four branch points. The latter can be represented as the curve y 3 = (x x+ 1 )(x x + ) (x x 1 )(x x ) (.1) in P 1 P 1. The genus two partition function of the CFT on the covering surface can be viewed as a correlation function of the 3-fold symmetric product CFT on the sphere: up to a conformal anomaly factor (dependent on the conformal frame), it is given by the 4-point function of Z 3 twist fields σ 3 and anti-twist fields σ 3, σ 3 (x + 1 )σ 3 (x + ) σ 3 (x 1 ) σ 3 (x ). 4
6 1 3 h 1 h σ 3 σ 3 σ 3 σ 3 h 3 Figure 1: Left: The 3-fold cover of the Riemann sphere with four branch points is a genustwo surface. The partition function of the CFT on the covering surface can be regarded as the four-point function of Z 3 twist fields in the 3-fold product CFT on the sphere. Right: The genus two conformal block associated with the σ 3 σ 3 OPE channel..1 OPE of Z 3 -twist fields in Sym 3 (CFT) We will begin by analyzing the OPE of the Z 3 twist field σ 3 and the anti-twist field σ 3. The 3-fold symmetric product CFT on the sphere with the insertion of σ 3 (z 1 ) and σ 3 (z ) can be lifted to a single copy of the CFT on the covering space Σ, which is also a Riemann sphere. Let t be the affine coordinate on the covering sphere. It suffices to consider the special case z 1 = 0, z = 1, where the covering map can be written as z = (t + ω) 3 3ω(1 ω)t(t 1), (.) where ω = e πi/3. The branch points z 1 = 0, z = 1 correspond to t = ω and t = 1 + ω respectively. We have chosen this covering map (up to SL(, C) action on Σ) such that the three points t 1 = 0, t = 1, and t 3 = on Σ are mapped to z =. Now let us compute the 3-point function of the pair of twist fields σ 3 (0), σ 3 (1), and a general Virasoro descendant operator in the 3-fold tensor product CFT of the form Φ = 3 L Ni φ i (.3) i=1 inserted at z = (as a BPZ conjugate operator). Here we will keep track of the holomorphic z-dependence only, and omit the anti-holomorphic sector. For each i = 1,, 3, φ i is a primary 5
7 of weight h i in a single copy of the CFT, N i = {n (i) 1,, n (i) k } is a partition of the integer N i in descending order, and L Ni is the Virasoro chain L (i) n L (i) 1 n. Following [14], we k can write σ 3 (0) σ 3 (1)Φ( ) = O σ 3 (0) σ 3 (1) 1(0)O (1)O 3( ). (.4) Here O i(t i ) is the conformally transformed operator of L Ni φ i on the i th covering sheet, O i(t i ) = (z (t i )) h i L t i N i φ i(t i ) = (z(t i )) h i [3ω(1 ω)] h i L t i N i φ i(t i ), (.5) where φ i(t i ) is the corresponding primary in the t-frame. L t N = Lt n 1 L t n k is the lift of L N (acting on an operator at z = ) to the t-plane. When acting on an operator at t = t i, L t n is given by L t du (z(u)) 1+n [ n = T C t πi z uu (u) c ] (u) 1 {z(u), u} ] = [3ω(1 ω)] n Res u t u 1 n (u 1) 1 n (u + ω) 1+3n (u + ω ) [T c uu (u), (u + ω) (u + ω ) (.6) where we used the Schwarzian derivative {z, t} = 1 (t + ω) (t + ω ). (.7) The contour integral in (.6) is taken on the t-plane, parameterized by the variable u. C ti is a small counterclockwise circular contour around t i for t 1 = 0 and t = 1. For t 3 =, C is taken to be a large clockwise circular contour on the t-plane. Note that the sign convention 1 for the residue at infinity is such that Res u = 1. The overall minus sign on the RHS u of (.6) is due to the orientation of the original z-contour (where we replace L n acting on an operator at z = by L n acting on the product operator σ 3 (0) σ 3 (1)). where We proceed by putting (.6) into the explicit form L t n = a t n,ml m + c b t n, m n a 0 n,m = [3ω(1 ω)] n Res u 0 u n m 1 (u 1) 1 n (u + ω) 1+3n (u + ω ), a 1 n,m = [3ω(1 ω)] n Res u 1 u 1 n (u 1) n m 1 (u + ω) 1+3n (u + ω ), a n,m = [3ω(1 ω)] n Res u u n+m 1 (u 1) 1 n (u + ω) 1+3n (u + ω ), (.8) (.9) and b t n = [3ω(1 ω)] n Res u t u 1 n (u 1) 1 n (u + ω) 1+3n (u + ω ) 4, (.10) The factor z(t i ) hi drops out of the correlator (.4) due to the normalization convention of Φ( ). 6
8 for t = 0, 1,. On the RHS of (.8), L m is understood to be acting on an operator inserted at t = 0, 1, or. Putting these together, the 3-point function of interest is σ 3 (0) σ 3 (1)Φ( ) σ 3 (0) σ 3 (1) = C 13 [3ω(1 ω)] h 1 h h 3 ρ(l N 3 h 3, L 1 N h, L 0 N 1 h 1 ), (.11) where C 13 = φ 1 (0)φ (1)φ 3 ( ) is the structure constant of the primaries, and ρ(ξ 3, ξ, ξ 1 ) is the 3-point function of Virasoro descendants at, 1, 0 on the plane, defined as in [1,0]. We remind the reader that so far we have only taken into account the holomorphic part of the correlator, for the purpose of deriving the holomorphic Virasoro conformal block in the next section.. The conformal block decomposition of σ 3 (0) σ 3 (z)σ 3 (1) σ 3 ( ) Now we turn to the 4-point function of twist fields, σ 3 (0) σ 3 (z)σ 3 (1) σ 3 ( ), and compute the contribution from general untwisted sector descendants of the form Φ = 3 i=1 L N i φ i in the σ 3 (0) σ 3 (z) OPE, for a given triple of primaries φ 1, φ, φ 3. Again, we focus only on the holomorphic sector. This is given by 3 {N i },{M i } k=1 = {N i },{M i } = {N i },{M i } = {N i },{M i } G N km k h k σ 3 (0) σ 3 (z) [ 3 i=1l Ni φ i ] [ 3 i=1 φ i L M i ] σ 3 (1) σ 3 ( ) z hσ+ 3 i=1 (h i+ N i ) z hσ+ 3 i=1 (h i+ N i ) z hσ+ 3 i=1 (h i+ N i ) 3 k=1 3 k=1 k=1 G N km k h k σ 3 (0) σ 3 (1) 3 i=1 L Ni φ i ( ) 3 j=1l Mj φ j (0)σ 3 (1) σ 3 ( ) G N km k h k σ 3 (0) σ 3 (1) 3 i=1 L Ni φ i ( ) σ 3 (0)σ 3 (1) 3 j=1 L Mj φ j ( ) [ 3 ] 3 3 G N km k h k z (t k ) h k L Ni φ i (t i ) L M j φ j (t j ). (.1) Here the summation is over integer partitions in descending order N i and M i, for i = 1,, 3, and G NM h are the inverse Gram matrix elements for a weight h Verma module (nontrivial only for N = M ). L M is defined as the complex conjugation of L M (not to be confused with the adjoint operator), which simply amounts to replacing ω by ω in (.8)-(.10). The appearance of the complex conjugate factors is due to the exchange of σ 3 with σ 3 in the last two factors in the third line of (.1). h σ is the holomorphic conformal weight of the Z 3 twist field, given by ( h σ = 3 1 ) c 3 4 = c 9. (.13) 7 i=1 j=1
9 Using the covering map in the previous section, we arrive at the genus two conformal block for the Renyi surface in the twist field frame F c (h 1, h, h 3 z) = i=1 h i {N i },{M i } z hσ+ 3 i=1 (h i+ N i ) 3 k=1 G N km k h k ρ(l N 3 h 3, L 1 N h, L 0 N 1 h 1 )ρ(l M 3 h 3, L 1 M h, L 0 M 1 h 1 ), (.14) where L 0,1, N are given by (.8)-(.10). Let us comment on the h i 0 limit, which is rather delicate. If one of the h i vanishes, say h 1 = 0, corresponding to the vacuum channel in one of the three handles of the genus two surface, then the only conformal blocks that appear in the genus two partition function involve h = h 3. For h = h 3 > 0, the h 1 = 0 block is given by the h 1 0 limit of (.14). This is not the case however for the vacuum block, where all three weights h i vanish: in fact the vacuum block differs from the simultaneous h i 0 limit of (.14). This is because the latter contains nonvanishing contributions from null descendants of the identity operator that are absent in the vacuum block..3 Recursive representation As already mentioned in the introduction, the genus two conformal block (.14) admits a recursive representation in the central charge of the form (1.4), (1.5). The recursion formula is useful in computing the z-expansion to high orders efficiently, and can be derived by essentially the same procedure as in [1]. The only new feature is that the twist field frame considered here is different from the plumbing frame of [1], which leads to the conformal anomaly factor exp [ cf cl (z) ] in (1.4). While in principle F cl (z) can be determined by evaluating a suitable classical Liouville action as in [14], we find it more convenient to compute F cl (z) by directly inspecting the large c limit of log F c (h 1, h, h 3 z). Indeed, the latter is linear in c in the large c limit (with a leading coefficient that is independent of the internal weights), with the following series expansion in z F cl (z) = ( z ) ( z ) 3 ( z ) 4 ( z ) 5 ( z 6 9 log(z) ) ( z ) ( z ) 8 ( z ) ( z ) + O(z 11 ). (.15) Note that cf cl (z) agrees with the semiclassical Virasoro sphere 4-point conformal block of 3 central charge c in the vacuum channel with four external primaries of weight hσ = c [1,]. 3 7 For numerical computations, we can pass to the elliptic nome parameter q = e πiτ, where τ 8
