Under the watchful eyes of Father Silvano Bentivoglio (Carmen Argenziano) and Dr. Vittoria Vetra (Ayelet Zurer), the European Organization for Nuclear Research (CERN) initiates the Large Hadron Collider and creates three vials of antimatter particles larger than any ever been produced before. Almost immediately, Father Silvano is killed and one of the vials of antimatter goes missing. At the same time, the Roman Catholic Church is mourning the sudden death of Pope Pius XVI in Rome and prepares for the papal conclave to elect the next Pope. The Camerlengo, Father Patrick McKenna (Ewan McGregor), assumes temporary control of the Vatican. Before the conclave enters seclusion, four of the "preferiti" (the favorite cardinals to be elected pope) are kidnapped by a man claiming to represent the Illuminati. In a video message, he threatens to kill one candidate every hour starting at 8pm and to destroy all of Vatican City at midnight, using the missing vial of antimatter as a bomb. The Vatican summons symbologist Robert Langdon (Tom Hanks) from Harvard University and Vetra to help them save the four preferiti and locate the vial.
Langdon listens to the Illuminati's message and deduces the four cardinals will die at the four altars of the "Path of Illumination", marked by statues of angels in locations relevant to the four classical elements. Over the objections of Commander Maximilian Richter, head of the Swiss Guard, but with McKenna's consent, Langdon is granted access to the Vatican Secret Archives. He examines Galileo Galilei's banned book with Vetra. Following the clues and accompanied by Inspector General Ernesto Olivetti and Claudio Vincenzi of the Vatican Gendarmerie Corps, they arrive at the Chigi Chapel in the Church of Santa Maria del Popolo. There they find Cardinal Ebner dead, suffocated with soil and branded with an ambigrammatic word "Earth". They verify the second location is the crowded Saint Peter's Square and arrive just as a bloody Cardinal LamassĂŠ emerges with his chest branded with "Air". Vetra unsuccessfully attempts to perform CPR on him, only to discover his lungs had been punctured, while the assassin slips through the crowd.
While Vetra studies Silvano's diaries, Langdon, Olivetti and Vincenzi locate the third church, Santa Maria della Vittoria, and arrive to see Cardinal Guidera, branded "Fire" and suspended above a burning pile of wood. A gunfight erupts between the assassin and the officers, with Olivetti and Vincenzi being killed in the process. Langdon manages to escape, but not before being spotted by the assassin. Langdon convinces two Carabinieri officers to take him to the next location and the trio race to the Water altar, the Fountain of the Four Rivers, just as the assassin arrives in a van. The assassin murders the officers and drops a bound and weighted Cardinal Baggia into the fountain before bidding farewell to Langdon and driving off. With the help of bystanders, Langdon rescues the cardinal, who tells him the Illuminati's lair is Castel Sant'Angelo. There, Langdon and Vetra discover a hidden passageway leading to the Vatican, being used as a hideout for the assassin. Discovering a case with marks for five branding irons, they realize the fifth brand is for the Camerlengo but are confronted by the assassin before they can alert McKenna. The assassin spares their lives once more, stating killing them is not a part of his mission unless they pursue him. He then cryptically warns them to be careful as his contractors are "men of God" before departing. He escapes to a car left for him by his contractor, but is killed immediately when it explodes upon ignition.
Langdon and Vetra rush back to the Vatican where they find Commander Richter hovering over McKenna with a gun, the Vatican symbol branded into McKenna's chest. Richter and Archbishop Simeon are shot by the Guards. As Richter dies before Langdon, he opens his hand to reveal a key which Langdon takes. The stolen antimatter vial is found in St. Peter's Tomb below the church, but the battery life is too low to risk re-connecting it to a battery. McKenna, a former military pilot, seizes the vial and uses an awaiting helicopter to fly above the Vatican. At a high altitude, he parachutes out as the antimatter bomb explodes overhead. McKenna is hailed a hero and savior, and the cardinals move to elect him pope. Langdon and Vetra use Richter's key to watch a security video showing McKenna speaking to Richter before the attack. The video reveals it is McKenna, not the Illuminati, who masterminded the scheme. McKenna reveals he murdered the previous pope because he felt the pope had betrayed the church by trying to bridge the gap between science and religion. Once McKenna had killed the pope, he intended to have himself elected to the papacy whilst rallying the most conservative Cardinals to his side. The recording is shown to the Papal conclave, and when it dawns on McKenna he has been exposed. He flees to a remote recess in the building where he is able to commit suicide by setting himself on fire.
The Vatican officially announces McKenna died due to internal injuries suffered during his parachute landing, and Cardinal Baggia is named Pope Luke, with Cardinal Strauss as the new Camerlengo. Strauss thanks Langdon for his assistance and gives Langdon Galileo's "Diagramma Veritatis" for his research as a gift from himself and Pope Luke, requesting only it be returned to the Vatican once he is finished, and any future references he may make about the Catholic Church in his future publications be done gently, to which Langdon replies, "I'll try."

Question: What do Langdon and Vetra study in the archives? 
Answer: Galilieo Galilei's banned book.