Beyond Standard Models and Grand Unifications: anomalies, topological terms, and dynamical constraints via cobordisms

Abstract

A bstract We classify and characterize fully all invertible anomalies and all allowed topo- logical terms related to various Standard Models (SM), Grand Unified Theories (GUT), and Beyond Standard Model (BSM) physics. By all anomalies, we mean the inclusion of (1) perturbative local anomalies captured by perturbative Feynman diagram loop calculations, classified by ℤ free classes, and (2) nonperturbative global anomalies, classified by finite group ℤ N torsion classes. Our work built from [31] fuses the math tools of Adams spectral sequence, Thom-Madsen-Tillmann spectra, and Freed-Hopkins theorem. For example, we compute bordism groups $$ {\Omega}_d^G $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>Ω</mml:mi><mml:mi>d</mml:mi><mml:mi>G</mml:mi></mml:msubsup></mml:math> and their invertible topological field theory invariants, which characterize d d topological terms and ( d − 1)d anomalies, protected by the following symmetry group G : $$ \mathrm{Spin}\times \frac{\mathrm{SU}(3)\times \mathrm{SU}(2)\times \mathrm{U}(1)}{{\mathrm{\mathbb{Z}}}_q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>Spin</mml:mtext><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mi>SU</mml:mi><mml:mfenced><mml:mn>3</mml:mn></mml:mfenced><mml:mo>×</mml:mo><mml:mi>SU</mml:mi><mml:mfenced><mml:mn>2</mml:mn></mml:mfenced><mml:mo>×</mml:mo><mml:mi>U</mml:mi><mml:mfenced><mml:mn>1</mml:mn></mml:mfenced></mml:mrow><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:mfrac></mml:math> for SM with q = 1 , 2 , 3 , 6; $$ \frac{\mathrm{Spin}\times \mathrm{Spin}(n)}{{\mathrm{\mathbb{Z}}}_2^F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:mtext>Spin</mml:mtext><mml:mo>×</mml:mo><mml:mtext>Spin</mml:mtext><mml:mfenced><mml:mi>n</mml:mi></mml:mfenced></mml:mrow><mml:msubsup><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn><mml:mi>F</mml:mi></mml:msubsup></mml:mfrac></mml:math> or Spin × Spin( n ) for SO(10) or SO(18) GUT as n = 10 , 18; Spin × SU( n ) for Georgi-Glashow SU(5) GUT as $$ n=5;\frac{\mathrm{Spin}\times \frac{\mathrm{SU}(4)\times \left(\mathrm{SU}(2)\times \mathrm{SU}(2)\right)}{{\mathrm{\mathbb{Z}}}_{q^{\prime }}}}{{\mathrm{\mathbb{Z}}}_2^F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mo>;</mml:mo><mml:mfrac><mml:mrow><mml:mtext>Spin</mml:mtext><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:mi>SU</mml:mi><mml:mfenced><mml:mn>4</mml:mn></mml:mfenced><mml:mo>×</mml:mo><mml:mfenced><mml:mrow><mml:mi>SU</mml:mi><mml:mfenced><mml:mn>2</mml:mn></mml:mfenced><mml:mo>×</mml:mo><mml:mi>SU</mml:mi><mml:mfenced><mml:mn>2</mml:mn></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow><mml:msub><mml:mi>ℤ</mml:mi><mml:msup><mml:mi>q</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:msub></mml:mfrac></mml:mrow><mml:msubsup><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn><mml:mi>F</mml:mi></mml:msubsup></mml:mfrac></mml:math> for Pati-Salam GUT as q ′ = 1 , 2; and others. For SM with an extra discrete symmetry, we obtain new anomaly matching conditions of ℤ 16 , ℤ 4 and ℤ 2 classes beyond the familiar Witten anomaly. Our approach offers an alternative view of all anomaly matching conditions built from the lower-energy (B)SM or GUT, in contrast to high-energy Quantum Gravity or String Theory Landscape v.s. Swampland program, as bottom-up/top-down complements. Symmetries and anomalies provide constraints of kinematics, we further suggest constraints of quantum gauge dynamics, and new predictions of possible extended defects/excitations plus hidden BSM non-perturbative topological sectors.

Topics & Concepts