Monday, 09 November 2026
The Batalin--Vilkovisky (BV) and Becchi--Rouet--Stora--Tyutin (BRST) formalisms provide a conceptual framework for understanding gauge symmetries and their role in classical and quantum field theory. At their heart lie three interconnected ideas: the cohomological description of symmetries, the geometry of odd symplectic manifolds, and the mathematical structures underlying perturbative quantization. This lecture series introduces the BRST complex and its relation to Lie algebra cohomology, then develops the geometric foundations of the BV formalism, including the BV bracket, the classical master equation, and the role of fields and antifields. We discuss the distinction between on-shell and off-shell gauge symmetries and explain how the BV formalism accommodates gauge algebras that fail to close off-shell. Finally, we turn to the quantum master equation and the challenges of extending the formalism to infinite-dimensional spaces of fields, highlighting the role of regularization, heat-kernel methods, and renormalization. Examples throughout the lectures will illustrate how these ideas fit together to provide a unified approach to gauge fixing and perturbative quantum field theory.
Will discuss the algebraic and geometric basis for string field theory, focusing on closed strings. Will explain the bar-cobar construction of universal quadratic equations.
Open string field theory (OSFT) provides non-manifestly background independent formulation of open string theory. Its classical solutions correspond to admissible D-branes in the theory. We shall review the developments in the field over the past 20 years, focusing mostly on the algebraic and analytic tools for Witten's OSFT required to construct and understand such solutions starting with a solution for the tachyon vacuum whose properties have been famously conjectured by Ashoke Sen in the late nineties. We will touch upon other approaches such as construction of numerical solutions which are useful for nontrivial matter CFT dependence, as well as analytic perturbative approaches.
These lectures are a step-by-step guide for constructing explicit 4d N=1 compactifications of type IIB string theory. We will introduce Calabi-Yau hypersurfaces, their holomorphic orientifolds, and develop methods for enumerating consistent flux backgrounds. We then discuss moduli stabilization and explain how to identify solutions that yield exponentially small cosmological constants, and, potentially, de Sitter vacua along the lines of the KKLT proposal.
Tuesday, 10 November 2026
The Batalin--Vilkovisky (BV) and Becchi--Rouet--Stora--Tyutin (BRST) formalisms provide a conceptual framework for understanding gauge symmetries and their role in classical and quantum field theory. At their heart lie three interconnected ideas: the cohomological description of symmetries, the geometry of odd symplectic manifolds, and the mathematical structures underlying perturbative quantization. This lecture series introduces the BRST complex and its relation to Lie algebra cohomology, then develops the geometric foundations of the BV formalism, including the BV bracket, the classical master equation, and the role of fields and antifields. We discuss the distinction between on-shell and off-shell gauge symmetries and explain how the BV formalism accommodates gauge algebras that fail to close off-shell. Finally, we turn to the quantum master equation and the challenges of extending the formalism to infinite-dimensional spaces of fields, highlighting the role of regularization, heat-kernel methods, and renormalization. Examples throughout the lectures will illustrate how these ideas fit together to provide a unified approach to gauge fixing and perturbative quantum field theory.
Will discuss the algebraic and geometric basis for string field theory, focusing on closed strings. Will explain the bar-cobar construction of universal quadratic equations.
Open string field theory (OSFT) provides non-manifestly background independent formulation of open string theory. Its classical solutions correspond to admissible D-branes in the theory. We shall review the developments in the field over the past 20 years, focusing mostly on the algebraic and analytic tools for Witten's OSFT required to construct and understand such solutions starting with a solution for the tachyon vacuum whose properties have been famously conjectured by Ashoke Sen in the late nineties. We will touch upon other approaches such as construction of numerical solutions which are useful for nontrivial matter CFT dependence, as well as analytic perturbative approaches.
These lectures are a step-by-step guide for constructing explicit 4d N=1 compactifications of type IIB string theory. We will introduce Calabi-Yau hypersurfaces, their holomorphic orientifolds, and develop methods for enumerating consistent flux backgrounds. We then discuss moduli stabilization and explain how to identify solutions that yield exponentially small cosmological constants, and, potentially, de Sitter vacua along the lines of the KKLT proposal.
