Deep and surprising relations between physics and geometry

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1 Twist fields, the elliptic genus, hidden symmetry Arthur Jaffe Harvard University, Cambridge, MA 138 Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, approved November, 1999 received for review September 1, 1999 We combine infinite dimensional analysis in particular a priori estimates twist positivity with classical geometric structures, supersymmetry, noncommutative geometry. We establish the existence of a family of examples of two-dimensional, twist quantum fields. We evaluate the elliptic genus in these examples. We demonstrate a hidden SL, symmetry of the elliptic genus, as suggested by Witten. Deep surprising relations between physics geometry emerged from the attempt to formulate an appropriate mathematical framework to describe physics. Supersymmetry quantized loops are basic ingredients of proposed theories combining quantum mechanics with general relativity. String theory was born in an effort to regularize the otherwise non-renormalizable aspects of gravitation as a quantum field. The present paper focuses on the mathematics underlying a formulation of the fundamental laws of physics, but within a restricted context. We investigate infrared ultraviolet convergence in certain nonlinear quantum field theories. Our nonperturbative analysis establishes the existence of solutions to the field equations, without being able to express the solutions in closed form. It also allows us to evaluate certain geometric invariants in the resulting field theory. We describe the results contained in a series of related papers 1 6. Time-zero bosonic fields ϕ x on a circle are called loops. The fields we consider are multi-valued maps from a circle of period l into a target n. Transporting x once around the circle yields the original value of ϕ multiplied by a phase e iχ, namely ϕ x + l =e iχ ϕ x. The angle χ characterizes the field ϕ. In what follows we study a family of fields with χ = φ, where is a strictly positive, diagonal matrix, where φ is a real parameter. We call ϕ x a twist field depending on φ; it reduces to a periodic field as φ. Twist fields have an intrinsic interest. In addition, the parameter φ provides an infrared regularization; in the next section we also introduce an ultraviolet regularization. A space of functionals of fields ϕ is called a loop space. Our loop space will be a space of quantum mechanical states, so it will also be a Hilbert space. To study analysis geometry on loop space, we desire to have an exterior derivative D that acts on loop space. Some time ago, Witten suggested that natural cidates for D arise in supersymmetric physics. We study some such examples, where D has the structure D = D + D I, with D an infinite-dimensional generalization of the de Rham derivative, with D I a connection determined by a holomorphic potential function V. In this context, Dirac fields are oneforms over loop space. The Fock space H is the exterior algebra over loop space defined by D, it inherits a natural Hilbert space structure. In our examples, D arises as a densely defined quadratic form. After regularization, D D also determine densely defined operators, so for each D we introduce a supercharge operator Q on H as the self-adoint closure of Q = D + D. This supercharge Q is a Dirac operator on loop space, is a generator of supersymmetry. The theories we study possess partial supersymmetry. The nontrivial twist χ has the effect that we obtain only half the number of invariant charges Q from what one expects when χ =. Each operator Q that we consider is also invariant under a U 1 3 U 1 group of translations twists. We use this Lie symmetry to evaluate a fundamental geometric invariant for the dynamics on loop space, namely the elliptic genus. We ustify a formula for the genus as a ratio of theta functions. The elliptic genus fits into the framework of noncommutative geometry, it has the interpretation as an equivariant index of the supercharge Q. The elliptic genus is constant on each universality class of potentials V defined in Furthermore, as a function of φ the two parameters of the symmetry group, the elliptic genus displays an SL symmetry. Because this symmetry appears, even though the underlying quantum field theory is not conformal, we call the SL symmetry hidden. 