10 is related to z via (1.1). The q-expansion converges much faster than the z-expansion 3 evaluated at the crossing symmetric point z = 1, which corresponds to q = e π. After factoring out exp [ cf cl (z) ], the remaining part of the conformal block G c (h 1, h, h 3 z) as a function of the central charge c has poles at c rs (h) = 1 + 6(b rs (h) + b rs (h) 1 ), with b rs (h) = rs 1 + h + (r s) + 4(rs 1)h + 4h, 1 r (.16) where r and s 1, for h = h i, i = 1,, 3. The residue at a pole c = c rs (h i ) is proportional to the conformal block with central charge c rs (h i ) and the weight h i shifted to h i + rs. The precise recursion formula is G c (h 1, h, h 3 z) = G (h 1, h, h 3 z) + where A c rs is the constant r,s 1 [ c ] A crs(h 1) rs rs(h 1 ) h 1 ( P rs c rs(h 1 ) c c rs (h 1 ) + ( cyclic permutations on h 1, h, h 3 ), A c rs = 1 r s m=1 r n=1 s for c = 1 + 6(b + b 1 ), and P rs c P rs c [ d1 d ] = r 1 p=1 r step q=1 s step [ h h 3 ]) G crs(h i )(h i h i + rs z) (.17) (mb + nb 1 ) 1, (m, n) (0, 0), (r, s), (.18) is the fusion polynomial s 1 where λ i are related to the weights d i by d i = 1 4 (b + b 1 ) 1 4 λ i. λ 1 + λ + pb + qb 1 λ 1 λ + pb + qb 1, (.19) The remaining undetermined piece in the formula (.17) is the c limit G (h 1, h, h 3 z). It was shown in [1] that G (h 1, h, h 3 z) is the product of the vacuum block and SL(, C) global block in the plumbing frame. The vacuum block is given by the holomorphic part of the gravitational 1-loop free energy of the genus two hyperbolic handlebody, computed in [3]. To translate the result of [3] into the vacuum part of our G requires expressing the Schottky parameters of the Renyi surface in terms of z; this can be achieved through the map between Schottky parameters and the period matrix (1.). Furthermore, the global 3 As explained in [], the q-expansion of F cl in general need not have unit radius of convergence, due to possible zeroes of the conformal block. In the present example, the radius of convergence nonetheless appears to be 1. We thank Y.-H. Lin for pointing out this subtlety and providing numerical verifications. 9
11 block of [1] is naturally expressed in terms of the plumbing parameters, whose map to z is nontrivial. The implementation of an efficient recursive computational algorithm for the genus two conformal blocks in the twist field frame will require knowing G, which is in principle computable given the above ingredients, based on the map from z to the Schottky parameters and the plumbing parameters of the Renyi surface. Here we simply evaluate the z-expansion (.14) directly, strip off the conformal anomaly factor and then take the c limit, giving the result G (h 1, h, h 3 z) = ( z 7) h1 +h +h 3 { 1 + [ h1 + h + h 3 + (h h 3 ) + (h 3 h 1 ) + (h ] 1 h ) z 54h 1 54h 54h 3 + z 916h 1 (1+h 1 )h (1+h )h 3 (1+h 3 ) [ 4h h h 3 h h h h h 3 h h 3 h h6 1 16h3 h5 1 16h3 3 h h h h h 3 h h 3 h h h h h 3 h h h 3 h h 3 h 5 1 3h h4 h h4 3 h h3 h4 1 08h h 3 3 h h3 3 h h h h h 3 h h h 3 h h 3 h h h h3 h 3 h h h 3 h h h 3 h h 3 h h5 h3 1 16h5 3 h h4 h3 1 08h h 4 3 h h4 3 h h3 h h h3 3 h h h 3 3 h h3 3 h3 1 84h h h3 h 3 h h h 3 h h h 3 h3 1 84h 3 h h h h4 h 3 h h3 h 3 h h h 3 h h h 3 h h 3 h h h6 h 1 + 4h6 3 h h5 h h h 5 3 h h5 3 h h4 h h h4 3 h h h 4 3 h h4 3 h 1 84h3 h h3 h3 3 h h h3 3 h h h 3 3 h 1 84h3 3 h 1 6h h h 4 h 3 h h3 h 3 h h h 3 h h h 3 h 1 6h 3 h 1 h h h5 h 3 h h4 h 3 h h3 h 3 h h h 3 h h h 3 h 1 h 3 h 1 + 6h6 h 1 + 8h h 6 3 h 1 + 6h 6 3 h h 5 h h h5 3 h h h 5 3 h h 5 3 h h 4 h 1 08h 3 h4 3 h h h4 3 h h h 4 h h 4 h h 3 h 1 08h 4 h3 h h 3 h3 h h h3 h h h 3 h h 3 h 3 1 h h h 5 h h h 4 h h h 3 h h h h h h h h 3 1 h h h 6 h 3 h h 5 h 3 h h 4 h 3 h h 3 h 3 h h h 3 h 1 + 6h h 3 h 1 + h 6 ] } +O(z 3 ). +4h h h h h6 3 3h5 16h3 h h h h h 5 3 3h h4 h h3 h h h h h h3 16h5 h h4 h h3 h3 3 84h h h h h h6 h h5 h h4 h 3 84h3 h 3 6h h 3 h h 3 + 6h6 h h 5 h h 4 h h 3 h 3 h h 3 (.0) As already noted, the analog of G for the vacuum block, G (z), 0 is not the same as the simultaneous h i 0 limit of (.0). The first few terms in the z-expansion of G (z) 0 is given explicitly by ( z ) 4 ( z ) 5 ( z ) 6 ( z ) 7 ( z ) 8 G (z) 0 = O(z 9 ) (.1).4 Mapping to the 3-fold-pillow In this section we consider the Renyi surface in the 3-fold-pillow frame, which makes obvious certain positivity properties of the genus two conformal block. Following [4], the map from the plane (parameterized by w) to the pillow (parameterized by v) is given by v = 1 (θ 3 (τ)) w 0 dx x(1 x)(z x). (.) The four branch points on the plane at 0, z, 1,, where the Z 3 twist fields and anti-twist fields are inserted, are mapped to v = 0, π, π(τ + 1), πτ respectively, where τ is given by (1.1). The covering surface is turned into a 3-fold cover of the pillow, with the twist fields inserted at the four corners. The Renyi surface conformal block in the twist field frame can be mapped to the pillow frame as F c (h 1, h, h 3 z) = (z(1 z)) c 8 hσ θ 3 (τ) 3c 16hσ q h 1+h +h 3 c 8 A n (h 1, h, h 3 )q n, (.3) 10 n=0
12 { 048 [16 (c + 8h 3 ) h 3 + ( 18h 3 + 4(c 5)h 3 + c(c + 3) ) h + ( 18h (c 5)h3 + 4 ( c 6c + 5 ) h 3 + c(3c 10) ) h + c (h 3 + 1) ( 16h 3 + (c 5)h 3 + c ) ] [ h (c + 8h 3 ) h 4 16 ( 56h 3 + 8(7c 3)h 3 + c(3c + 1) ) h 3 + ( 048h 33 18(3c + 1)h3 8 ( c + 66c 195 ) h 3 + c ( c 34c 63 )) h + ( 4096h 43 18(7c 3)h33 16 ( c + 66c 195 ) h ( c 3 43c + 47c 55 ) h 3 + c ( 3c 73c + 10 )) h + c ( 51h 43 16(3c + 1)h33 + ( c 34c 63 ) h 3 + ( 3c 73c + 10 ) h 3 + (c 1)c ) ] [ h (c + 8h 3 ) h 5 64 ( 64h (c )h 3 + (c 1)c ) h ( 104h 33 3(3c + 0)h3 + 4 ( 3c 96c + 7 ) h 3 + c ( c 15c 49 )) h 3 + ( 048h 43 56(3c + 0)h33 64(33c 103)h3 + 8 ( c 3 7c + 4c 70 ) h 3 + c ( 7c 103c + 13 ) ) h 16 ( 3768h (3c + 0)h53 56 ( c 57c + 75 ) h ( 6c 3 31c 1495c ) h 33 ( c c 6448c ) h 3 + ( 6c c c 4000 ) h 3 + 4c ( 3c 40c )) h 3 + ( 13107h (3c + 1)h (33c 103)h ( c 3 5c 180c ) h ( c c 6448c ) h ( 5c 4 47c c c 3000 ) h 3 + c ( 75c c 997c ) h c (5 c) ) h + ( 4915(c 5)h ( c + 66c 195 ) h ( c 3 7c + 4c 70 ) h ( 1c 3 35c + 81c 950 ) h ( 6c c c 4000 ) h 33 + c ( 75c c 997c ) h 3 + ( 5c 15c 15c 51 ) h 3 64c 3 ) ( h + 64ch 3 3(c + 3)h ( c 34c 63 ) h 53 ( 7c 103c + 13 ) h 43 + ( 5c 13c + 00 ) h 33 + ( 3c + 40c 100 ) h ( c + 66c 195 ) h ( c 3 7c + 4c 70 ) h ( 1c 3 35c + 81c 950 ) h ( 6c c c 4000 ) h 33 + c ( 75c c 997c ) h 3 + ( 5c 15c 15c 51 ) h 3 64c 3 ) h + h ( ( c 6c + 5 ) h ( c 3 43c + 47c 55 ) h ( 7c 3 186c + 779c 900 ) h ( 9c 3 19c + 370c 50 ) h c ( 16c + 761c 800 ) h 3 + ( 10c 15c 15c 51 ) h 3 + 3c 3 (5c + 56) ) h + 18ch 3 ( 16(3c 10)h 5 + ( 3c 73c + 10 ) h 43 3 ( 3c 6c + 10 ) h 33 + ( 3c 38c + 50 ) h c h 3 c ) ] h c ( 16h + (c 5)h + c ) (h h 3 ) ( 16h 3 + (c 5)h 3 + c ) ( (h 3 + 1) h 3 ( h 3 + 1) h + ( h ) h + (h 3 1) ) } h 3. v = πτ σ 3 σ 3 T (v) + σ 3 σ 3 v = 0 v = π Figure : Left: The pillow geometry is the quotient T /Z. The four branch points on the plane 0, z, 1, are mapped to the Z fixed points v = 0, π, π(τ + 1), πτ respectively. Right: The pillow with the Z 3 twist fields inserted at the corners. In Section 4 we will obtain the full set of genus two modular crossing equations by inserting the stress-energy tensor or more generally arbitrary Virasoro descendants of the identity at the front center on each sheet of the 3-fold-pillow. where q = e πiτ, h σ = c 9. For instance, the first few coefficients A 0, A 1 and A are given by A 0 = c A 1 = c 1 ( 16 7 