Wednesday, 11 November 2026
The Batalin--Vilkovisky (BV) and Becchi--Rouet--Stora--Tyutin (BRST) formalisms provide a conceptual framework for understanding gauge symmetries and their role in classical and quantum field theory. At their heart lie three interconnected ideas: the cohomological description of symmetries, the geometry of odd symplectic manifolds, and the mathematical structures underlying perturbative quantization. This lecture series introduces the BRST complex and its relation to Lie algebra cohomology, then develops the geometric foundations of the BV formalism, including the BV bracket, the classical master equation, and the role of fields and antifields. We discuss the distinction between on-shell and off-shell gauge symmetries and explain how the BV formalism accommodates gauge algebras that fail to close off-shell. Finally, we turn to the quantum master equation and the challenges of extending the formalism to infinite-dimensional spaces of fields, highlighting the role of regularization, heat-kernel methods, and renormalization. Examples throughout the lectures will illustrate how these ideas fit together to provide a unified approach to gauge fixing and perturbative quantum field theory.
Open string field theory (OSFT) provides non-manifestly background independent formulation of open string theory. Its classical solutions correspond to admissible D-branes in the theory. We shall review the developments in the field over the past 20 years, focusing mostly on the algebraic and analytic tools for Witten's OSFT required to construct and understand such solutions starting with a solution for the tachyon vacuum whose properties have been famously conjectured by Ashoke Sen in the late nineties. We will touch upon other approaches such as construction of numerical solutions which are useful for nontrivial matter CFT dependence, as well as analytic perturbative approaches.
These lectures are a step-by-step guide for constructing explicit 4d N=1 compactifications of type IIB string theory. We will introduce Calabi-Yau hypersurfaces, their holomorphic orientifolds, and develop methods for enumerating consistent flux backgrounds. We then discuss moduli stabilization and explain how to identify solutions that yield exponentially small cosmological constants, and, potentially, de Sitter vacua along the lines of the KKLT proposal.
I will explain a general technique to evaluate the integrals appearing in string one-loop calculations in closed form by turning them into infinite sums that make many of their properties manifest. I will discuss what physical lessons we can extract from these calculations.
Thursday, 12 November 2026
The Batalin--Vilkovisky (BV) and Becchi--Rouet--Stora--Tyutin (BRST) formalisms provide a conceptual framework for understanding gauge symmetries and their role in classical and quantum field theory. At their heart lie three interconnected ideas: the cohomological description of symmetries, the geometry of odd symplectic manifolds, and the mathematical structures underlying perturbative quantization. This lecture series introduces the BRST complex and its relation to Lie algebra cohomology, then develops the geometric foundations of the BV formalism, including the BV bracket, the classical master equation, and the role of fields and antifields. We discuss the distinction between on-shell and off-shell gauge symmetries and explain how the BV formalism accommodates gauge algebras that fail to close off-shell. Finally, we turn to the quantum master equation and the challenges of extending the formalism to infinite-dimensional spaces of fields, highlighting the role of regularization, heat-kernel methods, and renormalization. Examples throughout the lectures will illustrate how these ideas fit together to provide a unified approach to gauge fixing and perturbative quantum field theory.
Will discuss the algebraic and geometric basis for string field theory, focusing on closed strings. Will explain the bar-cobar construction of universal quadratic equations.
These lectures are a step-by-step guide for constructing explicit 4d N=1 compactifications of type IIB string theory. We will introduce Calabi-Yau hypersurfaces, their holomorphic orientifolds, and develop methods for enumerating consistent flux backgrounds. We then discuss moduli stabilization and explain how to identify solutions that yield exponentially small cosmological constants, and, potentially, de Sitter vacua along the lines of the KKLT proposal.
I will explain a general technique to evaluate the integrals appearing in string one-loop calculations in closed form by turning them into infinite sums that make many of their properties manifest. I will discuss what physical lessons we can extract from these calculations.
Friday, 13 November 2026
Will discuss the algebraic and geometric basis for string field theory, focusing on closed strings. Will explain the bar-cobar construction of universal quadratic equations.
I will explain a general technique to evaluate the integrals appearing in string one-loop calculations in closed form by turning them into infinite sums that make many of their properties manifest. I will discuss what physical lessons we can extract from these calculations.
Open string field theory (OSFT) provides non-manifestly background independent formulation of open string theory. Its classical solutions correspond to admissible D-branes in the theory. We shall review the developments in the field over the past 20 years, focusing mostly on the algebraic and analytic tools for Witten's OSFT required to construct and understand such solutions starting with a solution for the tachyon vacuum whose properties have been famously conjectured by Ashoke Sen in the late nineties. We will touch upon other approaches such as construction of numerical solutions which are useful for nontrivial matter CFT dependence, as well as analytic perturbative approaches.
I will explain a general technique to evaluate the integrals appearing in string one-loop calculations in closed form by turning them into infinite sums that make many of their properties manifest. I will discuss what physical lessons we can extract from these calculations.