The two mayoral candidates, both Democrats, are on opposite sides of the debate over crime and policing. Republicans, with an eye toward 2024, are watching closely.CHICAGO — For nearly three years, since the ebbing of the George Floyd protests of 2020, almost nothing has divided the Democratic Party like the issues of crime, public safety and policing, much to the delight of Republicans eager to center urban violence in the nation’s political debate.Now, an unanticipated mayoral runoff in the country’s third largest city between Paul Vallas, the white former Chicago public schools chief running as the tough-on-crime candidate, and Brandon Johnson, a Black, progressive Cook County commissioner questioning traditional methods of policing, will elevate public safety on the national stage and test how ugly the Democratic divide might get in a city known for bare-knuckled politics and racial division.Mr. Vallas, as Chicago’s schools chief, expanded charter schools, then virtually eliminated neighborhood public schools in the New Orleans school system.Taylor Glascock for The New York Times“Oh, it’s going to be good,” Christopher Z. Mooney, professor emeritus of political science at the University of Illinois at Chicago, said of the runoff contest, which will culminate on April 4. “It’s going to get pretty rancorous, and underlying all of it will be the racial subtext.”The mayoral runoff pits two Democrats against each other, divided not only by ideology but also by race in a city where racial politics have been prominent since it elected its first Black mayor, Harold Washington, 40 years ago.A Republican hasn’t controlled City Hall since William H. “Big Bill” Thompson left office in 1931, with an open alliance with Al Capone and three safe deposit boxes containing almost $1.6 million.But this year’s Chicago election will be watched by Republicans intently. Crime has already emerged as a potent weapon for a G.O.P. eager to win back the suburbs and chip away at Democratic gains among urban professionals.It has also highlighted the Democrats’ divide between a liberal left that coined the phrase “defund the police” and a resurgent center insisting the party does “back the blue.”In New York City, a moderate Democrat, Eric Adams, harnessed the surge of violence that hit cities across the country, exacerbated by the pandemic, to win the mayoral race in 2021. A Republican-turned-Democrat, Rick Caruso, leaned on the issue of crime last year to force a runoff in the nation’s second largest city, Los Angeles, though he ultimately lost the mayoralty to the more liberal candidate, Karen Bass.In San Francisco, Chesa Boudin, the liberal district attorney of a city once synonymous with liberalism, was recalled last year by voters infuriated by rising disorder, and similarly progressive prosecutors from Philadelphia to Chicago have become lightning rods in conservative campaigns against supposedly “woke” law enforcement. Michelle Wu, the newly elected mayor of Boston, was forced just this week to respond to criticism of her handling of violence, after Black leaders accused her of ignoring their safety.And while the G.O.P. was disappointed with its showing in November’s congressional elections, one bright spot for Republicans came in victories in New York and California that were fueled by advertisements portraying Democratic cities as lawless. Former Speaker Nancy Pelosi has said her Democratic Party may well have held onto control of the House in November if candidates had a better answer to Republican attacks on crime, especially in New York.Add to that the issue of education, another sharp divide between Mr. Johnson and Mr. Vallas, and the mayor’s race in the city of broad shoulders may play out exactly as Republican presidential candidates would want. Ever since Glenn Youngkin recaptured Virginia’s governorship for his party in 2021 with an education-focused campaign, Republicans have made problems in the nation’s schools a centerpiece of their attempted national comeback, especially in the suburbs.And that has included a pitch for more school choice, whether through charter schools or vouchers to help public school students attend private schools. Again, Mr. Vallas and Mr. Johnson represent polar opposite positions on the issue: Mr. Vallas, as Chicago’s schools chief, expanded charter schools, then virtually eliminated neighborhood public schools when he took over the New Orleans school system after Hurricane Katrina. Mr. Johnson, a former schoolteacher and teachers union leader, stands firmly against that movement.Mr. Johnson, a former schoolteacher and teachers union leader, has fought against charter schools.Jamie Kelter Davis for The New York Times“This is a microcosm of a larger battle for the soul of the nation,” said Delmarie Cobb, a progressive political consultant in Chicago, “and being the third largest city, it’s going to get all the national coverage. This is going to be an intense five weeks.”For the national parties, those five weeks will be tricky. The runoff between Ms. Bass and Mr. Caruso in Los Angeles forced the Democratic establishment to get behind Ms. Bass, a known quantity with a long career in the House of Representatives. If the Democratic establishment rallies around Mr. Johnson, the outcome of the Chicago mayor’s race could mirror Los Angeles, come Election Day.But Mr. Johnson’s ardent progressivism, including his outspoken skepticism of policing as the answer to rising crime, could make him toxic to Democrats with national ambitions, including Illinois’ billionaire governor, J.B. Pritzker.Likewise, Mr. Vallas’s pledge to beef up Chicago’s police force and unshackle officers from the controls put on them after high-profile police shootings like the killing of Laquan McDonald could make him a hero of Republicans eying a run at the White House next year. But their endorsements would run counter to Mr. Vallas’s efforts in the nonpartisan mayoral race to persuade Chicagoans that he really is a Democrat.Rodney Davis, a former Republican House member from central Illinois, said that he had no doubt Mr. Vallas was a Democrat, but that the ideological divide in the mayoral contest was no less important because the contestants are from the same party.“Are voters going to think about whether Brandon Johnson calls Paul Vallas a Republican, or are they going to think, ‘Do I feel safe when I leave my kid in the car to go back inside and grab something? Do I feel like the public school system is getting better or worse?’” he said, adding, “This has set up a fight that really is less about politics and more about issues.”National Republicans, eager to make the crime debate central as they joust with each other for their party’s presidential nomination in 2024, are not likely to stay quiet.“They may want to exploit the situation,” said Marc H. Morial, a former New Orleans mayor who now heads the National Urban League.Last week, Gov. Ron DeSantis of Florida swung through New York City, the Philadelphia suburbs and a bedroom community outside Chicago to speak to police unions about crime, and to lambaste what he called “woke” urban officials who he contends have eased up on policing and criminal prosecutions.“They just get put right back on the streets and they commit more crimes and it’s like a carousel,” Mr. DeSantis, an as-of-yet undeclared candidate for president, said Tuesday night during a speech in The Villages, a heavily Republican retirement community in Central Florida.Next, Mr. DeSantis is taking his critique of big cities on a national tour, including stops in states with the first three Republican primary contests, and will promote his new book, “The Courage to Be Free: Florida’s Blueprint for America’s Revival.”Though Donald J. Trump erected a gleaming skyscraper on the Chicago River, he has made the city his No. 1 example of what is wrong with urban America.“It’s embarrassing to us as a nation,” Mr. Trump said on a visit in 2019. “All over the world, they’re talking about Chicago.”Criminal justice could be a centerpiece in the coming fight between Mr. Trump and Mr. DeSantis for the 2024 nomination. As president, Mr. Trump signed the “First Step Act,” a bipartisan criminal justice law that has freed thousands of inmates from federal prison. As a House lawmaker from Florida, Mr. DeSantis supported Mr. Trump’s bill in Congress in 2018, but as governor in 2019, when the state passed its own version of that federal legislation, he opposed a measure that would have allowed certain prisoners convicted of nonviolent felonies to be released after serving at least 65 percent of their sentences.The Trump measure was opposed by some Republicans, including Mr. Trump’s own attorney general at the time, Jeff Sessions, and the former president has since appeared eager to distance himself from the law.The race in Chicago will pit a candidate with the police union’s support against one who has the teachers’ union on his side.Taylor Glascock for The New York TimesDuring the past two years, Mr. Trump has spoken more about the need for tougher criminal justice laws, renewing his widely criticized proposal to execute drug dealers, and less about the benefits or outcomes of the First Step Act. Speaking to New Hampshire Republicans in late January, in the first public event of his latest presidential campaign, Mr. Trump said he would have a tougher response to civil rights protests if elected to a second term.“Next time, it’s one thing I would do different,” Mr. Trump said.A Republican intervention in the mayoral runoff here would not be helpful to Mr. Vallas. He was forced to denounce Mr. DeSantis’s appearance in Elmhurst, Ill., last week lest he be tied to the polarizing Florida governor ahead of Tuesday’s voting.But Mr. Johnson almost certainly represents too perfect a target for Republicans to sit this one out. He may have walked back earlier comments on “defunding” the police, but last month, he was the only mayoral candidate who refused to say he would fill the growing number of vacancies in the Chicago Police Department.“Spending more on policing per capita has been a failure,” Mr. Johnson said at a news conference outside City Hall last month.“Look, I get it,” he continued. “People are talking about policing as a strategy. But, keep in mind, that is the strategy that has led to the failures we are experiencing right now.”A substantive debate on the best approach to public safety could be good for Chicago and the country — if it stays substantive, Mr. Morial said. Policing is not only about the number of officers, he said, but about the accountability of the force and the trust of the citizens.Mr. Morial expressed doubt that Mr. Trump or Mr. DeSantis would keep the debate focused that way. But the nation will be watching, starting with the Chicago mayoral runoff, he said.“I’m watching this race closely,” he said. “I think it’s going to become a national conversation, which I think is going to be good.”Jonathan Weisman reported from Chicago, and Michael C. Bender from Washington. More