A = ( ) h1 +h 16 +h 3, 7 ) h1 +h +h 3 +1 [ (h1 h ) h 3 + (h h 3 ) h 1 + (h 3 h 1 ) ( ) c 9 16 h1 +h +h h 1 (c + h 1 (c + 8h 1 5))h (c + h (c + 8h 5))h 3 (c + h 3 (c + 8h 3 5)) h ], (.4) + ( 188h (c )h ( 3c 96c + 7 ) h ( c 3 7c + 4c 70 ) h ( 7c 3 186c + 779c 900 ) h 3 + 3c ( 3c 6c + 10 ) ) h + c ( 1536h 53 64(c 1)h ( c 15c 49 ) h 33 + ( 7c 103c + 13 ) h ( 3c 6c + 10 ) h 3 + 3(c 1)c ) ] [ h (c + 8h 3 ) h ( 64h (c )h 3 + (c 1)c ) h ( 1536h 33 64(c + 5)h3 + 8 ( 5c 106c + 41 ) h 3 + c ( 3c 38c 93 )) h 4 + ( 307h 43 56(c + 7)h ( c 57c + 75 ) h 3 + ( c 3 9c 19c + 5 ) h 3 + c ( c 19c 39 ) ) h 3 + ( 048h 53 56(c + 5)h ( c 57c + 75 ) h ( c 3 5c 180c ) h 3 + ( 1c 3 35c + 81c 950 ) h 3 + c ( 5c 13c + 00 )) h + ( 4096h (c )h ( 5c 106c + 41 ) h ( c 3 9c 19c + 5 ) h 33 + ( 1c 3 35c + 81c 950 ) h 3 + ( 9c 3 19c + 370c 50 ) h 3 + c ( 3c 38c + 50 ) ) h + c ( 51h (c 1)h ( 3c 38c 93 ) h 43 + ( 4c 38c 78 ) h 33 + ( 5c 13c + 00 ) h 3 + ( 3c 38c + 50 ) h 3 5c ) ] [ h (c + 8h 3 ) h ( 56h 3 + 8(7c 3)h 3 + c(3c + 1) ) h ( 104h 33 3(3c + 0)h3 + 4 ( 3c 96c + 7 ) h 3 + c ( c 15c 49 )) h 5 3( 307h 43 56(c + 7)h ( c 57c + 75 ) h 3 + ( c 3 9c 19c + 5 ) h 3 + c ( c 19c 39 ) ) h ( 819h (c + 7)h ( 6c 68c 347 ) h 33 + ( 6c 3 31c 1495c ) h 3 + ( c 3 469c + 8c 600 ) h 3 + c ( 3c 10c + 40 )) h 3 + ( 3768h (3c + 0)h53 56 ( c 57c + 75 ) h ( 6c 3 31c 1495c ) h 33 ( c c 6448c ) h 3 + ( 6c c c 4000 ) h 3 + 4c ( 3c 40c ) ) h ( 16384h (7c 3)h63 51 ( 3c 96c + 7 ) h ( c 3 9c 19c + 5 ) h ( c 3 469c + 8c 600 ) h 33 ( 6c c c 4000 ) h 3 + c ( 16c + 761c 800 ) h 3 + 3c 3 ) h + 8c ( 56h (3c + 1)h ( c 15c 49 ) h 53 + ( 8c + 76c ) h 43 + ( 6c 04c + 80 ) h 33 + ( 3c 40c ) h 3 4c h 3 c ) ] [ h ( 18h 3 + 4(c 5)h 3 + c(c + 3) ) h ( 048h 33 18(3c + 1)h3 8 ( c + 66c 195 ) h 3 + c ( c 34c 63 )) h 6 18( 048h 43 56(3c + 0)h33 64(33c 103)h3 + 8 ( c 3 7c + 4c 70 ) h 3 + c ( 7c 103c + 13 ) ) h ( 048h 53 56(c + 5)h ( c 57c + 75 ) h ( c 3 5c 180c ) h 3 + ( 1c 3 35c + 81c 950 ) h 3 + c ( 5c 13c + 00 )) h 4 +4c(5 c)h 3 c )] [ h ( 18h (c 5)h3 + 4 ( c 6c + 5 ) h 3 + c(3c 10) ) h ( 4096h 43 18(7c 3)h33 16 ( c + 66c 195 ) h ( c 3 43c + 47c 55 ) h 3 + c ( 3c 73c + 10 )) h 6 18 ( 188h (c )h ( 3c 96c + 7 ) h ( c 3 7c + 4c 70 ) h ( 7c 3 186c + 779c 900 ) h 3 + 3c ( 3c 6c + 10 )) h ( 4096h (c )h ( 5c 106c + 41 ) h ( c 3 9c 19c + 5 ) h 33 + ( 1c 3 35c + 81c 950 ) h 3 + ( 9c 3 19c + 370c 50 ) h 3 + c ( 3c 38c + 50 ) ) h ( 16384h (7c 3)h63 51 ( 3c 96c + 7 ) h ( c 3 9c 19c + 5 ) h ( c 3 469c + 8c 600 ) h 33 ( 6c c c 4000 ) h 3 + c ( 16c + 761c 800 ) h 3 + 3c 3 ) h 3 + ( 4915(c 5)h 73 We also record here the first few coefficients A 0 n in the q-expansion of the vacuum block in the pillow frame analogous to (.3), which, as already emphasized, differ from the h i 0 11
13 limit of (.4), A 0 0 = c, A 0 1 = 0, A 0 = c 1 5c 43, A0 3 = 0, A 0 4 = c c c , A 0 5 = c , A 0 6 = c c c c c (.5) Importantly, all of the coefficients A n (h 1, h, h 3 ) are non-negative, as they can be interpreted as inner products of level n descendant states created by pairs of twist-anti-twist fields on two corners of the pillow, similarly to the sphere 4-point block analyzed in [4]. Indeed, we have explicitly verified the positivity of A n (h 1, h, h 3 ) with c > 1 and h i > 0, for n 5. 3 The genus two modular crossing equation 3.1 Some preliminary analysis Now we consider the genus two modular crossing equation restricted to the Renyi surface, as given by (1.6). Some crude but rigorous constraints on the structure constants in unitary CFTs can be deduced even without appealing to the details of the z-expansion of the genus two conformal block. First, let us write the twist field 4-point function (1.3) in the pillow coordinates, σ 3 (0) σ 3 (z, z)σ 3 (1) σ 3 ( ) = (z(1 z)) c 8 hσ θ 3 (τ) 3c 16hσ q c 8 CijkA n (h i, h j, h k )A m ( h i, h j, h k )q h i+h j +h k +n q h i + h j + h k +m i,j,k n,m=0 = (z(1 z)) c 8 hσ θ 3 (τ) 3c 16hσ q c 8 C qh q h. h, h (h, h) J (3.1) In the last line, we simply grouped terms of the same powers of q and q together in the sum. The index set J is by construction the union of ( 3 i=1 h i+z 0, 3 h i=1 i +Z 0 ) for all triples of conformal weights {(h i, h i ), i = 1,, 3} that appear in nonzero structure constants, including the case where one of the primaries is the identity and the structure constant reduces to the two-point function coefficient. It follows from the non-negativity of the coefficients A n that C are non-negative quantities in a unitary CFT. h, h Let us now apply (3.1) to a unitary noncompact CFT, where the SL()-invariant vacuum is absent and the identity is not included in the spectrum of (δ-function) normalizable 1
14 operators. C now only receives contributions from the structure constants of nontrivial h, h primaries. Applying first order derivatives in z and z to the crossing equation, and evaluating at z = z = 1, we have (h, h) J C h, h z z= 1 [ ] (z(1 z)) c 4 4hσ θ 3 (τ) 3c 3hσ q h+ h c 4 = 0. (3.) In the above equation, the factor multiplying C is negative for h + h below a certain h, h critical dimension crit and positive for > crit. It follows immediately that there must be a nonzero C for < h, h crit, i.e. there must be a triple of primaries with nonzero structure constant, whose total scaling dimension is less than crit, in any unitary noncompact CFT of central charge c. The value of (or rather, an upper bound on) the critical dimension is easily computed from (3.) to be crit = ( π ) c + 8 π h σ = 9π + 5 c 0.941c. (3.3) 36π As a consistency check, the Liouville CFT of central charge c has nonzero structure constants for triples of primaries of total scaling dimension above the threshold c 1, which is indeed 4 less than (3.3). Although rigorous, the bound (3.3) is quite crude. To deduce similar results in compact CFTs, it will be important to distinguish the contributions of Virasoro descendants from those of the primaries in (3.1). We will refine our analysis in the next subsection by computing the z or q-expansion of the genus two conformal block to higher orders. 3. Critical surfaces As is standard in the numerical bootstrap [5 7], we can turn the genus two modular crossing equation (1.6) into linear equations for Cijk by acting on it with the linear functional α = n+m=odd a n,m n z m z z= z= 1, (3.4) where a n,m are a set of real coefficients, and obtain constraints on the structure constants of the general form CijkF c α (h i, h j, h k ; h i, h j, h k ) = 0, (3.5) i,j,k I where Fc α is a function of a triple of left and right conformal weights. For typical choices of the linear functional α, Fc α will be negative on a domain D in the space of triples of 13
15 conformal weights, and positive on the complement of the closure of D. A critical surface S is defined to be the boundary of D where Fcα vanishes. With an appropriate choice of sign in α, the domain D consists of triples of low lying weights (we will see that the critical surface is often compact), and the equation (3.5) implies that the structure constants outside of D are bounded by those that lie within D. c=4 c=1 c=5 h3 h3 h h1 =0.01 h1 =0.03 h1 =0.05 h1 =0.07 h1 = h1 =0.06 h1 =0.14 h1 =0. h1 =0.3 h1 = h1 =0.5 h1 =1 h1 =1.5 h1 = h1 = h h h (3) Figure 3: Top: Three-dimensional plots of the domain Dh for c = 1, 4, 5. Bottom: Plots of the cross-sections of these domains for various values of h1. The structure constants of primaries with twists (τ1, τ, τ3 ) = (h1, h, h3 ) outside these critical domains are bounded by those whose twists lie within the domains. Clearly, the critical surface S depends on the choice of α. It is of interest to find critical surfaces that bound a domain D that is as small as possible, so that we can bound as many structure constants as possible based on the knowledge of a small set of structure constants of low dimension operators in any unitary CFT. Here we will consider the simplest nontrivial linear functional α which involves only first order derivatives in z or in z. In this case, the 14
