Title: Field theory expansions of string theory amplitudes
When: Oct 3, 2024, 4:00 PM- 5:00 PM IST
Organizers: Gaurav Bhatnagar (Ashoka University) , Atul Dixit (IIT, Gandhinagar) and Krishnan Rajkumar (JNU). Contact: sfandnt@gmail.com
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Happy new year.
Function
Speaker: Frank Garvan (University of Florida, USA)the mock theta functions analogous to Hecke's theory of modular forms. From this much surprising symmetry and q-series identities arise in joint work with Rishabh Sarma and Connor Morrow.
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Between 2006 and 2008, using various methods coming from representation theory (Westbury), gauge theory (Nekrasov--Okounkov) and combinatorics (Han), several authors proved the so-called Nekrasov–Okounkov formula which involves hook lengths of integer partitions.
This formula does not only cover the generating series for P, but more generally gives a connection between powers of the Dedekind η function and integer partitions. Among the generalizations of the Nekrasov--Okounkov formula, a (q, t)-extension was proved by Rains and Warnaar, by using refined skew Cauchy-type identities for Macdonald polynomials. The same result was also obtained independently by Carlsson–Rodriguez-Villegas by means of vertex operators and the plethystic exponential. As mentioned in both of these papers, the special case q=t of their formula correspond to a q version of the Nekrasov--Okounkov formula, which was already obtained by Dehaye and Han (2011) and Iqbal et al. (2012).
Motivated by the work of Han et al. around the generalizations of the Nekrasov--Okounkov formula, one way of deriving Nekrasov--Okounkov formula is by using the Macdonald identities for infinite affine root systems (Macdonald 1972), which can be thought as extension of the classical Weyl denominator formula.
In this talk, I will try to explain how some reformulations of the Macdonald identities (Macdonald 1972, Stanton 1989, Rosengren and Schlosser 2006) can be decomposed in the basis of characters for each infinite of the 7 infinite affine root systems by the Littlewood decomposition. This echoes a representation theoretic interpretation of the Macdonald identities (see the book of Carter for instance) and an ongoing project with Cédric Lecouvey, I will mention some partial results we get.
At last, I will briefly explain how to go from these reformulations of Macdonald identities to q Nekrasov--Okounkov type formulas.
Dear all, sorry for the late announcement. The next talk is by Seema Kushwaha of IIIT, Allahabad. We are back to our usual time now. Hope to see you later this week.