1. Twist Constructive Quantum Field Theory In constructive quantum field theory, the time-zero fields are operator-valued distributions on a Fock Hilbert space H. We study scalar fields ϕ x with n components ϕ i x, where 1 i n, a Dirac field ψ x with n components ψ α i x, where α = 1. The space H = H b H f is a tensor product of a bosonic Fock space H b = exp s with a fermionic Fock space H f = exp. The one-particle space is = n L S 1 dx L S 1 dx. A unitary group e ith on H generated by a self-adoint Hamiltonian H determines the time evolution, ϕ x t =e ith ϕ x e ith [1.1] ψ x t =e ith ψ x e ith [1.] We also denote the bosonic conugate time-zero field by π x, π x = ih ϕ x [1.3] The self-adoint momentum operator P commutes with H implements spatial translations. For example, e iσp ϕ x e iσp = ϕ x + σ [1.4] with similar action on π x ψ x. Furthermore, the unitary twist group e iθj has a generator J commuting with H P. For all x S 1 so consequently e iθj ϕi x e iθj = e iθ b i ϕ i x [1.5] e iθj π i x e iθj = e iθ b i π i x [1.6] the fermionic field satisfies e iθj ψ α i x e iθj = e iθ f α i ψ α i x [1.7] The twisting angles = i b α i f are given constants that characterize the twist generator J, up to an additive constant. We choose this additive constant ĉ/ so that 5J have the same spectrum. Then the zero-particle vector H satisfies J = 1 ĉ with ĉ = f i f 1 i [1.8] This paper was submitted directly Track II to the PNAS office PNAS February 15, vol. 97 no. 4

2 Definition 1.1: Twist quantum fields on a circle S 1 of length l are quantum fields of the type above, such that the initial data ϕ i x π i x ψ α i x satisfy ϕ i x + l =e iχb i ϕ i x π i x + l =e iχb i π i x [1.9] ψ α i x + l =e iχf α i ψ α i x [1.1] for all x S 1. The set of twisting angles χ = χ b i χ f α i is taken so that no twisting phase equals one, e iχb i 1 e iχf α i 1 for all i α [1.11] We study twist fields from two complementary points of view: as canonical quantum theories or via probability theories. Such interplay is stard in constructive quantum field theory without twists. The canonical quantum theory involves the direct study of linear transformations on Hilbert space; it is the traditional approach to quantum theory. It leads to harmonic analysis nonlinear hyperbolic equations. The quantum fields satisfy systems of nonlinear equations with canonical constraints on their initial data. On the other h, the probability approach relies on expectations over a classical configuration space, a fundamental representation of expectations of the heat kernel of the Hamiltonian as a functional integral. This approach for the bosonic fields involves the definition of non-gaussian, quasiinvariant measures on the space of functionals of classical fields on a torus. The inclusion of fermionic fields requires the extension of the classical space to include a tensor product with an infinite dimensional Grassmann algebra equipped with a Gaussian functional that defines a Gaussian integral. These two points of view are unified through the Feynman Kac representation, which shows the equality of moments of the functional integral as expectations of time-ordered products of fields. The nonlinearity of the systems we study is determined by a holomorphic polynomial V n, called the superpotential. Denote the degree of this polynomial by ñ = degree V we assume ñ [1.1] We have shown elsewhere the existence of solutions to these equations under certain assumptions on V z on the twisting angles that we detail below. These assumptions include the fact that V z is a holomorphic, quasihomogeneous polynomial of degree at least two. In other words, there exist n rational numbers i called weights, with i 1, such that V z = i z i V i z [1.13] where V i z = V z / z i. Each set of weights i determines a universality class of potentials V. In these examples, the Hamiltonian H = H V takes the form H = H + l H I x dx [1.14] where H = H denotes the free Hamiltonian, H I x = V ϕ x =1 + + ψ i 1 x ψ x V i ϕ x i =1 ψ i x ψ 1 x V i ϕ x [1.15] i =1 The phrase Wess Zumino equations or sometimes Lau Ginzburg equations identifies these examples in the literature. For cubic V, the equations reduce to the coupling of a nonlinear boson