Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T09:22:01.425Z Has data issue: false hasContentIssue false
  • English
  • Français

PyCFTBoot: A Flexible Interface for the Conformal Bootstrap

Part of: Quantum field theory; related classical field theories Algorithms - Computer Science

Published online by Cambridge University Press:  03 May 2017

Connor Behan
Connor Behan*
Affiliation:
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11790, USA
*
*Corresponding author. Email address:[email protected] (C. Behan)
Get access

Abstract

We introduce PyCFTBoot, a wrapper designed to reduce the barrier to entry in conformal bootstrap calculations that require semidefinite programming. Symengine and SDPB are used for the most intensive symbolic and numerical steps respectively. After reviewing the built-in algorithms for conformal blocks, we explain how to use the code through a number of examples that verify past results. As an application, we show that the multi-correlator bootstrap still appears to single out the Wilson-Fisher fixed points as special theories in dimensions between 3 and 4 despite the recent proof that they violate unitarity.

Keywords

CFTconformal bootstrappythonsemidefinite programming

MSC classification

Secondary: 65Z05: Applications to physics 68W30: Symbolic computation and algebraic computation 81T80: Simulation and numerical modeling 90C34: Semi-infinite programming
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 1 - 38
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ferrara, S., Grillo, A. F. and Gatto, R., Tensor representations of conformal algebra and conformally covariant operator product expansion, Ann. Phys., 76 (1973), 161–188.Google Scholar
[2] Polyakov, A. M., Non-Hamiltonian approach to conformal quantum field theory, J. Exp. Theor. Phys., 66 (1974), 23–42.Google Scholar
[3] Maldacena, J., The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys., 38 (1998), 1113–1133, arXiv:9711.200.Google Scholar
[4] Rattazzi, R., Rychkov, S., Tonni, E. and Vichi, A., Bounding scalar operator dimensions in 4D CFT, J. High Energy Phys., 12(31) (2008), arXiv:0807.0004.Google Scholar
[5] Rychkov, S. and Vichi, A., Universal constraints on conformal operator dimensions, Phys. Rev. D, 80(4) (2009), arXiv:0905.2211.Google Scholar
[6] Caracciolo, F. and Rychkov, S., Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D, 81(8) (2010), arXiv:0912.2726.Google Scholar
[7] Rattazzil, R., Rychkov, S. and Vichi, A., Central charge bounds in 4D conformal field theory, Phys. Rev. D, 83(4) (2011). arXiv:1009.2725.Google Scholar
[8] Rattazzil, R., Rychkov, S. and Vichi, A., Bounds in 4D conformal field theories with global symmetry, J. Phys. A, 44(3) (2011), arXiv:1009.5985.Google Scholar
[9] El-Showk, S., Paulos, M. F., Poland, D., Rychkov, S., Simmons-Duffin, D. and Vichi, A., Solving the 3D Isingmodelwith the conformal bootstrap, Phys. Rev. D, 86(2) (2012), arXiv:1203.6064.CrossRefGoogle Scholar
[10] Kos, F., Poland, D. and Simmons-Duffin, D., Bootstrapping the vector models, J. High Energy Phys., 6(91) (2014), arXiv:1307.6856.Google Scholar
[11] El-Showk, S., Paulos, M. F., Poland, D., Rychkov, S., Simmons-Duffin, D. and Vichi, A., Solving the 3D Ising model with the conformal bootstrap II: c-minimization and precise critical exponents, J. Stat. Phys., 157 (2014), 869–914, arXiv:1403.4545.Google Scholar
[12] Nakayama, Y. and Ohtsuki, T., Approaching conformal window of symmetric Landau-Ginzburg models from conformal bootstrap, Phys. Rev. D, 89(12) (2014), arXiv:1404.0489.CrossRefGoogle Scholar
[13] Nakayama, Y. and Ohtsuki, T., Five dimensional symmetric CFTs fromconformal bootstrap, Phys. Lett. B, 734 (2014), 193–197, arXiv:1404.5201.CrossRefGoogle Scholar
[14] Nakayama, Y. and Ohtsuki, T., Bootstrapping phase transitions in QCD and frustrated spin systems, Phys. Rev. D, 91(2) (2015), arXiv:1407.6195.Google Scholar
[15] Bae, J. B. and Rey, S. J., Conformal bootstrap approach to fixed points in five dimensions, (2014), arXiv:1412.6549.Google Scholar