16 critical surface is the locus a 1,0 W c (h 1, h, h 3 ) + a 0,1 W c ( h 1, h, h 3 ) = 0, (3.6) where W c (h 1, h, h 3 ) z log F c (h 1, h, h 3 z) z= 1. For instance, we can choose a 0,1 = 0, and the critical surface W c (h 1, h, h 3 ) = 0 bounds a compact domain D h in R 3 0 parameterized by the holomorphic weights (h 1, h, h 3 ), and bound structure constants of triples of primaries of higher twists by those of lower twists. From (.3), we have W c (h 1, h, h 3 ) = π K( 1 ) [ h 1 + h + h 3 ( π ) c + n=1 na ] n(h 1, h, h 3 )e nπ n=0 A. n(h 1, h, h 3 )e nπ (3.7) The last term in the bracket is always positive (assuming c > 1 and h i > 0), thus the domain W c < 0 lies within the region h 1 + h + h 3 < ( π) c and is compact. This is what we have seen in the previous subsection. h 3 c=4, h 1= N= N=3 N= h Figure 4: A slice of the c = 4 critical domain D (N) h, which converges quickly with the truncation order N of the q-expansion. For numerical evaluation we may work with the truncated version [ ( 1 h 1 + h + h ) c + 7π W (N) c (h 1, h, h 3 ) = π K( 1 ) ] N n=1 na n(h 1, h, h 3 )e nπ N n=0 A. n(h 1, h, h 3 )e nπ (3.8) The domain D (N) h = {(h 1, h, h 3 ) R 3 0 : W c (N) (h 1, h, h 3 ) < 0} becomes smaller with increasing N (and of course, converges to D h in the N limit). In Figure 3 we plot some examples of the critical domain D (N) h with N = 3, for central charges c = 1, 4, and 5. The location of the critical surface converges rather quickly with the q-expansion order: an example of a slice of the critical domain in the c = 4 case is shown in Figure 4. In particular, in the limit h 1 0, with h, h 3 fixed at generic positive values, the coefficients A n diverge like h 1 1 P n (h, h 3 ), where P n is a rational function of h, h 3 that 15
17 0.7 h 3 c=8 h 3 c= h 1 =10-4 h 1 =10-5 h 1 =10-6 h 1 = h 1 =10-4 h 1 =10-5 h 1 =10-6 h 1 = h h Figure 5: Slices of the critical domain D (3) h for c = 8 and c = 10 at small values of h 1. For c = 8, the critical domain D h intersects the h 1 = 0 plane only along a segment of the diagonal line h = h 3. This is not the case for c = 10. vanishes quadratically along h = h 3 (> 0). For h h 3 > 0, for instance, we have lim h1 0 W c (1) [ (h 1, h, h 3 ) = π K( 1 h + h 3 ( 1 ] 5 ) 8 7π) c + 1, which is always positive for c < A slightly more intricate analysis of lim h1 0 W c () shows that it is positive for c < Consequently, for this range of the central charge c, the domain D () h (and thereby D h ) meets the h 1 = 0 plane along a segment of the line h = h 3 only. This is demonstrated in Figure 5. For c > 1, we observe that W c (h 1, h, h 3 ) is minimized in the limit h 1 = h = h 3 0, where it approaches a negative value r c (note that in the simultaneous h i 0 limit W c depends on the ratios of the h i s). For a 1,0 and a 0,1 both positive, the domain D bounded by the critical surface S then lies strictly within the domain W c (h 1, h, h 3 ) < a 0,1 a 1,0 r c, W c ( h 1, h, h 3 ) < a 1,0 a 0,1 r c. (3.9) Let us choose a 0,1 = a 1,0, and define D as the domain W c (h 1, h, h 3 ) < r c in R 3 0. Now the compact domain D = D+ D (the set of sums of vectors from each set) in R 3 0 may be viewed as a critical domain in the triple of scaling dimensions ( 1,, 3 ), with i = h i + h i, in the sense that structure constants of triples of primaries of dimensions ( 1,, 3 ) outside 16
18 c=1 c=5 c=4 Δ3 Δ3 Δ3 0.4 Δ1 =0.08 Δ1 =0.14 Δ1 =0. Δ1 =0.6 Δ1 = Δ1 =0. Δ1 =0.55 Δ1 =0.9 Δ1 =1.5 Δ1 = Δ1 =1 Δ1 =3 Δ1 =5 Δ1 =7 Δ1 = Δ Δ Δ (3) Figure 6: Top: Three-dimensional plots of the domain D for c = 1, 4, 5. Bottom: Plots of the cross-sections of these domains for various values of 1. D are bounded by those that lie within D.4 Some examples of D are shown in Figure 6. A subtlety pointed out at the end of section. is that the simultaneous hi 0 limit of the genus two conformal block with three positive internal weights is distinct from the vacuum block. If we define Wc,0 to be (3.7) computed using the vacuum block, we would find a result that is slightly below limh1 =h =h3 0+ Wc (h1, h, h3 ) = rc. Since we seek critical surfaces such that the structure constants of heavy primaries outside are bounded by those of the light primaries that lie inside the surface, the vacuum block which enters the genus two partition function with coefficient 1 is not relevant, and thus the result (3.9) suffices. 4 e is convex, then D is simply D e rescaled by a factor of, but in fact D e is generally not Note that if D convex in the region where one of the weights is small. 17
19 4 Beyond the Z 3 -invariant surface In order to write the modular crossing equation for the partition functions on genus two Riemann surfaces of general moduli in a computationally useful manner, we will still work at the Z 3 -invariant Renyi surface and expand around the crossing-invariant point z = 1, but with extra insertions of stress-energy tensors T (z j ) and T ( z j ) on any of the three sheets. Under the crossing z 1 z, the transformation of the stress-energy tensors is simple. For instance, it suffices to work with the insertion of V = L N L Ñ 1 on one of the sheets at the point w. Here L N L n1 L nk is a Virasoro chain, and L Ñ is defined similarly. The crossing transformation sends V to the operator ( ) N + Ñ L N L Ñ 1 inserted at the position 1 w. The point w is mapped to the pillow coordinate via (.). In particular, with z = 1 1±i, τ = i, the points w = are mapped to v = ±1+i π (up to monodromies), i.e. the center on the front and back of the pillow. We can now define the modified conformal blocks with L Ri (x) insertion on the i-th sheet, F(h 1, h, h 3 ; R 1, R, R 3 ; w z) = i=1 h i z hσ+ 3 i=1 (h i+ N i ) w 3 k=1 ( M k N k R k ) {N i },{M i } ρ(l N 3 h 3, L 1 N h, L 0 N 1 h 1 )ρ(l M 3 h 3, L 1 M h, L 0 M 1 h 1 ) 3 G N kp k h k G M kq k h k ρ(l Qk h k, L Rk id, L Pk h k ). P i = N i, Q i = M i k=1 (4.1) Here the level sum takes the form of a series expansion in w and z/w. For numerical evaluation, it is far more efficient to reorganize the sum as an expansion in q 1 e i(πτ v) and q e iv instead, where τ and v are given by (1.1) and (.). As is evident from the pillow frame, evaluating at z = 1 1+i and w =, the effective expansion parameters are q 1 = q = e π/, with unit radius of convergence. Explicitly, we have z w = 4q 8q + 8q 1 q + 1q 3 3q 1 q + 4q 1q 16q q 1 q 3 48q 1q +..., (4.) and w is given by the same series expansion with q 1 and q exchanged. For example, the conformal block with a single stress-energy tensor inserted in the first 18
20 sheet, up to total level in q 1 and q, is given by F(h 1, h, h 3 ; R 1 = {}, R =, R 3 = ; q 1, q ) ( ) h1 +h 16q1 q +h 3 { (16q 1 q ) 9 c h1 = 7 q ( 18h 1 q 1 i ) 3(h h 3 )(q 1 q ) 36 [ 1 ( + h 1 h h 3 q1 8c + 413h 1 4(h + h 3 ) 48i ) 3(h h 3 ) 16h 1 h h 3 ( ( ) + 4q q 1 (h 1 6(c + 9)h3 h + h 3 + h3) 3 + h 4 1 (h + h 3 ) h 3 1 h + h 3 ( + h 1 h h 3 (h h 3 ) h h 3 + 1i ) ) ] 3 + h h 3 (h h 3 ) + h 1 h h 3 q (8c + 35h 1 4h 4h 3 ) } (4.3) If we symmetrize (4.1) with respect to R 1, R, R 3, we recover the conformal block of the Z 3 -invariant Renyi surface considered in the previous section, differentiated with respect to z, up to a conformal anomaly factor. In particular, summing over insertions of a single stress-energy tensor on one of the three sheets, we find F(h 1, h, h 3 ; R 1 = {}, R =, R 3 = ; q 1, q ) + ( cyclic permutations on R 1, R, R 3 ) = C(q 1, q ) q F(h 1, h, h 3 ; q) q=q1 q + cb(q 1, q )F(h 1, h, h 3 ; q = q 1 q ), (4.4) where the first term on the RHS is due to deformation of the modulus z or q and the second term is due to the conformal anomaly (from a Weyl transformation that flattens out the pillow geometry after the insertion of the stress-energy tensor). The functions C and B are independent of h i and c; they admit series expansions in q 1 and q of the form C(q 1, q ) = q + q 16q 1 + q ( ) 15q 1 + 8q + q q 1 q ( 9q q 1 q + 3q) +..., B(q 1, q ) = (4.5) + 19q 1 + 4q 1 q + 5q + 17q 1 + 8q 1 q + 11q q1 9q 1 36q1 9q 1 A complete set of genus two modular crossing equations can now be written as ( ) 3 j=1 ( R j + R j ) C h 1,h,h 3 ; h 1, h, h 3 F(h 1, h, h 3 ; R 1, R, R 3 ; w z)f( h 1, h, h 3 ; R 1, R, R 3 ; w z) = (h i, h i ) (h i, h i ) C h 1,h,h 3 ; h 1, h, h 3 F(h 1, h, h 3 ; R 1, R, R 3 ; 1 w 1 z)f( h 1, h, h 3 ; R 1, R, R 3 ; 1 w 1 z). (4.6) If we take into account all possible choices of integer partitions R j and R j, it suffices to evaluate this equation at the crossing-invariant point z = z = 1 1+i, with the choice w =, 19