field to the Dirac field by a Yukawa interaction, so occasionally these equations are also called generalized Yukawa equations. Denote the fermion number operator on H f by N f, let Ɣ = I Nf denote a -grading on H. AsƔ commutes with H, it follows from 1.15 that Ɣ commutes with H V. Werequire above that the Hamiltonian is invariant under both the translation group e iσp the twist group e iθj. The free Hamiltonian H the first term in 1.15 have this property as long as the bosonic twisting angles χ b the bosonic twisting parameters b in 1.5 are both proportional to the weights i in We obtain all possible twisting phases by considering χ b b modulo π. We choose a normalization for θ such that the bosonic parameters exactly equal the weights. Thus we choose b i = i χ b i = i φ [1.16] for convenience we restrict the parameter φ to lie in the interval φ π. The boson fermion interaction occurs in the two last terms in We want the Hamiltonian to be invariant both under the J-twist group the translation group. If we have e iθj H I x e iθj = H I x for all θ H I x + l = H I x for all x, then it follows that J H I = P H I =. Inserting into 1.15 shows that J-twist invariance requires f 1 i f + 1 i π [1.17] for all 1 i n. Similarly substituting into 1.15, translation invariance requires that χ f 1 i χf + φ iφ φ π [1.18] for all 1 i n. Because we may also reduce χ f f modulo π, the right sides of the constraints may be taken to equal zero. The solution to these constraints is f 1 i = i 1 + ɛ f i = i ɛ [1.19] χ f α i = f α iφ + µ [1.] where ɛ µ are real parameters that may depend on φ, but are independent of i α. The normalization constant ĉ defined in 1.8 is independent of ɛ µ, it equals ĉ = 1 i [1.1] The choices above yield a translation-invariant Hamiltonian, as well as one invariant under J-twists. A further restriction on the possible values of µ ɛ allows us also to define a translation-invariant supercharge Q. The two possibilities are ɛ = 5 1, both with µ =, leading to self-adoint supercharge operators Q + or Q that anticommute with Ɣ. These restrictions are necessary even in case V =. They correspond to the occurrence of ψ 1 i π i or ψ i πi in Q 5, respectively. The supersymmetry relations are Q + = H + P or Q = H P [1.] No single choice of ɛ leads to both relations 1.,sotheχ-twist cuts in half the number of translation-invariant supercharges. It turns out that each charge Q 5 also commutes with a respective MATHEMATICS Jaffe PNAS February 15, vol. 97 no

3 J-twist J 5. We study the case ɛ = 1, call this supercharge Q = Q + Q I V, with Q independent of V with Q I V linear in V. The J-twist parameters the χ-angles are in this case b i f 1 i f i = i i 1 i χ b i χf 1 i χf i = iφ i φ 1 i φ [1.3] We require an analytic condition on the polynomial V z, to ensure that the spectrum of the Hamiltonian 1.14 is discrete that the eigenvalues increase sufficiently rapidly. We call this an elliptic bound, assume that given + ɛ, there exists M +: such that the function V satisfies α V ɛ V + M z + V M V + 1 [1.4] Here α V denotes any multi-derivative of V, while z denotes the magnitude of z, V = n V/ z i is the squared magnitude of the gradient of V. We begin by stating our stard hypotheses the fundamental existence result. Definition 1.: The Stard Hypotheses SH. The potential V is a holomorphic, quasihomogeneous polynomial , satisfying the elliptic bound 1.4. The relations 1.3 for J-twists for χ-twists hold, yielding 1.1. Theorem 1.3. Assume SH. There is a self-adoint Q = Q V commuting with the two-parameter unitary group e iσp iθj, anticommuting with Ɣ, such that H = Q P. The Hamiltonian H V is bounded from below, the heat kernel e βh is trace class for all β,.. The Elliptic Genus Noncommutative Geometry Definition.1: The elliptic genus is defined as the partition function V = Tr H Ɣe iθj iσp βh [.1] We investigate the elliptic genus by representing it as a functional integral, as embodied in the following: Theorem.. Assume SH. Then there exists a positive, non- Gaussian, countably additive Borel measure dµ on the space = of distributions on the -torus, an integrable, regularized Fredholm determinant det 3 arising from the boson fermion interaction, such that V = det 3 dµ [.] The remarkable positivity of the measure dµ is a feature of the bosonic