[16] Chester, S.M., Pufu, S. S. and Yacoby, R., Bootstrapping vector models in 4<d<6, Phys. Rev. D, 91(8) (2015), arXiv:1412.7746.Google Scholar
[17] Poland, D. and Simmons-Duffin, D., Bounds on 4D conformal and superconformal field theories, J. High Energy Phys., 5(17) (2010), arXiv:1009.2087.Google Scholar
[18] Poland, D., Simmons-Duffin, D. and Vichi, A., Carving out the space of 4D CFTs, J. High Energy Phys., 5(110) (2012), arXiv:1109.5176.Google Scholar
[19] Beem, C., Rastelli, L. and van Rees, B. C., The superconformal bootstrap, Phys. Rev. Lett., 111 (2013), 071601, arXiv:1304.1803.Google Scholar
[20] Chester, S. M., Lee, J., Pufu, S. S. and Yacoby, R., The superconformal bootstrap in three dimensions, J. High Energy Phys., 9(143) (2014), arXiv:1406.4814.Google Scholar
[21] Beem, C., Lemos, M., Liendo, P., Rastelli, L. and van Rees, B. C., The superconformal bootstrap, J. High Energy Phys., 3(183) (2016), arXiv:1412.7541.Google Scholar
[22] Paulos, M. F., JuliBootS: A hands-on guide to the conformal bootstrap, (2014), arXiv:1412.4127.Google Scholar
[23] Simmons-Duffin, D., A semidefinite program solver for the conformal bootstrap, J. High Energy Phys., 6(174) (2015), arXiv:1502.02033.Google Scholar
[24] Kos, F., Poland, D. and Simmons-Duffin, D., Bootstrapping mixed correlators in the 3D Ising model, J. High Energy Phys., 11(109) (2014), arXiv:1406.4858.Google Scholar
[25] Swinarski, D., Software for computing conformal block divisors on , (2014), http://faculty.fordham.edu/dswinarski/conformalblocks/ConformalBlocksAug2014.pdf.Google Scholar
[26] Kos, F., Poland, D., Simmons-Duffin, D. and Vichi, A., Bootstrapping the archipelago, J. High Energy Phys., 11(106) (2015), arXiv:1504.07997.Google Scholar
[27] Chester, S. M., Giombi, S., Iliesiu, L. V., Klebanov, I. R., Pufu, S. S. and Yacoby, R., Accidental symmetries and the conformal bootstrap, J. High Energy Phys., 1(110) (2016), arXiv:1507.04424.Google Scholar
[28] Beem, C., Lemos, M., Rastelli, L. and van Rees, B. C., The (2,0) superconformal bootstrap, Phys. Rev. D, 93(2) (2016), arXiv:1507.05637.Google Scholar
[29] Iliesiu, L. V., Kos, F., Poland, D., Pufu, S. S., Simmons-Duffin, D. and Yacoby, R., Bootstrapping 3D fermions, J. High Energy Phys., 3(120) (2016), arXiv:1508.00012.Google Scholar
[30] Poland, D. and Stergiou, A., Exploring the minimal 4D SCFT, J. High Energy Phys., 12(121) (2015), arXiv:1509.06368.Google Scholar
[31] Lemos, M. and Liendo, P., Bootstrapping chiral correlators, J. High Energy Phys., 1(25) (2016), arXiv:1510.03866.Google Scholar
[32] Lin, Y., Shao, S., Simmons-Duffin, D., Wang, Y. and Yin, X., superconformal bootstrap of the K3 CFT, (2015), arXiv:1511.04065.Google Scholar
[33] Chester, S.M., Iliesiu, L. V., Pufu, S. S. and Yacoby, R., Bootstrapping vector models with four supercharges in 3≤d≤4, J. High Energy Phys., 5(103) (2016), arXiv:1511.07552.Google Scholar
[34] Chester, S. M. and Pufu, S. S., Towards bootstrapping QED3, J. High Energy Phys., 8(19) (2016), arXiv:1601.03476.Google Scholar
[35] van der Walt, S., Colbert, S. C. and Varoquaux, G., The NumPy array: A structure for efficient numerical computation, Comput. Sci. Eng., 13 (2011), 22–30.Google Scholar
[36] SymPy Development Team, SymPy: Python Library for Symbolic Mathematics, 2014, http://www.sympy.org.Google Scholar
[37] El-Showk, S., Paulos, M. F., Poland, D., Rychkov, S., Simmons-Duffin, D. and Vichi, A., Conformal field theories in fractional dimensions, Phys. Rev. Lett., 112 (2014), 141601, arXiv:1309.5089.Google Scholar
[38] Costa, M. S., Penedones, J., Poland, D. and Rychkov, S., Spinning conformal correlators, J. High Energy Phys., 11(71) (2011), arXiv:1107.3554.Google Scholar
[39] Costa, M. S., Penedones, J., Poland, D. and Rychkov, S., Spinning conformal blocks, J. High Energy Phys., 11(154) (2011), arXiv:1109.6321.Google Scholar
[40] Simmons-Duffin, D., Projectors, shadows and conformal blocks, J. High Energy Phys., 4(146) (2014), arXiv:1204.3894.Google Scholar