21 w = 1 i. The consequence of (4.6) in constraining structure constants in unitary CFTs is currently under investigation. 5 Discussion The main results of this paper are the formulation of genus two modular crossing equations in an explicitly computable manner, by working on the Renyi surface as well as expanding around it. As an application, we found compact critical surfaces that bound domains D R 3 0 such that structure constants C ijk involving a triple of primaries whose dimensions ( i, j, k ) or twists (τ i, τ j, τ k ) are outside of D are bounded by those that lie within D. The existence of the compact critical surface is a nontrivial consequence of genus two modular invariance that does not follow easily from the analysis of individual OPEs: roughly speaking, the crossing equation for the sphere 4-point function bounds light-light-heavy structure constants in terms of light-light-light ones, but the genus two modular crossing equation also bounds light-heavy-heavy and heavy-heavy-heavy structure constants in terms of light-lightlight ones. In deriving the critical surface, we have used merely a tiny part of the genus two crossing equation, namely the first order z and z derivatives of the Renyi surface crossing equation evaluated at the crossing invariant point z = z = 1. Clearly, stronger results for the critical surfaces (that bound smaller domains) should be obtained by taking into account higher order z and z derivatives of the crossing equation. This is rather tricky to implement numerically through semidefinite programming, simply due to the fact the genus two conformal block decomposition involves 3 continuously varying scaling dimensions and 3 spins. To implement the crossing equation through [8], for instance, one may attempt to vary the sum of the 3 scaling dimensions, and sample over their differences as well as truncating on the spins, but such a sampling would involve a huge set of conformal blocks that is hard to handle numerically. At the moment this appears to be the main technical obstacle in optimizing the genus two modular bootstrap bounds. 5 Many more constraints on the structure constants C ijk can in principle be obtained by consideration of higher order derivatives of the genus two crossing equation. For instance, combining first and third order derivatives, analogously to [8, 5], one can deduce the existence of structure constants C ijk with say the dimensions ( i, j, k ) lying within a small domain (typically, such a domain is strictly larger than one that is bounded by a critical surface). The genus two modular invariance potentially has the power to constrain CFTs with 5 A potentially more efficient numerical approach would be based on sum-of-squares optimization, as is explained to us by D. Simmons-Duffin. 0
22 approximately conserved currents (i.e. primaries with very small twist): if such a current operator propagates through one of the three handles of the genus two surface, modular invariance should constrain the pairs of operators propagating through the other two handles according to representations of an approximate current algebra or W -algebra. Typically, when OPE bounds or (genus one) modular spectral bounds are close to being saturated [11], one finds that there are necessarily low twist operators in the spectrum. For instance, this strategy may be used to severely constrain (and possibly rule out) unitary compact CFTs with central charge c slightly bigger than 1. There is another genus two conformal block channel (the dumbbell channel ) that we have not discussed so far, namely one in which the genus two surface is built by plumbing together a pair of 1-holed tori. The conformal block decomposition of the genus two partition function in this channel involves the torus 1-point functions, or the structure constants C ijj where a pair of primaries are identified. The modular covariance of the torus 1-point function cannot be used by itself to constrain C ijj in a unitary CFT, since C ijj does not have any positivity property in general. In the dumbbell channel decomposition of the genus two partition function, the structure constants appear in the combination C ijj C ikk, allowing for the implementation of semidefinite programming. In our present approach via expansion around the Renyi surface, it appears rather difficult to perform the conformal block decomposition in the dumbbell channel explicitly. How to incorporate this channel in the genus two modular bootstrap is a question left for future work. Acknowledgements We are grateful to Alex Maloney for pointing out to us the significance of the Z 3 -invariant Renyi surface and for sharing his notes on the subject. We would also like to thank Ying- Hsuan Lin and David Simmons-Duffin for discussions and comments on a preliminary draft. XY thanks Simons Collaboration Workshop on Numerical Bootstrap at Princeton University and Quantum Gravity and the Bootstrap conference at Johns Hopkins University for their hospitality during the course of this work. This work is supported by a Simons Investigator Award from the Simons Foundation and by DOE grant DE-FG0-91ER MC is supported by Samsung Scholarship. SC is supported in part by the Natural Sciences and Engineering Research Council of Canada via a PGS D fellowship. Explicit computations of the conformal blocks in this work were performed with Mathematica using the Weaver package [9]. 1
23 References [1] S. Ferrara, A. F. Grillo, and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) [] A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz 66 (1974), no [3] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B41 (1984) [4] D. Friedan and S. H. Shenker, The Analytic Geometry of Two-Dimensional Conformal Field Theory, Nucl. Phys. B81 (1987) [5] G. W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 13 (1989) 177. [6] Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang, and X. Yin, N=4 Superconformal Bootstrap of the K3 CFT, arxiv: [7] Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, (,) Superconformal Bootstrap in Two Dimensions, arxiv: [8] S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (011) 130, [arxiv: ]. [9] C. A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compactifications, Commun. Math. Phys. 34 (013) , [arxiv: ]. [10] D. Friedan and C. A. Keller, Constraints on d CFT partition functions, JHEP 10 (013) 180, [arxiv: ]. [11] S. Collier, Y.-H. Lin, and X. Yin, Modular Bootstrap Revisited, arxiv: [1] M. Cho, S. Collier, and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, arxiv: [13] G. Mason and M. P. Tuite, On genus two Riemann surfaces formed from sewn tori, Commun. Math. Phys. 70 (007) , [math/ ]. [14] O. Lunin and S. D. Mathur, Correlation functions for M**N / S(N) orbifolds, Commun. Math. Phys. 19 (001) , [hep-th/ ]. [15] A. Maloney, Notes on the renyi surface, Private Communication. [16] P. Calabrese, J. Cardy, and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech (009) P11001, [arxiv: ]. [17] P. Calabrese, J. Cardy, and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech (011) P0101, [arxiv: ].