theory; we call it twist positivity 3. We recognized this property established it for untwisted fields in ref. 3, we generalized this to twist fields in ref. 5, we abstract this property in a forthcoming oint work with O. Grean J. Tyson. The positivity of dµ arises from the fact that the measure dµ has the structure dµ = e S dµ, where S is a measurable real action functional, where dµ is a Gaussian measure. In fact dµ is a Gaussian with mean zero, with covariance 1, with an appropriate normalization. Here is the twisted Laplacian on the torus with periods l β; the Laplacian acts on the vector bundle n L, being uniquely determined by its action on the domain of smooth, n-component functions f x t satisfying the twist relations, f i x t + β =e i iθ f i x + σ t [.3] While earlier methods required a massive bosonic free Hamiltonian, see for example 7 9, twist fields allow a massless one. This provides a big advantage in preserving other symmetries. Even though twists partially break supersymmetry, our computation of invariants requires only one charge, but not both. It is a remarkable feature of these examples that they do not require infinite renormalization, as long as the twist relations 1.3 hold. Natural cancellations of divergences occur between the bosonic the fermionic degrees of freedom. f i x + l t =e i iφ f i x t [.4] The functions e k E i x t =δ i e ikx+iet+ikσt/β, in the case that lk π φ, that βe π θ, that 1 n, form an orthogonal basis of eigenfunctions for τ. Another aspect of our work involves understing the interaction introduced by non-zero potentials V, yielding the action S in the measure dµ the regularized Fredholm determinant det 3 in the representation.. We need to identify study the cancellations that occur in this representation in the derivative of this representation with respect to some parameter. Some of these cancellations occur directly in the heat kernel itself: these are renormalization cancellations can be hled by extensions of known methods. Other cancellations are more delicate only occur in estimates on differences of partition functions. We begin with some basic operator estimates. Let N denote the total number operator on H. Theorem.3. Assume SH. Then there exist constants M 1 = M 1 V φ M = M V φ M = M β V φ that are independent of independent of λ 1 such that N + H 1/ M 1 H λv +M [.5] Tr H e βh λv M [.6] Also, for fixed V, fixed λ 1 fixed τ fixed θ fixed φ π = [.7] : In ref. 6, we show that this estimate results in a priori estimates on the operators P J. For a fixed V fixed φ π, there exist constants M 1 M depending on φ, such that 5P M 1 H + M 5J M 1 H + M. It follows that Q, defined as the regularized Q, can be estimated in terms of H, Q e βh is trace class, uniformly in, for each β,. The bounds satisfied by Q the convergence of as :, lead to the analyticity of the elliptic genus. Let denote the interior of the upper-half plane. Define τ in terms of the spacetime parameters σ β to be τ = σ + iβ [.8] l Theorem.4 6. Assume SH. Then, for fixed real θ φ, the elliptic genus V τ θ φ defined in.1 is a holomorphic function of τ. The elliptic genus is one member in a family of K-theory invariants arising from non-commutative geometry entire cyclic cohomology, as explained in IX of ref. 1 based on ref. 1. It is the equivariant index of a Dirac operator Q on loop space. The elliptic genus, however, is only one invariant from a whole family of invariants, resulting from pairing the JLOcocycle 11. Therefore, it may be possible, within the framework of the Wess Zumino examples that we study here, to find closed form expressions for some other invariants given in ref. 1. We formulated various representations for such invariants in refs. 4, these might be useful in computation. As a computational tool, we wish to know under what hypotheses the partition function V λ is independent of a parameter λ in the potential function V λ z. The answer to this question depends both on analytic as well as on geometric data. The geometric requirement is satisfied if one symmetry group e iθj with fixed is applicable to the family V λ of potentials that 14 Jaffe

