Three Avatars of Mock Modularity
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Workshop on Black Holes September 2020 Three Avatars of Mock Modularity ATISH DABHOLKAR International Centre for Theoretical Physics Trieste 1 2 Mock Modularity Holography Topology Duality ❖ Dabholkar, Murthy, Zagier (arXiv:1208.4074) ❖ Dabholkar, Jain, Rudra (arXiv:1905.05207) ❖ Dabholkar, Putrov, Witten (arXiv:2004.14387) Mock Theta Functions ❖ In Ramanujan's famous last letter to Hardy in 1920, he gave 17 examples (without any definition) of mock theta functions that he found very interesting with hints of modularity. For example, 2 ∞ qn f(τ) = − q−25/168 ∑ n 2n−1 n=1 (1 − q )…(1 − q ) ❖ Despite much work by many eminent mathematicians, this fascinating `mock’ or `hidden' modular symmetry remained mysterious for a century until the thesis of Zwegers in 2002. A mock theta function is `almost modular’ As difficult to characterize as a figure that is `almost circular’ Three Avatars of Mock Modularity 5 Atish Dabholkar Modular Forms A modular form f(τ) of weight k on SL(2,ℤ) is a holomorphic function on the upper half plane ℍ that transforms as aτ + b a b (cτ + d)−k f( ) = f(τ) ∀ ∈ SL(2,ℤ) cτ + d (c d) (a, b, c, d, k integers and ad-bc =1 ) In particular, f(τ + 1) = f(τ). Hence, n f(τ) = ∑ anq (q := exp(2πiτ)) n Three Avatars of Mock Modularity 6 Atish Dabholkar Modularity in Physics ❖ Holography: In AdS3 quantum gravity the modular group SL(2,ℤ) is the group of Boundary Global Diffeomorphisms ❖ Duality: For a four-dimensional quantum gauge theory, a congruence subgroup of SL(2,ℤ) is the S-duality Group ❖ Topology: For a two-dimensional conformal field theory SL(2,ℤ) is the Mapping Class Group ❖ Physical observables in all these three physical contexts are expected to exhibit modular behavior. Three Avatars of Mock Modularity 7 Atish Dabholkar Power of Modularity Modular forms are highly symmetric functions. ❖ Ring of modular forms generated by the two Eisenstein Series E4(τ) and E6(τ). For example, for a weight 12 form we have 3 2 n f12(τ) = aE4 (τ) + bE6 (τ) = ∑ anq n ❖ It’s enough to know the first few Fourier coefficients to determine a modular form completely. ❖ Modular symmetry relates f(τ) to f(−1/τ). High temperature to low temperature or Strong coupling to Weak coupling. Cardy formula or Hardy-Ramanujan-Rademacher expansion. Three Avatars of Mock Modularity 8 Atish Dabholkar Mock Modular Forms The definition of a mock modular form involves a pair ( f, g) where f(τ) is a holomorphic function on ℍ with at most exponential growth at all cusps and g(τ) is a holomorphic modular form of weight 2 − k such that the sum f(̂ τ, τ¯) := f(τ) + g*(τ, τ¯) transforms like a holomorphic modular form of half-integer weight k of some congruence subgroup of SL(2,ℤ) (Zwegers, Zagier) • f is a mock modular form holomorphic ✓ modular × • g is its shadow holomorphic ✓ modular ✓ • f ̂ is its modular completion holomorphic × modular ✓ Three Avatars of Mock Modularity 9 Atish Dabholkar Holomorphic Xor Modular ❖ The function f is holomorphic but not modular. ❖ The function f ̂ is modular but not holomorphic This incompatibility between holomorphy and modularity is the essence of mock modularity. Three Avatars of Mock Modularity 10 Atish Dabholkar Holomorphic Anomaly The modular completion of a mock modular form is related to the shadow by a holomorphic anomaly equation: ∂f(̂ τ, τ¯) (4πτ )k = − 2πg(τ) 2 ∂τ¯ ❖ The shadow is both holomorphic and modular but by itself does not determine the modular completion. ❖ The non-holomorphic correction g*(τ, τ¯) by itself is not modular. ❖ The holomorphic mock modular form by itself is not modular. ❖ The completion contains all the information but is not holomorphic. Three Avatars of Mock Modularity 11 Atish Dabholkar Incomplete Gamma Function Given the -expansion n one can obtain q g(τ) = ∑ bnq n≥0 (4πτ )−k+1 ¯ 2 k−1 ¯ −n g*(τ, τ¯) = b0 + ∑ n bn Γ(1 − k,4πnτ2) q k − 1 n>0 where the incomplete Gamma function is defined by ∞ Γ(1 − k, x) = t−k e−t dt ∫x The incomplete gamma function will make its appearance again Three Avatars of Mock Modularity 12 Atish Dabholkar Holomorphic Modular ∼Blue Circular 13 Jacobi Forms A holomorphic function φ(τ, z) of two variables on ℍ × ℂ is a Jacobi form of weight k and index m if it is modular in τ and elliptic (doubly periodic up to phase) in z : 2 aτ + b z k 2πimcz ❖ φ , = (cτ + d) e cτ + d φ(τ, z) ( cτ + d cτ + d ) 2 ❖ φ(τ, z + λτ + μ) = e−2πim(λ τ+2λz)φ(τ, z) ∀ λ, μ ∈ ℤ with Fourier expansion n r φ(τ, z) = ∑ c(n, r)q y (y := exp(2πiz)) n,r Three Avatars of Mock Modularity 14 Atish Dabholkar Elliptic Genera and Jacobi Forms Consider a conformally invariant nonlinear sigma model with target space X with (2,2) superconformal symmetry with left and right super Virasoro algebra. One can define χ(τ, z) = Tr (−1)FR+FL e2πiτHL e−2πiτ¯HR e2πizFL . Elliptic genus thus defined is a Jacobi form. • By conformal symmetry, the path integral is modular. • By spectral flow symmetry, the path integral is elliptic. Right-moving Witten index that counts right-moving ground states. Three Avatars of Mock Modularity 15 Atish Dabholkar Examples of Jacobi forms ❖ Modified Elliptic genus of T4 weight -2 and index 1 ϑ2(τ, z) A(τ, z) = 1 (k = − 2 , m = 1) η6(τ) Since ϑ1(τ, z) has a simple zero, A(τ, z) has a double zero at z = 0. ❖ Elliptic genus of K3 weight 0 and index 1 ϑ2(τ, z) ϑ2(τ, z) ϑ2(τ, z) B(τ, z) = 8 2 + 3 + 4 (k = 0 , m = 1) 2 2 2 ( ϑ2(τ) ϑ3(τ) ϑ4(τ) ) where η(τ) is the Dedekind eta and ϑi(τ, z) are the Jacobi theta function. Three Avatars of Mock Modularity 16 Atish Dabholkar Ring of Jacobi forms ❖ Ring of Jacobi forms is generated by A(τ, z) and B(τ, z) with ordinary modular forms as coefficients. Index and weight both add when you multiply Jacobi forms. ❖ For example, a k = 4 , m = 2 Jacobi form is given by 2 2 2 a E4 (τ) A (τ, z) + b E6(τ) AB(τ, z) + c E4(τ) B (τ, z) ❖ Jacobi forms in are equivalent to vector valued modular forms in a single variable τ Three Avatars of Mock Modularity 17 Atish Dabholkar Theta Decomposition Using elliptic property, the z dependence goes into theta functions 2m−1 φ(τ, z) = ∑ ϑm,ℓ hℓ(τ) ℓ=0 where ϑm,ℓ are the standard level m theta functions. (ℓ+2mn)2/4m ℓ+2mn ϑm,ℓ(τ, z) := ∑ q y n∈ℤ A Jacobi form φ(τ, z) of weight k and index m is equivalent to a vector- 1 valued modular form {h (τ)} of weight (k − ) ℓ 2 Three Avatars of Mock Modularity 18 Atish Dabholkar “My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions . But before this can happen, the purely mathematical exploration of the mock-modular forms and their mock-symmetries must be carried a great deal further.” Freeman Dyson (1987 Ramanujan Centenary Conference) I. A Puzzle With Black Holes The counting function for the degeneracies of quarter-BPS black 2 holes is given in Type-II string compactified on in K3 × T is a meromorphic Jacobi form (weight −10 , index m) Bm+1(τ, z) + … ψ (τ, z) = m η24(τ) A(τ, z) The quantum degeneracies are given by the Fourier coefficients of this counting function. However, given the double pole in z, the Fourier coefficients depend on the choice of the contour and are not uniquely defined. What’s wrong with the counting? Three Avatars of Mock Modularity 20 Atish Dabholkar II. A Puzzle With Instantons ❖ Consider the Vafa-Witten partition function Z(τ) of � = 4 four-dimensional supersymmetric gauge theory on ℂℙ2 for gauge group SO(3) with coupling 4π constant τ = θ + i . g2 ❖ The twisted supercharge Q squares to zero Q2 = 0. By a general argument, the partition function is holomorphic because d Z(τ) = ⟨{Q, Λ}⟩ = 0 dτ¯ ❖ The holomorphic partition function receives contributions only from instantons but does not have any modular properties as expected from S- duality. What’s wrong with duality? Three Avatars of Mock Modularity 21 Atish Dabholkar III. A Puzzle With Elliptic Genus ❖ The elliptic genus of a SCFT like a sigma model is again expected to be holomorphic by a general argument. But for a noncompact SCFT (for example, a sigma model with infinite cigar as the target space) the holomorphic function is not modular as expected by conformal invariance. ❖ What’s wrong with conformal invariance? In all three cases the answer involves mock modularity from a mathematical point of view and a noncompact path integral from a physical point of view. Three Avatars of Mock Modularity 22 Atish Dabholkar Poles and Walls ❖ The poles in the counting function is a reflection of the wall- crossing phenomenon. The moduli space of the theory is divided into chambers separated by walls. The quantum degeneracies jump upon crossing walls. pole wall residue at the pole jump in the degeneracy How come? The quantum degeneracy of a black hole is a property of its horizon and not of asymptotic moduli.