Recursive Representations of Arbitrary Virasoro Conformal Blocks
Recursive Representations of Arbitrary Virasoro Conformal Blocks arxiv:1703.09805v4 hep-th 29 Nov 2018 Minjae Cho, Scott Collier, Xi Yin Jefferson Physical Laboratory, Harvard University, Cambridge, MA
More informationEntanglement Entropy for Disjoint Intervals in AdS/CFT
Entanglement Entropy for Disjoint Intervals in AdS/CFT Thomas Faulkner Institute for Advanced Study based on arxiv:1303.7221 (see also T.Hartman arxiv:1303.6955) Entanglement Entropy : Definitions Vacuum
More informationHIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY
HIGHER SPIN CORRECTIONS TO ENTANGLEMENT ENTROPY JHEP 1406 (2014) 096, Phys.Rev. D90 (2014) 4, 041903 with Shouvik Datta ( IISc), Michael Ferlaino, S. Prem Kumar (Swansea U. ) JHEP 1504 (2015) 041 with
More informationDyon degeneracies from Mathieu moonshine
Prepared for submission to JHEP Dyon degeneracies from Mathieu moonshine arxiv:1704.00434v2 [hep-th] 15 Jun 2017 Aradhita Chattopadhyaya, Justin R. David Centre for High Energy Physics, Indian Institute
More informationOne-loop Partition Function in AdS 3 /CFT 2
One-loop Partition Function in AdS 3 /CFT 2 Bin Chen R ITP-PKU 1st East Asia Joint Workshop on Fields and Strings, May 28-30, 2016, USTC, Hefei Based on the work with Jie-qiang Wu, arxiv:1509.02062 Outline
More informationarxiv: v1 [hep-th] 16 May 2017
Bootstrapping Chiral CFTs at Genus Two arxiv:1705.05862v1 [hep-th] 16 May 2017 Christoph A. Keller a, Grégoire Mathys b, and Ida G. Zadeh a a Department of Mathematics, ETH Zurich, CH-8092 Zurich, Switzerland
More information1 Unitary representations of the Virasoro algebra
Week 5 Reading material from the books Polchinski, Chapter 2, 15 Becker, Becker, Schwartz, Chapter 3 Ginspargs lectures, Chapters 3, 4 1 Unitary representations of the Virasoro algebra Now that we have
More informationRényi Entropy in AdS 3 /CFT 2
Rényi Entropy in AdS 3 /CFT 2 Bin Chen R Peking University String 2016 August 1-5, Beijing (a) Jia-ju Zhang (b) Jiang Long (c) Jie-qiang Wu Based on the following works: B.C., J.-j. Zhang, arxiv:1309.5453
More informationBootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ
CALT-TH 016-040 arxiv:170.0043v1 [hep-th] 1 Feb 017 Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ Scott Collier 1, Petr Kravchuk, Ying-Hsuan Lin, Xi Yin
More informationHolomorphic Bootstrap for Rational CFT in 2D
Holomorphic Bootstrap for Rational CFT in 2D Sunil Mukhi YITP, July 5, 2018 Based on: On 2d Conformal Field Theories with Two Characters, Harsha Hampapura and Sunil Mukhi, JHEP 1601 (2106) 005, arxiv:
More informationVirasoro and Kac-Moody Algebra
Virasoro and Kac-Moody Algebra Di Xu UCSC Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 1 / 24 Outline Mathematical Description Conformal Symmetry in dimension d > 3 Conformal Symmetry in dimension
More informationPhase transitions in large N symmetric orbifold CFTs. Christoph Keller
Phase transitions in large N symmetric orbifold CFTs Christoph Keller California Institute of Technology work in progress with C. Bachas, J. Troost 15-th European Workshop on String Theory, Zürich September
More informationS ps S qs S rs S 0s. N pqr = s. S 2 2g
On generalizations of Verlinde's formula P. Bantay Inst. for Theor. Phys., Eotvos Univ. July, 000 Abstract It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae.
More informationTREE LEVEL CONSTRAINTS ON CONFORMAL FIELD THEORIES AND STRING MODELS* ABSTRACT
SLAC-PUB-5022 May, 1989 T TREE LEVEL CONSTRAINTS ON CONFORMAL FIELD THEORIES AND STRING MODELS* DAVID C. LEWELLEN Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 ABSTRACT.*
More informationA Proof of the Covariant Entropy Bound
A Proof of the Covariant Entropy Bound Joint work with H. Casini, Z. Fisher, and J. Maldacena, arxiv:1404.5635 and 1406.4545 Raphael Bousso Berkeley Center for Theoretical Physics University of California,
More informationLecture 8: 1-loop closed string vacuum amplitude
Lecture 8: 1-loop closed string vacuum amplitude José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 5, 2013 José D. Edelstein (USC) Lecture 8: 1-loop vacuum
More informationKnot Homology from Refined Chern-Simons Theory
Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones
More informationConformal Field Theories Beyond Two Dimensions
Conformal Field Theories Beyond Two Dimensions Alex Atanasov November 6, 017 Abstract I introduce higher dimensional conformal field theory (CFT) for a mathematical audience. The familiar D concepts of
More informationarxiv: v5 [math-ph] 6 Mar 2018
Entanglement entropies of minimal models from null-vectors T. Dupic*, B. Estienne, Y. Ikhlef Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005 Paris, France
More informationMutual Information in Conformal Field Theories in Higher Dimensions
Mutual Information in Conformal Field Theories in Higher Dimensions John Cardy University of Oxford Conference on Mathematical Statistical Physics Kyoto 2013 arxiv:1304.7985; J. Phys. : Math. Theor. 46
More informationarxiv: v2 [hep-th] 10 Aug 2015
Entanglement Scrambling in 2d Conformal Field Theory Curtis T. Asplund, Alice Bernamonti, Federico Galli, and Thomas Hartman arxiv:1506.03772v2 [hep-th] 10 Aug 2015 Department of Physics, Columbia University,
More informationOn sl3 KZ equations and W3 null-vector equations
On sl3 KZ equations and W3 null-vector equations Sylvain Ribault To cite this version: Sylvain Ribault. On sl3 KZ equations and W3 null-vector equations. Conformal Field Theory, Integrable Models, and
More informationOverview: Conformal Bootstrap
Quantum Fields Beyond Perturbation Theory, KITP, January 2014 Overview: Conformal Bootstrap Slava Rychkov CERN & École Normale Supérieure (Paris) & Université Pierre et Marie Curie (Paris) See also David
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More informationFour Point Functions in the SL(2,R) WZW Model
Four Point Functions in the SL,R) WZW Model arxiv:hep-th/719v1 1 Jan 7 Pablo Minces a,1 and Carmen Núñez a,b, a Instituto de Astronomía y Física del Espacio IAFE), C.C.67 - Suc. 8, 18 Buenos Aires, Argentina.