4 we consider. Complementary to that, the analytic input involves whether the function λ V λ [.9] is a priori continuous or even differentiable in λ. We would like precise conditions under which V λ is differentiable, for which the derivative equals the expression obtained by exchanging the order of differentiation taking traces. In this paper, we study the case V λ = λv, so variation of λ does not change the quasihomogeneous weights of V.Asa consequence, if Ɣe iσp iθj commutes with H λv for one value of λ 1, it commutes for all λ 1. Thus we study = Tr H Ɣe iθj iσp βh λv [.1] For V =, the heat kernel e βh is also trace class, on account of the non-zero twisting parameter φ, we choose in J the twisting parameters associated with V. We let the partition function have an implicit dependence on V, brought about through the choice that J be appropriate for the associated family H λv. We approach the study of the properties of as a function of λ by introducing an approximating family of regularized partition functions. We obtain this family by replacing Q I V by Q I V, Q V by Q V =Q +Q I V, leading to a regularized Hamiltonian H V =Q V P. The approximating operators satisfy certain basic a priori estimates, some of which are uniform in, while others are not. At a technical level, the family of approximations we use involves mollifiers with slow decrease at infinity, as we introduced in ref. 1 to study a related problem, as implemented in ref. 6 for this problem. We write the elliptic genus as a function of the invariant charge Q λv, likewise we write as a function of Q V, which is also invariant. We establish certain properties of depending on estimates that are not uniform in. Theorem.5 6. Assume SH. Then, the map λ τ θ φ [.11] is differentiable in λ for λ 1 the order of λ- differentiation the trace in.1 can be interchanged. Furthermore, with ñ given in 1.1 with α + / ñ 1 there exists a constant M = M α β V φ such that for λ 1 Mλ α [.1] These two analytic results lie at the very heart of evaluating the elliptic genus, as they establish the existence of a homotopy between the genus V. Generically the genus is piecewise constant in λ, but not globally constant. We show that is differentiable on the open interval + λ, we evaluate the derivative by showing the existence of identifying the it λ λv = λ λ λ V λ λ = Tr H ƔQ λv Q I V e iθj iσp βh V Tr H ƔQI V Q λv e iθj iσp βh V [.13] We can then use a priori bounds on Q λv, Q I V, H λv to show that / λ =, for λ 1. An important fact is that we can use cyclicity of the trace Tr H AB = Tr H BA, in the case that A is trace class, namely A 1 +:, that B is bounded. The trace norm A 1 is the Schatten norm A p with p = 1, defined by A p p = Tr H A A p/ for p 1. In case A p +:, then A = inf p p A p. These norms satisfy Hölder s inequality with A : = A see ref. 13. The behavior of at the endpoint λ = is trickier. The second statement of the theorem claims that is Hölder continuous at the origin, with an exponent that may be arbitrarily small for potentials V of large degree. In ref. 6, we prove the identity 1 = βλ Tr H Ɣe iσp iθj e sβh λv / 3 Q I V e 1 s H Q I V e sβh λv / ds [.14] Choose α so that + α ñ 1 / + 1. We also establish in ref. 6 the existence of M 3 = M 3 α β V φ such that e sβh λv /4 Q I V e 1 s H /4 M 3 λ 1+α/ s 1/+α/4 1 s α ñ 1 /4 [.15] Although this bound is not uniform in, it serves our purpose. We estimate.14 using Tr H A A 1. We then apply a Hölder inequality in the Schatten norms. We conclude that there is a constant M 4 = M 4 α β V φ such that 1 βm 3 λα s 1+α/ 1 s α ñ 1 / 3 Tr H e βh λv /4 s TrH e βh /4 1 s ds M 4 λ α [.16] establishing the Hölder continuity of at λ =. Combining the vanishing of / λ for λ,, with continuity at λ = where is independent of, we infer the following from Theorem.4. Theorem.6. Assume SH. Then the map λ is constant for λ Hidden Symmetry In a seminal paper 14, Witten suggested that one could calculate the elliptic genus of these examples in closed form. He gave a proposed formula for φ = based on a free field computation pointed out why one expects that answer. Kawai, Yamada, Yang 15 elaborated on the algebraic aspects Witten s work made contact with related proposals of Vafa 16. These insights require further elaboration, as the representation.1 is ill-defined if both V = φ =. We prove here the representation for the elliptic genus V for φ π, relying on the analysis above to reduce the problem to the case V =, to a calculation carried out in ref. 6, similar to Witten s consideration for φ =. Define the variables q = e πiτ so q + 1 y = e iθ [3.1] so y =1, z = e iφτ so z + 1 [3.] Consider partition functions as functions of τ, θ, φ, related to q, y, z as above. The Jacobi theta function of the first kind ϑ 1 τ θ, defined for τ for θ, is given by ϑ 1 τ θ =iq 1 8 y 1 1 y 3 : 1 q n 1 q n y 1 q n y 1 [3.3] n=1 This function is odd in the second variable, namely ϑ 1 τ θ = ϑ 1 τ θ. We use the notation in 1.3 of Whittaker Watson 17. MATHEMATICS Jaffe PNAS February 15, vol. 97 no