[41] Siegel, W., Embedding vs 6D twistors, (2012), arXiv:1204.5679.Google Scholar
[42] Osborn, H., Conformal blocks for arbitrary spins in two dimensions, Phys. Lett. B, 718 (2012), 169–172, arXiv:1205.1941.Google Scholar
[43] Dymarsky, A., On the four-point function of the stress-energy tensors in a CFT, J. High Energy Phys., 10(75) (2015), arXiv:1311.4546.Google Scholar
[44] Costa, M. S. and Hansen, T., Conformal correlators of mixed-symmetry tensors, J. High Energy Phys., 2(151) (2015), arXiv:1411.7351.Google Scholar
[45] Elkhidir, E.; Karateev, D. and Serone, M., General three-point functions in 4D CFT, J. High Energy Phys., 1(133) (2015), arXiv:1412.1796.Google Scholar
[46] Echeverri, A. C., Elkhidir, E., Karateev, D. and Serone, M., Deconstructing conformal blocks in 4D CFT, J. High Energy Phys., 8(101) (2015), arXiv:1505.03750.Google Scholar
[47] Rejon-Barrera, F. and Robbins, D., Scalar-vector bootstrap, J. High Energy Phys., 1(139) (2016), arXiv:1508.02676.Google Scholar
[48] Iliesiu, L. V., Kos, F., Poland, D., Pufu, S. S., Simmons-Duffins, D. and Yacoby, R., Fermion-scalar conformal blocks, J. High Energy Phys., 4(74) (2016), arXiv:1511.01497.Google Scholar
[49] Echeverri, A. C., Elkhidir, E., Karateev, D. and Serone, M., Seed conformal blocks in 4D CFT, J. High Energy Phys., 2(183) (2016), arXiv:1601.05325.Google Scholar
[50] Dolan, F. A. and Osborn, H., Conformal four point functions and the operator product expansion, Nuclear Phys. B, 599 (2001), 459–496, arXiv:hep-th/0011040.Google Scholar
[51] Dolan, F. A. and Osborn, H., Conformal partial waves and the operator product expansion, Nuclear Phys. B, 678 (2004), 491–507, arXiv:hep-th/0309180.Google Scholar
[52] Fitzpatrick, A. L., Kaplan, J. and Poland, D., Conformal blocks in the large D limit, J. High Energy Phys., 8(107) (2013), arXiv:1305.0004.Google Scholar
[53] Hogervorst, M. and Rychkov, S., Radial coordinates for conformal blocks, Phys. Rev. D, 87(10) (2013), arXiv:1303.1111.Google Scholar
[54] Penedones, J., Trevisani, E. and Yamazaki, M., Recursion relations for conformal blocks, J. High Energy Phys., 9(70) (2016), arXiv:1509.00428.Google Scholar
[55] Dijkgraaf, R., Maldacena, J., Moore, G. and Verlinde, E., A black hole Farey tail, (2000), arXiv:hep-th/0005003.Google Scholar
[56] Hogervorst, M., Osborn, H. and Rychkov, S., Diagonal limit for conformal blocks in d dimensions, J. High Energy Phys., 8(14) (2013), arXiv:1305.1321.Google Scholar
[57] El-Showk, S. and Paulos, M. F., Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett., 111 (2013), 241601, arXiv:1211.2810.CrossRefGoogle ScholarPubMed
[58] Dolan, F. A. and Osborn, H., Conformal Partial Waves: Further Mathematical Results, 2011, arXiv:1108.6194.Google Scholar
[59] Tyrtyshnikov, E. E., A Brief Introduction to Numerical Analysis, Springer, 2012.Google Scholar
[60] Gautschi, W., ORTHPOL–A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules, ACM Transactions on Mathematical Software, 20 (1994), 21–62, arXiv:math/9307212.Google Scholar
[61] Dolan, F. A. and Osborn, H., Superconformal symmetry, correlation functions and the operator product expansion, Nuclear Phys. B, 629 (2002), 3–73, arXiv:hep-th/0112251.Google Scholar
[62] Wilson, K. G. and Fisher, M. E., Critical exponents in 3.99 dimensions, Phys. Rev. Lett., 28(4) (1972), 240–243.CrossRefGoogle Scholar
[63] Le Guillo, J. C. and Zinn-Justin, J., Accurate critical exponents for Ising like systems in non-integer dimensions, Journal de Physique, 48 (1987), 19–24.Google Scholar
[64] Hogervorst, M., Rychkov, S. and van Rees, B. C., Univarity violation at Wilson-Fisher fixed point in 4-epsilon dimensions, Phys. Rev. D, 93(12) (2016), arXiv:1512.00013.Google Scholar
[65] Gliozzi, F., Constraints on conformal field theories in diverse dimensions from the bootstrap mechanism, Phys. Rev. Lett., 111 (2013), 161602, arXiv:1307.3111.Google Scholar
[66] Golden, J. and Paulos, M. F., No unitary bootstrap for the fractal Ising model, J. High Energy Phys., 3(167) (2015), arXiv:1411.7932.Google Scholar