More informationExtended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function
CFT and integrability in memorial of Alexei Zamolodchikov Sogan University, Seoul December 2013 Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function Yutaka Matsuo (U. Tokyo)
More informationPart II. Riemann Surfaces. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised
More informationN = 2 heterotic string compactifications on orbifolds of K3 T 2
Prepared for submission to JHEP N = 2 heterotic string compactifications on orbifolds of K3 T 2 arxiv:6.0893v [hep-th 7 Nov 206 Aradhita Chattopadhyaya, Justin R. David Centre for High Energy Physics,
More informationLecture A2. conformal field theory
Lecture A conformal field theory Killing vector fields The sphere S n is invariant under the group SO(n + 1). The Minkowski space is invariant under the Poincaré group, which includes translations, rotations,
More information1 Introduction. or equivalently f(z) =
Introduction In this unit on elliptic functions, we ll see how two very natural lines of questions interact. The first, as we have met several times in Berndt s book, involves elliptic integrals. In particular,
More informationHolographic Entanglement Beyond Classical Gravity
Holographic Entanglement Beyond Classical Gravity Xi Dong Stanford University August 2, 2013 Based on arxiv:1306.4682 with Taylor Barrella, Sean Hartnoll, and Victoria Martin See also [Faulkner (1303.7221)]
More informationBMS current algebra and central extension
Recent Developments in General Relativity The Hebrew University of Jerusalem, -3 May 07 BMS current algebra and central extension Glenn Barnich Physique théorique et mathématique Université libre de Bruxelles
More information31st Jerusalem Winter School in Theoretical Physics: Problem Set 2
31st Jerusalem Winter School in Theoretical Physics: Problem Set Contents Frank Verstraete: Quantum Information and Quantum Matter : 3 : Solution to Problem 9 7 Daniel Harlow: Black Holes and Quantum Information
More informationConformal Field Theory and Combinatorics
Conformal Field Theory and Combinatorics Part I: Basic concepts of CFT 1,2 1 Université Pierre et Marie Curie, Paris 6, France 2 Institut de Physique Théorique, CEA/Saclay, France Wednesday 16 January,
More informationThe Non-commutative S matrix
The Suvrat Raju Harish-Chandra Research Institute 9 Dec 2008 (work in progress) CONTEMPORARY HISTORY In the past few years, S-matrix techniques have seen a revival. (Bern et al., Britto et al., Arkani-Hamed
More informationhigher genus partition functions from three dimensional gravity
higher genus partition functions from three dimensional gravity Henry Maxfield 1601.00980 with Simon Ross (Durham) and Benson Way (DAMTP) 21 March 2016 McGill University 1 motivation entanglement and geometry
More informationRepresentation theory of vertex operator algebras, conformal field theories and tensor categories. 1. Vertex operator algebras (VOAs, chiral algebras)
Representation theory of vertex operator algebras, conformal field theories and tensor categories Yi-Zhi Huang 6/29/2010--7/2/2010 1. Vertex operator algebras (VOAs, chiral algebras) Symmetry algebras
More informationComplex Analysis Math 185A, Winter 2010 Final: Solutions
Complex Analysis Math 85A, Winter 200 Final: Solutions. [25 pts] The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant (u, v) J = (x, y) = u x u y v x v y.
More informationExact Solutions of 2d Supersymmetric gauge theories
Exact Solutions of 2d Supersymmetric gauge theories Abhijit Gadde, IAS w. Sergei Gukov and Pavel Putrov UV to IR Physics at long distances can be strikingly different from the physics at short distances
More informationarxiv: v1 [hep-th] 20 Mar 2012
SFT Action for Separated D-branes arxiv:20.465v [hep-th] 20 Mar 202 Matheson Longton Department of Physics and Astronomy University of British Columbia Vancouver, Canada Abstract We present an action for
More informationCurrent algebras and higher genus CFT partition functions
Current algebras and higher genus CFT partition functions Roberto Volpato Institute for Theoretical Physics ETH Zurich ZURICH, RTN Network 2009 Based on: M. Gaberdiel and R.V., arxiv: 0903.4107 [hep-th]
More informationChern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action,
Lecture A3 Chern-Simons gauge theory The Chern-Simons (CS) gauge theory in three dimensions is defined by the action, S CS = k tr (AdA+ 3 ) 4π A3, = k ( ǫ µνρ tr A µ ( ν A ρ ρ A ν )+ ) 8π 3 A µ[a ν,a ρ
More information20 Entanglement Entropy and the Renormalization Group
20 Entanglement Entropy and the Renormalization Group Entanglement entropy is very di cult to actually calculate in QFT. There are only a few cases where it can be done. So what is it good for? One answer
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationThree-Charge Black Holes and ¼ BPS States in Little String Theory I
Three-Charge Black Holes and ¼ BPS States in Little String Theory I SUNGJAY LEE KOREA INSTITUTE FOR ADVANCED STUDIES UNIVERSITY OF CHICAGO Joint work (1508.04437) with Amit Giveon, Jeff Harvey, David Kutasov
More informationLecture 24 Seiberg Witten Theory III
Lecture 24 Seiberg Witten Theory III Outline This is the third of three lectures on the exact Seiberg-Witten solution of N = 2 SUSY theory. The third lecture: The Seiberg-Witten Curve: the elliptic curve
More informationarxiv: v1 [math.dg] 28 Jun 2008
Limit Surfaces of Riemann Examples David Hoffman, Wayne Rossman arxiv:0806.467v [math.dg] 28 Jun 2008 Introduction The only connected minimal surfaces foliated by circles and lines are domains on one of
More informationUniversal Dynamics from the Conformal Bootstrap
Universal Dynamics from the Conformal Bootstrap Liam Fitzpatrick Stanford University! in collaboration with Kaplan, Poland, Simmons-Duffin, and Walters Conformal Symmetry Conformal = coordinate transformations
More informationarxiv: v2 [math.ds] 13 Sep 2017
DYNAMICS ON TREES OF SPHERES MATTHIEU ARFEUX arxiv:1406.6347v2 [math.ds] 13 Sep 2017 Abstract. We introduce the notion of dynamically marked rational map and study sequences of analytic conjugacy classes
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationEntanglement in Quantum Field Theory
Entanglement in Quantum Field Theory John Cardy University of Oxford Landau Institute, June 2008 in collaboration with P. Calabrese; O. Castro-Alvaredo and B. Doyon Outline entanglement entropy as a measure
More information14 From modular forms to automorphic representations
14 From modular forms to automorphic representations We fix an even integer k and N > 0 as before. Let f M k (N) be a modular form. We would like to product a function on GL 2 (A Q ) out of it. Recall
More informationarxiv: v2 [hep-th] 2 Nov 2018
KIAS-P17036 arxiv:1708.0881v [hep-th Nov 018 Modular Constraints on Conformal Field Theories with Currents Jin-Beom Bae, Sungjay Lee and Jaewon Song Korea Institute for Advanced Study 8 Hoegiro, Dongdaemun-Gu,
More informationRefined Chern-Simons Theory, Topological Strings and Knot Homology
Refined Chern-Simons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: 1105.5117 arxiv: 1202.4456 Chern-Simons theory played
More informationBootstrap Program for CFT in D>=3
Bootstrap Program for CFT in D>=3 Slava Rychkov ENS Paris & CERN Physical Origins of CFT RG Flows: CFTUV CFTIR Fixed points = CFT [Rough argument: T µ = β(g)o 0 µ when β(g) 0] 2 /33 3D Example CFTUV =
More informationHigher Spin AdS/CFT at One Loop
Higher Spin AdS/CFT at One Loop Simone Giombi Higher Spin Theories Workshop Penn State U., Aug. 28 2015 Based mainly on: SG, I. Klebanov, arxiv: 1308.2337 SG, I. Klebanov, B. Safdi, arxiv: 1401.0825 SG,
More informationThe moduli space of binary quintics
The moduli space of binary quintics A.A.du Plessis and C.T.C.Wall November 10, 2005 1 Invariant theory From classical invariant theory (we refer to the version in [2]), we find that the ring of (SL 2 )invariants
More informationHere are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.
Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More informationOne Loop Tests of Higher Spin AdS/CFT
One Loop Tests of Higher Spin AdS/CFT Simone Giombi UNC-Chapel Hill, Jan. 30 2014 Based on 1308.2337 with I. Klebanov and 1401.0825 with I. Klebanov and B. Safdi Massless higher spins Consistent interactions
More informationTalk at the International Workshop RAQIS 12. Angers, France September 2012
Talk at the International Workshop RAQIS 12 Angers, France 10-14 September 2012 Group-Theoretical Classification of BPS and Possibly Protected States in D=4 Conformal Supersymmetry V.K. Dobrev Nucl. Phys.