5 Theorem Assume SH. Then the elliptic genus V depends on V only through its universality class, as determined by the weights i, it equals V ϑ τ θ φ 1 τ 1 i θ φτ =zĉ/ [3.4] ϑ i=i 1 τ i θ φτ Remark: Theorem 3.1 shows that V τ θ φ extends to a holomorphic function for τ, θ, φ. If a b c d, ad bc = 1, then a b c d SL. Let τ = aτ + b cτ + d θ = θ [3.5] cτ + d φ = φτ [3.6] aτ + b The analytic continuation of the partition function V τ θ φ obeys the transformation law V τ θ φ =e πi ĉ8 c θ φτ cτ+d V τ θ φ [3.7] One obtains iting values from the representation 3.4 as the parameters φ, θ, orq vanish; these its are not uniform do not commute. Define the integer-valued index of the selfadoint operator Q restricted to the subspace H = with respect to the grading Ɣ as the difference in the dimension of the kernel the dimension of the cokernel of Q as a map from the +1 eigenspace of Ɣ to the 1 eigenspace of Ɣ. Denote this integer by Index Ɣ Q. Corollary 3.. We have the following its. i As φ tends to zero, the partition function converges to ϑ φ V = 1 τ 1 i θ [3.8] ϑ i=i 1 τ i θ As θ, the partition function converges to θ V = zĉ/ ϑ 1 τ 1 i φτ [3.9] ϑ 1 τ i φτ ii For θ π, we may take the iterated it as φ then q to obtain the equivariant, quantum-mechanical index studied in ref. 4, q φ V = sin 1 i θ/ [3.1] sin i θ/ The existence of a field theory for φ = requires special analysis. For λ, the existence of a φ = field theory not ust the partition function is a consequence of the assumption 1.4 for V, the φ = theory is also the φ it of the twist field theory. The elliptic genus of the iting theory is the it 3.8, it agrees with the formula proposed in ref. 15. In the case λ =, the elliptic genus also has a φ it as long as + θ + π, but this it is not the genus of a iting theory. iii The integer-valued index Index Ɣ Q can be obtained as Index Ɣ Q = θ φ V iv On the other h, Example 1: Then θ q V = φ = θ = θ V q 1 i For any n, let V z = n i = 1 k i ĉ = Index Ɣ Q = φ V 1 [3.11] = q θ V = 1 [3.1] zk i i, with k i. k i k i [3.13] k i 1 [3.14] Example : For n =, let V z =z k z 1 z k. In this case, 1 = 1 k = k 1 1 ĉ = k 1 1 k 1 [3.15] 1 k 1 k k 1 k Index Ɣ Q =k 1 k 1 +1 [3.16] Remark: The integer-valued index 3.11 is stable under a class of perturbations of V that are not necessarily quasihomogeneous. Briefly, we require that V = V 1 + V, where V 1 satisfies the hypotheses above. While V is a holomorphic polynomial, it is not necessarily quasi-homogeneous. In place of this, we assume that the perturbation V is small with respect to V 1 in the following sense: given + ɛ, there exists a constant M +:such that for any multi-derivative α of total degree α, α V ɛ V 1 +M [3.17] This work was supported in part by the Department of Energy under Grant DE-FG-94ER-58. A.J. performed this research in part for the Clay Mathematics Institute. 1. Jaffe, A Adv. Math. 143, Jaffe, A. Commun. Math. Phys. 9, Jaffe, A Ann. Phys. 78, Jaffe, A. Ann. Phys. 79, Grean, O. & Jaffe, A. J. Math. Phys., in press. 6. Jaffe, A., Commun. Math. Phys., in press. 7. Gm, J. & Jaffe, A Quantum Physics Springer, New York, nd Ed. 8. Jaffe, A., Lesniewski, A. & Weitsman, J Commun. Math. Phys. 114, Jaffe, A. & Lesniewski, A Commun. Math. Phys. 114, Connes, A K-Theory 1, Jaffe, A., Lesniewski, A. & Osterwalder, K Commun. Math. Phys. 118, Jaffe, A., Lesniewski, A. & Osterwalder, K Commun. Math. Phys. 185, Schatten, R. 197 Norm Ideals of Completely Continuous Operators Springer, New York, nd Ed. 14. Witten, E Internat. J. Modern Phys. A9, Kawai, T., Yamada, Y. & Yang, S.-K Nuclear Phys. B 414, Vafa, C Modern Phys. Lett. A4, Whittaker, E. T. & Watson, G. N. 196 A Course of Modern Analysis Cambridge Univ. Press, Cambridge, U.K., 4th Ed Jaffe