More informationEric Perlmutter, DAMTP, Cambridge
Eric Perlmutter, DAMTP, Cambridge Based on work with: P. Kraus; T. Prochazka, J. Raeymaekers ; E. Hijano, P. Kraus; M. Gaberdiel, K. Jin TAMU Workshop, Holography and its applications, April 10, 2013 1.
More informationarxiv: v2 [hep-th] 12 Apr 2016
Prepared for submission to JHEP S-duality, triangle groups and modular anomalies in = SQCD arxiv:1601.0187v [hep-th] 1 Apr 016 S. K. Ashok, a E. Dell Aquila, a A. Lerda b and M. Raman a a Institute of
More information16. Local theory of regular singular points and applications
16. Local theory of regular singular points and applications 265 16. Local theory of regular singular points and applications In this section we consider linear systems defined by the germs of meromorphic
More informationMathieu Moonshine. Matthias Gaberdiel ETH Zürich. String-Math 2012 Bonn, 19 July 2012
Mathieu Moonshine Matthias Gaberdiel ETH Zürich String-Math 2012 Bonn, 19 July 2012 based on work with with S. Hohenegger, D. Persson, H. Ronellenfitsch and R. Volpato K3 sigma models Consider CFT sigma
More informationIsomorphisms between pattern classes
Journal of Combinatorics olume 0, Number 0, 1 8, 0000 Isomorphisms between pattern classes M. H. Albert, M. D. Atkinson and Anders Claesson Isomorphisms φ : A B between pattern classes are considered.
More informationQuantum Information and Entanglement in Holographic Theories
Quantum Information and Entanglement in Holographic Theories Matthew Headrick randeis University Contents 1 asic notions 2 1.1 Entanglement entropy & mutual information............................ 2 1.2
More informationA CHARACTERIZATION OF THE MOONSHINE VERTEX OPERATOR ALGEBRA BY MEANS OF VIRASORO FRAMES. 1. Introduction
A CHARACTERIZATION OF THE MOONSHINE VERTEX OPERATOR ALGEBRA BY MEANS OF VIRASORO FRAMES CHING HUNG LAM AND HIROSHI YAMAUCHI Abstract. In this article, we show that a framed vertex operator algebra V satisfying
More informationSymmetric Jack polynomials and fractional level WZW models
Symmetric Jack polynomials and fractional level WZW models David Ridout (and Simon Wood Department of Theoretical Physics & Mathematical Sciences Institute, Australian National University December 10,
More informationLanglands duality from modular duality
Langlands duality from modular duality Jörg Teschner DESY Hamburg Motivation There is an interesting class of N = 2, SU(2) gauge theories G C associated to a Riemann surface C (Gaiotto), in particular
More informationBlack Holes, Integrable Systems and Soft Hair
Ricardo Troncoso Black Holes, Integrable Systems and Soft Hair based on arxiv: 1605.04490 [hep-th] In collaboration with : A. Pérez and D. Tempo Centro de Estudios Científicos (CECs) Valdivia, Chile Introduction
More informationALF spaces and collapsing Ricci-flat metrics on the K3 surface
ALF spaces and collapsing Ricci-flat metrics on the K3 surface Lorenzo Foscolo Stony Brook University Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016 The Kummer construction
More informationNon-geometric states and Holevo information in AdS3/CFT2
Non-geometric states and Holevo information in AdS3/CFT2 Feng-Li Lin (NTNU) @KIAS workshop 11/2018 based on 1806.07595, 1808.02873 & 1810.01258 with Jiaju Zhang(Milano-Bicocca) & Wu-Zhong Guo(NCTS) 1 Motivations
More informationTHE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS
Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 225 259 S 0077-1554(06)00156-7 Article electronically published on December 27, 2006 THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL
More informationMath 145. Codimension
Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More informationThree-Charge Black Holes and ¼ BPS States in Little String Theory
Three-Charge Black Holes and ¼ BPS States in Little String Theory SUNGJAY LEE KOREA INSTITUTE FOR ADVANCED STUDIES Joint work (JHEP 1512, 145) with Amit Giveon, Jeff Harvey, David Kutasov East Asia Joint
More informationarxiv: v3 [hep-th] 5 Jul 2016
Higher spin entanglement entropy at finite temperature with chemical potential Bin Chen 1,2,3,4 and Jie-qiang Wu 1 arxiv:1604.03644v3 [hep-th] 5 Jul 2016 1 Department of Physics and State Key Laboratory
More informationPart II. Geometry and Groups. Year
Part II Year 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2014 Paper 4, Section I 3F 49 Define the limit set Λ(G) of a Kleinian group G. Assuming that G has no finite orbit in H 3 S 2, and that Λ(G),
More informationVARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN
VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN Abstract. For any field k and integer g 3, we exhibit a curve X over k of genus g such that X has no non-trivial automorphisms over k. 1. Statement
More informationLiouville Theory and the S 1 /Z 2 orbifold
Liouville Theory and the S 1 /Z 2 Orbifold Supervised by Dr Umut Gursoy Polyakov Path Integral Using Polyakov formalism the String Theory partition function is: Z = DgDX exp ( S[X; g] µ 0 d 2 z ) g (1)
More informationLecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)
Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries
More informationERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX
ERRATUM TO AFFINE MANIFOLDS, SYZ GEOMETRY AND THE Y VERTEX JOHN LOFTIN, SHING-TUNG YAU, AND ERIC ZASLOW 1. Main result The purpose of this erratum is to correct an error in the proof of the main result
More informationFROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS
FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.
More informationConformal bootstrap at large charge
Conformal bootstrap at large charge Daniel L. Jafferis Harvard University 20 Years Later: The Many Faces of AdS/CFT Princeton Nov 3, 2017 DLJ, Baur Mukhametzhanov, Sasha Zhiboedov Exploring heavy operators
More informationNull Cones to Infinity, Curvature Flux, and Bondi Mass
Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,
More information1 Polyakov path integral and BRST cohomology
Week 7 Reading material from the books Polchinski, Chapter 3,4 Becker, Becker, Schwartz, Chapter 3 Green, Schwartz, Witten, chapter 3 1 Polyakov path integral and BRST cohomology We need to discuss now
More informationString Theory I GEORGE SIOPSIS AND STUDENTS
String Theory I GEORGE SIOPSIS AND STUDENTS Department of Physics and Astronomy The University of Tennessee Knoxville, TN 37996-2 U.S.A. e-mail: siopsis@tennessee.edu Last update: 26 ii Contents 4 Tree-level
More information2D CFTs for a class of 4D N = 1 theories
2D CFTs for a class of 4D N = 1 theories Vladimir Mitev PRISMA Cluster of Excellence, Institut für Physik, THEP, Johannes Gutenberg Universität Mainz IGST Paris, July 18 2017 [V. Mitev, E. Pomoni, arxiv:1703.00736]
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationThe analytic bootstrap equations of non-diagonal two-dimensional CFT
The analytic bootstrap equations of non-diagonal two-dimensional CFT Santiago Migliaccio and Sylvain Ribault arxiv:7.0896v [hep-th] 5 May 08 Institut de physique théorique, Université Paris Saclay, CEA,
More informationD-Brane Conformal Field Theory and Bundles of Conformal Blocks
D-Brane Conformal Field Theory and Bundles of Conformal Blocks Christoph Schweigert and Jürgen Fuchs Abstract. Conformal blocks form a system of vector bundles over the moduli space of complex curves with
More informationIntegrability of Conformal Fishnet Theory
Integrability of Conformal Fishnet Theory Gregory Korchemsky IPhT, Saclay In collaboration with David Grabner, Nikolay Gromov, Vladimir Kazakov arxiv:1711.04786 15th Workshop on Non-Perturbative QCD, June
More informationConformal blocks from AdS
Conformal blocks from AdS Per Kraus (UCLA) Based on: Hijano, PK, Snively 1501.02260 Hijano, PK, Perlmutter, Snively 1508.00501, 1508.04987 1 Introduction Goal in this talk is to further develop understanding
More informationGeneralised Moonshine in the elliptic genus of K3
Generalised Moonshine in the elliptic genus of K3 Daniel Persson Chalmers University of Technology Algebra, Geometry and the Physics of BPS-States Hausdorff Research Institute for Mathematics, Bonn, November
More informationElliptic Curves and Elliptic Functions
Elliptic Curves and Elliptic Functions ARASH ISLAMI Professor: Dr. Chung Pang Mok McMaster University - Math 790 June 7, 01 Abstract Elliptic curves are algebraic curves of genus 1 which can be embedded
More informationHiggs Bundles and Character Varieties
Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character
More informationThe Dirac-Ramond operator and vertex algebras
The Dirac-Ramond operator and vertex algebras Westfälische Wilhelms-Universität Münster cvoigt@math.uni-muenster.de http://wwwmath.uni-muenster.de/reine/u/cvoigt/ Vanderbilt May 11, 2011 Kasparov theory
More informationHomological mirror symmetry via families of Lagrangians
Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants
More informationThe Canonical Sheaf. Stefano Filipazzi. September 14, 2015
The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over
More information